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JULIAN NEWMAN

Julian Newman

EPSRC Doctoral Prize Fellow

 
 

Room: 635 Huxley Building

Research

I am a researcher in the Dynamical Systems group, working on random dynamical systems.

What is a random dynamical system? A random dynamical system (RDS) is a time-homogeneous rule determining the time-evolution of the state of a system given both its current state and the realised behaviour of some stationary random process affecting the system. Now in many contexts where a "homogeneous Markov process" appears, such a process really arises as one trajectory of an underlying RDS; thus a RDS model can be used to describe a system more fully than the Markov transition probabilities describe it. The theory of random dynamical systems has been employed in the study of a wide variety of contexts, such as finance and economics (e.g. here), filtering theory (e.g. here), neural networks (e.g. here), chemical reaction rates (e.g. here), statistical hydrodynamics (e.g. here), and more.

Moreover, random dynamical systems are often the right framework in which to study the phenomenon of noise-induced synchronisation (e.g. as in here). Discovered in the 1980s by Arkady Pikovsky, this is the phenomenon that two or more processes starting at different states are caused to synchronise in state with each other over time, due to being exposed to the same source of random perturbations. Provided that the processes do not interact with each other, and that they evolve according to the same (time-homogeneous) laws, affected equally and simultaneously by the same source of (statistically time-homogeneous) random perturbations, these processes can be regarded as different simultaneous trajectories of one random dynamical system.

My current research is primarily focused on precisely this topic of synchronisation of trajectories of a random dynamical systems.

When I have time, I am also involved in research on a problem in contact dynamics - namely, a disc placed between the walls of a frictional V-shaped groove, subject to gravity and a turning moment - in collaboration with Dr Oleg Makarenkov, Dr Wolfgang Stamm, and Prof Alexander Fidlin. Some results on this can be found in the paper Regularization of a disk in a frictionable wedge below. [The approach described in section 5 of this paper is potentially problematic, if collision of eigenvalues can occur. However, Dr Makarenkov and I are preparing a further paper on the disc in a wedge, in which this potential problem is overcome.]

Papers

  • Thai Son Doan, Jeroen S.W. Lamb, Julian Newman, and Martin Rasmussen, Classification of random circle homeomorphisms up to topological conjugacy. DynamIC Preprint 2018-3
  • Julian Newman, A note on the conditional triviality property for regular conditional distributions. DynamIC Preprint 2015-13
  • Julian Newman, Synchronisation of almost all trajectories of a random dynamical system. DynamIC Preprint 2015-12
  • Julian Newman, Synchronisation in Invertible Random Dynamical Systems on the Circle. DynamIC Preprint 2015-11
  • Julian Newman, Necessary and sufficient conditions for stable synchronization in random dynamical systems, Ergodic Theory and Dynamical Systems 38, 5 (2018), 1857−1875. Article MathSciNet Preprint
  • Julian Newman and Oleg Makarenkov, Resonance oscillations in a mass-spring impact oscillator, Nonlinear Dynamics 79, 1 (2015), 111−118. Article MathSciNet Preprint
  • Julian Newman, Regularization of a disk in a frictionable wedge, in: Mathematical Modelling (Inge Troch and Felix Breitenecker), 830−835, Vienna University of Technology, Vienna, 2013. Article

Other Documents

  • Ergodic Theory for Semigroups of Markov Kernels (last updated 05/07/05). The evolution of a homogeneous Markov process may be described statistically by a "family of transition probabilities", sometimes known as a transition function or a semigroup of Markov kernels. These notes concern the abstract theory of stationary distributions and invariant sets of such semigroups.