Lyapunov exponent

本项目旨在更好地理解随噪声强度增加，噪声依赖随机动力系统从微不足道到出现混沌的分岔情形。众所周知，这种转变是通过称为李亚普诺夫指数的统计量的符号变化来观察的，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，我们的目的是检查随机设置中是否存在类似的对象，或者确定类似对象的存在。理解控制混沌动力系统的机制的重要性在所有科学中都很重要，因为许多现象，例如气候和医学现象，在对初始条件非常敏感的意义上表现出混沌行为：某个地方的小修改会导致结果的大变化。本项目的第一个目的是对具有正李雅普诺夫指数的随机系统的行为进行拓扑表征，特别是试图建立由同斜交产生的马蹄形的随机版本的存在性。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平衡状态，但收敛速度缓慢且在状态空间中不均匀。我们对瞬态混沌现象所知甚少，所以我们的目标是更好地理解它。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存在马蹄形的文献，其结果也比确定性类比弱得多。主要的问题是，当随机性出现时，当前的技术无法估计返回时间。此外，人们对瞬态混沌现象的认识也很差，已知的主要结果都是与典型例子有关的，不可能发展出某种普遍的理论。事实上，目前的技术是由随机微分方程生成的随机系统的特权，其中布朗运动的性质允许通过一般不可计算的Kolmogorov方程计算一些统计数据。为了克服这一问题，我们开发了随机双曲时间的工具，并采用大偏差技术来控制状态空间中扩展速率较慢的点的度量，并利用概率技术建立了一类具有正李雅普诺夫指数的非均匀双曲随机系统的马蹄形随机版本的存在性度量。据我们所知，我们是第一个得到这类结果的并且我们期望这类结果能推广到更大的一类系统。这个项目有合作者Jeroen Lamb博士和Dimitry Turaev博士。本项目属于EPSRC统计与应用概率研究领域。

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.