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:
- (i): $\phi_{t} ( x_{0} )$ is bounded for $t \ge 0$.
- (ii): At least one of the Lyapunov exponents of $\phi_{t} ( x_{0} )$ is positive.
- (iii): The omega limit set $\omega \left( x_{0} \right)$ is either not periodic or not composed solely of a single fixed point or arcs connecting fixed points.
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
Yorke. (1996). CHAOS: An Introduction to Dynamical Systems: p385~386. ↩︎