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|
| Name |
Title |
Date |
Time |
Room |
| Sigurður Freyr Hafstein (University of Iceland
) | Lyapunov functions for stochastic differential equations: Theory and computationAbstract: Attractors and their basins of attraction in deterministic dynamical systems are most commonly studied using the Lyapunov stability theory.
Its centerpiece is the Lyapunov function, which is an energy-like function from the state-space that is decreasing along all solution trajectories.
The Lyapunov stability theory for stochastic differential equations is much less developed and, in particular,
numerical methods for the construction of Lyapunov functions for such systems are few and far between.
We discuss the general problem and present some novel numerical methods. |
Tuesday, 2 March 2021 |
1:00pm |
Online |
| Erik Bollt (Clarkson University) | Geometry and Good Dictionaries for Koopman Analysis of Dynamical Systems
Abstract: In the spirit of optimal approximation and reduced order modelling the goal of DMD methods and variants is to describe the dynamical evolution as a linear evolution in an appropriately transformed lower rank space, as best as possible. That Koopman eigenfunctions follow a linear PDE that is solvable by the method of characteristics yields several interesting relationships between geometric and algebraic properties. We focus on contrasting cardinality, algebraic multiplicity and other geometric aspects with the introduction of an equivalence class, “primary eigenfunctions,” for those eigenfunctions with identical sets of level sets. We present a construction that leads to functions on the data surface that yield optimal Koopman eigenfunction DMD, (oKEEDMD). We will also describe that disparate systems can be “matched” transformed by a diffeomorphism constructed via eigenfunctions from each system, a reinterpretation of integrability, computationally stated by our “matching extended dynamic mode decomposition (EDMD)” (EDMD-M). |
Tuesday, 9 March 2021 |
1:00pm |
Online |
| Yoshito Hirata (University of Tsukuba) | Unified time series analysis for nonlinear deterministic/stochastic systems
Abstract: Distinguishing deterministic systems from stochastic systems has been
discussed for a long time. But, such analysis had been qualitative or
contrasting nonlinear deterministic systems with linear stochastic
systems. Thus, we could not identify nonlinear stochastic systems with
some hypothesis tests. Here, we propose to use permutations or
recurrence plots for distinguishing stochastic systems from
deterministic systems with a hypothesis test. Therefore, permutations
and recurrence plots can be used also for analyzing a time series
generated from nonlinear stochastic systems. |
Tuesday, 16 March 2021 |
1:00pm |
Online |
| Christoph Kawan (University of Munich) | Control of chaos with minimal information transferAbstract: Networked control systems violate standard assumptions of classical control theory.
One of the many challenges in their analysis and design concerns information constraints
present in the communication between sensors, controllers and actuators. A fundamental
question in this field is thus concerned with the smallest rate of information flowing
from the sensors to the controller, above which a given control task can be solved.
In this talk, we address this question in the context of set-stabilization for nonlinear
systems. To obtain exact results, we impose the assumption of uniform hyperbolicity on
the sets under consideration, which provides us with powerful tools such as shadowing
and hyperbolic volume estimates. As we can then see, the minimal information rate is
closely related to quantities extensively studied in smooth dynamical systems: escape
rates, topological pressure, measure-theoretic entropy and Lyapunov exponents. |
Tuesday, 23 March 2021 |
1:00pm |
Online |
| Woojin Kim (Duke University) | The Persistent Topology of Dynamic DataAbstract: This talk introduces a method for characterizing the dynamics of
time-evolving data within the framework of topological data analysis
(TDA), specifically through the lens of persistent homology. Popular
instances of time-evolving data include flocking or swarming behaviors
in animals, and social networks in the human sphere. A natural
mathematical model for such collective behaviors is that of a dynamic
metric space. In this talk I will describe how to extend the well-known
Vietoris-Rips filtration for metric spaces to the setting of dynamic
metric spaces. Also, we extend a celebrated stability theorem on
persistent homology for metric spaces to multiparameter persistent
homology for dynamic metric spaces. In order to address this stability
property, we extend the notion of Gromov-Hausdorff distance between
metric spaces to dynamic metric spaces. This talk will not require any
prior knowledge of TDA. This talk is based on joint work with Facundo
Memoli and Nate Clause. |
Tuesday, 23 March 2021 |
2:15 pm |
Online |
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Date |
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