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Chaos in Systems Described by Differential Equations 📂Dynamics

Chaos in Systems Described by Differential Equations

Definition 1

Consider the following vector field given as a differential equation for the function $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$. $$ \dot{x} = f(x) $$ The orbit $\phi_{t} ( x_{0} )$ of this system at a point $x_{0} \in X$ is said to be chaotic if it satisfies the following conditions:

Explanation

Condition (iii) in the definition essentially excludes the conditions stated in the Poincaré–Bendixson theorem. A positive Lyapunov exponent indicates sensitivity to initial conditions, which is an indispensable element in the concept of chaos.

See Also


  1. Yorke. (1996). CHAOS: An Introduction to Dynamical Systems: p385~386. ↩︎