Home Publications Undergraduates Postgraduates Postdocs Calendar Contact

  Jeroen Lamb  
  Martin Rasmussen  
  Dmitry Turaev  
  Sebastian van Strien  
Mike Field
Fabrizio Bianchi
Trevor Clark
Nikos Karaliolios
Sofia Trejo Abad
Paul Verschueren
Alex Athorne
Sajjad Bakrani Balani
Andrew Clarke
Maximilian Engel
Michael Hartl
Giuseppe Malavolta
Guillermo Olicón Méndez
Cezary Olszowiec
Christian Pangerl
Matteo Tanzi
Kalle Timperi
Anna Maria Cherubini
Carlos Siquiera
Bill Speares
Mauricio Barahona
Davoud Cheraghi
John Elgin
Darryl Holm
Greg Pavliotis

DynamIC Seminars (back)

Daniel Thompson (Ohio State University)Abstract: TBA TBA4 July 2017
Vasileios Basios (Université Libre de Bruxelles)Linear and nonlinear Arabesgues: Negative 2-element Circuits and Chaos21 June 2017
Rene Medrano (Universidade Federal de São Paulo)Abstract: TBA Shrimps, cockroaches and some other strange structures in chaotic systems20 June 2017
Maciej Capinski (AGH University of Science and Technology)Abstract: TBA A fast diffusion mechanism with application to the Neptune-Triton elliptic restricted three body problem20 June 2017
De-Qi Zhang (National University of Singapore)Abstract: We consider automorphism g of positive entropy on a compact Kahler manifold X. The existence of such g imposes strong constraints on the geometrical structure of X. We also show that a projective manifold X has a maximal number of commutative automorphisms of positive entropy only when X is a complex torus or its quotient. The minimal model program of algebraic geometry will be employed. Automorphisms of positive entropy on compact Kahler manifolds13 June 2017
Artur Oscar Lopes (Universidade Federal do Rio Grande do Sul)New results on Thermodynamic Formalism: entropy, pressure and the involution kernel11 April 2017
Dmitry Treschev (Steklov Institute, Moscow)Arnold diffusion in a priori unstable case23 March 2017
Ilya Goldsheid (Queen Mary University of London)Abstract: The asymptotic behaviour of products of independent identically distributed $m\times m$ random matrices is now relatively well understood (if $m$ is fixed!). A long standing natural problem is: what part of the corresponding theory can be extended to the case of products of non-identically distributed matrices and, more generally, transformations? Perturbation theory is a very natural example of a situation where such a question arises. In my talk, I'll try to answer this question. Products of random transformations and Lyapunov exponents21 March 2017
Manuela Aguiar (University of Porto)Abstract: A coupled cell system is a dynamical system distributed over the nodes (cells) of a network. Each cell is an individual dynamical system (which we are going to assume continuous) and the coupling structure of the network indicates the interactions between those cell dynamics. One of the key aspects in the theory of coupled cell networks concerns the existence of synchrony subspaces - subspaces defined in terms of equalities between cell coordinates which are flow-invariant. Synchrony subspaces (flow-invariant subspaces) can have a major impact on both global and local dynamics and are important from the point of view of the study of that dynamics. Surprisingly, synchrony subspaces are independent of the specific individual dynamics at the nodes and are determined only by the network structure. That is, all the coupled cell systems admissible by a given network structure share the same structure of patterns of synchrony. In my talk, I will review recent concepts and results concerning and related with synchrony subspaces in coupled cell networks. The talk includes joint work with Peter Ashwin (University of Exeter), Ana Dias (Univer- sity of Porto), Flora Ferreira (University of Porto), Mike Field (Rice University, Imperial College), Marty Golubitsky (The Ohio State University), Maria Leite (University of South Florida) and Haibo Ruan (Universität Hamburg). About Patterns of Synchrony in Coupled Cell Networks21 March 2017
Tomoo Yokoyama (Kyoto University of Education)Abstract: In this talk, we present a necessary and sufficient condition for the existence of dense orbits of continuous flows on compact connected surfaces, which is a generalization of a necessary and sufficient condition on area-preserving flows obtained by H. Marzougui and G. Soler López. We also consider what class of flows on compact surfaces can be characterized by finite labeled graphs. In particular, a class of surface flows, up to topological conjugacy, which contains both the set of Morse Smale flows and the set of area-preserving flows with finite singular points is classified by finite labeled graphs. Finally, we discuss applications to fluid dynamics. Topological transitivity and representability of surfaces flows15 March 2017
Oliver Jenkinson (Queen Mary University of London)Abstract: The Lagarias-Wang finiteness conjecture asserted that the joint spectral radius of a set of matrices is always realised by a finite matrix product. Although this conjecture has been disproved, counterexamples are not easy to find. In this talk I will describe a new approach to generating finiteness counterexamples, making links with ergodic optimization and Sturmian measures. This is joint work with Mark Pollicott. Joint spectral radius, Sturmian measures, and the finiteness conjecture14 March 2017
Ursula Hamenstädt (Rheinische Friedrich-Wilhelms-Universität Bonn )Abstract: We begin with looking at a cocycle over the geodesic flow on a negatively curved surface with values in Rn, where the flow is equipped with an invariant measure obtained from a harmonic measure of a random walk. We summarize work of Guivarch Raugy and Benoist Quint which gives a simple criterion for simplicity of the Lyapunov spectrum. We then explain why this covers a very general class of invariant measures and then show how this viewpoint largely generalizes to many other flows which are not necessariy Anosov flows on a compact manifold. Simplicity of the Lyapunov spectrum for cocycles of flows14 March 2017
Maik Gröger (University of Jena)Abstract: Studying low-complexity notions and possible concepts of long-rage order are two closely linked endeavours. With this in mind we investigate the relations of two complexity notions in the zero entropy regime: mean equicontinuity and amorphic complexity. As it turns out, there is a close relationship in the minimal setting and for mean equicontinuous subshifts we show that amorphic complexity corresponds to the box dimension of the maximal equicontinuous factor. Further, for certain Toeplitz subshifts we show how to calculate amorphic complexity using the theory of iterated function systems. We will also elaborate on possible extensions to more general group actions and applications to the theory of quasicrystals. This is work in progress with Gabriel Fuhrmann, Tobias Jäger and Dominik Kwietniak. Amorphic complexity and long-range order8 March 2017
Ian Melbourne (University of Warwick)Abstract: Over the last 10 years or so, advanced statistical properties, including exponential decay of correlations, have been established for the classical Lorenz attractor. Many of the proofs use the smoothness of the stable foliation for the flow, or at least smoothness of the stable foliation for a Poincare map. In this talk, I will survey the recent results in this area for the classical Lorenz attractor (where the stable foliations are known to be at least $C^{1.278}$), and more generally for singular hyperbolic attractors when the stable foliations are assumed to be $C^{1+\epsilon}$. Then I will describe some new results, joint with Vitor Araujo, where we show that many statistical properties persist without these smoothness assumptions. Such properties include existence of SRB measures, central limit theorems and invariance principles, mixing, and (typically) superpolynomial decay of correlations. However exponential decay of correlations remains an open question in this generality. Mixing etc for Lorenz attractors with nonsmooth stable foliations28 February 2017
Alexey Kazakov (Nizhny Novgorod University)Variety of strange attractors in nonholonomic mechanics28 February 2017
Bassam Fayad (Institut de Mathématiques de Jussieu-Paris Rive Gauche)Abstract: TBA Dynamics and the distribution of $n\alpha$ on the torus22 February 2017
Jordi-Lluis Figueras (Uppsala University)Abstract: In this talk we will present a numerical algorithm for the computation of (hyperbolic) periodic orbits of the 1-D Kuramoto-Sivashinsky equation u_t+v*u_{xxxx}+u_{xx}+u*u_x = 0, with v>0. This numerical algorithm consists on applying a suitable quasi-Newton scheme. In order to do this, we need to rewrite the invariance equation that must satisfy a periodic orbit in a form that its linearization around an approximate solution is a bounded operator. We will also show how this methodology can be used to compute a-posteriori estimates of the errors of the solutions computed, leading to the rigorous verification of the existence of the periodic orbit. If time permits, we will finish showing some numerical outputs of the algorithms presented along the talk. This is a joint work with Rafael de la Llave, School of Mathematics, Georgia Institute of Technology. Numerical algorithms and a-posteriori verification of periodic orbits of the Kuramoto-Sivashinsky equation21 February 2017
Matteo Ruggiero (Paris 7)Abstract: We consider the local dynamical system induced by a non-invertible selfmap f of C^2 fixing the origin. Given a modification (composition of blow-ups) over the origin, the lift of f on the modified space X defines a meromorphic map F. We say that F is algebraically stable if for every compact curve E in X, its orbit through F does not intersect the indeterminacy set of F. We show that, starting from any modification, we can also blow-up some more and obtain another modification for which the lift F is algebraically stable. The proof relies on the study of the action f_* induced by f on a suitable space of valuations V. In particular we construct a distance on V for which f_* is non-expanding. This allows us to deduce fixed point theorems for f_*. If time allows, I will comment on the recent developments about local dynamics on normal surface singularities. Joint work with William Gignac. Local dynamics of non-invertible selfmaps on complex surfaces14 February 2017
Stefan Ruschel (TU Berlin)Abstract: Infectious diseases are among the most prominent threats to mankind. When an outbreak is not foreseen, preventive healthcare cannot be provided. In this case, the unanimous means of control is isolation of infected individuals, as implemented in the 2014 Ebola outbreak in West Africa. We investigate how isolation of identified individuals impacts the spread of an otherwise endemic disease. We model this effect in a homogenous population with mass-action-type contacts. We obtain a mean field description obeying a system of Delay Differential Equations, assuming the identification and isolation time are the same for all individuals. Our analysis reveals that isolation before a critical identification time prevents the outbreak. When isolation is implement after this critical time the disease is endemic, but its severity depends on the time spend in isolation. At first, increasing the isolation time reduces the number of infected individuals. However, long isolation causes the disease to reappear periodically with severe outbreaks. SIQ epidemiological model (Impact of isolation on endemic diseases) 7 February 2017
Matthieu Astorg (Université d'Orleans)Abstract: Finite type maps are a class of analytic maps on complex 1-manifolds introduced by Epstein, that notably include rational maps and entire functions with a finite singular set. Each of those maps possess a natural finite-dimensional moduli space, and one can define a dynamical Teichmüller space parametrizing their quasiconformal conjugacy class. Using the fact that this Teichmüller space immerses into the moduli space, we will generalize rigidity results of Avila, Dominguez, Makienko and Sienra under an assumption of expansion along the critical orbits. Summability condition and rigidity for finite type maps31 January 2017
Nikolaos Karaliolios (Imperial College London)Abstract: We revisit M. Herman's KAM theorem on perturbations of Diophantine translations in tori of arbitrary dimension. We give a partially optimal condition for the perturbation to be smoothly conjugated to the exact model and relate this rigidity result with a Denjoy-like construction of McSwiggen. This is part of an on-going project joint with A. Kocsard, where we try to blow up the rotation set by perturbing a pseudo-rotation, and with S. van Strien, where we explore the possibility of generalizing Denjoy's theory to tori of dimension higher than one. Local rigidity for Diophantine translations on tori of arbitrary dimension24 January 2017
Disheng Xu (Institut de Mathématiques de Jussieu-Paris Rive Gauche)Abstract: We study smooth group actions and random walks on surface under a mild assumption called "weakly expanding". In particular, we prove that some statistical properties (large deviation, equidistribution, etc.) for non-abelian semigroup of linear action on the torus persist under C^1 conservative perturbation of the generators. In addition we give a sufficient condition and an example for robust minimality of the action of semigroup generated by conservative diffeomorphisms on the surfaces. This is a joint work with X. Liu. Statistical properties and robust minimality for smooth random walks on surfaces24 January 2017
Stefano Marmi (Scuola Normale Superiore di Pisa)Abstract: We introduce a coupled deterministic/slow - random/fast discrete time dynamical system to investigate the role of expectation feedbacks on systemic stability of financial markets. In our stylized world financial institutions are subject to value at risk constraints, follow standard mark-to-market and risk management rules and invest in some risky assets whose prices evolve stochastichally and are endogenously driven by the impact of portfolio decisions of financial institutions.When the traders become more myopic and build their expectations giving more weight to short term volatility estimates the equilibrium solution becomes unstable, leverage cycles arise and a cascade of bifurcations leading to chaos follows. This is joint work with Fabrizio Lillo and Piero Mazzarisi. When Panic Makes You Blind: a Chaotic Route to Systemic Risk18 January 2017
Pierre Berger (CNRS-LAGA, Université Paris 13, UPSC)Abstract: Recently we showed that some degenerate bifurcations can occur robustly. Such a phenomena enables ones to prove that some pathological dynamics are not negligible and even typical in the sens of Arnold-Kolmogorov. More precisely, we proved: Theorem: For every $\infty>r\ge 1$, for every $k\ge 0$, for every manifold of dimension $\ge 2$, there exists an open set $\hat U$ of $C^r$-$k$-parameters families of self-mappings, so that for every topologically generic family $(f_a)_a\in \hat U$, for every $\|a\|\le 1$, the mapping $f_a$ displays infinitely many sinks. We will introduce the concept of Emergence which quantifies how wild is the dynamics from the statistical viewpoint, and we will conjecture the local typicality of super-polynomial ones in the space of differentiable dynamical systems. For this end, we will develop the theory of Para-Dynamics, by giving a negative answer to the following problem of Arnold (1992): Theorem: For every $\infty>r\ge 1$, for every $k\ge 0$, for every manifold of dimension $\ge 2$, there exists an open set $\hat U$ of $C^r$-$k$-parameters families of self-mappings, so that for every topologically generic family $(f_a)_a\in \hat U$, for every $\|a\|\le 1$, the map $f_a$ displays a fast increasing number of periodic points: $$\limsup \frac{\log Card \; Per_n \, f_a}n = \infty$$ We also give a negative answer to questions asked by Smale 1967, Bowen in 1978 and by Arnold in 1989, for manifolds of any dimension $\ge 2$: Theorem: For every $\infty\ge r\ge 1$, for every manifold of dimension $\ge 2$, there exists an open set $U$ of $C^r$-diffeomorphisms, so that a generic $f\in U$ displays a fast growth of the number of periodic points. The proof involves a new object, the $\lambda$-$C^r$-parablender, the Renormalization for hetero-dimensional cycles, the Hirsh-Pugh-Shub theory, the parabolic renormalization for parameter family, and the KAM theory. Emergence and Para-Dynamics17 January 2017
Mark Pollicott (University of Warwick)Abstract: TBA Analytic Cantor sets and stationary measures13 December 2016
Julian Newman (University of Bielefeld)Abstract: Suppose we have a parameter-dependent orientation-preserving circle homeomorphism, together with a probability measure $\nu$ on the parameter space. This naturally generates a "random dynamical system" on the circle, where at each time step a parameter is randomly chosen with distribution $\nu$ (independently of all previous time steps) and the associated homeomorphism is applied. Given two parameter-dependent orientation-preserving homeomorphisms defined over the same parameter space (with the same measure), one can define a notion of "topological conjugacy" between the random dynamical systems that they generate. Under certain assumptions, we will classify such parameter-dependent homeomorphisms up to topological conjugacy. Topological conjugacy of iterated random orientation-preserving homeomorphisms of the circle13 December 2016
Raj Prasad (University of Massachussetts)Multitowers, conjugacies and codes13 December 2016
Ricardo Pérez-Marco (CNRS, IMJ-PRG, Paris 7)Abstract: We review the current knowledge on the structure of hedgehogs and its dynamics and present some of the the main open problems. Open problems in hedgehogs dynamics13 December 2016
André de Carvalho (USP)Abstract: The purpose of this talk is not to set a new record for the number of noun modifiers in mathematical definitions, but to present a construction which applies to graph maps in general and yields interesting surface homeomorphisms as follows: 1) a pseudo-Anosov map if the graph map is a train track map; 2) a generalized pseudo-Anosov map if the graph map is post-critically finite and has an irreducible aperiodic transition matrix; 3) an interesting type of surface homeomorphisms which generalizes both the previous classes otherwise. In particular, this produces a unified construction of surface homeomorphisms whose dynamics mimics that of the tent family of interval endomorphisms, completing an earlier construction of unimodal generalized pseudo-Anosov maps in the post-critically finite case. This is joint work with Phil Boyland and Toby Hall. TBSpAgs: Super generalized pseudo-Anosov maps6 December 2016
Johan Taflin (Université de Bourgogne)Abstract: In complex dynamics in several variables, the classical dichotomy between Julia and Fatou sets is insufficient to describe the dynamics. On the other hand, attractors are fondamental objets in the general theory of dynamical systems. In this talk we will explain how to start the ergodic study of attractors in the projective space using pluripotential theory. In particular, to each attractor we will associated an attracting current and an equilibrium measure which give geometric and dynamical information about the attractor. Currents associated to attractors in complex dynamics30 November 2016
Liviana Palmisano (University of Bristol)Abstract: We prove that circle maps with a flat interval and degenerate geometry are an example of a dynamical system for which the topological classes don't coincide with the rigidity classes. Contrarily to all the well-known examples in one-dimensional dynamics (such as circle diffeomorphisms, unimodal interval maps at the boundary of chaos, critical circle maps) we show that the class of functions with Fibonacci rotation numbers is a C^1 manifold which is foliated with finite codimension rigidity classes. This is a joint work with M. Martens. Foliations of Rigidity Classes29 November 2016
Sanju Velani (University of York)Abstract: The aim is to initiate a ``manifold'' theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite Kleinian group framework. Also, we consider the natural analogue of Davenport's problem regarding badly approximable limit points in a given subset of the limit set. Beyond extremality, we discuss potential Khintchine-type statements for subsets of the limit set. These can be interpreted as the conjectural ``manifold'' strengthening of Sullivan's logarithmic law for geodesics. Diophantine approximation in Kleinian groups: singular, extremal and bad limit points22 November 2016
Jimmy Tseng (University of Bristol)Abstract: A theme in ergodic theory is exploring how much information ergodicity itself, without using stronger properties such as mixing, can give about an ergodic dynamical system. In this talk, we will focus on the system given by diagonal flows on the space of unimodular lattices and use the Birkhoff ergodic theorem, the Siegel mean value theorem, and an approximation argument to give an alternative proof of a form of Schmidt’s theorem on the number of integer lattice solutions of a system of inequalities where the solutions have uniformly bounded height. Joint work with J. Athreya and A. Parrish. Applications of the Birkhoff ergodic theorem and the Siegel mean value theorem to Diophantine approximation15 November 2016
Oleg Smolyanov (Moscow University)Feynman path integrals and quantum anomalie10 November 2016
Toby Hall (University of Liverpool)Abstract: In joint work with Phil Boyland and Andre de Carvalho we show that, for every unimodal map $f$ with topological entropy $h(f)\ge \frac{1}{2}\log2$, the natural extension of $f$ is semi-conjugate to a sphere homeomorphism. I will discuss some of the ideas of the proof and the dynamics of the resulting sphere homeomorphisms. Sphere homeomorphisms from unimodal maps8 November 2016
Vjacheslav Grines (Novgorod State University)Abstract: This talk is devoted to finding sufficient conditions for the existence of closed and heteroclinic trajectories of Morse-Smale flows on a smooth closed orientable three-dimensional manifold $M^3$. Let $f^t$ Morse-Smale flow on $M^3$, whose non-wandering set consists of $k$ saddles and $ \ell $ nodes (sinks and sources). Set $g=\frac{k-\ell + 2}{2}$. We represent two results: 1. There exists a Morse-Smale flow without closed and heteroclinic trajectories on $M^3$ if and only if $M^3$ is the sphere $S^3$ and $k = \ell - 2$, or $M^3$ is the connected sum of $g$ copies of $S^2\times S^1$. 2. If the Heegaard genus of $ M ^3 $ more than the number $g$ then non-wandering set of a Morse-Smale flow $ f ^ t $ has at least one closed trajectory. On inter-relation between existence of heteroclinic and closed trajectories of Morse-Smale flows and the topology of the ambient manifold.1 November 2016
Olga Pochinka (Novgorod State University)Abstract: In this talk I will discuss the problem of topological classification of simple structural stable system, from Andronov-Potryagin to the present time. Classification of Morse-Smale dynamical systems1 November 2016
Nikolaos Karaliolios (Imperial College London)Abstract: After having introduced some definitions, we will present a classification holding true in an open set of the total space of such dynamical systems. The phenomena that occur comprise the foliation of the phase space in smooth KAM tori, in tori of lower regularity down to simply measurable ones, and (for systems where the tori break down completely) Weak Mixing in the fibres and Unique Ergodicity in the space of Distributions. This classification is based on works of H. Eliasson, R. Krikorian and the author, among others. The KAM regime for Quasi-Periodic cocycles in T^dxSU(2)25 October 2016
Laurent Stolovitch (Université de Nice-Sophia Antipolis)Abstract: We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension $n\geq 3$. Based on Belitskii's work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensure that the normalizing transformation is holomorphic at the fixed point. We shall show that this sufficient condition is a {\it nilpotent version} of Bruno's condition (A). In dimension 2, no condition is required since, according to Str{\'o}{\.z}yna-{\.Z}o{\l}adek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton's method and $\frak{sl}_2(\Bbb C)$-representations. Holomorphic normal form of nonlinear perturbations of nilpotent vector fields13 October 2016
Erik Bollt (Clarkson University)Identifying Interactions in Complex Networked Dynamical Systems through Causation Entropy14 September 2016
Isabel Rios (UFF)Uniqueness of equilibrium states for a family of partially hyperbolic systems14 September 2016
Björn Winckler (Stony Brook University)Abstract: Two infinitely renormalizable unimodal maps of bounded combinatorics which belong to the same topological conjugacy class have Cantor attractors which on small scales look the same. Colloquially, we say that topology determines geometry. This is what we have come to expect for interval maps. In this talk I will discuss recent surprising results that show that topology does not always determine geometry in the case of bounded combinatorics Lorenz maps. Topology determines geometry?23 August 2016
Rafael Obaya (University of Valladolid)Abstract: In this talk the dissipativity of a family of non-autonomous linear-quadratic control processes is studied using methods of topological dynamics. The application of the Pontryaguin Maximum Principle to this problem give rise to a family of linear Hamiltonian systems for which the existence of an exponential dichotomy is assumed, but no condition of controllability is imposed. As a consequence, some of the systems of this family could be abnormal and some of their dynamical properties are given. Sufficiente conditions for the dissipativity of the processes are provided assuming the existence of global positive solutions of the Riccati equation induced by the family of linear hamiltonian sytems or by a conveniente disconjugate perturbation of it. Abnormal linear hamiltonian systems with application in non-autonomous linear-quadratic control processes22 July 2016
Peter Hazard (University of Toronto)Abstract: In the early 1980's it was observed that period-doubling cascades in families of area-preserving planar diffeomorphisms occur at a universal rate. As in the one-dimensional case it was conjectured that there exists a renormalization operator, acting on some class of area-preserving planar maps, possessing a hyperbolic fixed-point, with codimension-one stable manifold and one-dimensional unstable manifold. Later, a computer-assisted proof of the existence of the fixed point was given by Eckmann, Koch and Wittwer. However, a conceptual proof is still missing. I will discuss recent progress on this problem. This is joint work with D. Gaidashev. Period-Doubling Renormalization of Area-Preserving Planar Maps23 June 2016
Arnaldo Nogueira (Institut de Mathématiques de Marseille)Abstract: Let I be the unit interval [0,1) and -1 < λ < 1. Let f : I → R be a piecewise λ-affine map, that is, there are real numbers b_1;,..., b_u and a sequence of points 0 = c_1 < c_2 <...< c_{u-1} < c_u = 1 such that f(x)= λx + b_i, for every x in the interval [c_{i-1} , c_i). In the talk, we examine the class of maps f_ρ = f + ρ mod 1, where ρ is a real parameter. We prove that, for Lebesgue almost every real parameter \rho, the map f_ρ is asymptotically periodic. More precisely, f_ρ has at most 2n periodic orbits and the ω-limit set of every x in the unit interval I is a periodic orbit. Our theorem is an extension of the result obtained for the much-studied case where λ is a positive constant and f is the continuous map x → λx . This is a joint work with Benito Pires and Rafael Rosales. Topological dynamics of piecewise λ-affine maps16 June 2016
Ale Jan Homburg (University of Amsterdam)Abstract: I will discuss iterated function systems on the unit interval. The iterated function systems are generated by diffeomorphisms or by logistic maps. I will discuss possible dynamics such as intermittency in these contexts. Random interval maps14 June 2016
Huaizhong Zhao (Loughborough University)Abstract: TBA Ergodicity of Random Dynamical Systems Under Random Periodic Regimes 2 June 2016
Chunrong Feng (Loughborough University)Abstract: TBA Random periodic solutions of SDEs2 June 2016
