Ordinary differential equations Definition

Synchronization of chaos

Builds on Autonomous system Chaos theory

Definition

Let $f \colon \R^n \to \R^n$ be a locally Lipschitz vector field whose autonomous system $x' = f(x)$ has chaotic solutions. Let $\varepsilon > 0$ and let $\Gamma$ be an $n \times n$ matrix. Consider the coupled system $x' = f(x) + \varepsilon \Gamma (y - x), \qquad y' = f(y) + \varepsilon \Gamma (x - y).$ The diagonal $\set{(x, y) : x = y}$ is invariant under this system, and on it each component follows the chaotic system $x' = f(x)$ . The coupled system shows complete synchronization from a set $U \subseteq \R^n \times \R^n$ of initial states if every solution with $(x(0), y(0)) \in U$ satisfies $\lim_{t \to \infty} \norm{x(t) - y(t)} = 0.$ Synchronization of chaos refers to such behaviour, with $U$ a neighbourhood of a chaotic attractor on the diagonal, and to its weaker forms.

Remarks

The difference $e = x - y$ satisfies, to first order, $e' = \big(Df(x) - 2 \varepsilon \Gamma\big) e$ . Synchronization is stable when the largest Lyapunov exponent of this linear equation along the synchronized orbit, the transverse Lyapunov exponent, is negative.

Example: let $f(x) = a x$ be scalar with $a > 0$ , which is not chaotic but shows the mechanism, and $\Gamma = 1$ . Then $e' = (a - 2\varepsilon) e$ exactly, so $e(t) \to 0$ for every initial state exactly when $\varepsilon > a/2$ .

Non-example: for $\varepsilon = 0$ the two copies are uncoupled, and for a chaotic $f$ nearby initial states separate, so $\norm{x(t) - y(t)}$ does not tend to $0$ for most initial states.

Weaker forms are phase synchronization, where only the phases of oscillations lock, and generalized synchronization, where $y(t) - \psi(x(t)) \to 0$ for some function $\psi$ .

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Tag	`NZBKMKB8CV`
URI	`m2://NZBKMKB8CV`
Kind	definition
Writing	W0 stub
MSC	34D06, 37D45
Defines	Synchronization of chaos
Uses	Autonomous system, Chaos theory

Synchronization of chaos

Definition

Remarks

Reading path

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