Local phenomena in random dynamical systems: bifurcations, synchronisation, and quasi-stationary dynamics
Creators: name: Engel, Maximilian
Publisher: name: Imperial College London
Year: 2017

description: We consider several related topics in the bifurcation theory of random dynamical systems: synchronisation by noise, noise-induced chaos, qualitative changes of finite-time behaviour and stability of systems surviving in a bounded domain. Firstly, we study the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise. Depending on the deterministic Hopf bifurcation parameter and a phase-amplitude coupling parameter called shear, three dynamical phases can be identified: a random attractor with uniform synchronisation of trajectories, a random attractor with non-uniform synchronisation of trajectories and a random attractor without synchronisation of trajectories. We prove the existence of the first two phases which both exhibit a random equilibrium with negative top Lyapunov exponent but differ in terms of finite-time and uniform stability properties. We provide numerical results in support of the existence of the third phase which is characterised by a so-called random strange attractor with positive top Lyapunov exponent implying chaotic behaviour. Secondly, we reduce the model of the Hopf bifurcation to its linear components and study the dynamics of a stochastically driven limit cycle on the cylinder. In this case, we can prove the existence of a bifurcation from an attractive random equilibrium to a random strange attractor, indicated by a change of sign of the top Lyapunov exponent. By establishing the existence of a random strange attractor for a model with white noise, we extend results by Qiudong Wang and Lai-Sang Young on periodically kicked limit cycles to the stochastic context. Furthermore, we discuss a characterisation of the invariant measures associated with the random strange attractor and deduce positive measure-theoretic entropy for the random system. Finally, we study the bifurcation behaviour of unbounded noise systems in bounded domains, exhibiting the local character of random bifurcations which are usually hidden in the global analysis. The systems are analysed by being conditioned to trajectories which do not hit the boundary of the domain for asymptotically long times. The notion of a stationary distribution is replaced by the concept of a quasi-stationary distribution and the average limiting behaviour can be described by a so-called quasi-ergodic distribution. Based on the well-explored stochastic analysis of such distributions, we develop a dynamical stability theory for stochastic differential equations within this context. Most notably, we define conditioned average Lyapunov exponents and demonstrate that they measure the typical stability behaviour of surviving trajectories. We analyse typical examples of random bifurcation theory within this environment, in particular the Hopf bifurcation with additive noise, with reference to whom we also study (numerically) a spectrum of conditioned Lyapunov exponents. Furthermore, we discuss relations to dynamical systems with holes.