Tuomas Sahlsten (University of Bristol)Abstract: Quantum Ergodicity Theorem of Shnirelman, Zelditch and Colin de Verdière is an equidistribution result of eigenfunctions of the Laplacian in large frequency limit on a Riemannian manifold with an ergodic geodesic flow. We complement this work by introducing a Quantum Ergodicity theorem on hyperbolic surfaces, where instead of taking high frequency limits, we fix an interval of frequencies and vary the geometric parameters of the surface such as volume, injectivity radius and genus. In particular, we are interested in such results under Benjamini-Schramm convergence of hyperbolic surfaces. This work is inspired by analogous results for holomorphic cusp forms and eigenfunctions for large regular graphs. The proof uses mixing properties of the geodesic flow together with a wave propagation approach recently considered by Brooks, Le Masson and Lindenstrauss on discrete graphs, making it quite different from the usual proof of quantum ergodicity in the large eigenvalue limit, in the sense that it does not use any microlocal analysis. Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces26 May 2016
Esmerelda Sousa Dias (IST, Lisbon)Abstract: Cluster maps are birational maps arising from mutation-periodic quivers. These maps preserve a log-canonical presymplectic form $\omega$ (defined in terms of a skew-symmetric matrix B which simultaneously defines the map) and so they can be reduced to symplectic maps. On the other hand, the existence of certain quadratic Poisson structures for which the cluster map is a Poisson map, leads to another reduction of the same map. It will be explained how the null foliation of $\omega$, the symplectic foliation of the referred Poisson structures, and the dynamics of the reduced maps, offer a complete understanding of the geometry underlying the (discrete) dynamics of some cluster maps. This is joint work with Inês Cruz (FCUP) and Helena Mena-Matos (FCUP) Poisson structures and the dynamics of cluster maps24 May 2016
Jérémy Blanc (Universität Basel)Abstract: "One way to study the iterations or birational maps is to compute the sequence of degrees we obtain. The type of growth ( bounded / polynomial / exponential) is well-known in dimension 2 and is useful to study the dynamics of the map (entropy, growth of fixed points of iterates,...) but also the conjugacy class, the centraliser, … I will describe the results one knows in this direction and some of the open questions in higher dimension." Degree growth of birational maps19 May 2016
Davoud Cheraghi (Imperial College London)Abstract: In the 1970’s, physicists, working numerically, observed “universal scaling laws” in the bifurcation loci of generic families of one-dimensional analytic transformations. To explain this phenomena, they conjectured that a non-linear (renormalisation) operator acting on an infinite-dimensional function-space is hyperbolic which has a one-dimensional unstable direction and a co-dimension-one stable direction. This was the focus of research in the 90’s, leading to a rigorous proof of the conjecture on subspaces where the operator is compact, while a successful study of the problem on subspaces where the operator is not compact has been obtained only recently. In an introductory talk, we present the motivations and some of the key ideas employed in the subject. Hyperbolicity of renormalization operators6 May 2016
Stefano Luzzatto (ICTP)SRB measures for nonuniformly hyperbolic surface diffeomorphisms19 April 2016
John Smillie (Univeristy of Warwick)Abstract: I will begin this talk with a general introduction to the dynamics of the complex Henon family of diffeomorphisms. What might we learn by studying them? What techniques have proven effective? I will discuss joint work with Eric Bedford which addresses the question of what is the natural two dimensional analogue of the Misiurewicz property for polynomial maps. This work addresses the connection between regularity of stable and unstable manifolds and uniformity of expansion and the presence of tangence’s. Notions of regularity for Complex Henon Diffeomorphisms17 March 2016
Dalia Terhesiu (University of Exeter)Abstract: For non-uniformly expanding maps inducing w.r.t. a general return time to Gibbs Markov maps, we provide sufficient conditions for obtaining sharp estimates for the correlation function. This applies to both, finite and infinite measure setting. The results are illustrated by non Markov interval maps with an indifferent fixed point. This is joint work with H. Bruin. Sharp mixing rates via inducing w.r.t. general return times17 March 2016
Stefano Marmi (Scuola Normale Superiore di Pisa)Regularity of solutions of the cohomological equation for interval exchange maps11 March 2016
Katsutoshi Shinohara (Hitotsubashi University)Abstract: beta-encoders are Analog-to-Digital encoders based on beta-expansions. In order to give estimates of their quantization errors, the analysis of corresponding piecewise-linear maps plays an important role. In this talk, I will talk about this interplay between electronic circuit design theory and dynamical systems. beta-encoders and Fredholm determinant of generalized beta-maps11 March 2016
Henna Koivusalo (University of York)Abstract: We study local properties of fractal sets, their tangent sets. These are the limiting patterns in the Hausdorff metric when zooming in to a point, along a sequence of scales converging to 0. Tangent sets give a good description of the fine structure of the fractal set, and are well-understood for self similar sets. We investigate tangent sets of self affine sets in the plane. We prove that under some natural assumptions on the self affine set, almost everywhere the tangent sets have a fibered structure; that is, they are the product of a line segment with a Cantor set in a suitably chosen basis. This is in stark contrast to the self similar case, where the tangent sets are deformations of the self-similar set itself. Self affine sets with a fibered tangent structure10 March 2016
John Mackay (University of Bristol)Abstract: Groups can be investigated by considering how they can act on suitable spaces. For example, the notion of Kazhdan's property (T), relating to how groups can act on Hilbert spaces, has been used very successfully for many applications over the last fifty years. More recently, similar definitions have been used to study actions on other L^p spaces, with motivations from dynamics amongst other fields. After outlining some of this story, I'll explain why actions of certain random groups on L^p spaces have fixed points. (Joint work with Cornelia Drutu. Fixed point properties for groups acting on L^p spaces3 March 2016
Paul Verschueren (Open University)Abstract: We discuss an important class of functions, denoted Quasiperiodic Sums and Products, which link the study of critical phenomena in diverse fields such as the birth of Strange Non-Chaotic Attractors, Critical KAM Theory, and q-series (much used in String Theory). They also link study of pure topics in Power Series analysis, Partition Theory, and Diophantine Approximation. The graphs of these functions form intriguing geometrically strange and self-similar structures. They are easy and rewarding to investigate numerically, and suggest many avenues for investigation. However they prove remarkably resistant to rigorous analysis. In this talk we will review results on the most heavily studied example, Sudler's product of sines. We will also report on our own new work towards developing a rigorous theoretical foundation in this area. In particular this allows us to settle negatively a conjecture of Erd?s & Szekeres from 1959, and to prove a number of experimental results reported recently by Knill & Tangerman (2011). Quasiperiodic sums and products2 March 2016
Charles Walkden (University of Manchester)Abstract: We define and explore the notion of finitely coupled iterated function system that satisfy a notion of contraction on average. We study the existence and uniqueness of an invariant probability measure for such systems by defining an appropriate transfer operator. Using arguments of Keller and Liverani on properties of perturbations of linear operators, the continuity of the invariant measure as the coupling tends to zero is proved. Other limit theorems, including a central limit theorem, can also be proved. This is joint work with Anthony Chiu. Stochastic stability and limit theorems for coupled iterated function systems that contract on average 25 February 2016
Neil Dobbs (University of Geneva)Abstract: Free energy is not semi-continuous, but we show that in some contexts, it nearly is. This engenders proofs of existence of equilibrium states and (almost) continuity of equilibrium states as one varies the potential and the map. We do this in the context of one-dimensional dynamics. Joint with M. Todd. Free energy jumps up22 February 2016
Masayuki Asaoka (University of Kyoto)A C-infinity closing lemma for Hamiltonian diffeomorphisms on surfaces10 February 2016
Péter Bálint (Budapest University of Technology and Economics )Abstract: TBA Mean field coupling of doubling maps5 February 2016
Stergios Antonakoudis (Cambridge University)Abstract: TBA The complex geometry of Teichmüller spaces and bounded symmetric domains4 February 2016
Oleg Smolyanov (Moscow University)Quasi-invariant measures on infinite-dimensional spaces4 February 2016
Diana Tolstyga (Moscow University)Feynman formulas for stochastic and quantum dynamics of particles in multidimensional domains3 February 2016
Oleg Ivrii (University of Helsinki)Abstract: We examine several characteristics of conformal maps that resemble the variance of a Gaussian: (1) asymptotic variance, (2) the constant in Makarov’s law of iterated logarithm and (3) the second derivative of the integral means spectrum at the origin, amongst others. While these quantities need not be equal in general, they agree for domains whose boundaries are regular fractals such as Julia sets or limit sets of quasi-Fuchsian groups. We show these characteristics have the same universal bounds over various collections of conformal maps. As an application, we show that the maximal Hausdorff dimension of a k-quasicircle is strictly less than 1 + k^2. (Part of this work is joint with I. Kayumov.) On Makarov’s principle in conformal mapping28 January 2016
Tiago Pereira (University of Sao Paulo, Sao Carlos)Abstract: TBA Improving Network connectivity can lead to Functional Failures28 January 2016
Fabrizio Bianchi (University of Toulouse)Abstract: Starting from the basics in holomorphic dynamics in one variable, I will review the classical theory by Mané-Sad-Sullivan about stability of rational maps and briefly present a generalization of this theory in higher dimension. I will focus on the arguments that do not readily generalize in the second case, and introduce the tools and ideas that allow to overcome these problems. Holomorphic motions of Julia sets13 January 2016
Dmitry Todorov (Centre de Physique Théorique (Aix-Marseille Université))Abstract: There is a strong and long-lasting interest in chaotic dynamical systems as mathematical models of various processes in different areas of science. Like for any other mathematical models for chaotic systems to be useful it is desirable that they have stability properties. There are exist different stability properties. In particular, there exist two notions of stability with respect to small per-iteration perturbations – shadowing property and stochastic stability. System is said to have shadowing property if every (pseudo)trajectory with small errors can be uniformly approximated by a trajectory without errors. System is stochastically stable if the noise perturbing the system is considered to be random and invariant measures for the stochastic process corresponding to the perturbed system converge to an invariant measure of the unperturbed system. Although conceptually these properties are somewhat similar and it is known that some chaotic systems have both properties, no direct relations between shadowing and stochastic stability were established so far. I will discuss some of these relations both for qualitative and quantitative versions of shadowing and stochastic stability. Shadowing and stochastic stability10 December 2015
Anton Solomko (University of Bristol)Abstract: A probability preserving action T is called simple if the set of its self-joinings is the smallest possible and arise from the centralizer C(T) of T. For simple actions, there is an interplay between algebraic properties of the group C(T) and ergodic properties of T. First I will explain (C,F)-construction of measure preserving actions, which is an algebraic counterpart of classical cutting-and-stacking technique. This method combined with Ornstein's technique of 'random spacers' allows to produce simple actions with centralizers prescribed in advance. Then I will focus on some new examples, obtained via (C,F)-construction, including mixing actions with uncountably many prime factors and mixing transformations with infinitely many non isomorphic embeddings into a flow. Based on joint works with Alexandre Danilenko and Joanna Ku?aga-Przymus. On centralizers of simple mixing actions10 December 2015
Caroline Series (University of Warwick)Abstract: A Kleinian group is a discrete group of isometries of hyperbolic 3-space. Its limit set, contained in the Riemann sphere, is the set of accumulation points of any orbit. In particular the limit set of a hyperbolic surface group F is the unit circle. If G is a Kleinian group abstractly isomorphic to F, there is an induced map, known as a Cannon-Thurston (CT) map, between their limit sets. More precisely, the CT-map is a continuous equivariant map from the unit circle into the Riemann sphere. Suppose now F is fixed while G varies. We discuss work with Mahan Mj about the behaviour of the corresponding CT-maps, viewed as maps from the circle to the sphere. We explain how a simple criterion for the existence of a CT-map can be adapted to establish conditions on convergence of a sequence of groups G_n under which the corresponding sequence of CT-maps converges uniformly to the expected limit. We also discuss an example which shows that under certain circumstances, CT-maps may not even converge pointwise. Continuous motions of limit sets26 November 2015
David Simmons (University of York)Abstract: We consider a class of measures from Diophantine approximation known as \emph{extremal} measures. The class of measures known to be extremal has expanded in recent years to include not only the Lebesgue measures of nondegenerate manifolds, but also various measures defined using conformal dynamics. In this talk I will describe this history as well as describing a new geometric condition which implies extremality, giving examples of dynamical measures satisfying this condition which could not previously be proven to be extremal. This work is joint with Tushar Das, Lior Fishman, and Mariusz Urba?ski. Extremality and dynamically defined measures19 November 2015
Simon Baker (University of Reading)Abstract: Expansions in non-integer bases is a natural generalisation of the well know binary/tertiary/decimal expansions. Associated to these expansions is a natural class of dynamical systems. In this talk I will introduce these dynamical systems and discuss their first return dynamics. In particular, I will discuss allowable sequences of return times, and when the first return map is a generalised Luroth series transformation. Induced random beta transformations 12 November 2015
Zemer Kosloff (University of Warwick)Abstract: Markov partitions introduced by Sinai and Adler and Weiss are a tool that enables transfering questions about ergodic theory of Anosov Diffeomorphisms into questions about Topological Markov Shifts and Markov Chains. This talk will be about a reverse reasonning, that gives a construction of $C^{1}$ conservative (satisfy Poincare\textquoteright s reccurrence) Anosov Diffeomorphism of $\mathbb{T}^{2}$ without a Lebesgue absolutely continuous invariant measure. By a theorem of Gurevic and Oseledec, this can\textquoteright t happen if the map is $C^{1+\alpha}$ with $\alpha>0$. Our method relies on first choosing a nice Toral Automorphism with a nice Markov partition and then constructing bad conservative Markov measure on the symbolic space given by the Markov partition. We then push this measure back to the Torus to obtain a bad measure for the Toral automorphism. The final stage is to find by smooth realization a conjugating map $H$ such that $H\circ f\circ H^{-1}$ with Lebesgue measure is metric equivalent to $\left(\mathbb{T}^{2},\mathcal{B},f,\ Bad\ measure\right)$. Conservative Anosov diffeomorphisms of the two torus without an absolutely continuous invariant measure5 November 2015
Carlos Siqueira (Imperial College London)Abstract: We introduce for the first time the notion of hyperbolicity and structural stability for the one parameter family of (multi-valued) complex maps f_c(z) =z^r +c, where r > 1 is a rational number. Such a family is particularly important not only as a generalisation of the quadratic family but also because it is a conformal extension (as a Riemann surface) of some unimodal maps. We show that its Julia set is the projection of a solenoid and consists of uncountably many quasi-conformal arcs for parameters close to the origin. We also estimate the Hausdorff dimension of the Julia set using the formalism of Gibbs states. The general structural stability is proved for every hyperbolic parameter using holomorphic motions on Banach spaces. For such parameters the limit set (accumulation points out of pre-orbits starting at the basin of infinity) splits into two disjoint, compact and invariant sets: the standard Julia set and the dual Julia set. The former is the closure of repelling periodic orbits and the latter is the closure of attracting periodic orbits. Surprisingly, the dual Julia set is typically a Cantor set associated with a Conformal Iterated Function System. Therefore, we have infinitely many attracting periodic points! This set is always finite for rational maps. Cantors sets and structural stability for holomorphic correspondences29 October 2015
Alexey Korepanov (University of Warwick)Abstract: We consider a family of Pomeau-Manneville type interval maps $T_\alpha$, parametrized by $\alpha \in (0,1)$, with the unique absolutely continuous invariant probability measures $\nu_\alpha$, and rate of correlations decay $n^{1-1/\alpha}$. We show that despite the absence of a spectral gap for all $\alpha \in (0,1)$ and despite nonsummable correlations for $\alpha \geq 1/2$, the map $\alpha \mapsto \int \varphi \, d\nu_\alpha$ is continuously differentiable for $\varphi \in L^{q}[0,1]$ for $q$ sufficiently large. Linear response for intermittent maps with summable and nonsummable decay of correlations22 October 2015
Bastien Fernandez (Laboratoire de Probabilités et Modèles Aléatoires (LPMA) CNRS - Université Paris 7 Diderot)Abstract: The Kuramoto model is the archetype of heterogeneous systems of (globally) coupled oscillators with dissipative dynamics. In this model, the order parameter that quantifies the population synchrony decays to 0 in time, as long as the interaction strength remains small (so that the uniformly distributed stationary solution remains stable). While this phenomenon has been identified since the first studies of the model, its proof remained to be provided (most studies in the literature are limited to the linearized dynamics). The goal of this talk is to address this issue. I will present rigorous results on the Kuramoto dynamics, and in particular, I will sketch a proof of nonlinear damping of the order parameter in the incoherent phase. Joint work with D. Gérard-Varet and G. Giacomin Landau damping in the Kuramoto model8 October 2015
Sina Tureli (ICTP)Abstract: We are going to give a new theorem for integrability of rank 2 continuous tangent sub-bundles (distributions) in 3 dimensional manifolds. The theorem depends on approximations and a natural notion of asymptotic involutivity. We will then state some applications to PDE, ODE and Dynamical Systems. A Frobenius Theorem for Continuous Distributions2 September 2015
Gary Froyland (University of New South Wales)Abstract: Transfer operators are global descriptors of ensemble evolution under nonlinear dynamics and form the basis of efficient methods of computing a variety of statistical quantities and geometric objects associated with the dynamics. I will discuss two related methods of identifying and tracking coherent structures in time-dependent fluid flow; one based on probabilistic ideas and the other on geometric ideas. Applications to geophysical fluid flow will be presented. Transfer operators and dynamics23 July 2015
Christian Kühn (Technical University of Vienna)Abstract: In this talk I am going to report on the geometric decomposition of nonlinear dynamics in the Olsen model. Although this model has been proposed by Olsen already in the late 1970s and has been investigated many times with di fferent methods, a full understanding of the mechanisms that lead to oscillatory patterns was not available. Nonlinearity, several small parameters, higher-dimensionality and wide parameter ranges are the key difficulties in this context. However, using methods from the geometric theory of multiple time scale dynamical systems, it is possible to identify the main mechanisms. In particular, I am going to illustrate the main steps to prove the existence of non-classical relaxation oscillations and explain how one may deal with mixed-modes and chaotic solutions from the same viewpoint. Multiscale Oscillations in the Olsen Model16 July 2015
Arno Berger (University of Alberta)Abstract: The study of numbers generated in one way or another by dynamical systems is a classical, multifaceted field. A notorious gem in this field is the wide-spread, unexpected emergence of a particular logarithmic distribution, commonly referred to as Benford's Law (BL). This talk will describe how dynamical systems may conform to BL, and what this in turn may tell about the dynamics in question. As one illustrative example, a characterization of BL in finite-dimensional linear systems, recently obtained in joint work with G. Eshun, will be discussed in some detail. While this result is quite general and implies, for instance, that such systems typically conform to BL in a very strong sense, it also raises intriguing new questions. Digit distributions in dynamics25 June 2015
Ken Palmer (Providence University, Taiwan)Abstract: This is joint work with Kaijen Cheng. There are two parts. In the first part, we study unimodal maps on the closed unit interval, which have a stable period 3 orbit and an unstable period 3 orbit, and give conditions under which all points in the open unit interval are either asymptotic to the stable period 3 orbit or land after a finite time on an invariant Cantor set $\Lambda$ on which the dynamics is chaotic. For the particular value of $\mu=3.839$, Devaney, following ideas of Smale and Williams, shows that the logistic map $f(x)=\mu x(1-x)$ has this property. In this case the stable and unstable period 3 orbits appear when $\mu=1+\sqrt{8}$. We use our theorem to show that the property holds for all values of $\mu>1+\sqrt{8}$ for which the stable period 3 orbit remains stable. \medskip In the second part we study a model for masting, that is, the intermittent production of flowers and fruit by trees. This model is an asymmetric unimodal piecewise linear map with one parameter $k>0$. When $k<1$ all orbits are attracted to a stable fixed point. On the other hand, when $k>\kappa_0=(1+\sqrt{5})/2$, the map is chaotic on $[1-k,1]$. As $k$ increases through $1$, chaos arises immediately. We find a strictly decreasing sequence $\kappa_p$, $p\ge 0$, with $\kappa_p\to 1$ as $p\to\infty$ such that when $\kappa_p <= k < \kappa_{p-1}$, $p\ge 1$, $g$ has an invariant set $\Lambda(k,p)$ consisting of $2^p$ disjoint closed intervals on which the dynamics is chaotic and to which all points except for a countable set are attracted; as $k$ decreases through $\kappa_p$ each of the $2^p$ intervals splits into two (the middle part drops out). $\Lambda(k,p)$ is squeezed towards the two point set $\{0,1\}$ as $p\to\infty$. Period 3 and Chaos for Unimodal Maps18 June 2015
Olga Pochinka (Novgorod State University)Abstract: http://wwwf.imperial.ac.uk/~tclark/Pochinka-abstract.pdf On topological classification of diffeomorphisms of surfaces with a finite number of moduli of stability5 June 2015
Vjacheslav Grines (Novgorod State University)Abstract: http://wwwf.imperial.ac.uk/~tclark/Grines-abstract.pdf On topological classification of structurally stable cascades with two-dimensional basic sets on 3-manifolds5 June 2015
Thomas Kaijser (Linköping University)Abstract: I shall consider three examples: 1) Stochastic perturbations of iterations of an analytic homeomorphism of the circle having irrational rotation number; 2) Stochastic iterations of the two functions f(x)= ax(1-x) and g(x) = bx(1-x); and 3) Stochastic perturbation of iterations of the function f(x)= x/2 +17/30 (mod 1). I will mainly spent time on the first example. On stochastic iterations of circle maps and interval maps28 May 2015
Alexandre De Zotti (University of Liverpool)Abstract: We recall some results on the rigidity of dynamics of transcendental entire functions in the class B near infinity, obtained by Lasse Rempe-Gillen and Gwyneth Stallard. Those results are based on quasiconformal rigidity of the dynamics of class B functions near infinity. Different dimension quantities are known to be invariant in affine equivalence classes of maps. Poincaré maps shows that this invariance may not hold for quasiconformal classes. This is a joint work with Lasse Rempe-Gillen. The eventual hyperbolic dimension of entire functions14 May 2015
Giancarlo Benettin (University of Padova)Abstract: In 1954, Fermi, Pasta and Ulam for the first time used a computer to understand the ergodic behavior of a dynamical system with many degrees of freedom, interesting to investigate the very foundations of Statistical Mechanics. Several baranches of physical and mathematical investigations started from that paper. The aim of the talk is to revisit, in the light of some recent numerical results, some significant ideas and conjectures on the model. In particular: (i) The presence, in the model, of (at least) two well separated time-scales: a short one, where only a few normal coordinates share energy, and a larger one, where energy equipartition among all normal modes occurs and the behavior of the model, in view of Statistical Mechanics, is regular. (ii) The fact that in the short time scale the dynamics of FPU, in spite of the partial energy sharing, is essentially integrable and closely follows the dynamics of the Toda model, while in the large time scale nonintegrability becomes manifest. The stability of results in the limit of large N (ideally, the search of uniformity in N) will play a central role. The gap between numerical insights and mathematical results, unfortunately, is large. The Fermi-Pasta-Ulam problem: old ideas, recent results, open problems11 May 2015
Christian Bick (University of Exeter)Abstract: Phase oscillators are interacting oscillatory units whose state is solely determined by a phase variable taking values on the circle. Probably the most widely studied system of interacting phase oscillators is the Kuramoto model; here, the interaction between two oscillators is given by the sine of the phase difference. What happens if the interaction between pairs of oscillators is more general, for example if the interaction contains more than a single Fourier component? We discuss the impact of generalized coupling on phase oscillator dynamics. Moreover, we show how generalized couplings can be useful in applications; for example, it allows to control spatially localized states in non-locally coupled oscillator systems. Dynamics of Coupled Phase Oscillators with Generalized Coupling30 April 2015
Michael Tsiflakos (University of Vienna)Abstract: TBA Chaotic behaviour of bouncing balls20 April 2015
