Lyapunov exponent

本项目旨在更好地理解噪声相关随机动力系统的分岔场景，随着噪声强度的增加，该系统显示出从平凡到混沌出现的过渡。众所周知，这种转变是通过称为李雅普诺夫指数的统计量符号的变化来观察的，但对产生这种转变的几何机制知之甚少。众所周知，在确定性系统中，这种向混沌的过渡伴随着拉伸和折叠机制以及称为马蹄铁的混沌集的出现。在这个项目中，我们的目标是检查是否存在类似的对象的随机设置或建立类似的存在。理解的重要性，规则混沌动力系统的机制是很重要的，在所有的科学作为许多现象，例如气候和医学，显示混沌行为的意义上，他们是非常敏感的初始条件：这个项目的第一个目标是从拓扑上描述具有正李雅普诺夫指数的随机系统的行为，特别是试图建立由同宿相交产生的随机版本的马蹄铁的存在性。另一个目标是建立瞬态混沌现象的定性表征，这意味着轨迹仍然收敛到一个平衡点，但收敛速度很慢，并且在状态空间中是不均匀的。人们对瞬态混沌现象知之甚少，所以我们的目的是更好地理解它。目前，除了一些非常特殊的情况外，几乎没有关于随机动力系统马蹄铁存在性的文献，而且其结果比确定性类比弱得多。主要的问题是，当随机性开始时，当前的技术无法估计返回时间。此外，对瞬态混沌现象的理解还很有限，主要的已知结果都与原型例子有关，不可能发展出一些通用的理论。实际上，目前的技术是由随机微分方程产生的随机系统的特权，其中布朗运动的性质允许通过Kolmogorov方程计算一些统计数据，这些统计数据通常是不可计算的。为了克服这一问题，我们开发了随机双曲时间工具，采用大偏差技术来控制状态空间中扩张速率较慢的点的测度，并利用概率技术建立了一类具有正Lyapunov指数的非一致双曲随机系统的随机马蹄铁的测度存在性.据我们所知，我们是第一个获得这种结果，我们希望这种结果可以推广到更大的一类系统。这个项目有合作者，博士Jeroen兰姆和博士Dimitry Turaev。这个项目福尔斯EPSRC统计和应用概率研究领域。

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.