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.
- Pikovsky, A.; Rosemblum, M.; Kurths, J. (2001). Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press. ISBN 0-521-53352-X.
- González-Miranda, J. M. (2004). Synchronization and Control of Chaos. An introduction for scientists and engineers. Imperial College Press. ISBN 1-86094-488-4.