Christian Pötzsche (Alpen Adria University of Klagenfurt)Abstract: The dichotomy spectrum (also known as dynamical or Sacker-Sell spectrum) is a crucial notion in the theory of dynamical systems. It contains information on stability and robustness properties, and is moreover fundamental to establish a geometric theory including invariant manifolds, linearization and normal forms. However, recent applications in nonautonomous bifurcation theory showed that a detailed insight into the fine structure of this spectral notion is necessary. On this basis, we explore a helpful connection between the dichotomy spectrum and operator theory. It relates the asymptotic behavior of linear nonautonomous equations to the (approximate) point, surjectivity and Fredholm spectra of weighted shifts. This link yields several dynamically meaningful subsets of the dichotomy spectrum, which (a) allow to classify nonautonomous bifurcations on a linear basis already (b) simplifies proofs for results on the long term dynamics of difference and differential equations with explicitly time-dependent right-hand side (c) yield sufficient conditions for a continuous (rather than merely an upper-semicontinuous) behavior of the dichotomy spectrum under perturbation On the dichotomy spectrum9 April 2015
Dierk Schleicher (Jacobs University)Abstract: We all know that Newton’s method is efficient as a local root finder, but has a reputation of being globally unpredictable. We present an algorithm that makes it globally efficient and predictable, and (news of a few days ago) make it possible to find, in practice, all roots of polynomials of degree one million in just a few seconds on a standard personal computer. (Party joint work with Robin Stoll.) Newton’s method as a practical root finder30 March 2015
Jonathan Fraser (University of Manchester)Abstract: I will consider the abstract problem of ‘zooming in’ on a shift ergodic measure. This is motivated by recent and influential developments in the ergodic theory of the so called ‘scenery flow’, largely due to Mike Hochman and his collaborators. In order to capture the zoom in dynamics one defines a sequence of ‘scenery distributions’, which are measures on the space of measures defined by summing up Dirac masses along the orbit of the original measure under the zooming in process. I will show that at almost every point the scenery distributions converge to a common distribution on the space of measures and show how to characterise this distribution in terms of an appropriately defined reverse Jacobian. This is based on joint work with Mark Pollicott. Blowing up ergodic measures19 March 2015
Vito Latora (Queen Mary University of London)Abstract: TBA The growth of cities and neural networks19 March 2015
Liviana Palmisano (Institute of Mathematics Polish Academy of Sciences)Abstract: We study C^2 weakly order preserving circle maps with a flat interval. We prove that, if the rotation number is of bounded type, then there is a sharp transition from the degenerate geometry to the bounded geometry depending on the degree of the singularities at the boundary of the flat interval. The general case of functions with rotation number of unbounded type is also studied. The situation becomes more complicated due to the presence of underlying parabolic phenomena. Moreover, the results obtained for circle maps allow us to study the dynamics of Cherry flows. In particular we analyze their metric, ergodic and topological properties. On circle endomorphisms with a flat interval and Cherry flows13 March 2015
Shin Kiriki (Tokai University)Abstract: We give an answer to a version of the open problem of F. Takens 2008 which is related to historic behavior of dynamical systems. To obtain the answer, we show the existence of non-trivial wandering domains near a homoclinic tangency, which is conjectured by Colli and Vargas 2001. Concretely speaking, it is proved that any Newhouse open set in the C^r topology of two-dimensional diffeomorphisms with 2 \leq r < \infty is contained in the closure of the set of diffeomorphisms which have non-trivial wandering domains whose forward orbits have historic behavior. Moreover, this result implies an answer in the C^r category to one of the open problems of van Strien 2010 which is concerned with wandering domains for H\'enon family. Takens' last problem and existence of non-trivial wandering domains12 March 2015
Zin Arai (Hokkaido University)Abstract: We discuss the structure of the parameter space of the Henon family. Our main tool is the monodromy representation that assigns an automorphism of the full shift to each loop in the hyperbolic parameter locus of the complex Henon family. We show that the monodromy carries the information of the bifurcations taking place inside the loop, and this enables us to construct pruning fronts, a generalization of kneading theory to the real Henon family. Furthermore, assuming that there exist infinitely many non-Wieferich prime numbers (it suffices to assume "abc conjecture"), we show that monodromy automorphisms must satisfy a certain algebraic condition, which imposes geometric restrictions on the structure of the parameter space. On the monodromy and bifurcations of the Henon map12 March 2015
Remus Radu (Stony Brook University)Abstract: We study the global dynamics of complex Hénon maps with a semi-parabolic fixed point that arise as small perturbations of a quadratic polynomial p with a parabolic fixed point. We prove that this family of semi-parabolic Hénon maps is structurally stable on the sets J and J^+; the Julia set J is homeomorphic to a quotiented solenoid (hence connected), while the Julia set J^+ inside a polydisk is a fiber bundle over the Julia set of the polynomial p. We then exhibit certain paths in parameter space for which the semi-parabolic structure can be deformed into a hyperbolic structure, and show that the parametric region of semi-parabolic Hénon maps with small Jacobian lies in the boundary of a hyperbolic component of the Hénon connectedness locus. This is joint work with Raluca Tanase. Semi-parabolic Hénon maps and perturbations12 March 2015
Peter Ashwin (University of Exeter)Abstract: A chimera state in a coupled oscillator system is a dynamical state that combines regions of coherence (or synchrony) with regions if incoherence (or asynchrony). However exactly how one defines these terms affects which states can be identified as chimeras and which not, especially for small groups of phase oscillators. In this talk I will discuss some joint work with O. Burylko (Kiev) where propose a definition of a weak chimera based on partial frequency synchrony. This allows one to explore the existence and stability of weak chimeras in small networks of phase oscillators. In particular we find that the usual coupling considered when investigating chimeras (Kuramoto-Sakaguchi type) leads to rather degenerate sets of neutrally stable chimeras, while more generic coupling unfolds this degeneracy. Chimera states for minimal systems of phase oscillators5 March 2015
Jochen Bröcker (University of Reading)Abstract: The filtering process is the conditional probability of the state of a Markov process (the signal process), given a series of observations which are conditionally independent given the signal process. Stability means that the distance between the true filtering process and a wrongly initialised filter converges to zero (at an exponential rate in our case) as time progresses. In the present setting, the signal process arises through iterating an iid series of random and uniformly expanding maps on a Riemannian manifold. We will see that the problem bears a strong similarity to the problem of constructing random invariant measures for these mappings, and hence the connection with dynamical systems. A fruitful approach in both cases is to ensure that the Frobenius Perron operator (or the filtering operator) is a contraction wrt to Hilbert's projective metric on suitably defined cones of functions. Stability of the nonlinear filter for random expanding maps26 February 2015
Rafael Labarca (Universidad de Santiago de Chile)Abstract: TBA Bubbles of entropy in the Lexicographical world and applications to the quadratic family of Lorenz maps19 February 2015
Paul Glendinning (University of Manchester)Abstract: TBA Dimension and geometry for piecewise smooth maps12 February 2015
Oscar Bandtlow (Queen Mary University of London)Abstract: In a seminal paper Ruelle showed that the long time asymptotic behaviour of analytic hyperbolic systems can be understood in terms of the eigenvalues of a certain operator acting on a suitable Banach space of holomorphic functions. Ruelle also showed that these eigenvalues, also known as Pollicott-Ruelle resonances, are bounded from above by a decaying exponential. In this talk I will focus on lower bounds for the Ruelle eigenvalues. More precisely, I will explain how to prove that there exists a dense set of analytic expanding circle maps for which the Ruelle eigenvalues enjoy exponential lower bounds. This is based on work with W. Just, J. Slipantschuk, and F. Naud. Lower bounds for the Ruelle eigenvalues of analytic circle maps5 February 2015
Mike Field (Imperial College London)Abstract: The talk is intended to be a general (and gentle) introduction to models of network dynamics that are applicable to contemporary problems in biology, engineering and technology. In particular, we will discuss the limitations of classical models and how to deal with networks where, for example, connection structure may vary in time and nodes may stop and later restart. Asynchronous networks and event driven dynamics29 January 2015
Alex Clark (University of Leicester)Abstract: TBA An invitation to the Pisot Conjecture29 January 2015
Wael Bahsoun (Loughborough University)Abstract: We study a class of random transformations built over finitely many intermittent maps sharing a common indifferent fixed point. Using a Young-tower technique, we show that the map with the fastest relaxation rate dominates the asymptotics. In particular, we prove that the rate of correlation decay for the annealed dynamics of the random map is the same as the sharp rate of correlation decay for the map with the fastest relaxation rate. Decay of correlation for random intermittent maps22 January 2015
Bruno Ziliotto (Toulouse School of Economics)Abstract: A stochastic game is described by a set of states, an action set for each player, a payoff function and a transition function. At each stage, the two players simultaneously choose an action, and receive a payoff determined by the actions and the current state. Then a new state is drawn from a distribution depending on the actions and on the former state. We consider stochastic games with long duration, and investigate the existence of a concept of long-term equilibrium payoff, called the asymptotic value. A counterexample to a long-standing conjecture concerning the existence of the asymptotic value will be presented. Knowledge of game theory is no prerequisite for this talk. All the basic concepts will be redefined and illustrated by examples. Asymptotic Value in Two-Player Zero-Sum Stochastic Games and the Mertens conjecture15 January 2015
Peter Giesl (University of Sussex)Abstract: In this talk we consider two methods of determining the basin of attraction. In the first part, we discuss the construction of a Lyapunov function using Radial Basis Functions. The basin of attraction of equilibria or periodic orbits of an ODE can be determined through sublevel set of a Lyapunov function. To construct such a Lyapunov function, i.e. a scalar-valued function which is decreasing along solutions of the ODE, a linear PDE is solved approximately using Radial Basis Functions. Error estimates ensure that the approximation is a Lyapunov function. For the construction of a Lyapunov function it is necessary to know the position of the equilibrium or periodic orbit. A different method to analyse the basin of attraction of a periodic orbit without knowledge of its position uses a contraction metric and is discussed in the second part of the talk. A Riemannian metric with a local contraction property can be used to prove existence and uniqueness of a periodic orbit and determine a subset of its basin of attraction. In this talk, the construction of such a contraction metric is achieved by formulating it as an equivalent problem, namely a feasibility problem in semidefinite optimization. The contraction metric, a matrix-valued function, is constructed as a continuous piecewise affine function, which is affine on each simplex of a triangulation of the phase space. Construction of Lyapunov functions and Contraction Metrics to determine the Basin of Attraction15 December 2014
Mark Holland (University of Exeter)Abstract: For dynamical systems we discuss the statistics of extremes, namely the statistical limit laws that govern the process $M_{n}=\max\{X_1,X_2,\ldots,X_n\}$ , where $X_i$ correspond to a stationary time series of observations generated by the dynamical system. We discuss extreme statistics for a range of examples of interest to those working in ergodic theory and chaotic dynamical systems. In a work in progress, we discuss almost sure growth rates of $M_n$, and the statistics of records: namely the distribution of times $n$ such that $X_{n}=M_n$. On Extremes, recurrence and record events in dynamical systems11 December 2014
Mike Todd (University of St. Andrews)Abstract: Fernandez and Demers studied the statistical properties of the Manneville-Pomeau map with the physical measure when a hole is put in the system, overcoming some of the problems caused by subexponential mixing. I’ll discuss the same setup, but with a class of natural equilibrium states. We find conditionally invariant measures and give precise information on the transitions between the fast exponentially mixing, the slow exponentially mixing and the subexponentially mixing phases. This is joint work with Mark Demers. Dynamical systems with holes: slow mixing cases11 December 2014
Gabriel Paternain (University of Cambridge)Abstract: I will discuss the inverse problem of recovering a unitary connection from the parallel transport along geodesics of a compact Riemannian manifold of negative curvature and strictly convex boundary. The solution to this geometric inverse problem is based on a range of techniques, including energy estimates and regularity results for the transport equation associated with the geodesic flow. Recovering a connection from parallel transport along geodesics5 December 2014
Han Peters (Universiteit van Amsterdam)Abstract: Sullivan's non-wandering domains theorem from 1985 showed that rational functions do not have wandering Fatou components, which completed the classification of Fatou components in the Riemann sphere. In this talk I will discuss recent work with Matthieu Astorg, Xavier Buff, Romain Dujarin and Jasmin Raissy showing that there exist polynomial maps in two variables with wandering Fatou components. While our methods are complex, we also obtain real polynomial maps with wandering domains. The main idea, suggested by Misha Lyubich, is to apply parabolic implosion techniques to polynomial skew products. A polynomial map in two variables with a wandering Fatou component5 December 2014
Nikita Sidorov (University of Manchester)Abstract: In this talk I will consider a natural two-parameter family of self-affine iterated function systems in the plane and provide a detailed analysis of their attractors. In particular, I will describe new results on the set of parameters for which the attractor has a non-empty interior and on the connectedness locus for this family. I will also talk about the set of uniqueness and simultaneous signed beta-expansions. This talk is based on a joint paper with Kevin Hare. On a family of two-dimensional self-affine sets27 November 2014
Nadia Sidorova (University College London)Abstract: The parabolic Anderson problem is the Cauchy problem for the heat equation on the d-dimensional integer lattice with random potential. It describes the behaviour of branching random walks in a random environment (represented by the potential) and is being actively studied by mathematical physicists. One of the most important situations is when the potential is time-independent and is a collection of independent identically distributed random variables. We discuss the intermittency effect occurring for such potentials and consisting in increasing localisation and randomisation of the solution. We also discuss the ageing behaviour of the model showing that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time. Localisation and ageing in the parabolic Anderson model20 November 2014
James Meiss (University of Colorado Boulder)Abstract: Turbulent fluid flows have long been recognized as a superior means of diluting initial concentrations of scalars due to rapid stirring. Conversely, experiments have shown that the structures responsible for this rapid dilution can also aggregate initially distant reactive scalars and thereby greatly enhance reaction rates. Indeed, chaotic flows not only enhance dilution by shearing and stretching, but also organize initially distant scalars along transiently attracting regions in the flow. We demonstrate that Lagrangian coherent structures (LCS), as identified by ridges in finite time Lyapunov exponents, are directly responsible for this coalescence of reactive scalar filaments. When highly concentrated filaments coalesce, reaction rates can be orders of magnitude greater than would be predicted in a well-mixed system. This is further supported by an idealized, analytical model that quantifies the competing effects of scalar dilution and coalescence. Chaotic flows, known for their ability to efficiently dilute scalars, therefore have the competing effect of organizing initially distant scalars along the LCS at timescales shorter than that required for dilution, resulting in reaction enhancement. This work is joint with K. Pratt and J. Crimaldi. Reaction Enhancement of Initially Distant Scalars by Lagrangian Coherent Structures13 November 2014
Guillaume Lajoie (Max Planck Institute for Dynamics and Self-Organization Göttingen)Abstract: In this talk, I will discuss the use of Random Dynamical Systems (RDS) Theory as a framework to approach problems involving network dynamics in the brain. I will briefly outline the recent efforts to use RDS methods in the field of theoretical neuroscience before presenting recent results. Large networks of sparsely coupled, excitatory and inhibitory cells occur throughout the brain. they support very complex computations, the exact mechanisms of which are poorly understood. For many models of these networks, a striking feature is that their dynamics are chaotic and thus, are sensitive to small perturbations. How does this chaos manifest in the neural code? Specifically, how variable are the spike patterns that such a network produces in response to an input signal? To answer this, we derive a bound for a general measure of variability - spike-train entropy. The analysis is based on results from RDS theory and is complemented by detailed numerical simulations aimed at computing quantitative attributes of high-dimensional random strange attractors. This leads to important insights about the variability of multi-cell spike pattern distributions in large recurrent networks of spiking neurons responding to fluctuating inputs. Moreover, we show how spike pattern entropy is controlled by temporal features of the inputs. Our findings provide insight into how neural networks may encode stimuli in the presence of inherently chaotic dynamics. A Random Dynamical Systems framework to study encoding and variability in large neural networks13 November 2014
Alexander Gorodnik (University of Bristol)Abstract: In this talk we discuss dynamics on flag manifolds, and in particular, we will be interested in describing factors of such dynamical systems. A well-known theorem of Margulis classifies measurable factors, and an analogous result in the continuous category has been established by Dani. We explain a classification of smooth factors under some hyperbolicity assumptions. This is a joint work with R. Spatzier. Dynamics on flag manifolds6 November 2014
Richard Sharp (University of Warwick)Abstract: A beautiful theorem of Brooks says that, for a wide class of Riemannian manifolds, the bottom of the spectrum of the Laplacian on a regular cover is equal to the bottom of the spectrum of the base if and only if the covering group is amenable. In the case where the base manifold is a quotient of a simply connected manifold with pinched negative curvatures by a convex co-compact group, we will give a analogous results for critical exponents and for the growth of closed geodesics. This is joint work with Rhiannon Dougall. Critical exponents, growth and amenability30 October 2014
Laurent Stolovitch (University of Nice)Abstract: We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the ``small divisors'' are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth $\al$-Gevrey vector field with an hyperbolic linear part admits a smooth $\be$-Gevrey transformation to a smooth $\be$-Gevrey normal form. The Gevrey order $\be$ depends on the rate of accumulation to $0$ of the small divisors. We show that a formally linearizable Gevrey smooth germ with the linear part satisfies Brjuno's small divisors condition can be linearized in the same Gevrey class. Smooth Gevrey normal forms of vector fields near a fixed point23 October 2014
Claude Baesens (University of Warwick)Abstract: Bleher, Ott and Grebogi found numerically an interesting chaotic phenomenon in 1989 for the scattering of a particle in a plane from a potential field with several peaks of equal height. They claimed that when the energy E of the particle is slightly less than the peak height E_c there is a hyperbolic suspension of a topological Markov chain from which chaotic scattering occurs, whereas for E > E_c there are no bounded orbits. They called the bifurcation at E = E_c an abrupt bifurcation to chaotic scattering. Our aim in this work is to establish a rigorous mathematical explanation for how chaotic orbits occur via the bifurcation, from the viewpoint of the anti-integrable limit, and to do so for a general range of chaotic scattering problems. Abrupt bifurcations in chaotic scattering: view from the anti-integrable limit16 October 2014
Ana Rodrigues (University of Exeter)Abstract: In this talk, we will study the existence and non-existence of periodic orbits and limit cycles for planar polynomial differential systems of degree $n$ having $n$ real invariant straight lines taking into account their multiplicities. Periodic orbits for real planar polynomial vector fields of degree n having n invariant straight lines.9 October 2014
Gerhard Keller (University of Erlangen-Nürnberg)Abstract: We study dynamical systems forced by a combination of random and deterministic noise and provide criteria, in terms of Lyapunov exponents, for the existence of random attractors with continuous structure in the bres. For this purpose, we provide suitable random versions of the semiuniform ergodic theorem and also introduce and discuss some basic concepts of random topological dynamics. Random minimality and a theorem of Jaroslav Stark29 September 2014
Oleg Makarenkov (University of Texas at Dallas)Abstract: Consider a couple of vector fields (F1,F2) and a couple of switching manifolds (S1,S2). Each trajectory x(t) follows the vector field F1 until x(t) crosses S1, where the system switches to the vector field F2 that governs the trajectory until it reaches S2 where the switch back to S1 occurs. The existence and stability of limit cycles in such a system is known since the classical work of Barbashin (1967) whose approach employs a Lyapunov-like technique. In this talk I show that the aforementioned cycles can be seen as a bifurcation from (0,0) when a suitably defined parameter crosses its bifurcation value. Bifurcation of limit cycles from a fold-fold singularity in switching systems16 September 2014
Jun-nosuke Teramae (Osaka University, Japan)Abstract: Neurons in the brain sustain spontaneous ongoing activity with highly irregular spike trains. While the irregular activity has been regarded as ignorable background noise, recent experiments reveal that the spontaneous activity is significant player of computation in the brain. In this talk, introducing a recently proposed deterministic model of cortical network of spiking neurons that stably generates and maintains spontaneous ongoing activity with highly irregular spike trains, we study stochastic dynamics of the ongoing activity and discuss possible roles of the stochastic spiking for neural computation. Stochastic dynamics and computation in network models of cortical spiking neurons15 September 2014
Tiago Pereira (Imperial College London)Abstract: Recent results reveal that typical real-world networks have various levels of connectivity. These networks exhibit emergent behaviour at various levels. Striking examples are found in the brain, where synchronisation between highly connected neurons coordinate and shape the network development. These phenomena remain a major challenge. I will discuss a probabilistic dimension reduction principle to describe the network dynamics. I show that, at large levels of connectivity, the high-dimensional network dynamics can be reduced to a few macroscopic equations. The strategy is to describe ensembles of random networks, and the dynamics almost every initial state. This reduction provides the opportunity to explore the coherent properties at various network connectivity scales. This is a joint work with Sebastian van Strien and Jeroen Lamb. Dynamics in Heterogeneous Networks: Emergence at Various Scales 15 September 2014
Tsuyoshi Chawanya (Osaka University, Japan)Abstract: Quasi-periodically forced logistic map system is one of the representative systems where strange non-chaotic attractors(SNAs) are observed. In this talk, we show that the numerical analysis on the bifurcation phenomena in QPLM can be carried out efficiently using a map describing the evolution of "line segment" in the logistic map system, and exhibit some of the obtained results including phase diagram with fine resolution indicating the existence of variety of bifurcation senario from a stable torus to a chaotic attractor with or without SNA in the middle. On bifurcation phenomena in the quasi-periodically driven logistic map system12 September 2014
Hiroshi Teramoto (Hokkaido University, Japan)Abstract: Nonlinear resonance is one of the most efficient mechanisms for vibrational energy transfers among vibrational modes. In this talk, we propose a method to extract vibrational modes from a given time series to understand the underlying mechanism behind the energy transfers in terms of nonlinear resonances. Extracting coherent molecular vibrational modes by nonlinear time series analysis11 September 2014
Yuzuru Sato (Hokkaido University)Abstract: Dynamics of dice roll on heterogeneous environments is studied in terms of random basin strucuture. Uncertainty exponents are numerically estimated in a random dynamical system and final state sensitivity under the presence of noise is investigated. Extracting random maps from experimental data of dice roll is also briefly discussed. Random basin in dice roll11 September 2014
Hiroki Sumi (Osaka University, Japan)Abstract: In this talk, we consider random dynamical systems of complex polynomial maps on the Riemann sphere. It is well-known that for each rational map $f$ on the Riemann sphere with $\deg (f)\geq 2$, the Hausdorff dimension of the set of points $z$ in the Riemann sphere for which the Lyapunov exponent of the dynamics of $f$ is positive, is positive. However, we show that for generic i.i.d. random dynamical systems of complex polynomials, the following$B!!(Bholds. For all but countably many points $x$ in the Riemann sphere, for almost every sequence $\gamma =(\gamma _{1}, \gamma _{2}, \gamma _{3},\cdots )$ of polynomials, the Lyapunov exponent along $\gamma $ starting with $x$ is negative. Note that the above statement cannot hold in the usual iteration dynamics of a single rational map $f$ with $\deg (f)\geq 2.$ Therefore the picture of the random complex dynamics is completely different from that of the usual complex dynamics. We remark that even if the chaos of the random system disappears in the $C^{0}$ sense, the chaos of the system may remain in the $C^{1}$ sense, and we have to consider the ``gradation between chaos and order''. Negativity of Lyapunov exponents of generic random dynamical systems of complex polynomials9 September 2014
