On attractors, spectra and bifurcations of random dynamical systems

摘要

In this thesis a number of related topics in random dynamical systems theory are studied: local attractors and attractor-repeller pairs, the exponential dichotomy spectrum and bifurcation theory.ududWe review two existing theories in the literature on local attractors for random dynamical systems on compact metric spaces and associated attractor-repeller pairs and Morse decompositions, namely, local weak attractors and local pullback attractors. We extend the theory of past and future attractor-repeller pairs for nonautonomous systems to the setting of random dynamical systems, and define local strong attractors, which both pullback and forward attract a random neighbourhood. Some examples are given to illustrate the nature of these different attractor concepts. For linear systems considered on the projective space, it is shown that a local strong attractor that attracts a uniform neighbourhood is an object with sufficient properties to prove an analogue of Selgrade's Theorem on the existence of a unique finest Morse decomposition.ud udWe develop the dichotomy spectrum for random dynamical systems and investigate its relationship to the Lyapunov spectrum. We demonstrate the utility of the dichotomy spectrum for random bifurcation theory in the following example. Crauel and Flandoli [Journal of Dynamics and Differential Equations, 10(2):259–274, 1998] studied the stochastic differential equation formed from the deterministic pitchfork normal form with additive noise. It was shown that for all parameter values this system possesses a unique invariant measure given by a globally attracting random fixed point with negative Lyapunov exponent, and hence the deterministic bifurcation scenario is destroyed by additive noise. Here, however, we show that one may still observe qualitative changes in the dynamics at the underlying deterministic bifurcation point, in terms of: a loss of hyperbolicity of the dichotomy spectrum; a loss of uniform attractivity; a qualitative change in the distribution of finite-time Lyapunov exponents; and that whilst for small parameter values the systems are topologically equivalent, there is a loss of uniform topological equivalence.