The Lyapunov exponent and the first return time in a chaotic flow

Abstract

In the first part of the thesis, we present a new numerical algorithm based on high precision computation to estimate the largest Lyapunov exponent $L_{\max}$ of a chaotic flow $F_t(x)$, $t \ge 0$, $x \isin \mathbb{R}^m$. Our method makes use of the divergence speed, which is the minimal time for two nearby trajectories to diverge beyond a given distance from each other. Take $x, \hat{x} \isin \mathbb{R}^m$ with $|| x-\hat{x}|| = 10^{-D}$ for a fixed integer $D\gg1$. The divergence speed $V(n)$ for $n\ge 1$ is defined to be the minimal time for two trajectories $\{F_t(x)}_{t\ge 0}$ and $\{F_t(\hat{x})}_{t\ge 0}$ starting from $x$ and $\hat{x}$, respectively, to diverge until they are away from each other with the distance of $10^{-D+n}$. With probability 1 the divergence speed does not depend on the direction of $x - \hat{x}$. The key idea is to employ enough number of significant digits in order to ensure that the distance of $10^{-D}$ makes sense in a numerical scheme, which is a discretized version of the flow $F_t$. It is shown that $L_{\max}$ is approximated by $\It{n/V(n)}$ for sufficiently large $\It{n}$. The result can be used to investigate the cumulative effect of nonlinearity of dynamical systems, which is due to imprecise initial data. We apply the divergence speed $\It{n}$ to find $\It{L_{\max}}$ for chaotic flows $F_t$ arising from differential equations such as Lorenz and $R\ddot{o}umlssler$ equations. The second part, we consider a forward limit set in $\mathbb{R}^m$ where a forward limit set is fractal arising from differential equation such as Lorenz and $R\ddot{o}umlssler$ equation. Let $\It{Y}$ be the forward limit set, which is a fractal set. Let $\It{X}$ be the $Poincar\acute{e}$ section of $\It{Y}$ by a $(m-1)$ dimensional plane $\It{H}$ transversal to $\It{Y}$, i.e., $\It{X=Y\cap H}$. For a $Poincar\acute{e}$ mapping $T:X\to X$ and $x\in X$, we define the $n$th metric version of the first return time on $\It{X}$ by $R_n(x) = \min{ k...