Masayuki Asaoka (Kyoto University, Japan)Abstract: Theory of random dynamics studies properties of almost every path of iterated function systems. In this talk, we discuss how bad the dynamics can be for a special path and how does it effect the average of dynamical quantity of iterated function systems. More precisely, we prove the following theorem: There exists an open subset of the set of smooth iterated function systems on the unit interval with three generators such that (a) For all most path, the n-th iteration is uniform contraction, and hence it has unique fixer point for any large n, (b) For any generic element of U, the limsup of the average of the number of fixed points of n-th iterations grows superexpoentially. The result illustrates that the behavior of the average in finite time is different from that of a.e. paths. We also proves that for generic element in U there exists a path along which `any' dynamics is realized. Arbitrary growth of the number of periodic points and universal dynamics in one-dimensional semigroups9 September 2014
Christian Rodrigues (Max Planck Institute, Leipzig, Germany)Abstract: Amongst the main concerns of Dynamics one wants to decide whether asymptotic states are robust under random perturbations. For practical applications, the randomly perturbed dynamics is given by a Markov chain model. Nevertheless, in order to derive conclusions from the mathematical perspective about stability, asymptotic states, invariance, etc., one is always enforced to use a model based on iteration of maps chosen at random with a given probability. The two approaches however are in general not equivalent. In this talk we systematically investigate the problem of representing Markov chains by families of random maps, and which regularity of these maps can be achieved depending on the properties of the probability measures. Our key idea is to use techniques from optimal transport to select optimal such maps. From this scheme, we not only deduce the representation by measurable and continuous random maps, but also obtain conditions for the to construct random diffeomorphisms from a given Markov chain. This is a joint work with Jost, and Kell. Stochastic stability and the representation of Markov chains by random maps 8 September 2014
Rainer Klages (Queen Mary University of London)Abstract: Dynamical systems having many coexisting attractors present interesting properties from both fundamental theoretical and modelling points of view. When such dynamics is under bounded random perturbations, the basins of attraction are no longer invariant and there is the possibility of transport among them. Here we introduce a basic theoretical setting which enables us to study this hopping process from the perspective of anomalous transport using the concept of a random dynamical system with holes. We apply it to a simple model by investigating the role of hyperbolicity for the transport among basins. We show numerically that our system exhibits non-Gaussian position distributions, power-law escape times, and subdiffusion. Our simulation results are reproduced consistently from stochastic Continuous Time Random Walk theory. Diffusion in randomly perturbed dissipative dynamics 8 September 2014
Jennifer Creaser (University of Auckland)Abstract: The well-known Lorenz system is classically studied via its reduction to the one-dimensional Lorenz map, which captures the full behaviour of the dynamics of the system. The reduction requires the existence of a stable foliation. We study a parameter regime where this so-called foliation condition fails and the Lorenz map no longer accurately represents the dynamics. Hence, we study how the three-dimensional phase space is organised by the global invariant manifolds of saddle equilibria and saddle periodic orbits. We consider a previously unexplained phenomenon, found by Sparrow in the 1980's where the one-dimensional stable manifolds of the secondary equilibria flip from one side to the other. We characterise this geometrically as a bifurcation in the alpha-limit of these one-dimensional stable manifolds which we call an alpha-flip. We find many such alpha-flips and, by following them in two parameters, show that each ends at a different co-dimension two bifurcation point, known as a T-point, many of which have not been found before. Moreover, we argue that the alpha-flip is a precursor to the loss of the foliation condition. The Lorenz system near the loss of the foliation condition17 July 2014
Gary Froyland (University of New South Wales)Abstract: Transfer operators provide a global description of dynamics and carry information that is di?fficult to extract by trajectory simulation. We describe theory and numerical methods for analysing (possibly time-dependent or random) chaotic dynamical systems, and applications in the natural and physical sciences. Transfer operators and dynamics14 July 2014
Nguyen Dinh Cong (Vietnam Academy of Science and Technology)Abstract: Lyapunov exponents and Lyapunov spectrum are important tools in the theory of dynamical systems. They characterize basic qualitative properties of dynamical systems: stability, hyperbolicity, chaos, etc. In this talk I will present some results on generic properties of Lyapunov spectrum in the space of linear random dynamical systems. Generic properties of Lyapunov spectrum of linear random dynamical systems14 July 2014
Sanjeeva Balasuriya (University of Adelaide)Abstract: Within the context of fluid flows at scales ranging from the geophysical to the microfluidic, one might ask the complementary questions: (1) How does one define a flow barrier? and (2) Is it possible to quantify transport (i.e., a flux) across such barriers? In steady (autonomous) flows, these questions can be answered in terms of stable and unstable manifolds---whose intersection can only occur in restricted ways---across which there is zero flux. When the velocity field is unsteady (nonautonomous), such invariant manifolds move with time, and general intersections between them are possible. Defining these manifolds in realistic (unsteady, finite-time, discrete-valued) data sets is a challenge; accordingly, many heuristic definitions for flow barriers continue to be developed. Theoretical advances on nonautonomous stable and unstable manifolds are therefore of interest. Here, I address the question of obtaining these manifolds explicitly in a nonautonomously perturbed 2D situation, and in a nonautonomous nonchaotic 3D flow. The former requires the quantification of the tangential displacement of these manifolds under nonautonomous perturbations. The latter approach addresses a class of flows in which the time-variation of the manifolds can be explicitly established using exponential dichotomies. These models offer testbeds for the many proposed heuristics for identifying flow barriers in genuinely unsteady flows. Flow barriers and flux in unsteady flows2 July 2014
Jason Gallas (UFPB)Abstract: Chaotic oscillations were recently discovered for some specific parameter values in the de Pillis-Radunskaya model of cancer growth, a model including interactions between tumor cells, healthy cells, and activated immune system cells. I present high-resolution phase diagrams from a wide-ranging systematic numerical classification of "all" dynamical states of the model and their relative abundance. I characterize cell dynamics by two independent and complementary types of stability diagrams: Lyapunov and isospike diagrams. The cancer model is shown to display stability phases regularly organized in old and in many novel ways. In addition to spirals of stability, the model displays very long sequences of zig-zag accumulations and intertwined cascades of two- and three-chaos flanked stability islands previously observed only in systems governed by delay-differential equations. We also characterize a spike-adding mechanism underlying the systematic complexification of regular wave patterns in generic flows when control parameters are tuned continuously. Complex oscillations and chaos in a three-cell population model of cancer23 June 2014
James Robinson (University of Warwick)Abstract: Suppose that we have a Lipschitz continuous differential equation on a Banach space $X$: $\dot x=f(x)$, where $\|f(x)-f(y)\|\le L\|x-y\|$. Using a geometric argument, Yorke showed that if $X$ is $R^n$ equipped with the usual Euclidean norm then any non-constant periodic orbit must have period at least $2\pi/L$. Busenberg, Fisher, and Martelli gave an analytical proof of the same result valid in any Hilbert space, and showed that in an arbitrary Banach space the minimal period is at least $6/L$. I will give proofs of these results, show that the minimal period is strictly larger than $6/L$ in any strictly convex Banach space (e.g. in $R^n$ with the $\ell^p$ norm, and discuss some related open problems. Minimal periods in Lipschitz differential equations19 June 2014
Dayal Strub (University Of Warwick)Abstract: A transition state for a Hamiltonian system is a closed, invariant codimension-2 submanifold of an energy-level, spanned by two compact codimension-1 surfaces of unidirectional flux whose union locally separates the energy-level. This union, called a dividing surface, has locally minimal geometric flux through it and can therefore be used to find an upper bound on the rate of transport in Hamiltonian systems. We shall first recall the basic transport scenario about an index-1 critical point of the Hamiltonian, and find transition states diffeomorphic to spheres for energies just above the critical one. This leads naturally to the question of what qualitative changes in the transition state may occur as the energy is increased further. We shall find that there is a class of systems for which the transition state changes diffeomorphism class via Morse bifurcations, and consider a number of examples. This is joint work with Robert MacKay. Bifurcations of transition states12 June 2014
Alexandre Rodrigues (University of Porto)Abstract: In this talk, we present a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a Bykov cycle where trajectories turn in opposite directions near the two nodes - we say that the nodes have different chirality. We show that in a $C^2$-open class of vector fields defined on a three-dimensional compact manifold, tangencies of the invariant manifolds of two hyperbolic saddle-foci occur at a full Lebesgue measure set of the parameters that determine the linear part of the vector field at the equilibria. This has important consequences: the global dynamics is persistently dominated by heteroclinic tangencies and by Newhouse phenomena, coexisting with hyperbolic dynamics arising from transversality. The coexistence gives rise to linked suspensions of Cantor sets, with hyperbolic and non-hyperbolic dynamics, in contrast with the case where the nodes have the same chirality. Dense heteroclinic tangencies near a Bykov cycle 10 June 2014
Alexandre Rodrigues (University of Porto)Abstract: In this talk, we characterise the set of $C^1$ area-preserving maps on a surface displaying a reversing isometry $R$ of degree 2 (involution). We show that $C1$-generic $R$-reversible area-preserving maps are Anosov or else Lebesgue almost every orbit displays zero Lyapunov exponents. This result generalizes Bochi-Mañé Theorem for the class of reversing-symmetric maps. Generic Area preserving reversible diffeomorphisms9 June 2014
Vassilis Rothos (Aristotle University of Thessaloniki)Bifurcation of Travelling Waves in Implicit Nonlinear Lattices: Applications in magnetic metamaterials4 June 2014
Claudia Wulff (University of Surrey)Abstract: Relative equilibria and relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems and occur for example in celestial mechanics, molecular dynamics and rigid body motion. Relative equilibria are equilibria and RPOs are periodic orbits of the symmetry reduced system. Relative Lyapunov centre bifurcations are bifurcations of relative periodic orbits from relative equilibria corresponding to Lyapunov centre bifurcations of the symmetry reduced dynamics. In this talk we prove a relative Lyapunov centre theorem by combining recent results on persistence of RPOs in Hamiltonian systems with a symmetric Lyapunov centre theorem of Montaldi et al. We then develop numerical methods for the detection of relative Lyapunov centre bifurcations along branches of RPOs and for their computation. We apply our methods to Lagrangian relative equilibria of the N-body problem. Relative Lyapunov centre bifurcation29 May 2014
Jason Gallas (UFPB)Abstract: A central problem in modern cryptography is the factorization of large integers involved in theoretically breakable but computationally secure mechanisms used to protect data. I will discuss a method to solve an analogous but more general problem for functions, not numbers: the factorization of exceptionally large polynomials defining the orbital points of periodic orbits of the quadratic (logistic) map, a paradigmatic discrete-time dynamical system of algebraic origin. The method is based on an infinite set of nonlinear transformations whose zeros are all preperiodic points of the dynamics. Forward iteration from such preperiodic points opens access to orbits with arbitrarily long periods which cannot be reached using standard multivalued inverse functions. As a concrete example, I show how to extract expeditiously a long periodic orbit buried in a polynomial with degree larger than a billion. Even rough numerical approximations of the zeros suffice to obtain long orbits precisely. Factorization of exceptionally long periodic orbits of the quadratic map22 May 2014
Daniel Karrasch (ETH Zurich)Abstract: In this talk I give an overview on the geodesic theory of Lagrangian Coherent Structures (LCS) in two-dimensional, finite-time nonautonomous dynamical systems. LCS are exceptional material lines that act as cores of observed tracer patterns in finite-time (fluid) flows. The talk is organized in two parts. In the first part, I briefly present the variational derivation of hyperbolic LCS, as due to Farazmand, Blazevski & Haller. I then describe an attraction-based approach to hyperbolic LCS (joint work with M. Farazmand and G. Haller), which correponds to a paradigm shift in hyperbolic LCS theory. In the second part, I briefly present the variational derivation of elliptic LCS, which are buidling blocks of coherent Lagrangian vortices, as due to Haller & Beron-Vera. I then present an automated detection method based on index theory for line fields (joint work with F. Huhn and G. Haller). Overview on geodesic Lagrangian Coherent Structures22 May 2014
Kathrin Padberg-Gehle (TU Dresden)Abstract: Numerical methods involving transfer operators have only recently been recognized as powerful tools for analyzing and quantifying transport and mixing in time-dependent systems. This talk discusses several different constructions that allow us to extract coherent structures and dynamic transport barriers in nonautonomous dynamical systems. Apart from exploiting spectral properties of the transfer operator, we pinpoint transport barriers as regions of low predictability. Predictability can be very efficiently measured via the growth of entropy that is experienced by a small localised density under the evolution of the numerical transfer operator. Transport, mixing and predictability15 May 2014
Julia Slipantschuk (Queen Mary University)Abstract: Spectral data of transfer operators yield insight into fine statistical properties of the underlying dynamical system, such as rates of mixing. In this talk, I will describe the spectral structure of transfer operators associated to analytic expanding circle maps. For this, I will first derive a natural representation of the respective adjoint operators. For expanding circle maps arising from finite Blaschke products, this representation takes a particularly convenient form, allowing to deduce the entire spectra of the corresponding transfer operators. These spectra are completely determined by the multipliers of attracting fixed points of the Blaschke products. Spectral structure of transfer operators for expanding circle maps8 May 2014
Heather Reeve-Black (Queen Mary University)Abstract: We consider a one-parameter family of invertible maps of a two-dimensional lattice, obtained by applying round-off to planar rotations. We let the angle of rotation approach $\pi/2$, and show that the system exhibits a failure of shadowing: the limit of vanishing discretisation corresponds to a piecewise-affine Hamiltonian flow, whereby the plane foliates into invariant polygons with an increasing number of sides. Considered as perturbations of the piecewise-affine flow, the lattice maps assume a different character, described in terms of strip maps, a variant of those found in outer billiards of polygons. Furthermore the flow is nonlinear (unlike the rotation) and a suitably chosen Poincare return map is a twist map. We show that the motion at infinity, where the invariant polygons approach circles, is a dichotomy: there is one regime in which the nonlinearity tends to zero, leaving only the perturbation, and a second where the nonlinearity dominates. In the domains where the nonlinearity remains, numerical evidence suggests that the distribution of the periods of orbits is consistent with that of random dynamics, whereas in the absence of nonlinearity, the fluctuations result in intricate discrete resonant structures. Near-Integrability in a Family of Discretised Rotations1 May 2014
Boumediene Hamzi (Yildiz Technical University)Abstract: We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control and random dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems - with a reasonable expectation of success - once the nonlinear system has been mapped into a high or infinite dimensional Reproducing Kernel Hilbert Space. In particular, we develop computable, non-parametric estimators approximating controllability and observability energy functions for nonlinear systems, and study the ellipsoids they induce. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system. We also apply this approach to the problem of model reduction of nonlinear control systems. In all cases the relevant quantities are estimated from simulated or observed data. These results collectively argue that there is a reasonable passage from linear dynamical systems theory to a data-based nonlinear dynamical systems theory through reproducing kernel Hilbert spaces. This is joint work with J. Bouvrie (MIT). On Control and Random Dynamical Systems in Reproducing Kernel Hilbert Spaces24 April 2014
Sebastian Hage-Packhäuser (University of Paderborn)Abstract: Numerous dynamical systems describing real world phenomena exhibit a characteristic fine structure which stems from the interaction of many dynamic instances. Furthermore, since reality crucially depends on time, such dynamical system networks are generally subject to temporal changes. Particularly in applications involving technology, this temporal evolution often occurs as a consequence of instantaneously time-varying network structures. In this talk, time-varying networks of dynamical systems are discussed in terms of hybrid dynamical systems with a special consideration of symmetries which are naturally due to the network structures involved. By means of the recent notion of hybrid symmetries, a hybrid symmetry framework is presented and symmetry-induced switching strategies are investigated - from a structural point of view, but also with regard to stability. Switching Dynamical System Networks in the Light of Hybrid Symmetry2 April 2014
Eduardo Garibaldi (UNICAMP)Abstract: In this talk, we discuss the existence of minimizing configurations associated with generalized Frenkel-Kontorova models on quasi-crystals. This is a joint work with Samuel Petite (Université de Picardie) and Philippe Thieullen (Université de Bordeaux). The Frenkel-Kontorova model for almost-periodic environments31 March 2014
Frits Veerman (University of Oxford)Abstract: The study of localised structures in systems of reaction-diffusion equations is a very active field of research. The nonlinear nature of the equations often makes it difficult to go beyond standard analysis, i.e. Turing patterns. The presence of small parameters, in particular the diffusion ratio, can be used to successfully exploit these nonlinearities and to invoke geometric singular perturbation theory to prove, by construction, the existence of localised patterns. Even the stability of these patterns can be addressed: using Evans function techniques, the spectrum of the pulse can be determined to leading order. This method, which can be applied to a very general class of reaction-diffusion systems, will be applied to a example system, where previously unobserved behaviour is found and analysed. This is joint work with A. Doelman, University of Leiden. Pulses in singularly perturbed reaction-diffusion systems28 March 2014
Evamaria Ruß (Alpen Adria University Klagenfurt)Abstract: The dichotomy spectrum (also known as Sacker-Sell or dynamical spectrum) is a crucial spectral notion in the theory of dynamical systems. In this talk we study the dichotomy spectrum for linear difference equations with an infinite-dimensional state space. In general we cannot expect a nice structure of the dichotomy spectrum like in the finite dimensional case, but compactness properties of the transition operator provide a more regular spectrum. Finally, we have a look at various evolutionary differential equations in order to illustrate possible applications. Dichotomy Spectrum in Infinite Dimensions27 March 2014
Ian Morris (University of Surrey)Abstract: The binary Euclidean algorithm is a variation on the classical Euclidean algorithm which is designed to take advantage of the efficiency of division by two on a binary computer. Whereas the classical Euclidean algorithm can be understood in terms of the dynamics of the continued fraction transformation on the unit interval, the analysis of the binary Euclidean algorithm requires the use of a random dynamical system. I will describe some of my recent work on the analysis of this algorithm via the Ruelle transfer operator associated to this random dynamical system and its connection with a conjecture of D. E. Knuth. A random dynamical model for the binary Euclidean algorithm20 March 2014
Masayuki Asaoka (Kyoto University)Abstract: In 1999, Kaloshin showed that persistence of homoclinic tangency implies local genericity of arbitrary fast growth of the number of periodic orbits. Local genericity of universal dynamics (Bonatti-Diaz, Turaev) also implies local genericity of such pathological behavior. But, up to now, all known mechanism of abnormal growth is based on homoclinic tangency. A natural question is whether generic partially hyperbolic system can exhibit arbitrary fast growth of the number of periodic orbit or not. As a step to answer this problem, we give a solution to a corresponding problem for 1-dimensional iterated function systems: Theorem: There exists an open set of C^r 1-dimensional iterated function system in which generic system exhibits arbitrary fast growth of the number of periodic points. If possible, we also discuss a work in progress which extends the above theorem to 3-dimensional partially hyperbolic systems. This is a joint work with K.Shinohara and D.Turaev. Arbitrary fast growth of the number of periodic orbits in 1-dimensional iterated function systems14 March 2014
Piotr Slowinski (University of Warwick)Abstract: In this talk I will present bifurcation analysis of a semiconductor laser receiving delayed filtered optical feedback from two filter loops (2FOF). Specifically, we compute the basic cw-solutions (called external filtered modes EFM) of an underlying delay differential equation model. The EFMs as represented as surfaces in the space of filter phase difference versus frequency and inversion of the laser. In this way, I am able to present a comprehensive geometric picture of how the EFM structure and stability depends on parameters, including the filter detunings and delays. Overall, the study of the EFM-surface is a geometric tool for the multi-parameter analysis of the 2FOF laser, which provides comprehensive insight into the solution structure and dynamics of the system. Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops13 March 2014
Sofia Castro (University of Porto)Abstract: It is well-known that some cycles in heteroclinic networks appear more frequently in simulations than others. Making use of a convenient notion of stability and of the stability index introduced by Podvigina and Ashwin [3], I shall report on the possible relative stability of two cycles of type B inside a network with no other cycles. This is recent work with Alexander Lohse [2]. Another illustrative example of the importance of relative stability of cycles in networks is provided by the quotient network arising in the Rock-Scissors-Paper. This consists of older work with Manuela Aguiar [1] and work in progress with Yuzuru Sato. [1] M.A.D. Aguiar and S.B.S.D. Castro, Chaotic switching in a two-person game, Physica D: Nonlinear Phenomena, vol. 239 (16), 1598-1609 (2010) [2] S.B.S.D. Castro and A. Lohse, Stability in simple heteroclinic networks in R^4, arXiv:1401.3993 [math.DS] (2014) [3] O. Podvigina and P. Ashwin, On local attraction properties and a stability index for heteroclinic connections, Nonlinearity, Vol. 24, 887-929 (2011) Stability of cycles in heteroclinic networks13 March 2014
Robert Szalai (University of Bristol)Abstract: Friction and impact are two non-smotth nonlinearities that are of great interests to engineers. The problem with non-smooth phenomena is that they introduce a great deal of uncertainty of into model predictions. Specifically it is possible to lose forward time uniqueness and have generic models that are structurally unstable. In this talk I will introduce a modelling framework that hopefully resolves these problems. First I demonstrate how non-unique solutions arise in mechanical systems composed of rigid bodies. It turns out that the major component of such singular behaviour is the rigid body assumption. In reality nothing is ideally rigid. To resolve such singularities the elasticity needs to be taken into account. These elastic systems are described by partial differential equations (PDE), which make the analysis complicated. Hence to resolve the problem I introduce a model reduction technique that turns those PDEs into low dimensional delay equations. It turns out that the only condition for regular dynamics is to have a finite wave speed within the mechanism. The method is general and does not rely on the existence of travelling wave solutions. How to resolve singularities in non-smooth systems using better physical models6 March 2014
Hans-Henrik Rugh (University of Paris-Sud)Abstract: (Work in progress with O. Bandtlow). We consider uniformly expanding interval maps that are Markov. Introducing a finite number of holes, restricting the map to the remaining set. We show that the entropy of the restricted map is Hölder continuous with respect to the hole position and size under a non-degeneracy condition. The Hölder exponent is related to the entropy itself and expansion rates of the map. Examples and counter-examples, with numerics illustrates the result. On the Hölder continuity of the entropy for expanding Interval maps with holes6 March 2014
Stefano Marmi (Scuola Normale Superiore, Pisa)Abstract: Addressing a question raised by Kolmogorov and Herman, we show that KAM curves of area-preserving twist maps are uniquely determined by their knowledge on a set of positive 1-dimensional Hausdorff measure in frequency space. This result is obtained by complexifying the rotation number and by an extension of the classical theory of quasianalytic functions. The parametrization of KAM curves is naturally defined in a complex domain containing the real Diophantine frequencies and real frequencies constitute a natural boundary for the analytic continuation from the Weierstrass point of view because of the density of the resonances. Natural boundaries and uniqueness of KAM curves6 March 2014
Jimmy Tseng (University of Bristol)Abstract: We show that, for two commuting automorphisms of the d-torus, many points have drastically different orbit structures for the two maps. Specifically, using measure rigidity and the Ledrappier-Young formula, we show that the set of points that have dense orbit under one map and nondense orbit under the other has full Hausdorff dimension. This mixed case, dense orbit under one map and nondense orbit under the other, is much more delicate than the other two possible cases. Our technique can also be applied to other settings. For example, we show the analogous result for two elements of the Cartan action on compact higher rank homogeneous spaces. This is joint work with V. Bergelson and M. Einsiedler. Simultaneous dense and nondense orbits for commuting maps27 February 2014
