Chaos In Higher Dimensions

Lyapunov Exponents

Maps in ${\mathbb{R}}^m$ will have $m$ lyapunov numbers measuring hte rate of separation from the current orbit along $m$ orthogonal directions.

Direction 1: largest rate of expansion ( least contracting if the map contracts in all directions )

Direction 2: largest remaining rate in all directions perpendicular to direction 1

Let $\vec{f}$ be a smooth map on ${\mathbb{R}}^n$ and let $J_n=D{\vec{f}}^n({\vec{v}}_0)$ be the Jacobian of the $n^{th}$ iterate of $\vec{f}$.

For $K=1,2,\dots ,m$, let $r_k^n$ be the length of the $k^{th}$ longest orthogonal axis of the ellipsoid $J_{n}U$ ( where $U$ is the unit disk ) for an orbit with initial point ${\vec{v}}_0$

Then $r_k^n$ measures the contraction/expansion near ${\vec{v}}_0$ during the first $n$ iterates. The $k^{th}$ Lyapunov Number of ${\vec{v}}_0$ is:

$$L_K=\underset{n\to \infty }{\lim }(r_k^n{)}^{\frac{1}{n}}$$

if it exists and the $k^{th}$ Lyapunov Exponent of ${\vec{v}}_0$ is:

$$h_k=\ln (L_k)$$

Note

$$L_1\ge L_2\ge \cdots \ge L_m$$

and

$$h_1\ge h_2\ge \cdots \ge h_m$$

Let $\vec{f}$ be a map on ${\mathbb{R}}^m$, $m\ge 1$ and let $\{{\vec{v}}_0,{\vec{v}}_1,\dots \}$ be a bounded orbit of $\vec{f}$. The orbit is chaotic if:

1. it is not Asymptotically Periodic
2. no Lyapunov Number is 1
3. $L_1({\vec{v}}_0)>1$ , or $h_1>0$

Definition

Skinny Bakers Map (cutting out the middle third) on ${\mathbb{R}}^3$:

Let $S=[0,1]\times [0,1]$

For $0\le x_2\le \frac{1}{2}$:

$$B(x_1,x_2)=\left(\begin{array}{cc}\frac{1}{3}& 0\\ 0& 2\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$$

For $\frac{1}{2}\le x_2\le 1$:

$$B(x_1,x_2)=\left(\begin{array}{cc}\frac{1}{3}& 0\\ 0& 2\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)+\left(\begin{array}{c}\frac{2}{3}\\ -1\end{array}\right)$$

The invariant set $A$ of the map $B$ consists of points that lie in $B^n(S)\forall n\in \mathbb{Z}$, a middle third Cantor Set of x-coords.

The Lyapunov Exponents: we need to find

$$J=\mathcal{D}B(\vec{v})=\left(\begin{array}{cc}\frac{1}{3}& 0\\ 0& 2\end{array}\right)$$

which yields LE: $\ln (2),-\ln (3)$

$⟹$ every