📂Dynamics

Multidimensional Map Chaos

Definition¹

A map’s orbit $f : \mathbb{R}^{m} \to \mathbb{R}^{m}$ is considered chaotic if it satisfies the following for its bounded orbit $\left\{ \mathbf{v}_{0}, \mathbf{v}_{1}, \cdots \right\}$:

• (i): It is not asymptotically periodic.
• (ii): For all $i = 1,\cdots , m$, there is $h_{i} ( \mathbf{v}_{0} ) \ne 0$.
• (iii): $h_{1} ( \mathbf{v}_{0}) > 0$

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• An orbit being bounded means that for all $n \in \mathbb{N}_{0}$, there exists $\left\| \mathbf{v}_{n} \right\| < M$ which satisfies $M \in \mathbb{R}$.
• $h_{i}(\mathbf{v}_{0})$ refers to the Lyapunov exponent.

Description

The difference from the chaos of a $1$-dimensional map lies in the fact that with the domain’s dimension $m$, as many Lyapunov exponents as $m$ need to be calculated, and not all of the Lyapunov exponents should be $0$, and the largest Lyapunov exponent must be positive. Meanwhile, when more than two Lyapunov exponents are positive, the term hyper chaos is used.

See Also

• 1-dimensional map chaos