📂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:

• (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

• Chaos in invariant sets
• Chaos in multidimensional maps