Lyapunov exponent

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This project aims to get a better understanding of the bifurcation scenarios for noise-dependent random dynamical system which display a transition from trivial to emergence of chaos, as the noise strength increases. It is known that this transition is observed by the change of the sign of a statistical quantity called Lyapunov exponent, but little is known of the geometric mechanism generating this transition. It is known that in the deterministic system this transition to chaos is accompanied with a stretching and folding mechanism and the emergence of chaotic sets called horseshoes. In this project, we aim to check whether similar objects exist for the random settings or to establish the existence of analogous ones.The importance of understanding the mechanism that rule chaotic dynamical systems is important in all sciences as numerous phenomena, for example climatic and medical ones, display chaotic behaviour in the sense that they are very sensitive to initial condition: a small modification somewhere leads to a big change in the outcome.The first aim of this project is to characterize topologically the behaviour of random systems with positive Lyapunov exponent, in particular trying to establish the existence of random version of horseshoes generated by homoclinic intersections. Another goal is to establish a qualitative characterization of the phenomenon of transient chaos, which means that trajectories still converge to an equilibrium, but the convergence is slow and non uniform in the state space. Very little is known about the phenomenon of transient chaos, so the aim is to get a better understanding at it.Currently, there is almost no literature on the existence of horseshoe for random dynamical systems, except for some very specific cases, and the results are much weaker than the deterministic analogous. The main issue is that current technique fail at estimating return times when randomness kicks in. Also the phenomenon of transient chaos is poorly understood, and the main known results are related to prototypical examples and it is impossible to develop some general theory. Indeed, current techniques are privileged of random systems generated by stochastic differential equation, in which the property of Brownian motion allows to compute some statistics via Kolmogorov equations, which are not in general computable. In order to overcome this issue, we developed the tool of random hyperbolic times and adopt large deviations technique in order to control the measure of the points in the state space which slower expansion rates, and used probabilistic techniques to establish a measure one existence of a random version of an horseshoe for a class of non uniformly hyperbolic random systems with positive Lyapunov exponents. To our knowledge, we are the first to obtain this sort of result and we expect this kind of result to be generalized to a larger class of systems.This project has collaborators, Dr Jeroen Lamb and Dr Dimitry Turaev.This project falls within the EPSRC statistics and applied probability research area.