Synchronization of chaos
Synchronization of chaos is a phenomenon that may occur when two, or more, chaotic oscillators are coupled, or when a chaotic oscillator drives another chaotic oscillator. Because of the butterfly effect, which causes the exponential divergence of the trajectories of two identical chaotic system started with nearly the same initial conditions, having two chaotic system evolving in synchrony might appear quite surprising. However, synchronization of coupled or driven chaotic oscillators is a phenomenon well n, when t→∞. That means that for time large enough the dynamics of the two oscillators verifies x'i(t)=xi(t), for i=1,2,...,n, in a good approximation. This is called the synchronized state in the sense of identical synchronization.
- Generalized synchronization. This type of synchronization occurs mainly when the coupled chaotic oscillators are different, although it has also been reported between identical oscillators. Given the dynamical variables (x1,x2,,...,xn) and (y1,y2,,...,ym) that determine the state of the oscillators, generalized by a positive Lyapunov exponent of the system composed of the two oscillators, which becomes negative when chaotic synchronization is achieved.
Books
- Pikovsky, A.; Rosemblum, M.; Kurths, J. (2001). Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press. ISBN 0-521-53352-X.
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- González-Miranda, J. M. (2004). Synchronization and Control of Chaos. An introduction for scientists and engineers. Imperial College Press. ISBN 1-86094-488-4.