Consider $\mathcal{T}=S^{1}\times D^{2}$, where $S^{1}=[0,1]\mod 1$ and $D^{2}=\{(x,y)\in\mathbb{R}^{2}|x^{2}+y^{2}\le 1\}$. Fix $\lambda\in(0,\frac{1}{2})$ and define the map $F:\mathcal{T}\to\mathcal{T}$ by $$F(\phi,x,y)=\left(2\phi,\lambda x+\frac{1}{2}\cos 2\pi\phi,\lambda y+\frac{1}{2}\sin 2...

Identify the Lyapunov exponent for the cat map: $C(x,y) = (2x+y , x+y)$. I am very confused as to finding the Lyapunov exponent for a two-dimensional map. I've come across a resource that states $$\lambda(T,(x, y)) = \liminf_{n\to\infty} \frac1n\log\|DT^n(x, y)\|$$ but I have no idea how to...

I'm following an example from Nonlinear Dynamics and Chaos (Strogatz) that asks to show how if $f$ has a stable $p$-cycle then the Lyapunov exponent $\lambda<0$. I understand why this must be the case, and I follow the maths detailed in the example except for this part: How does the limit resolv...

Given a linear non autonomous dynamical system in two dimensions $x_{n+1}=A_{n}x_{n}$ started at $x_0\neq0$, $A_n\longrightarrow A$ invertible, assume it has zero Lyapunov exponent for all $x_0$, i.e. $\lim_{n\longrightarrow\infty}\log\Vert x_n \Vert/n=0$. Is this enough to conclude that the syst...

Consider a series of stochastic matrices $\{M_t\}_{t\in \mathbb{N}}$, where $M_t$ is strongly connected for all $t$, and the sum of each row is $1$. Let $\zeta_M$ be the top Lyapunov exponent corresponding to the sequence $$\zeta_M = \text{lim}_{t\to\infty}\frac{1}{t}\text{log}||M_t M_{t-1}\cdots...

The following is an exercise from Marcelo Viana's Lectures on Lyapunov Exponents. The goal is to calculate the extremal Lyapunov exponents. I am having trouble calculating the limit of the product of random matrices, which I believe should be done by applying the law of large numbers. Consider an...

Suppose $f(x)$ is periodic. Or even quasiperiodic. Does it follow that $f(x)$ has a zero top Lyapunov exponent? I'm thinking that if we say $f(x)$ is periodic, then $f(x)=f(x+2\pi)$ as an example. This implies that $f$ is bounded and thus if we consider; $$\lambda_{\mathrm{top}}=\lim_{x\to\infty}...

Suppose $x_{n+1}=A_nx_n$ is a two dimensional linear non-autonomous dynamical system and starting at $x_0\neq0$ and $x_0+\delta_0\neq 0$ we follow the dynamics of the distance between the two orbits, which will be $\delta_{n+1}=A_n\delta_n$ by linearity. Assume that the usual time average definin...