Ilies Zidane (Paul Sabatier University)Abstract: Yoccoz gave a sufficient arithmetical condition of linearization of fixed points of holomorphic germs with multiplier exp(2 i \pi a) where a is an irrational number: f(z)=exp(2 i \pi a) z+O(z^2). He also proved that this condition is optimal for quadratic polynomials. We will discuss this optimality for cubic polynomials and quadratic rational maps. We will see how is it related to the size of Siegel disks and parabolic implosion/renormalization. This leads to the study of slices of bifurcation locus where some surprising, unexpected and complicated phenomenons occur due to the interaction between the two critical points. We also investigate some virtual slices arising as geometric limits (parabolic enrichment) of dynamical systems. We seek analogues of Zakeri curves (the locus where the two critical points lie at the boundary of the Siegel disk) in these slices, when the rotation number is not of bounded type, and even, for Cremer slices. Given a Siegel slice, the logarithm of the conformal radius of the Siegel disk is a subharmonic function, whose Laplacian is therefore a measure which gives a new viewpoint as well as a lot of information. On the bifurcation locus of cubic polynomials and the size of Siegel disks25 February 2014
Oleg Kozlovski (University of Warwick)Abstract: In this talk we will discuss if maps in a "typical" family of maps of an interval or a circle have bounded number of attracting trajectories. The answer is not straightforward and depends on what "typical" means and on the smoothness of the maps. Hilbert-Arnold problem for 1D maps13 February 2014
David Lloyd (University of Surrey)Abstract: Experiments involving a plate of magnetic fluid in the presence of a magnetic field have shown that it is possible for stable localised spots on the surface of the magnetic to exist. Building on work analysing localised patterns in reaction-diffusion equations, we try to explain the nucleation of such spots for the magnetic fluid and some of their properties away from onset. In particular, I will show that the equations governing the free-surface of the magnetic fluid obey an energy principle. Furthermore, an approximate Hamiltonian formulation can be found near onset that simplifies a centre-manifold reduction for patterns in the one-dimensional case. Finally, we show some numerical continuation results for 2D and 3D localised Ferropatterns and relate the results to work on reaction-diffusion systems in the presence of a conserved quantity. Nucleation of localised Ferrosolitons and Ferro-patterns6 February 2014
Luciana Luna Anna Lomonaco (Roskilde University)Abstract: A polynomial-like mapping is a proper holomorphic map f : U' -> U, where U' and U isomorphic to a a disk, and U' compactly contained in U. This definition captures the behaviour of a polynomial in a neighbourhood of its filled Julia set. A polynomial-like map of degree d is determined up to holomorphic conjugacy by its internal and external classes, that is, the (conjugacy classes of) the restrictions to the filled Julia set and its complement. In particular the external class is a degree d real-analytic orientation preserving and strictly expanding self-covering of the unit circle: the expansivity of such a circle map implies that all the periodic points are repelling, and in particular not parabolic. We extended the polynomial-like theory to a class of parabolic mappings which we called parabolic-like mappings. A parabolic-like mapping is an object similar to a polynomial-like mapping, but with a parabolic external class; that is to say, the external map has a parabolic fixed point, whence the domain is not contained in the codomain. In this talk we present the parabolic-like mapping theory. We give a sketch of the proof of the Straightening Theorem for parabolic-like mappings, which states that every degree 2 parabolic-like mapping is hybrid equivalent to a member of the family of quadratic rational maps of the form P_A(z)=z+ 1/z+ A, where A is a complex number. Then we will consider families of parabolic-like mappings, state the main result in this setting and give an application. Parabolic-like mappings31 January 2014
Nick Sharples (Imperial College London)Abstract: In this talk I will outline the renormalization theory for irregular Ordinary Differential Equations, introduced by DiPerna & Lions, which provides existence and uniqueness of generalised flow solutions for vector fields that have limited (e.g. Sobolev) regularity. I will discuss a recent extension of these results to include vector fields with a set of 'bad' singularities, where the vector field is not locally of bounded variation, provided that this singular set is sufficiently small in a fractal sense (joint with E. Miot). I will consider the non-autonomous case where the singular set is a subset of the extended phase space. In this setting the appropriate notion of fractal dimension is that of an anisotropic 'codimension print' in which the spatial and temporal detail of the singular set can be distinguished. I will relate this esoteric notion of dimension to the more familiar box-counting dimension (joint with J.C. Robinson) providing straightforward criteria for the existence and uniqueness of non-autonomous generalised flows. Irregular ODEs: renormalization and geometry30 January 2014
Konstantin Khanin (Loughborough University)Abstract: In this talk we shall report on recent progress in rigidity theory for nonlinear interval exchange transformations corresponding to cyclic permutations. Such maps can be viewed as circle homeomorphisms with multiple break points. We shall discuss both recent results on renormalizations of such maps in case of one break point (joint with S. Kocic and A. Teplinsky), and extension to the multiple-break setting (based on work in progress with A. Teplinsky). On rigidity for cyclic nonlinear interval exchange transformations23 January 2014
Anna Cherubini (University of Salento)Abstract: This study deals with the identification of early-warning signals for desertification in fragile ecosystems such as arid or semi-arid ones. Literature on this topic shows that vegetation patchiness in semi-arid ecosystems can lead to the identification of indicators for an approaching transition to desertification. In particular, the analysis of the spatial fluctuations of vegetation patterns suggests that a significant deviation from power laws in vegetation patch size distributions is a reliable signal for an approaching desertification. In this work we analysed a model for semi-arid ecosystems based on a stochastic cellular automaton (CA) depending on a number of parameters (accounting for external stresses, soil quality, water, interactions between plants). We investigated the time fluctuations properties of the quantities associated with the steady states of the CA and we found that other possible and earlier signals are related to percolation thresholds and to the time fluctuations distribution of the biggest cluster size. Time fluctuations of vegetation patterns and early warnings for desertification21 January 2014
Raffaele Vitolo (University of Salento)Abstract: Homogeneous differential-geometric Poisson brackets were introduced by Dubrovin and Novikov in 1984. Such operators appear in many integrable systems. First order operators have been extensively studied so far. In this talk we will devote ourselves to the classification of third order operators with non-degenerate leading term. Starting from old results by one of us (GVP) we prove that such operators are completely characterized by their leading term, which is a Monge metric. Monge metrics are distinguished pseudo-Riemannian metrics which are in bijection with quadratic line complexes. Quadratic line complexes are algebraic varieties for which classification results are known in the 2 and 3 component cases. Using such results we are able to provide a classification of Hamiltonian operators in 2 and 3 components, together with some hints on how to solve the problem in a higher number of components. Joint work with E.V. Ferapontov, M.V. Pavlov, G.V. Potemin. On the classification of homogeneous differential-geometric third-order Poisson brackets16 January 2014
Rainer Klages (Queen Mary University)Abstract: My talk is about the impact of spatial [1] and temporal [2] random perturbations on diffusion in chaotic dynamical systems. As an example, I consider deterministic random walks on a one-dimensional lattice. The system is modeled by a piecewise linear map defined on the unit interval which depends on two control parameters and is lifted onto the whole real line. Computer simulations show a rich scenario in the diffusion coefficient of this model by increasing the perturbation strength. Typical signatures of the transition from small to large perturbations are multiple suppression and enhancement of diffusion by approaching basic asymptotic laws for large perturbation strength. These results are reproduced by simple approximations based on the parameter dependence of the unperturbed deterministic diffusion coefficient [3]. [1] R.Klages, Phys. Rev. E 65, 055203(R) (2002) [2] R.Klages, Europhys. Lett. 57, 796 (2002) [3] R.Klages, Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics, Advanced Series in Nonlinear Dynamics Vol.24 (World Scientific, Singapore, 2007), Part 1. Chaotic diffusion in randomly perturbed dynamical systems12 December 2013
Sigurður Hafstein (Reykjavik University)Abstract: Lyapunov functions can characterize many fundamental properties of dynamical systems as attractor-repeller pairs, basins of attraction, and the chain-recurrent set. We discuss an algorithm to compute continuous and piecewise affine (CPA) Lyapunov functions for continuous nonlinear systems by linear optimization. The algorithm can be adapted to compute Lyapunov functions for continuous differential inclusions and discrete systems. We further discuss how the algorithm can be combined with faster methods with less concrete bounds, e.g. the radial-basis-functions collocation method, to deliver true Lyapunov functions comparatively fast. Computation of CPA Lyapunov functions9 December 2013
Sebastian Wieczorek (University of Exeter)Abstract: Rate-induced bifurcations occur in forced systems where there is a stable state for every fixed level of forcing. When forcing varies in time, the position of the stable state changes and the system tries to keep pace with the changes. However, some systems fail to adiabatically follow the continuously changing stable state and destabilise above some critical rate of forcing. Scientists often find rate-induced bifurcations counter-intuitive because there is no obvious loss of stability and no obvious threshold. On the other hand, these non-autonomous instabilities cannot in general be described by classical bifurcations or asymptotic approaches and remain fairly unexplored. I will present an approach based on geometrical singular perturbation theory to study critical rates of change and non-obvious thresholds. I will also discuss repercussions for climate change policy making which currently focuses on critical levels of the atmospheric temperature whereas the critical factor may be the rate of warming rather than the temperature itself. Rate-induced bifurcations: critical rates, non-obvious thresholds and failure to adapt to changing external conditions5 December 2013
Jürgen Knobloch (TU Ilmenau)Abstract: Homoclinic snaking refers to the continuation curves of homoclinic orbits near a heteroclinic cycle, which connects an equilibrium and a periodic orbit in a reversible Hamiltonian system. We consider non-reversible perturbations of this situation and show analytically that such perturbations typically lead to either infinitely many closed continuation curves (isolas) or to two snaking continuation curves, which follow the primary sinusoidal continuation curves alternately (criss-cross snaking). Non-reversible perturbations of homoclinic snaking scenarios28 November 2013
Evelyne Miot (CNRS, Paris)Abstract: A system of equations combining the 1D Schrödinger equation and the point vortex system has been derived by Klein, Majda and Damodaran to modelize the evolution of nearly parallel vortex filaments in 3D incompressible fluids. In this talk I will describe some dynamics for this system such as travelling waves, collisions and finite-time blow-up. I will finally present the case of pairs of filaments. This is joint work with Valeria Banica and Erwan Faou. Some examples of dynamics for nearly parallel vortex filaments21 November 2013
Janosch Rieger (Goethe University Frankfurt)Abstract: The theory of finite element methods is one of the success stories of modern mathematics. It is therefore reasonable to ask what set-valued numerical analysis can learn from this field. The main aim of this talk is to popularise the concept of set spaces that are defined in terms of a generalization of the well-known support function. This concept seems to be the language in which the central ideas behind finite element methods can be transferred to the computation of sets. After a formal introduction to set spaces, some graphic examples of such spaces will be given, and the relationship between conventional representations of sets and set spaces will be explained. Some preliminary results will be shown and commented. Spaces of non-convex sets and their potential for set-valued numerical analysis14 November 2013
Ale Jan Homburg (University of Amsterdam)Abstract: I'll consider iterated function systems generated by finitely many diffeomorphisms on compact manifolds. I'll discuss aspects of their dynamics, in particular minimality and synchronization. These iterated function systems play a pivotal role in the study of dynamical systems: they correspond to dynamical systems of skew product type and provide examples of "partially hyperbolic dynamical systems". I'll discuss how iterated function systems are giving new results and insights in the study of partially hyperbolic dynamics. Iterated function systems, skew product dynamics, partially hyperbolic dynamics7 November 2013
Gioia Vago (Université de Bourgogne)Abstract: The Ogasa invariant is defined for any compact manifold in any dimension. Roughly speaking, it is computed on the largest regular level of a slimmest Morse function. The minimax procedure underlying its definition makes its exact computation extremely hard, not to say impossible, in the very general case. However, it is crucial to know as much as we can about its behaviour, because this invariant contains precious and sharp information about how much structure the manifold has, and therefore it can be used as a detector of singularities. In dimension 2, the computation of this invariant is straightforward. Dimension 3 is the first non-trivial case. As a result of a joint work with Michel Boileau (Univ. Aix-Marseille, France), now we have a global qualitative and quantitative understanding of what this dynamical invariant measures, and how it is related to other topological and algebraic invariants of the underlying manifold. The Ogasa invariant in dimension 331 October 2013
Jan Sieber (University of Exeter)Abstract: If one wants to perform bifurcation analysis for a given smooth low-dimensional dynamical system of interest, one can use a range of ready-made numerical tools such as AUTO, MatCont or CoCo. I will show which aspects of this analysis can be carried out also in physical experiments. One typical limitation in physical experiments is that one cannot set the internal state of the system at will. An example, tracking branches of unstable periodic orbits around a saddle-node, in a simple mechanical experiment will demonstrate the basic principle. Other potential applications are: * finding the flow on unstable parts of the slow manifold in slow-fast systems (the unstable parts of so-called canards), * study of the collapse of long chaotic transients in high-dimensional systems. [joint work with David Barton (Bristol), Oleh Omelchenko, Matthias Wolfrum (WIAS Berlin, Germany)] Bifurcation Analysis for Experiments17 October 2013
Dmitry Turaev (Imperial College London)Abstract: Let a real-analytic Hamiltonian system have a normally-hyperbolic cylinder such that the Poincare map on the cylinder has a twist property. Let the stable and unstable manifolds of the cylinder intersect transversely in a certain strong sense. The homoclinic channel is a small neighbourhood of the union of the cylinder and the homoclinic. We show that generically (in the real-analytic category) in the channel there always exist orbits which move from one end of the cylinder to the other. This opens a way of showing that given an integrable system with 3 or more degrees of freedom, for arbitrarily small generic Hamiltonian perturbations the change of action variables along resonances is bounded away from zero. Arnold diffusion in a priori chaotic systems10 October 2013
Davoud Cheraghi (Imperial College London)Abstract: The local and global dynamics of holomorphic maps near fixed points with asymptotic irrational rotation has been extensively studied through various methods over the last decades. The problem involves delicate interplay between the Diophantine approximation of the irrational rotation, "small divisors", and the distortion properties of holomorphic maps. In this talk we report on historical developments in the subject and discuss some recent breakthrough using renormalization techniques. Small divisors in holomorphic dynamics8 October 2013
Erik Bollt (Clarkson University)Abstract: Mixing, and coherence are fundamental issues at the heart of understanding fluid dynamics and other non- autonomous dynamical systems. Only recently has the notion of coherence come to a more rigorous footing, and particularly within the recent advances of finite-time studies of nonautonomous dynamical systems.. Here we define shape coherent sets which we relate to measure of coherence in differentiable dynamical systems from which we will show that tangency of finite time stable foliations (related to a forward time perspective) and finite time unstable foliations (related to a “backwards time" perspective) serve a central role. This perspective is agreeable with the recent theory of geodesics by Haller et. al. derived from a variational principle of geodesics. We develop zero-angle curvers, meaning non-hyperbolic splitting, by continuation methods in terms of the implicit function theorem, from which follows a simple ODE description of the boundaries of shape coherent sets. Differential Geometry Perspective of Shape Coherence and Curvature Evolution by Finite-Time Nonhyperbolic Splitting 3 October 2013
Sebastian van Strien (Imperial College London)Abstract: These lectures will review some results about the dynamics of interval maps, focusing especially on recent results which rely on tools from complex analysis. Topics covered will include: topological and topological properties of the Julia sets, density of maps with good behaviour (hyperbolic maps) There are no specific prerequisites for this course, apart from a basic course in complex analysis. A related, and more extensive MSc course, will be taught jointly with Davoud Cheraghi in the 2nd term (TCC). Holomorphic dynamics of real interval maps: chaos and hyperbolicity (Part III)3 October 2013
Sebastian van Strien (Imperial College London)Abstract: These lectures will review some results about the dynamics of interval maps, focusing especially on recent results which rely on tools from complex analysis. Topics covered will include: topological and topological properties of the Julia sets, density of maps with good behaviour (hyperbolic maps) There are no specific prerequisites for this course, apart from a basic course in complex analysis. A related, and more extensive MSc course, will be taught jointly with Davoud Cheraghi in the 2nd term (TCC). Holomorphic dynamics of real interval maps: chaos and hyperbolicity (Part II)2 October 2013
Sebastian van Strien (Imperial College London)Abstract: These lectures will review some results about the dynamics of interval maps, focusing especially on recent results which rely on tools from complex analysis. Topics covered will include: topological and topological properties of the Julia sets, density of maps with good behaviour (hyperbolic maps) There are no specific prerequisites for this course, apart from a basic course in complex analysis. A related, and more extensive MSc course, will be taught jointly with Davoud Cheraghi in the 2nd term (TCC). Holomorphic dynamics of real interval maps: chaos and hyperbolicity (Part I)1 October 2013
Bob Rink ( Vrije Universiteit Amsterdam )Abstract: A classical result of Aubry and Mather states that Hamiltonian twist maps have orbits of all rotation numbers. Analogously, one can show that certain ferromagnetic crystal models admit ground states of every possible mean lattice spacing. In this talk, I will show that these ground states generically form Cantor sets, if their mean lattice spacing is an irrational number that is easy to approximate by rational numbers. This is joint work with Blaz Mramor. Ferromagnetic crystals and the destruction of minimal foliations12 July 2013
Bob Rink (Vrije Universiteit Amsterdam)Abstract: Dynamical systems with a coupled cell network structure arise in applications that range from statistical mechanics and electrical circuits to neural networks, systems biology, power grids and the world wide web. A network structure can have a strong impact on the behaviour of a dynamical system. For example, it has been observed that networks can robustly exhibit (partial) synchronisation, multiple eigenvalues and degenerate bifurcations. In this talk I will explain how semigroups and their representations can be used to understand and predict these phenomena. As an application of our theory, I will discuss how a simple feed-forward motif can act as an amplifier. This is joint work with Jan Sanders. Using semigroups to study coupled cell networks10 July 2013
June Barrow-Green (The Open University)Abstract: In October 1912, the young American mathematician G. D. Birkhoff 'astonished the mathematical world' by providing a proof of Poincaré's last geometric theorem. The theorem, which was connected to Poincaré's long standing interest in the periodic solutions of the three-body problem, had been proposed by him only months before he died. Birkhoff continued to work on aspects of dynamical systems throughout his career, his aim being to create a general theory. Many of his ideas are contained in his book Dynamical Systems (1927), the first book to develop the qualitative theory of systems defined by differential equations and where he effectively 'created a new branch of mathematics separate from its roots in celestial mechanics and making broad use of topology'. G. D. Birkhoff and the development of dynamical systems theory 26 June 2013
Ken Palmer (Providence University, Taiwan)Abstract: Theoretical aspects: If a smooth dynamical system on a compact invariant set is structurally stable, then it has the shadowing property, that is, any pseudo (or approximate) orbit has a true orbit nearby. In fact, the system has the Lipschitz shadowing property, that is, the distance between the pseudo and true orbit is at most a constant multiple of the local error in the pseudo orbit. S. Pilyugin and S. Tikhomirov showed the converse of this statement for discrete dynamical systems, that is, if a discrete dynamical system has the Lipschitz shadowing property, then it is structurally stable. In this talk this result will be reviewed and the analogous result for flows, obtained jointly with S. Pilyugin and S. Tikhomirov, will be described. Numerical aspects: This is joint work with Brian Coomes and H\" useyin Ko\c cak. A rigorous numerical method for establishing the existence of an orbit connecting two hyperbolic equilibria of a parametrized autonomous system of ordinary differential equations is presented. Given a suitable approximate connecting orbit and assuming that a certain associated linear operator is invertible, the existence of a true connecting orbit near the approximate orbit and for a nearby parameter value is proved provided the approximate orbit is sufficiently ``good''. It turns out that inversion of the operator is equivalent to the solution of a boundary value problem for a nonautonomous inhomogeneous linear difference equation. A numerical procedure is given to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse (the latter determines how ``good'' the approximating orbit must be). Some theoretical and numerical aspects of shadowing26 June 2013
Pablo Guarino (Universidade de Sao Paulo)Abstract: The so-called critical circle maps are orientation-preserving smooth circle homeomorphisms having critical points (they are not diffeo- morphisms). In a recent joint work with Welington de Melo (avail- able at arXiv:1303.3470) we proved that any two C3 critical circle maps with the same irrational rotation number of bounded type and with a unique critical point of the same odd criticality are conjugate to each other by a C1+? circle diffeomorphism, for some universal ? > 0. This geometric rigidity was conjectured in the early eight- ies (after several works of Feigenbaum, Kadanoff, Lanford, Rand, Shenker among others) and has promoted a lot of previous results in the real-analytic category (Swiatek, Herman, de Faria-de Melo, Yampolsky, Khanin-Teplinsky among others). In this talk we will discuss the main ideas of the proof. Geometric Rigidity of critical circle maps26 June 2013
Andrey Shilnikov (Georgia State University)Abstract: We identify and describe the principal bifurcations of bursting rhythms in multi-functional central pattern generators (CPG) composed of three neurons connected by fast inhibitory or excitatory synapses. We develop a set of computational tools that reduce high-order dynamics in biologically relevant CPG models to low-dimensional return mappings that measure the phase lags between cells. We examine bifurcations of fixed points and invariant curves in such mappings as coupling properties of the synapses are varied. These bifurcations correspond to changes in the availability of the network's phase locked rhythmic activities such as periodic and aperiodic bursting patterns. As such, our findings provide a systematic basis for understanding plausible biophysical mechanisms for the regulation of, and switching between, motor patterns generated by various animals. Key bifurcations of bursting polyrhythms in three-cell central pattern generators 12 June 2013
Andrey Shilnikov (Georgia State University)Abstract: TBA Chaos: stirred not shaken 11 June 2013
Mason Porter (Oxford University)Abstract: I discuss "simple" dynamical systems on networks and examine how network structure affects dynamics of processes running on top of networks. I consider results based on "locally tree-like" and/or mean-field and pair approximations and examine when they seem to work well, what can cause them to fail, and when they seem to produce accurate results even though their hypotheses are violated fantastically. I'll also present a new model for multi-stage complex contagions--in which fanatics produce greater influence than mere followers--and examine dynamics on networks with hetergeneous correlations. (This talk discusses joint work with James Gleeson, Sergey Melnik, Peter Mucha, JP Onnela, Felix Reed-Tsochas, and Jonathan Ward.) Cascades and Social Influence on Networks30 May 2013
Michel Crucifix (Université catholique de Louvain)Abstract: Glacial-interglacial cycles have paced climate for over 3 million years. The phenomenon is thought to result from the interplay between non-linear internal dynamics and the extrenal forcing induced by the variations in the Earth's orbit and obliquity, commonly modelled as quasi-periodic functions. Here we concentrate to a number of different simple determinstic models of ice ages, available in the form of small systems of ordinary differential equations. Our objective is to explore and explain the sensitivity of these models to initial conditions and parameters that has been previously discussed in the literature, in terms of dynamical system theory. To this end, we use a series of tools and concepts, including the section of the pullback attractor, the Lyapunov exponent, and the phase sensitivity exponents. The most interesting behaviours appear to be linked to the emergence of strange nonchaotic attractors (SNA). The scenario corresponds to a negative long-term Lyaponov exponent, with sensitive dependence both to parameters and additive noise. This dependence is manifested in the form af trajectory shifts, which may be interpreted as a consequence of the non-smooth character of the attractor. We discuss the implications of these results on our understanding of palaeoclimate dynamics and ability to predict future glacial cycles. References: T. Mitsui and K. Aihara, Dynamics between order and chaos in conceptual models of glacial cycles, published online in Climate Dynamics (available at http://link.springer.com/article/10.1007/s00382-013-1793-x M. Crucifix, Why glacial cycles could be unpredictable ? accepted in Climate of the Past (available at http://arxiv.org/abs/1302.1492) Glacial cycles and strange non-chaotic attractors 22 May 2013
