Synchronization of chaos

M.A. Aziz-Alaoui
Labo. Applied Math., Le Havre University

(Invited paper for) Encyclopedia of Mathematical Physics, Elsevier, Vol. 5, pp :213-226, (2006).

Voir Encyclopedia MP


Chaotic systems can synchronize.

Synchronization is a ubiquitous phenomenon characteristic of many processes in natural systems and (nonlinear) science, it has permanently remained an objectif of intensive research and is today considered as one of the basic nonlinear phenomena studied in mathematics, physics, engineering or life science. This word has a greek root, {syn = common} and {chronos = time}, which means to share the common time or to occur at the same time, that is correlation or agreement in time of different processes (Boccaletti et al, 2002). Thus, synchronization of two dynamical systems generally means that one system somehow traces the motion of another. Indeed, it is well known that many coupled oscillators have the ability to adjust some common relation that they have between them due to weak interaction, which yields to a situation in which a synchronization-like phenomenon takes place.


Roughly speaking, a system is chaotic if it is deterministic, has a long-term aperiodic behavior, and exhibits sensitive dependence on initial conditions on a closed invariant set (the chaos theory has its own entry in the Encyclopedia).


Despite this, in the last decades, the search for synchronization has moved to chaotic systems. A lot of research has been done and, as a result, Yamada and Fujisaka (1983), Afraimovich et al. (1986) and Pecora and Carroll (1990) showed that two chaotic systems could be synchronized by coupling them : synchronization of chaos is actual and chaos could then be expoitable. Ever since, many researchers have discussed the theory, the design or applications of synchronized motion in coupled chaotic systems. A broad variety of applications have emerged, for example to increase the power of lasers, to synchronize the output of electronic circuits, to control oscillations in chemical reactions or to encode electronic messages for secure communications.


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