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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 *

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ABSTRACT

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.

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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).

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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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