Lev Lerman (Lobachevsky University of Nizhni Novgorod)Abstract: I’ll discuss results obtained in our group in Nizhny Novgorod (former Gorky) mostly in seventies of XX century when studying smooth nonautonomous flows on smooth closed manifolds. The main classifying relation for this study was taken the uniform equivalence for two nonautonomous flows, more precisely, for their foliations into integral curves in the extended phase space. This allowed us to give a notion of structurally stable nonautonomous flows and find a natural class of flows for which the invariant of its uniform equivalency was found was proved they to be structurally stable. Also it gave a possibility to find connections between the structure of flows and topology of the ambient manifold (like Morse inequalities). One more application of these notions was the relation of uniform homotopy equivalence for the maps that classified diffeomorphisms and gave a source of constructing nonautonomous flows with various structure. Nonautonomous flows and uniform topology1 May 2013
Jason Gallas (Universidade Federal da Paraíba)Abstract: The investigation of regular and irregular patterns in nonlinear oscillators is an outstanding problem in physics and in all natural sciences. In general, regularity is understood as tantamount to periodicity. However, there is now a flurry of works proving the existence of ``antiperiodicity'', an unfamiliar type of regularity. In this seminar, we present a report about the experimental observation and numerical corroboration of antiperiodic oscillations. In contrast to the isolated solutions presently known, we show several examples of infinite hierarchies of antiperiodic waveforms that can be tuned continuously and that form wide spiral-shaped stability phases in the control parameter plane (i.e. in phase diagrams). The waveform complexity increases towards the focal point common to all spirals, a key hub interconnecting them all. Antiperiodic oscillations25 April 2013
Vered Rom-Kedar (Weizmann Institute)Abstract: Simple models of the innate immune system teach us much about the development of infections when the bone-marrow function is damaged by chemotherapy. The results depend only on robust properties of the underlying modeling assumptions and not on the detailed models. Such models may lead to improved treatment strategies for neutropenic patients [1,2,3,4]. [1] Roy Malka, Baruch Wolach, Ronit Gavrieli, Eliezer Shochat and Vered Rom-Kedar, Evidence for bistable bacteria-neutrophil interaction and its clinical implications J. Clin Invest. doi:10.1172/JCI59832, 2012. See also commentary . [2] R. Malka and V. Rom-Kedar, Bacteria--Phagocytes Dynamics, Axiomatic Modelling and Mass-Action Kinetics, Mathematical Biosciences and Engineering, 8(2), 475-502, 2011. [3] E. Shochat and V. Rom-Kedar; Novel strategies for G-CSF treatment of high-risk severe neutropenia suggested by mathematical modeling, Clinical Cancer Research 14, 6354-6363, October 15, 2008. [4] E. Shochat, V. Rom-Kedar and L. Segel; G-CSF control of neutrophils dynamics in the blood, Bull. Math. Biology , 69(7), 2299-2338, 2007 The innate immune system: some theory, experiments and medical implications24 April 2013
Sergey Yakovenko (Weizmann Institute)Abstract: The original Hilbert 16th problem about the limit cycles of polynomial planar vector fields stays open for over 110 years; its centennial history was full of dramatic turns, as exposed in the survey "Centennial History of the Hilbert 16th Problem" by Yulij Ilyashenko. About 50 years ago Ilyashenko himself suggested a problem about zeros of the Poincare-Pontryagin integral which describes bifurcation of limit cycles in perturbations of integrable (Hamiltonian) systems. I will discuss various precise formulations of this problem and recent results by G. Binyamini, D. Novikov and the speaker giving explicit and existential upper bounds for the number of isolated zeros of Abelian and pseudoabelian integrals. The talk is intended for a broad audience. Semicentennial history of the Infinitesimal Hilbert 16th problem9 April 2013
Alejandro Passeggi (TU Dresden)Abstract: We give an introduction to the rotation theory in the two dimensional torus which can be seen as the generalization of Poincaré theory in the circle, and present the following result: "There exists an open and dense set of homeomorphisms homotopic to the identity (with respect the $C^0$ topology), such that the rotation set of its elements is a rational polygon". Rational Polygons as Rotation Sets of Generic Homeomorphisms of the two Torus21 March 2013
Mike Field (Rice University )Abstract: For dynamicists, a network consists of interconnected dynamical systems (or "nodes"). Classical networks encountered in dynamics are synchronous: nodes all run on the same clock and connectivity is fixed. However, most networks encountered in contemporary science and technology are asynchronous. In particular, biological networks, computer networks and distributed systems generally are asynchronous: nodes may run on different clocks, connectivity may vary in time and nodes may stop and then restart: the antithesis of 'analytic behaviour' expected of solutions of smooth differential equations. In this talk, we describe how a dynamicist might approach the definition and mathematics of asynchronous networks as well as describe and illustrate some recent results and observations about dynamics on asynchronous networks and possible mechanisms that allow them to work efficiently. Asynchronous Networks 14 March 2013
Ivan Wong (University of Manchester)Abstract: Piecewise smooth maps appear as models of various physical, economical and other systems. In such maps bifurcations can occur in when a fixed point (or periodic orbit) crosses or collides with the border between two regions of smooth behaviour. These bifurcations have little analogue in standard bifurcation theory for smooth maps and they are now known as border collision bifurcations. In this talk, we show that for piecewise smooth maps which allow area expansions, the dynamics of the system is not necessarily trivial. In particular, snap-back repellers and two-dimensional attractors can exist for appropriate parameter values. Border collision bifurcations – snap-back repellers and two-dimensional attractors7 March 2013
Thomas Bridges (University of Surrey )Abstract: The backbone of the talk is "modulation"; specifically what to modulate and how modulation generates geometry. The talk will be based on three examples. (1D modulation) how modulation gives a new viewpoint on elementary homoclinic bifurcation with curvature of modulation determining the coefficient of the nonlinear term. (2D modulation) the mythical origins of the KdV equation are given a new perspective, resulting in a universal form for emergence and how geometry of modulation determines the coefficients, and a dynamical systems argument determines the dispersion. Considering that the classical derivation of KdV is about the trivial solution, a by-product of the result is how to modulate the trivial solution! (3D modulation) The third example will show how modulation of periodic solutions leads to a sequence of multi-pulse planforms in PDEs like the Swift-Hohenberg equation. This theory is based on a new modulation equation in two space dimensions and time. How modulation generates geometry in dynamical systems and nonlinear waves28 February 2013
Zeng Lian (Loughborough University)Abstract: Lyapunov exponents play an important role in the study of the behavior of dynamical systems, which measure the average rate of separation of orbits starting from nearby initial points. They are used to describe the local stability of orbits and chaotic behavior of systems. Multiplicative Ergodic Theorem provides the theoretical fundation of Lyapunov exponents, which gives the fundamental information of Lyapunov Exponents and their associates invariant subspaces. In this talk, I will report the work on Multiplicative Ergodic Theorem (with Kening Lu), which is applicable to infinite dimensional random dynamical systems in a separable Banach space. The system could be generated by, for example, random partial differential equations. Lyapunov exponents and Multiplicative Ergodic Theorem for random systems in a separable Banach space21 February 2013
Sofia Trejo (Warwick University)Abstract: I will talk about the construction of complex box mappings with complex bounds for real analytic interval maps. More specifically, I will show that given a real analytic map and a point in its postcritical set it is possible to construct complex box mappings, associated to the real first return maps, with complex bounds for arbitrarily small scales. In the case the map is a non-renormalizable polynomial (not necessarily real) with only hyperbolic periodic points, the complex-box mapping can be constructed using the Yoccoz puzzle. For real analytic maps we can not guarantee the existence of a Yoccoz puzzle. For this reason the construction of the box mapping and the prove of complex bounds in this case requires more work. Complex bounds are fundamental for the prove of quasiconformal rigidity, renormalization results and ergodic properties. Complex Bounds for real analytic interval maps. 14 February 2013
Franco Vivaldi (Queen Mary)Abstract: We consider a model of planar rotations subject to round-off, which leads to dynamics on a lattice. We let the angle of rotation approach a low-order rational. There is a non-smooth integrable Hamiltonian system, featuring a foliation by polygonal invariant curves, which represents the limit of vanishing discretisation of the space. We prove that, for sufficiently small discretization, a positive fraction of those invariant curves survives, leading to a discrete space version of the KAM scenario. The surviving curves are characterised in terms of congruences and properties of Gaussian integers. joint work with Heather Reeve-Black Near-Integrable behaviour in a system with discrete phase space7 February 2013
Vasso Anagnostopoulou (Imperial College)Abstract: We study a class of model systems which exhibit the full two step scenario for the nonautonomous Hopf bifurcation, as proposed by Ludwig Arnold. The specific structure of these models allows for a rigorous and thorough analysis of the bifurcation pattern. In particular, we show the existence of an invariant 'torus' splitting off a previously stable central manifold after the second bifurcation point. A model for the nonautonomous Hopf bifurcation31 January 2013
John Lowenstein (New York University)Renormalization of Piecewise Isometries24 January 2013
Ole Peters (London Mathematical Laboratory)Abstract: Ensemble averages are frequently used as the basis for decision theories in economics and games. However, they are often applied in situations which do not correspond physically to ensembles. In many common problems, such as that of an individual deciding how to invest his wealth, it would be more appropriate to look at time averages. The question of whether these two averages are identical leads to the concept of ergodicity. In a non-ergodic system they may differ, making it vital to know which is relevant. Economists have largely failed to make this distinction and, as a result, are stuck with a confusing and incomplete mathematical framework. The problems caused by the inappropriate use of ensemble averages and the failure to recognise non-ergodicity are illustrated by a famous example: the St Petersburg Paradox. At best, these failings have spurred the development of arbitrary and unjustifiable techniques, such as utility theory. At worst, they have caused mathematical errors to lie hidden in the economics literature for decades. Proper acknowledgement of ergodicity and the role of time leads naturally to a simple resolution of the St Petersburg Paradox. We wish to explore how a deeper understanding of ergodic theory might shed light on other foundational problems in economics. In particular, we are keen to establish a standard nomenclature bringing together the different definitions of ergodicity that exist in dynamical systems, stochastic processes, and economics itself. Ergodicity in Economics 17 January 2013
Thomas Jordan (University of Bristol)Abstract: This is joint work with Godofredo Iommi. In the setting of one dimension expanding maps the topological pressure function is well known to depend analytically on suitably defined permutations. The same is true when we move from expanding maps to suspension flows. In the case where the expanding map has countably many branches (for example the Gauss map) the pressure function has been studied by Sarig and Mauldin and Urbanski. In this case if the system is modeled by the full shift then when finite the pressure function varies analytically. We will look at the case when suspension flows over countable state maps are considered and give certain conditions under which the pressure varies analytically and show examples where there are phase transitions in the pressure function. Phase transitions for suspension flows10 January 2013
Phil Boyland (University of Florida)Abstract: Given $\beta>0$, Renyi's beta-shift $Z_\beta$ encodes the collection of expansions base $\beta$ of all $x\in [0,1]$. The \textit{digit frequency vector} records the relative frequencies of the various digits in the beta-expansion of a given x, and the set of all such vectors for a given beta is its \textit{digit frequency set}. We show that this set is always compact, convex, $k-$dimensional (where $k \leq \beta \leq k+1$), and it varies continuously with $\beta$. When $k=2$ we give a complete description of the family of the digit frequency sets: roughly, it looks like nested convex-set valued devil's staircases. Considering the family of sets with the Hausdorff topology, the typical frequency set has countably infinite vertices with a single, completely irrational limit vertex. We then discuss how these results yield a near complete understanding of the rotation sets and their bifurcations of a family of two-torus homeomorphisms. Rotation sets for beta-shifts and torus homeomorphisms13 December 2012
Peter Hazard (University of Warwick )Abstract: Roughly speaking, a modulus of stability is a number which is invariant under topological conjugacy. For unimodal maps there is a basic invariant - the kneading invariant - which determines the topological class of a unimodal map if the map is sufficiently smooth. However, it is known that in dimension two there are maps with infinitely many moduli. We will discuss what happens `in between', in the setting of Henon-like maps which are small perturbations of unimodal maps. This is joint work with M. Martens and C. Tresser. Infinite Moduli of Stability for Henon-like Maps6 December 2012
Anatoly Neishtadt (Loughborough University )Abstract: We study a classical billiard of charged particles in a strong non-uniform magnetic field. We provide an adiabatic description for skipping motion along the boundary of the billiard. We show that a sequence of many changes of regimes of motion from skipping to motion without collisions with the boundary and back to skipping leads to destruction of the adiabatic invariance and chaotic dynamics in a large domain in the phase space. This is a new mechanism of the origin of chaotic dynamics for systems with impacts. Destruction of adiabatic invariance for billiards in strong non-uniform magnetic field 29 November 2012
Rajendra Bhansali (University of Liverpool)Long memory properties of stochastic intermittency maps22 November 2012
Jens Marklof (University of Bristol )Abstract: Despite their simple geometry, billiards in polygons give rise to a rich variety of dynamical phenomena. One example is the asymptotic distribution of closed billiard trajectories, which is still only partially understood. In the present lecture I will discuss a different natural problem: the distribution of eigenfunctions of the Dirichlet Laplacian of a polygon--- is the L^2 mass of the eigenfunctions highly localized or equidistributed on the billiard domain? The lecture is based on joint work with Zeev Rudnick (Tel Aviv). Eigenfunctions of polygonal billiards 15 November 2012
Wolfram Just (Queen Mary, University of London)Abstract: We apply time-delayed feedback control to stabilise unstable periodic orbits of an amplitude-phase oscillator. The control acts on both, the amplitude and the frequency of the oscillator.The model has been introduced by Fiedler and Sch"oll as a counterexample for the so called odd-number limitation of time-delayed feedback control. A comprehensive bifurcation analysis in terms of the control phase and the control strength reveals large stability regions of the target periodic orbit, as well as an increasing number of unstable periodic orbits caused by the time delay of the feedback loop. The theoretical results are illustrated by an experimental realisation of the time-delayed feedback scheme proposed by Sch"oll and Fiedler. The experimental control performance is in quantitative agreement with the bifurcation analysis. The results uncover some general features of the control scheme which are deemed to be relevant for a large class of setups. Pyragas-Schöll-Fiedler control8 November 2012
Gorka Zamora (Humboldt Universitaet)Abstract: Despite the significant differences in the sizes of brains in the animal kingdom, there is increasing evidence that they all share similar modular and hierarchical organization. Modelling the brain activity to reproduce the observed functional networks brings important challenges to solve. Beyond the purely computational limitations, we also need to face theoretical issues before we are able to reproduce brain-like dynamics with certain degree of confidence. Complex brain networks: structure, modeling and function1 November 2012
Natalia Janson (Loughborough University)Abstract: A single neuron is modelled as a highly non-linear dynamical system. Its crucial feature is excitability: when there is no input, or the input is below a threshold, the neuron is silent, and when the input is above the threshold, the neuron fires. The noise can play a highly counter-intuitive role in such systems: with the increase of the amount of noise, the amount of order in the system grows. Thus, the noise-induced spiking in such neurons becomes almost regular at the optimal intensity of the stimulus. Many neurons are coupled together in a network by various means, to model a biological neural network. Their collective behavior ranges from independent spiking, through partly and fully synchronised spiking, to the lack of any spiking, depending on the parameters of the coupling. Synchronized spiking in the brain is associated with epilepsy, Parkinsonian disease and tremor, so the ability to eliminate synchronization by non-invasive weak stimulation could be essential in treating such conditions. It appears that in the model of the stochastic neural network it is possible to partly control the collective behavior by a specially constructed feedback. Self-organisation and synchronization in stochastic neuron-like networks 24 October 2012
Jaap Eldering (Imperial College)Abstract: Normally Hyperbolic Invariant Manifolds (NHIMs for short) are an important tool to globally study perturbations of dynamical systems. I will first recall what NHIMs are (simply put, these are generalisations of hyperbolic fixed points) and then indicate how my result on persistence of noncompact NHIMs generalizes the classical compact case. Noncompactness requires us to introduce the concept of Riemannian manifolds of bounded geometry. These can be viewed as the class of uniformly C^k manifolds. I will assume no detailed knowledge of differential geometry, but illustrate this with images. Standard analysis techniques and results can be adapted to this setting. I will illustrate this with the example of constructing a uniformly sized tubular neighborhood of a (noncompact) submanifold, which is necessary to construct the persistent manifold. Normally Hyperbolic Invariant Manifolds the noncompact way18 October 2012
Yuzuru Sato (RIES, Hokkaido University)Abstract: Interaction between deterministic chaos and stochastic randomness has been an important problem in nonlinear dynamical systems studies. Noise-induced phe- nomena are understood as drastic change of natural invariant densities by adding external noise to a deterministic dynamical systems, resulting qualitative transition of observed nonlinear phenomena. Stochastic resonance, noise-induced synchronization, and noise-induced chaos are typical examples in this scheme. The simplest mathematical model for problem is one-dimensional map stochastically perturbed by noise. In this presentation, we discuss typical behavior of noised dynamical sys- tems based on numerically observed noise-induced phenomena in logistic map, Belousov-Zhabotinsky map and modified Lasota-Mackey map. Our observation indicates that (i) both noise-induced chaos and noise-induced order may coexist, and that (ii) asymptotical periodicity of densities varies according to noise amplitude. An application to time-series analysis of rotating fluid is also exhibited. Noise-induced phenomena in one-dimensional maps18 October 2012
Sebastian van Strien (Imperial College)Abstract: One of the best known dynamical systems with intermittency behaviour is the well-known Pomeau-Manneville circle map $x\mapsto x+x^{1+\alpha} \mod 1$. This map has a neutral fixed point at $0$ which causes orbits to linger there for long periods. Nevertheless this map has always a physical measure: for $\alpha\ge 1$ it is the Dirac measure at $0$ while for $\alpha\in (0,1)$ it is absolutely continuous. It was also known for quite a while that this map is stochastically stable when $\alpha\ge 1$. In this talk I will discuss a result which implies that this map is also stochastic stable when $\alpha\in (0,1)$. (joint with Weixiao Shen) Stochastic stability of expanding circle maps with neutral fixed points11 October 2012
Genadi Levin (Hebrew University)Abstract: We consider an infinitely renormalizable map f_c with all its renormalizations of non-primitive (satellite) type. Accosiated to it is a combinatorial data: a sequence of (rational non-zero) rotation numbers {t_m=p_m/q_m} of the dividing fixed points of the renormalizations of f_c. Equivalently, the parameter c is a limit point of a sequence of "satellite bifurcations" with the "internal arguments" {t_m}. For example, the stationary sequence {t_m=1/2} corresponds (on the real line) to the famous Feigenbaum parameter. I explain a procedure (Douady, Hubbard, Sorensen), which shows that, if the sequence {t_m} tends to zero fast enough, then M is locally connected at c while J_c is not. Then I describe a framework allowing to get a class of explicit combinatorics, for which this effect occurs. For instance, any sequence {1/q_m} with q_{m+1}>2^{q_m} fits. Rigid non-locally connected Julia sets4 October 2012
Janosch Rieger (Goethe-Universitaet Frankfurt)Abstract: The implicit Euler scheme for nonlinear ordinary differential inclusions was recently shown to possess favourable analytiv and convergence properties. As in the ODE case, its performance is substantially better than that of the explicit Euler scheme if the underlying differential inclusion is stiff. This effect is more pronounced than in the classical case, because the size of the explicit Euler images grows rapidly when stability is lost, which causes an exponential explosion of computational costs. The spatial discretization of the implicit Euler scheme, however, is problematic, because its construction involves explicit knowledge of the modulus of continuity of the right-hand side. The semi-implicit Euler schemes presented in this talk overcome this problem. In addition, their performance is significantly better than that of the implicit Euler scheme. Semi-Implicit Methods for Differential Inclusions1 October 2012
Katsutoshi Shinohara (Pontifícia Universidade Católica do Rio de Janeiro)Abstract: We consider (attracting) free semigroup actions on the interval with two generators. It is known that, if those two generators are sufficiently C^2-close to the identity, then there is a restriction on the shape of the (forward) minimal set. Namely, it must be the whole interval. (This statement is not accurate. I will give the precise statement in my talk.) In this talk, I will explain that the similar argument fails in C^1-topology. On minimality of free-semigroup actions on the interval C^1-close to the identity21 June 2012
Begoña Alarcón (University of Oviedo)Abstract: We consider sufficient conditions to determine the global dynamics for equivariant maps of the plane with a unique fixed point which is also hyperbolic. When the map is equivariant under the action of a subgroup of O(2), it is possible to describe the local dynamics and – from this – also the global dynamics. In particular, if the group contains a reflection, there is a line invariant by the map. We will see in this seminar that this invariant line allows us to use results based on the theory of free homeomorphisms of the plane to describe the global dynamical behaviour. Otherwise, in the absence of reflections, equivariant examples can be used to show that global dynamics may not follow from local dynamics near the unique fixed point. This is joint work with Isabel S. Labouriau (Centro de Matematica, Universidade do Porto). Global dynamics for symmetric planar maps10 May 2012
Vasso Anagnostopoulou (Technical University of Dresden)Abstract: We study the effect of external forcing on the saddle-node bifurcation pattern of interval maps. Replacing fixed points of unperturbed maps by invariant graphs, we obtain direct analogues to the classical result both in a topological and a measure-theoretic setting. As an interesting new phenomenon, a dichotomy appears for the behaviour at the bifurcation point, which allows the bifurcation to be either 'smooth' (as in the classical case) or 'non-smooth'. Non-autonomous saddle-node bifurcations26 April 2012
Peter Hazard (University of Warwick)Abstract: A well-know theorem of A. Katok states that if a sufficiently smooth diffeomorphism of a compact surface has positive topological entropy then it has a homoclinic point, and consequently infinitely many periodic orbits. A counterexample to this statement for homeomorphisms was later given by M. Rees who constructed a positive entropy homeomorphism of the 2-torus which was minimal. I will discuss new elementary constructions of this type which replace minimality with the 'no periodic orbits' property, with some other related results. This is joint work with E. de Faria and C. Tresser. Periodic Points and Entropy on Surfaces15 March 2012
Sergey Gonchenko (University of Nizhny Novgorod)Abstract: We study stable dynamics (both regular and chaotic) of the well-known mechanical system "celtic stone" (called sometimes as "celt" or "rattleback", or "wobblestone", or even "Russian stone"). Physically, it is a canoe-shaped rigid body with the curious property of spin asymmetry: it tends to have the stable vertical rotation in one direction only, independently on initial conditions for rotation. In the talk we try to explain this dynamical property by means of some mathematical idealization - the nonholonomic model of celtic stone. We study this model by numerical and qualitative methods and describe various stable regimes: permanent rotation, oscillations and, finally, chaotic dynamics. On regular and chaotic dynamics of "celtic stone"8 March 2012
