Synchronization of chaos

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 established experimentally and reasonably understood theoretically.

It has been found that chaos synchronization is quite a rich phenomenon that may present a variety of forms. When two chaotic oscillators are considered, these include:
* Identical synchronization. This is a straightforward form of synchronization that may occur when two identical chaotic oscillators are mutually coupled, or when one of them drives the other. If (x1,x2,,...,xn) and (x'1, x'2,...,x'n) denote the set of dynamical variables that describe the state of the first and second oscillator, respectively, it is said that identical synchronization occurs when there is a set of initial conditions [x1(0), x2(0),...,xn(0)] , [x'1(0), x'2(0),...,x'n(0)] such that, denoting the time by t, |x'i(t)-xi((t)|→0, for i=1,2,...,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 synchronization occurs when there is a functional, Φ, such that, after a transitory evolution from appropriate initial conditions, it is [y1(t), y2(t),...,ym(t)] =Φ [x1(t), x2(t),...,xn(t)] . This means that the dynamical state of one of the oscillators is completely determined by the state of the other. When the oscillators are mutually coupled this functional has to be invertible, if there is a drive-response configuration the drive determines the evolution of the response, and Φ does not need to be invertible. Identical synchronization is the particular case of generalized synchronization when Φ is the identity.

* Phase synchronization. This form of synchronization, which occurs when the oscillators coupled are not identical, is partial in the sense that, in the synchronized state, the amplitudes of the oscillator remain unsynchronized, and only their phases evolve in synchrony. Observation of phase synchronization requires a previous definition of the phase of a chaotic oscillator. In many practical cases, it is possible to find a plane in phase space in which the projection of the trajectories of the oscillator follows a rotation around a well-defined center. If this is the case, the phase is defined by the angle, φ(t), described by the segment joining the center of rotation and the projection of the trajectory point onto the plane. In other cases it is still possible to define a phase by means of techniques provided by the theory of signal processing, such as the Hilbert transform. In any case, if φ1(t) and φ2(t) denote the phases of the two coupled oscillators, synchronization of the phase is given by the relation nφ1(t)=mφ2(t) with m and n whole numbers.

* Anticipated and lag synchronization. In these cases the synchronized state is characterized by a time interval τ such that the dynamical variables of the oscillators, (x1,x2,,...,xn) and (x'1, x'2,...,x'n), are related by x'i(t)=xi(t+τ); this means that the dynamics of one of the oscillators follows, or anticipates, the dynamics of the other. Anticipated synchronization may occur between chaotic oscillators whose dynamics is described by delay differential equations, coupled in a drive-response configuration. In this case, the response anticipates de dynamics of the drive. Lag synchronization may occur when the strength of the coupling between phase-synchronized oscillators is increased.

* Amplitude envelope synchronization. This is a mild form of synchronization that may appear between two weakly coupled chaotic oscillators. In this case, there is no correlation between phases nor amplitudes; instead, the oscillations of the two systems develop a periodic envelope that has the same frequency in the two systems. This has the same order of magnitude than the difference between the average frequencies of oscillation of the two chaotic oscillator. Often, amplitude envelope synchronization precedes phase synchronization in the sense that when the strength of the coupling between two amplitude envelope synchronized oscillators is increased, phase synchronization develops.

All these forms of synchronization share the property of asymptotic stability. This means that once the synchronized state has been reached, the effect of a small perturbation that destroys synchronization is rapidly damped, and synchronization is recovered again. Mathematically, asymptotic stability is characterized by a positive Lyapunov exponent of the system composed of the two oscillators, which becomes negative when chaotic synchronization is achieved.

Books

*cite book | author=Pikovsky, A.; Rosemblum, M.; Kurths, J. | title=Synchronization: A Universal Concept in Nonlinear Sciences | publisher=Cambridge University Press | year=2001 | id=ISBN 0-521-53352-X

*cite book | author=González-Miranda, J. M. | title=Synchronization and Control of Chaos. An introduction for scientists and engineers | publisher=Imperial College Press | year=2004 | id=ISBN 1-86094-488-4


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Chaos communications — is an application of chaos theory which is aimed to provide security in the transmission of information performed through telecommunications technologies. By secure communications, one has to understand that the contents of the message… …   Wikipedia

  • Chaos theory — This article is about chaos theory in Mathematics. For other uses of Chaos theory, see Chaos Theory (disambiguation). For other uses of Chaos, see Chaos (disambiguation). A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3 …   Wikipedia

  • Synchronization — Synchrony redirects here. For linguistic synchrony, see Synchronic analysis. For the X Files episode, see Synchrony (The X Files). For similarly named concepts, see Synchronicity (disambiguation). Not to be confused with data… …   Wikipedia

  • Chaos in Optical Systems — Optical Chaos is observed in many non linear optical systems. One of the most common examples is a ring resonator. One of the most seminal works is published by Ikeda (Physical Review Letters, 1982) where chaotic behavior in a ring resonator was… …   Wikipedia

  • Chaos in optical systems — Optical Chaos is observed in many non linear optical systems. One of the most common examples is a ring resonator. One of the most seminal works is published by Ikeda (Physical Review Letters, 1982) where chaotic behavior in a ring resonator was… …   Wikipedia

  • Control of chaos — This article is about a non linear system. For the technique in the Sonic the Hedgehog games, see Chaos Control. In chaos theory, control of chaos is based on the fact that any chaotic attractor contains an infinite number of unstable periodic… …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • Cybernetical physics — is a scientific area on the border of Cybernetics and Physics which studies physical systems with cybernetics methods. Cybernetics methods are understood as methods developed within control theory, information theory, systems theory and related… …   Wikipedia

  • Cellular neural network — Cellular neural networks (CNN) are a parallel computing paradigm similar to neural networks, with the difference that communication is allowed between neighbouring units only. Typical applications include image processing, analyzing 3D surfaces,… …   Wikipedia

  • Coupled map lattice — A coupled map lattice (CML) is a dynamical system that models the behavior of non linear systems (especially partial differential equations). They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems.… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”