📂Dynamics

Multidimensional Map Chaos

Definition¹

An orbit of the map $f : \mathbb{R}^{m} \to \mathbb{R}^{m}$ is called chaotic if it satisfies the following conditions:

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

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• Saying an orbit is bounded indicates the existence of $M \in \mathbb{R}$ that satisfies $\left\| \mathbf{v}_{n} \right\| < M$ for every $n \in \mathbb{N}_{0}$.
• $h_{i}(\mathbf{v}_{0})$ refers to the Lyapunov exponent.

Explanation

The difference in chaos for a map of dimension $1$ is that, due to the domain dimension $m$, as many Lyapunov exponents $m$ are calculated, and it is not enough that all Lyapunov exponents be $0$; the largest Lyapunov exponent must be positive. In addition, if more than one Lyapunov exponent is positive, the term hyper chaos is used.

See Also

• Chaos of a 1-dimensional map
• Chaos of invariant sets
• Chaos in systems described by differential equations