Tiago Pereira (Imperial College)Abstract: A remarkable property of Hermitian ensembles is their universal behavior, that is, once properly rescaled the eigenvalue statistics does not depend on particularities of the ensemble. According to the theory of random matrices, the eigenvalue correlations in Hermitian are given by the determinant of an integral kernel. The limit integral kernel is well known to be universal for standard models of Hermitian ensembles. The scenery for normal ensembles, despite of certain efforts in this direction remains undisclosed. We study the integral kernel of a certain ensembles of normal matrices, and as a corollary we obtain the universality of eigenvalue statistics. Universality in Normal Random Matrix Ensembles1 March 2012
Thomas Berger (Technical University Ilmenau)Abstract: We study exponential stability and its robustness for time-varying linear index-1 differential-algebraic equations. The effects of perturbations in the leading coefficient matrix are investigated. A reasonable class of allowable perturbations is introduced. Robustness results in terms of the Bohl exponent and perturbation operator are presented. Finally, a stability radius involving these perturbations is introduced and investigated. In particular, a lower bound for the stability radius is derived. The results are presented by means of illustrative examples. On perturbations in the leading coefficient matrix of time-varying index-1 DAEs27 February 2012
Pablo Shmerkin (University of Surrey)Abstract: The Hausdorff dimension of sets invariant under conformal dynamical systems can often be realized as the zero of certain natural pressure equation (going back to Bowen). This pressure is usually continuous as a function of the defining dynamics, in the appropriate topology, and hence so is the Hausdorff dimension of the invariant set. The situation is dramatically more complicated in the non-conformal situation where, nevertheless, a subadditive version of the pressure equation, involving singular values of a matrix cocycle, is crucial. A natural question is therefore whether this subadditive pressure is also a continuous function of the dynamics (or, what is the same, of the associated cocycle). We resolve this in the affirmative in many important situations, in particular answering a question of Falconer and Sloan. This is joint work with De-Jun Feng (Chinese University of Hong Kong). Continuity of subadditive pressure23 February 2012
Stefanella Boatto (Universidade Federal do Rio de Janeiro )The Poisson equation, the Robin function and singularities' dynamics: an hydrodynamics approach 16 February 2012
Davoud Cheraghi (University of Warwick)Abstract: The local, semi-local, and global dynamics of the complex quadratic polynomials $P_\alpha(z):=e^{2\pi i \alpha}z+z^2: \mathbb{C}\to \mathbb{C}$, for irrational values of $\alpha$, have been extensively studied through various methods over the last decades. The difficulty comes from the interplay between the tangential movement created by the fixed point and the radial movement created by the critical point which brings in the arithmetic nature of $\alpha$. Using a renormalisation technique we analyse this interaction, and in particular, describe the topological behavior of the orbit of typical points under these maps. Typical orbits of complex quadratic polynomials with a neutral fixed point9 February 2012
André Caldas (Universidade de Brasilia)Product type dynamical systems and the variational principle26 January 2012
Vassilis Rothos (University of Thessaloniki)Abstract: The existence of quasi periodic travelling waves solutions in DNLS equation with nonlocal interactions and with polynomial type potentials will be considered. Calculus of variations and topological methods are employed to prove the existence of these type of solutions. Travelling waves in nonlocal lattice equations 19 January 2012
Yizhar Or (Technion Israel)Abstract: The motion of swimming microorganisms and robotic microswimmers is governed by low Reynolds number hydrodynamics where viscous effects dominate and inertial effects are negligible. The time-independence of Stokes equations augmented by structural geometric symmetries can lead to a reversing symmetry which governs the swimming dynamic equations. This is demonstrated in the talk for three different dynamic models: fixed-shape swimmer near a wall, Purcell’s three link swimmer near a wall, and torque control of the three-link swimmer. It is shown that breaking the geometric symmetries can lead to asymptotic stability of solutions which have clear physical meaning. Experimental results on macro-scale robotic swimmers will be reported, and the relation of the results to observations from swimming microorganisms’ behavior will be discussed. Bio: Dr. Yizhar Or earned his PhD in 2007 at the Technion, Israel, in the field of robot dynamics. During 2007-2009 he was a postdoctoral scholar with Prof. Richard Murray in the Dept. of Control and Dynamical Systems at Caltech, USA, funded by Fulbright Program and the Israeli Science Foundation (Bikura Program). He is currently a Senior Lecturer in the Faculty of Mechanical Engineering at the Technion, Israel. His research interests are in nonlinear dynamics, mechanics and control of locomotion, with applications to robotics and biology. Reversing symmetry and dynamic stability in low-Reynolds-number swimming24 November 2011
Jürgen Knobloch (TU Ilmenau)Abstract: Homoclinic snaking refers to the sinusoidal ``snaking'' continuation curve of homoclinic orbits near a heteroclinic cycle connecting an equilibrium $E$ and a periodic orbit $P$. Along this curve the homoclinic orbit performs more and more windings about the periodic orbit. Typically this behaviour appears in reversible Hamiltonian systems. Here we discuss this phenomenon in systems without any particular structure. We give a rigorous analytical verification of homoclinic snaking under certain assumptions on the behaviour of the stable and unstable manifolds of $E$ and $P$. We show how the snaking behaviour depends on the signs of the Floquet multipliers of $P$. Nonreversible Homoclinic Snaking24 November 2011
Vered Rom-Kedar (Weizmann Institute)Abstract: A geometrical model which captures the main ingredients governing atom-diatom collinear chemical reactions is proposed. This model is neither near-integrable nor hyperbolic, yet it is amenable to analysis using a combination of the recently developed tools for studying systems with steep potentials and the study of the phase space structure near a center-saddle equilibrium. The nontrivial dependence of the reaction rates on parameters, initial conditions and energy is thus qualitatively explained. Conditions under which the phase space transition state theory assumptions are satisfied and conditions under which these fail are derived. Extensions of these ideas to other impact-like systems and to other models of reactions will be discussed. Joint works w L. Lerman and M. Kloc. A saddle in a corner - a model of atom-diatom chemical reactions18 November 2011
Dmitry Turaev (Imperial College)Abstract: We prove that the attractor of the 1D quintic complex Ginzburg- Landau equation with a broken phase symmetry has strictly positive space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations in grids of weakly interacting solitons in a class of Ginzburg- Landau type equations. We provide an analytic proof for the existence of two- soliton configurations with Shilnikov-type chaotic temporal behavior, and construct solutions which are close to a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring ones for all time. The temporal evolution of the well-separated multi-soliton structures is described by a weakly coupled lattice dynamical system (LDS) for the coordinates and phases of the solitons. We develop a version of normal hyperbolicity theory for the weakly coupled LDS’s with continuous time and establish for them the existence of space-time chaotic patterns reminiscent of the Sinai-Bunimovich chaos in discrete-time LDS’s. While the LDS part of the theory may be of independent interest, the main difficulty concerns with lifting the space-time chaotic solutions of the LDS back to the initial PDE. The equations we consider are space-time autonomous, i.e. we impose no spatial or temporal modulation which could prevent the individual solitons in the grid from drifting towards each other and destroying the well-separated grid structure in a finite time. We however manage to show that the set of space-time chaotic solutions for which the random soliton drift is arrested is large enough, so the corresponding space-time entropy is strictly positive. This is a joint work with S. Zelik. Space-time chaos in Ginzburg-Landau equations10 November 2011
Sergey Zelik (University of Surrey)Abstract: We discuss various aspects of the theory of exponential attractors for autonomous and non-autonomous dissipative systems generated by PDEs. The possible extensions of this theory to the case of random dynamical system and the new difficulties arising here will be also discussed. Finally, some recent results on the convergence of stochastic exponential attractors to the limit deterministic one when the amplitude of white noise tends to zero will be presented. Exponential attractors: from non-autonomous to random dynamics3 November 2011
Julian Newman (Imperial College)Abstract: In traditional free body analysis, unknown contact forces are calculated by using Newton's Laws to form equations of motion, which can then be solved for these unknown contact forces. However, if there are more contact forces than there are equations of motion, then this will generally be impossible. This is particularly problematic when friction is involved, because it can give rise to the situation that one cannot, by traditional means, determine whether a system undergoes sliding or sticking. However, recent numerical simulations carried out by engineers in Germany suggest that the problem might be resolvable in the case of a disc in contact with two frictionable walls. The potential resolution comes from "regularising" the walls - that is, taking into account that the walls are not completely rigid. In this talk, results of an analytical study of the engineers' results are presented. Tracking Variations of Indeterminable Contact Forces between a Disc and a Frictionable Wedge: Part II3 November 2011
Julian Newman (Imperial College)Abstract: In traditional free body analysis, unknown contact forces are calculated by using Newton's Laws to form equations of motion, which can then be solved for these unknown contact forces. However, if there are more contact forces than there are equations of motion, then this will generally be impossible. This is particularly problematic when friction is involved, because it can give rise to the situation that one cannot, by traditional means, determine whether a system undergoes sliding or sticking. However, recent numerical simulations carried out by engineers in Germany suggest that the problem might be resolvable in the case of a disc in contact with two frictionable walls. The potential resolution comes from "regularising" the walls - that is, taking into account that the walls are not completely rigid. In this talk, results of an analytical study of the engineers' results are presented. Tracking Variations of Indeterminable Contact Forces between a Disc and a Frictionable Wedge: Part I27 October 2011
Philipp Düren (University of Augsburg)Abstract: When considering control systems in discrete time one can define invariant control sets as sets of "maximal controllability". On the other hand, the discussion of (also time-discrete) random diffeomorphisms (as for example used by H. Zmarrou, A. J. Homburg et al.) often uses the notion of stationary measures. We will work with a random diffeomorphism, called System A. When interpreting the stochastic noise of A as a arbitrary control we obtain a control system B. We will see that the supports of stationary measures of A correspond bijectively to the invariant control sets of B. This is extendable to open systems as well. Invariant control sets and stationary measures20 October 2011
Gabor Kiss (University of Exeter)Abstract: In many applications the rate of change of state variables depends on their states at prior times. When these processes are assumed to be deterministic, they are modelled by functional differential equations. In the simplest cases only one, time invariant time lag is considered. However, equations with multiple delays offer richer dynamics, thus they are of mathematical interest and potential models of real-world problems with complex oscillations. We present results on the coexistence of periodic solutions to equations with multiple delays. Furthermore, we report on the existence of pullback attractors to equations with multiple time-varying delays. A model for exchange-rate fluctuations is considered as a motivating example. Ideas for possible future work are outlined. Oscillating solutions of functional differential equations20 September 2011
Shangjiang Guo (Hunan University)Abstract: This talk deals with the existence, monotonicity, uniqueness, asymptotic behavior, and nonlinear stability of travelling wavefronts for temporally delayed spatially discrete reaction-diffusion equations. Based on the combination of the weighted energy method ad the Green function technique, we prove that all noncritical wavefronts are globally exponentially stable, and critical wavefronts are globally algebraically stable when the initial perturbations around the wavefront decay to zero exponentially near the negative infinity regardless of the magnitude of time delay. Wavefronts in Discrete Reaction-Diffusion Equations with Nonlocal Delayed Effects14 September 2011
Sergey Gonchenko (University of Nizhny Novgorod)On Newhouse regions with mixed dynamics28 March 2011
Lev Lerman (University of Nizhny Novgorod)Integrable Hamiltonian systems: structure and bifurcations21 March 2011
Thorsten Huels (University of Bielefeld)Computing dichotomy projectors and Sacker-Sell spectra in discrete time dynamical systems2 March 2011
Boumediene Hamzi (Imperial College)Model Reduction of Nonlinear Control Systems in Reproducing Kernel Hilbert Space1 February 2011
Martin Rasmussen (Imperial College)Morse decompositions of nonautonomous and set-valued dynamical systems 25 January 2011
Chris Warner (Imperial College)Long-time Asymptotics for a Classical Particle Interacting with a Scalar Wave Field 16 December 2010
Konstantinos Kourliouros (Imperial College)Typical Singularities of Constrained Systems II 25 November 2010
Konstantinos Kourliouros (Imperial College)Typical Singularities of Constrained Systems I18 November 2010
Lina Avramidou (University of Surrey)Abstract: Given a real-valued function f defined on the phase space of a dynamical system, ergodic optimization is the study of the orbits that maximize the ergodic f-time-average. It turns out that this is equivalent in maximizing the space average over all invariant probability measures. In this talk, we examine the question of maximization of non-conventional ergodic averages along square iterates for the doubling map and fairly simple functions f. Ergodic Optimization along the squares23 June 2010
Leonid Bunimovich (Georgia Tech)Abstract: A natural question on how a position of a 'hole" in a phase space seems to be never studied. The answer is interesting by itself and demonstrates that the dynamical systems theory can make finite time predictions of dynamics. Some new results for dynamical networks and even for Markov chains are obtained along these lines as well. Open Systems and Dynamical Networks14 June 2010
Andrei Vladimirov (WIAS Berlin)Dissipative localised structures of light and their interaction.27 April 2010
Oliver Jenkinson (QMUL)How to maximize long-term happiness? - Ergodic optimization of utility functions.24 March 2010
Fritz Colonius (Augsburg)Abstract: Control of digitally connected dynamical systems is a subject which has recently found considerable interest. In this talk, an abstract approach is presented intending to specify minimal data rates for control tasks. It is based on a concept which is motivated by the notion of topological entropy in the theory of dynamical systems. Entropy-like notions in control16 March 2010
Paulo Ruffino (UNICAMP)Abstract: Given two complementary distributions in the tangent bundle of a manifold, we find conditions to factorize an stochastic flow into a diffusion in the (infinite dimensional) Lie group of diffeomorphisms which preserve one distribution (horizontal), composed with a process in the Lie group of diffemorphisms which preserve the other distribution (vertical). This decomposition generalizes previous approach, e.g. using coordinate maps, by Ming Liao and others. Decomposition of stochastic flows along complementary distributions20 January 2010
Joseph Rosenblatt (University of Illinois at Urbana-Champaign)Abstract: We want to characterize which sequences of whole numbers $(n_m)$ admit a weakly mixing transformation $\tau$ such that $\tau$ is rigid along $(n_m)$, and which times $(n_m)$ admit a weakly mixing transformation $\tau$ such that $\tau$ is not recurrent along $(n_m)$. These questions are in opposition, but also related. The necessary properties are a mix of sparsity and combinatorial, and/or algebraic, structure of the sequence. Examples and counterexamples, as well as some general results, will be described. This talk is based on joint current work with V. Bergelson (Ohio State), A. del Junco (Toronto), and M. Lemanczyk (Torun). Rigidity and Recurrence for Dynamical Systems13 January 2010
Vered Rom-Kedar (The Weizmann Institute)Stability in high dimensional steep repelling potentials and the Boltzmann ergodic hypothesis. 18 November 2009
Wael Bahsoun (Loughborough University)Random maps, skew-products and existence of invariant measures 27 October 2009
Christian Poetzsche (TU Munich)Discrete Dynamics and Nonlinear Analysis: A Commensal Relationship! 7 October 2009
Guillaume Defrance (Jussieu)Using Matching Pursuit for estimating mixing time within Room Impulse Responses8 July 2009
Ole Peters (Imperial College)On time and risk8 June 2009
John Roberts (UNSW)Abstract: In a program joint with F Vivaldi (Queen Mary), we have shown that structural properties of discrete dynamical systems leave a universal signature on the reduced dynamics over finite fields (analogous to the division of quantum spectral statistics into those of certain random matrix ensembles). I will briefly review previous results, then consider reversible rational maps, i.e. those maps in d-dimensional space that can be written as the composition of 2 rational involutions. We study the reduction of such rational maps to finite fields and look to study the proportion of the finite phase space occupied by cycles and by aperiodic orbits and the length distributions of such orbits. We find that the dynamics of these low-complexity highly deterministic maps has some universal (i.e. map-independent) aspects. The distribution is well explained using a combinatoric model that averages over an ensemble of pairs of random involutions in the finite phase space. Universal Period Distribution for Reversible Rational Maps over Finite Fields21 May 2009
Alexandre Rodrigues (University of Porto)Switching near a Heteroclinic Network of Rotating Nodes7 May 2009
Renato Vitolo (Exeter)The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: resonance `bubbles' and routes to chaos24 March 2009
Jaap Eldering (Utrecht)The Perron method for invariant fibrations24 March 2009
Peter Giesl (University of Sussex)Abstract: The basin of attraction of equilibria or periodic orbits of an ODE can be determined through sublevel set of a Lyapunov function. To construct such a Lyapunov function, i.e. a scalar-valued function which is decreasing along solutions of the ODE, a linear PDE is solved approximately using Radial Basis Functions. Error estimates ensure that the approximation is a Lyapunov function. For the construction of a Lyapunov function it is necessary to know the position of the equilibrium or periodic orbit. A different method to analyse the basin of attraction of a periodic orbit without knowledge of its position is Borg's criterion. The sufficiency and necessity of this criterion in different settings will be discussed. Determination of the Basin of Attraction of Equilibria and Periodic Orbits24 March 2009
Andrey Shilnikov (Atlanta)TBC19 March 2009
Sergey Gonchenko (University of Nizhny Novgorod)Attractors and repellers in reversible maps with heteroclinic tangencies17 March 2009
Tony Samuel (St Andrews)Dynamics and noncommutative geometry11 March 2009
Pawel Pilarczyk (Universidade do Minho, Portugal)A method for automatic classification of global dynamics in multi-parameter systems10 March 2009
Tomás Lázaro (Universitat Politécnica de Catalunya)Abstract: Tchebycheff systems share most of the nice properties satisfied by the set of polynomials of a given degree and, in this sense, they become its natural extension. Despite the fact that there exist algorithms to build such a sets, there are no many examples in the literature. In this work we prove that a suitable type of functions forms a Tchebycheff system. This family of functions uses to appear when one deals with Abelian Integrals in some concrete problems associated to the Weak 16th Hilbert's Problem. This is a joint work (in progress) with A. Gasull and J. Torregrosa, Departament de Matemátiques, Universitat Autónoma de Barcelona, Spain. Using Tchebycheff systems to estimate the number of zeroes of some Abelian Integrals3 March 2009
Thomas Jordan (Bristol)Where is a topological conjugacy differentiable?24 February 2009
Peter Kloeden (Frankfurt)Random attractors and the preservation of synchronization in the presence of noise (AMMP colloquium) 17 February 2009
Henk Bruin (Surrey)Li-Yorke chaos and Cantor attractor of interval maps17 February 2009
Corinna Ulcigrai (Bristol)Abstract: We consider a class of area-preserving (locally Hamiltonian) flows on a surface of genus g. We are interested in their ergodic properties, especially mixing: it turns out that the presence/absence of mixing depends on the type of fixed points. We proved that the presence of centers in a generic such flow is enough to create mixing. Recently we showed that if the flow has only saddles, it is generically not mixing, but weakly mixing. The results uses the flows representation as suspensions over interval exchange transformations and the study of deviations of Birkhoff averages over interval exchanges. Mixing properties of area-preserving flows on surfaces 10 February 2009
Martin Andersson (ENS, Paris)Bifurcations of physical measures3 February 2009
Christian Rodrigues (University of Aberdeen)Emergent attractors for weakly dissipative systems27 January 2009
Jeroen Lamb (Imperial College)Abstract: In many examples of classical mechanics, reversibility (playing a film of the dynamics yielding a physically realistic event) arises simultaneously with the occurrence of a Hamiltonian structure of the equations of motion. In fact reversibility has often be identified as a useful tool to prove results for mechanical systems, and indeed many important results can be proven based on the assumption of a Hamiltonian structure or the presence of a time-reversal symmetry. In this talk I will review our understanding of the differences and similarities between reversible and Hamiltonian dynamics. Reversible versus Hamiltonian dynamical systems21 January 2009
Isabel Rios (Universidade Federal Fluminense)Twisted Horseshoes20 January 2009
Rafael Ortega (Granada)A property of stable fixed points of area-preserving maps15 January 2009
Martin Rasmussen (Imperial College)Three different approaches to the approximation of nonautonomous invariant manifolds (AMMP Colloquium)13 January 2009
Oliver Butterley (Imperial College)Transfer operators for suspension semiflows28 November 2008
Ian Melbourne (University of Surrey)Validity of the 0-1 test for chaos26 November 2008
Mark Holland (University of Exeter)Extreme Events in Chaotic Dynamical Systems19 November 2008
Sergey Zelik (University of Surrey)Dissipative dynamics in large and unbounded domains: attractors, entropies and space-time chaos12 November 2008
Oleg Makarenkov (Imperial College London)Nonsmooth bifurcation theory and mechanics (AMMP Colloquium)11 November 2008
Anatoly Neishtadt (Loughborough University)Separatrix crossings in slow-fast Hamiltonian systems5 November 2008
Vassili Gelfreich (University of Warwick)Fermi acceleration in non-autonomous billiards 29 October 2008
Lev Lerman (University of Nizhny Novgorod)Abstract: Let a 2 d.o.f. Hamiltonian system have an equilibrium with two double pure imaginary eigenvalues and non-semisimple Jordan form for the linearisation matrix. Such a degeneration is generically met in one parameter unfoldings, the related bifurcation is called to be the Hamiltonian Hopf Bifurcation. We prove the Lyapunov stability of the equilibrium when some coefficient of the 4th order normal form is positive (the equilibrium is unstable, if this coefficient is negative). The result is known since 1977, but both proofs having appeared are incorrect. The proof is based on the KAM theory and uses a work with Weierstrass elliptic functions, estimates of power series and methods of the theory of dynamical systems. On stability at the Hamiltonian Hopf Bifurcation15 October 2008
Bob Rink (Free University Amsterdam)Continuum equations for the Fermi-Pasta-Ulam chain8 October 2008
Juergen Knobloch (TU Ilmenau)Snaking near heteroclinic cycles17 September 2008
Zalman Balanov (Netanya)Application of Twisted Equivariant Degree to Symmetric Hopf Bifuraction7 July 2008
Wieslaw Krawcewicz ( Alberta)Variational Problems and Gradient Equivariant Degree 7 July 2008
Ramon Driesse (Amsterdam)Essential Asymptotic Stability Of A Homoclinic Cycle10 June 2008
Jens Rademacher (CWI)Unfolding heteroclinic networks of equilibria and periodic orbits using Liapunov-Schmidt reduction26 February 2008
Alexander Plakhov (University of Wales - Aberystwith)Abstract: A body moves through a rarefied medium of point particles. The particles hit the body in the absolutely elastic manner and do not mutually interact. Find the body, out of a given class of admissible bodies, such that the force of resistance of the medium to its motion is minimal. This is the Newton problem of minimal resistance. We will discuss several generalizations of this problem, as well as related questions of billiards theory and Monge-Kantorovich optimal mass transportation. Some possible applications (retroreflectors, Magnus effect) will also be discussed. Billiards, Newtonian aerodynamics, and optimal mass transportation22 January 2008
Marc Georgi (Berlin)Bifurcations of Travelling Waves in Lattice Differential Equations22 January 2008
Michael Field (University of Houston)Global dynamics and recurrence in (small) coupled dynamical systems.20 November 2007
James Meiss (University of Colorado)Dynamics and Bifurcations in Volume Preserving Maps 13 November 2007
Ian Melbourne (University of Surrey)Statistical and probabilistic properties of dynamical systems 30 October 2007
Andrew Stuart (University of Warwick)Probabilistic Inverse Problems in Differential Equations16 October 2007
Michael Jakobson (University of Maryland)Abstract: For quadratic-like families f we are reviewing constructions in the phase space and in the parameter space, and estimating the measure of parameter values such that f have absolutely continuous invariant measures. Estimating measures in the parameter space4 July 2007
Kie van Ivanky Saputra (La Trobe University Melbourne)Alternative walk in the parameter space20 June 2007
Raul Ures (Montevideo, Uruguay)Abstract: Abstract. In a previous work we proved the Pugh-Shub conjecture for partially hyperbolic diffeomorphisms with 1-dimensional center, i.e. stable ergodic diffeomorphism are dense among the partially hyperbolic ones. In this talk we shall address the issue of giving a more accurate description of this abundance of ergodicity. In particular, we shall give the first examples of manifolds in which al l conservative partially hyperbolic diffeomorphisms are ergodic. The talk will be based in a joint work with Federico Rodriguez Hertz and Mar ́ıa A. Rodriguez Hertz. Partial hyperbolicity and ergodicity in dimension three12 June 2007
Phil Boyland (University of Florida)Abstract: A covering space is a way of "unwinding" a manifold, transforming loops into translations. Lifting a dynamical system to a cover allows one to understand how the dynamicswraps around the manifold by examining how lifted orbits translate in the cover. After introducing the rotation set, the talk will focus on a kind of "hyper-transitivity" where a given map has a dense orbit when lifted to the largest Abelian cover. Roughly speaking, this implies that a single orbit explores repeatedly all the ways of traversing loops in the manifold. I will begin with the relatively simple circle case and then focus mainly on iterated surface homeomorphisms. Unraveling Dynamics via covering spaces16 May 2007
Henk Bruin (Surrey)Equilibrium States for One-Dimensional Maps19 February 2007
Jorge Freitas (Porto)Statistical stability for some non-hyperbolic systems19 February 2007
Oscar Bandtlow (Queen Mary University of London)Invariant measures for analytic maps with unbounded distortion19 February 2007
Jose' Alves (Porto)Mixing rates and hyperbolic structures for partially hyperbolic diffeomorphisms19 February 2007
Todd Young (Ohio & Warwick)Intermittency in Unimodal Maps19 February 2007
Y. Sinai (Princeton University)Limiting Behaviour for Large Frobenius Numbers31 January 2007
Oliver Butterley (Imperial College)Transfer operators for Anosov flows19 January 2007
Thomas Jordan (Warwick)Randomly perturbed self-affine sets19 January 2007
Ian Morris (Manchester)Estimation of the maximum ergodic average19 January 2007
Neil Dobbs (Paris Orsay)Markov maps with integrable return times19 January 2007
Mike Todd (Surrey)Hofbauer towers in ergodic theory19 January 2007
Paulo Ruffino (Campinas, Brazil)Abstract: Consider a cocycle of random orientation preserving diffeomorphisms on the circle $f(\omega)$, based on a probability space $(\Omega, F, P)$, with an ergodic shift $\theta: \Omega --> \Omega$. We present an ergodic theorem for the rotation number $R_{(f,\theta)}$ of the composition of the random sequence $(f(\theta^n \omega))$. If $R_{(f,\theta)}$ is irrational, we look for conditions for the existence of a random (measurable) homeomorphism $h(\omega)$ which provides the cocycle conjugacy with rotation: f(\omega)= h^{-1}(\theta \omega)\circ R_{(f(\omega))} \circ h(\omega). Yet, we investigate the existence of this cocycle conjugacy for a stochastic flow (non structurally stable) in $S^1$. Conditions for a random version of Denjoy Theorem5 December 2006
Guido Gentile (Rome)Abstract: Existence of lower-dimensional tori in quasi-integrable Hamiltonian systems can be proved by various means. In this talk I shall discuss a method based on renormalization group techniques, with the aim of showing that the Bryuno condition arises as a very natural Diophantine condition to be imposed on the frequency vectors of the persisting invariant tori. Renormalization group for lower-dimensional tori under the Bryuno condition28 November 2006
Guillaume James (Toulouse)Abstract: The propagation of nonlinear waves in spatially discrete media gives rise to interesting phenomena. An effect of spatial discreteness can be the formation of "hot spots" where vibrational energy remains localized without dispersion. The study of "discrete breathers" (time-periodic and spatially localized oscillations) in nonlinear Hamiltonian oscillator chains provides an interesting mathematical framework for studying this phenomenon. In this talk we consider the Fermi-Pasta-Ulam lattice, which consists in a chain of particles nonlinearly coupled to their nearest neighbours (here the chain is of infinite extent). When the system admits a hard interaction potential, the existence of weakly localized breathers has been predicted by Tsurui in the 70s, and the existence of strongly localized ones has been suggested by Sievers and Takeno in the 80s, both studies using formal approximations. A delicate mathematical question is to determine if such approximate solutions correspond to exact breather solutions for the oscillator chain. As we shall see, this question leads us to analyze the dynamics of an infinite-dimensional nonlinear map, whose linear part is an unbounded operator. Thanks to good spectral properties, the local dynamics of the map is shown to be finite-dimensional, which allows us to conclude on the existence or nonexistence of breathers in the small amplitude limit. The question of the existence of "breathers" in the Fermi-Pasta-Ulam model28 November 2006
Richard Sharp (Manchester)Abstract: We discuss ways of comparing lengths of pairs of closed geodesics on negatively curved surfaces. We focus on pairs where the difference of the lengths lies in an interval which is allowed to shrink. This is inspired by problems of pair correlations for eigenvalues in quantum chaos. If time permits, we will discuss analogous results for ergodic sums over hyperbolic systems. (Joint work with Mark Pollicott.) Refs: M Pollicott and R Sharp, Correlations for pairs of closed geodesics, Invent. math. 163 (2006), 1-24. M Pollicott and R Sharp, Distribution of ergodic sums for hyperbolic maps, in "Representation Theory, Dynamical Systems, and Asymptotic Combinatorics" (ed. V Kaimanovich and A Lodkin), American Mathematical Society (2006). Pair correlations and length spectra on negatively curved sufaces31 October 2006
Juergen Knobloch (TU Ilmenau)Chaotic behavior near homoclinic points with quadratic tangency3 October 2006
Martijn van NoortAbstract: We study a Hamiltonian parametrically forced pendulum system with two parameters, and show that for a large range of parameter values the system has a Cantor family of invariant tori in an annulus around its lower equilibrium, corresponding to quasiperiodic oscillations of the pendulum. The parametric forcing need not be very small, as long as it is smaller than the force of gravity, and the pendulum is short enough. Invariant oscillatory tori in the forced pendulum22 June 2006
Yakov Pesin (Penn State University)Is chaotic behavior typical among dynamical systems?20 June 2006
Anatole Katok (Penn State University)Abstract: We present an overview and some new applications of the approximation by conjugation method introduced by Anosov and the second author more than thirty years ago \cite{AK}. Michel Herman made important contributions to the development and applications of this method beginning from the construction of minimal and uniquely ergodic diffeomorphisms jointly with Fathi in \cite{FH} and continuing with exotic invariant sets of rational maps of the Riemann sphere \cite{H3}, and the construction of invariant tori with nonstandard and unexpected behavior in the context of KAM theory \cite{H1, H2}. Recently the method has been experiencing a revival. Some of the new results presented in the paper illustrate variety of uses for tools available for a long time, others exploit new methods, in particular possibility of mixing in the context of Liouvillean dynamics discovered by the first author \cite{F1, F2}. Liouvillean phenomena in dynamics and ergodic theory14 June 2006
Tobias Jaeger (University of Erlangen)General pattern in the formation of strange non-chaotic attractors (AMMP Colloquium) 25 April 2006
Frank Schilder (University of Bristol)Abstract: A famous phenomenon in circle-maps and synchronisation problems leads to a two-parameter bifurcation diagram commonly referred to as the Arnol'd tongue scenario. One considers a perturbation of a rigid rotation of a circle, or a system of coupled oscillators. In both cases we have two natural parameters, the coupling strength and a detuning parameter that controls the rotation number/frequency ratio. The typical parameter plane of such systems has Arnol'd tongues with their tips on the decoupling line, opening up into the region where coupling is enabled, and in between these Arnol'd tongues, quasi-periodic arcs. In this talk we present unified algorithms for computing both Arnol'd tongues and quasi-periodic arcs for both maps and ODEs. The algorithms generalise and improve on the standard methods for computing these objects. We illustrate our methods by numerically investigating the Arnold tongue scenario for two examples from electrical engineering: a parametrically forced network and a system of coupled Van der Pol oscillators. Computing Arnol'd tongue scenarios 12 April 2006
Cristina Ciocci (Imperial College London)Steady states for the tippe top 5 April 2006
Stavros Komineas (University of Cambridge)Abstract: Quasi-one-dimensional solitons that occur in an elongated Bose-Einstein condensate (BEC) are described by a nonlinear Schroedinger equation (NLS). These become unstable at high particle density. Within a nonlinear Gross-Pitaevskii model we study a basic mode of instability and the corresponding bifurcation to genuinely three-dimensional axisymmetric vortex rings. We calculate their profiles and examine their dependence on the velocity of propagation along a cylindrical trap. At sufficiently high velocity, the vortex ring transforms into an axisymmetric soliton. We also calculate the energy-momentum dispersions and show that a Lieb-type mode appears in the excitation spectrum for all particle densities. We further study interactions of solitons and vortex rings in a cylindrical BEC by simulating their head-on collisions. The results are compared against collisions of solitons in the NLS, and also against the dynamics of solitary waves in the three-dimensional homogeneous Bose gas. We discuss a related recent experiment [Ginsberg et al. Phys. Rev. Lett. 94, 040403 (2005)]. Solitons and vortex rings in a cylindrical Bose-Einstein condensate 22 March 2006
Martijn van Noort (Imperial College London)Abstract: In this talk we employ KAM theory to rigorously investigate the transition between quasiperiodic and chaotic dynamics in cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and superlattices, modelled by a parametrically forced Duffing equation that describes the spatial dynamics of the condensate. We will show the existence of KAM tori for lattices of arbitrary size, that is, for shallow-well, intermediate-well, as well as deep-well potentials. Hence one obtains a large measure of quasiperiodic dynamics for condensate wave functions of sufficiently large amplitude, with rotation number proportional to the amplitude. Our approach is applicable to periodic superlattices with an arbitrary number of rationally dependent wave numbers. Bose-Einstein Condensates in Optical Lattices and Superlattices22 March 2006
Gerton Lunter (University of Oxford)Abstract: To study the behaviour of Hamiltonian dynamical systems, singularity theory is often useful in reducing the system to a simple normal form. However, although singularity theory guarantees the existence of an appropriate normalizing coordinate transformation, in applications it is often desirable to be able to construct this transformation, for instance to pull back bifurcation curves to original parameters. In this talk I will show how Groebner basis techniques can be modified to give efficient algorithms for computing normalizing transformations. Groebner bases and constructing normalizing transformations8 March 2006
Ana Paula Dias (University of Porto)Abstract: In the analysis of stability in bifurcation problems it is often assumed that the (appropriate reduced) equations are in normal form. In the presence of symmetry, the truncated normal form is an equivariant polynomial map. Therefore, the determination of invariants and equivariants of the group of symmetries of the problem is an important step. In general, these are hard problems of invariant theory, and in most cases, they are tractable only through symbolic computer programs. Nevertheless, it is desirable to obtain some of the information about invariants and equivariants without actually computing them. In this work we obtain formulas for the number of linearly independent homogeneous invariants or equivariants for Hopf bifurcation degree by degree in terms of characters and we show that they are effectively computable in several concrete examples. This information allows to draw some predictions about the structure of the bifurcations. Invariants, Equivariants and Characters in Symmetric Bifurcation Theory3 March 2006
Alastair Rucklidge (University of Leeds)Quasipatterns in surface waves (Applied Colloquium)9 November 2004
Bjorn Sandstede (University of Surrey)Dynamics of coherent structures in oscillatory media (Applied Colloquium)26 October 2004
Alan Champneys (University of Bristol)Rock, rattle and slide; towards a bifurcation theory for piecewise smooth systems26 October 2004
Michael Field (Imperial/University of Houston)Geometry, symmetry and bifurcation (Applied Colloquium)5 October 2004
Stefanella Boatto (McMaster University)Periodic solutions of Euler equations on the sphere5 October 2004
Fima DinaburgOn statistical mechanics model built on sand23 June 2004
Victor Planas-Bielsa (INLN Nice)Leibniz manifolds and Lyapunov stability of Poisson equilibria4 June 2004
John Elgin (Imperial College)On the computation of multifractal spectra from time series data2 June 2004
Cristina Stoica (University of Surrey)Relative Equilibria of Systems with Configuration Space Isotropy2 June 2004
Marco Antonio Teixeira (University of Campinas)Invariant varieties for discontinuous vector fields10 May 2004
Konstantinos Efstathiou (University of Dunkerque)Metamorphoses of Hamiltonian systems with symmetries27 April 2004
Mike Field (University of Houston)Abstract: Recently Stewart, Golubitsky and coworkers have formulated a general theory of networks of coupled cells. Their approach depends on groupoids, graphs, balanced equivalence relations and 'quotient networks'. We present a combinatorial approach to coupled cell systems. While largely equivalent to that of Stewart et al., our approach is motivated by ideas coming from analog computers, is directed towards eventual applications in engineering, and avoids abstract algebraic formalism. Combinatorial Dynamics24 March 2004
Odo Diekmann (University of Utrecht)A crash course in physiologically structured population models (applied colloquium)17 February 2004
Rob Beardmore (Imperial College)Some toy models for the evolution of disease17 February 2004
James Montaldi (UMIST)Abstract: In the first part of the talk I will describe some recent results (joint work with Mark Roberts and Frederic Laurent-Polz) on the stability of configurations of point vortices on the sphere. In the second I will describe in the general context of symmetric Hamiltonian systems the different stability transitions that occur. Point vortices and stability transitions in symmetric Hamiltonian systems4 February 2004
Gerald Moore (Imperial College)Floquet theory as a computational tool4 February 2004
Rob Beardmore (Imperial College)An index-2 Kronecker normal form and singularities of DAEs21 January 2004
Vassilis Rothos (Queen Mary)Bifurcations of travelling breathers in the discrete NLS equation.21 January 2004
Dmitrii Sadovskii (University of Dunkerque)Qualitative analysis of internal molecular dynamics19 January 2004
Franz Gaehler (University of Stuttgart)Spaces of tilings and their topology3 December 2003
Oliver Jenkinson (Queen Mary)Sturmian orbits in ergodic optimization 3 December 2003
Uwe Grimm (Open University)Shelling of planar tilings with N-fold symmetry3 December 2003
Jean-Paul Allouche (University of Paris-Sud, Orsay)Iteration of continuous real maps, non-integer bases, and a fractal set of sequences 3 December 2003
Edmund Harriss (Imperial College)Canonical substitution tilings3 December 2003
Jens Marklof (Bristol)Ergodic theory and the distribution of n^2 alpha mod 119 November 2003
Roland Zweimueller (Imperial)Invariant measures for generalized induced transformations19 November 2003
Henk Bruin (Surrey)Existence of invariant densities for unimodal maps without growth conditions.19 November 2003
Peter Ashwin (University of Exeter)What is a random attractor and how do I know when I've got one? (Applied Colloquium)4 November 2003
Marcelo Viana (IMPA, Rio de Janeiro)Homoclinic Bifurcations and Fractal Invariants for High Dimensional Dynamical Systems (Applied Colloquium)14 October 2003
Vassili Gelfreich (University of Warwick)Homoclinic orbits near strong resonances in Hamiltonian systems14 October 2003
Ale Jan Homburg (University of Amsterdam)Multiple homoclinic orbits in conservative and reversible systems14 October 2003
Henk Broer (University of Groningen)Abstract: (joint work with Richard Cushman and Francesco Fass\`{o}): The classical Kolmogorov Arnold Moser (KAM) theory deals with persistence of Lagrangean invariant tori in nearly integrable Hamiltonian systems. The persistent tori have Diophantine frequencies and hence are parametrized over a nowhere dense set of almost full measure. The KAM theory provides conjugacies between the unperturbed, integrable tori and the perturbed, nearly integrable tori which are smooth in the sense of Whitney. However, these conjugacies in the action space are only locally defined. We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangean tori by glueing together the local KAM conjugacies with help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a nearly-integrable system and an integrable one. The global conjugacy at once is an isomorphism of torus bundles. This leads to preservation of geometry, which allows us to define all the nontrivial geometric invariants like monodromy or Chern classes of an integrable system also for nearly integrable systems. This result is relevant for semiclassical versions of nearly integrable systems. Geometry of KAM tori for nearly integrable Hamiltonian systems3 October 2003
Kevin Webster (Imperial College)Heteroclinic cycle bifurcation in 3D reversible vector fields3 October 2003
Carlangelo Liverani (University of Rome)Abstract: I consider a one-parameter family of area-preserving smooth maps that cross a non-uniformly hyperbolic situation into an elliptic one. I prove that exponentially close to such a family there are maps with positive metric entropy. Birth of an elliptic island in a chaotic sea16 September 2003
Yongluo Cao (Suzhou University, P.R. China)The basin problems of attractors5 September 2003
Bob Rink (University of Utrecht)Abstract: At Los Alamos in 1954, nobel prize winner Fermi, computer expert Pasta and mathematician Ulam performed a numerical experiment to study the ergodic properties of a one-dimensional continuum. They discretised this continuum by considering a lattice of material elements, each of which interacts with its neighbours. Statistical mechanics postulates that nonlinearities in the interparticle forces will then make the equations of motion ergodic such that the lattice reaches a thermal equilibrium. The numerical integration was therefore amazing, as it actually turned out that the lattice behaved quasi-periodically. This paradox is known as the Fermi-Pasta-Ulam problem. One possible and generally excepted explanation for this observation is based on the Kolmogorov-Arnol'd-Moser (KAM) theorem, which assures quasi-periodicity under certain restrictive conditions. But proofs that these conditions are satisfied have been absent for 50 years. In this talk I will present the first complete proof in this direction, which makes use of Birkhoff normal forms, symmetries and number theory. Dynamical and geometric properties of the equations of motion are emphasized. Geometry and dynamics in the Fermi-Pasta-Ulam lattice5 September 2003
Andrew Torok (University of Houston)Abstract: We show that Axiom A flows are generically stably mixing on each of their nontrivial basic sets. This is a consequence of our results concerning generic stable transitivity of smooth suspensions over a hyperbolic base (the fiber being either a compact Lie group or R^n). Stable mixing for hyperbolic flows, and stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets19 June 2003
Willy GovaertsMatCont: an interactive Matlab package for dynamical systems, continuation and bifurcation22 May 2003
Jaroslav Stark (Imperial College)Invariant Sets for Quasiperiodically Forced Maps.2 April 2003
Gerhard Keller (Erlangen)Some remarks on strange nonchaotic attractors for quasiperiodically driven systems2 April 2003
Jacques Fejoz (Paris VI)The problem of the stability of the solar system17 February 2003
Gabriel Paternain (Cambridge)An introduction to boundary rigidity for Lagrangian submanifolds and Aubry-Mather theory27 November 2002
Richard Sharp (Manchester)Periodic orbits of Anosov flows and homology.27 November 2002
Ian Stewart, FRS (Warwick)Dynamics on Networks20 November 2002
Henrik Jensen (Imperial)An introduction tothe Theory of Self-Organizing Critialities13 November 2002
Kevin Webster (Imperial)The Kupka-Smale Theorem for Differential Equations16 October 2002
Vincent Lynch (Warwick)Decay of correlations9 October 2002
Juergen Knobloch (TU Ilmenau)Bifurcation from degenerate homoclinic orbits12 July 2002
Oleg Stenkin (Imperial College)Conservative and non-conservative behaviour in Newhouse regions12 July 2002
Mike Field (University of Houston)Stable transitivity and ergodicity for compact abelian extensions over general hyperbolic basic sets29 May 2002
Mark Holland (University of Manchester)Slowly mixing systems and intermittency maps29 May 2002
Kenneth Meyer (University of Cincinnati)Abstract: We consider the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter, $\nu $. The eigenvalues of the linearized system are complex for $\nu < 0$ and pure imaginary for $\nu > 0$. Thus, for $ \nu < 0 $ the equilibrium has a two-dimensional stable manifold and a two-dimensional unstable manifold, but for $ \nu > 0 $ these stable and unstable manifolds are gone. If the sign of a certain term in the normal form is positive then for small negative $\nu$ the stable and unstable manifolds of the system are either identical or must have transverse intersection. Thus, either the system is totally degenerate or the system admits a suspended Smale horseshoe as an invariant set. Evolution of invariant manifolds15 May 2002
Joao da Rocha Medrado (Autonomous University of Barcelona)Symmetric singularities of reversible vector fields of Rn15 May 2002
Isabel Rios (Universidade Federal Fluminense)Abstract: We study one-parameter families $(f_\mu)_{\mu\in [-1,1]}$ of two dimensional diffeomorphisms unfolding critical saddle-node horseshoes (say at $\mu=0$) such that $f_\mu$ is hyperbolic for negative $\mu$. We describe the dynamics at some isolated secondary bifurcations which appear in the sequel of the unfolding of the initial saddle-node bifurcation. We construct two classes of open sets of such arcs. For the first class, we exhibit a collection of parameter intervals $I_n$, $I_n\subset (0,1]$, converging to the saddle-node parameter, $I_n\to 0$, such that the topological entropy of $f_\mu$ is a constant $h_n$ in $I_n$ and $h_n$ is an increasing sequence. So, for parameters in $I_n$, the topological entropy is upper bounded by the entropy of the initial saddle-node diffeomorphisms. This illustrates the following intuitive principle: a critical cycle of an attracting saddle-node horseshoe is a destroying dynamics bifurcation. In the second class, the entropy of $f_\mu$ does not depend monotonely on the parameter $\mu$. Finally, when the saddle-node horseshoe is not an attractor, we prove that the entropy may increase after the bifurcation. Critical saddle-node horseshoes: bifurcations and entropy30 April 2002
Oleg Stenkin (Imperial College)About boundaries of intervals of hyperbolicity at homoclinic Omega-explosion30 April 2002
Jeroen Lamb (Imperial College)Normal form theory for linear reversible equivariant vector fields21 March 2002
Keith Briggs (BTexact Technologies)Simultaneous Diophantine approximation and linearization of C2 maps6 February 2002
Mauricio Barahona (Imperial College)Synchronization in small-world systems6 February 2002
Oleg Kozlovski (University of Warwick)Real Koebe Lemma30 January 2002
Shaun Bullett (Queen Mary)Sturmian sequences and holomorphic correspondences30 January 2002
Thomas Wagenknecht (TU-Ilmenau)Homoclinic orbits to degenerate equilibria in reversible systems16 January 2002
Claudio Buzzi (UNESP-Rio Preto)Hamiltonian vector fields with symplectic time-reversing symmetry.16 January 2002
Miguel Mendes (University of Surrey)Some recent developments and open questions on the dynamics of piecewise isometries (slides)16 January 2002
Ben Mestel (Exeter University)A golden mean functional recurrence12 December 2001
Adam Epstein (Warwick University)Abstract: The moduli space of all quadratic rational maps up to M\"obius conjugacy is isomorphic to ${\Bbb C}^2$. It is possible, and also useful, to regard one of the coordinate axes as the moduli space of quadratic polynomials; the Mandelbrot set, parametrizing the quadratic polynomials with connected Julia set, thereby lies in this slice. Nearly twenty years ago, Douady conjectured that the rational maps in the central portion of the moduli space of quadratic rational maps might be understood as {\em matings} of pairs of quadratic polynomials. The proposed construction is purely topological: one glues filled-in Julia sets back-to-back along complex-conjugate prime ends to obtain a branched cover of the sphere. Work of Tan Lei and Mary Rees shows that under favorable circumstances, the resulting branched cover is topologically conjugate to an essentially unique quadratic rational map. According to Milnor, mating is an interesting operation because it possesses none of the usual good properties: it is not injective, surjective, continuous, or even everywhere defined. We will survey recent results concerning these issues - in particular, the discontinuity of mating. Matings of Quadratic Polynomials12 December 2001
Marcelo Viana (IMPA and College de France)Deterministic products of matrices, in Dynamics and other places7 December 2001
Konstantin Mishaikov (Georgia Institute of Technology)Rigorous Computations for Infinite Dimensional Dynamics.28 November 2001
Raymond Hide (Imperial College)Nonlinear quenching of current fluctuations in a self-exciting dynamo28 November 2001
Oliver Jenkinson (Queen Mary and Westfield College, London)Cohomology classes of dynamically non-negative c^k functions7 November 2001
Matt Nicol (Surrey University)Statistical properties of compact group extensions of chaotic systems7 November 2001
David Broomhead (UMIST)Dynamical models of digital channels - from the sublime to the ridiculous24 October 2001
Franco Vivaldi (Queen Mary)Hamiltonian round-off errors24 October 2001
Ian Melbourne (University of Surrey)Ginzburg-Landau theory of transitions in spatially extended systems10 October 2001
Edgar Knobloch (University of Leeds)New type of complex dynamics in the 1:2 spatial resonance10 October 2001
Fernando Sanchez-SalasMarkov towers for hyperbolic systems23 May 2001
Omri Sarig (Warwick)Abstract: We describe a new method for estimating the correlation functions for equilibrium measures for countable Markov shifts in situations when the tail of the first return time function to some partition set is O(1/n^b) for b>2. Under some aperiodicity condition, this method allows one to determine the second order asymptotics of the iterates of the transfer operator and therby obtain improved estimates for the ``rate of convergence to equilibrium''. These estimates are strong enough to yield sharp upper AS WELL AS LOWER bounds for the correlation functions, and show that at least in the polynomial case, LS Young's upper estimates are sharp. Polynomial Lower Bounds for Rates of Decay of Correlations23 May 2001
Adam EpsteinAbstract: The moduli space of all quadratic rational maps up to M\"obius conjugacy is isomorphic to ${\Bbb C}^2$. It is possible, and also useful, to regard one of the coordinate axes as the moduli space of quadratic polynomials; the Mandelbrot set, parametrizing the quadratic polynomials with connected Julia set, thereby lies in this slice. Nearly twenty years ago, Douady conjectured that the quadratic rational maps in the central portion of moduli space might be understood as {\em matings} of pairs of quadratic polynomials. The proposed construction is purely topological: one glues filled-in Julia sets back-to-back along complex-conjugate prime ends to obtain a branched cover of the sphere. Work of Tan Lei and Mary Rees shows that under favorable circumstances, the resulting branched cover is topologically conjugate to an essentially unique quadratic rational map. According to Milnor, mating is an interesting operation because it possesses none of the usual good properties: it is not injective, surjective, continuous, or even everywhere defined. We will survey recent results concerning these issues. Matings of quadratic polynomials24 April 2001
Henk Bruin (Groningen)Maximal automorphic factors and interval dynamics18 April 2001
Jose Alves (University of Oporto)Stochastic Dynamics18 April 2001
Y. PuriArithmetic of numbers of periodic points20 March 2001