Synchronization of chaos is a phenomenon that may occur when two or more dissipative chaotic systems are coupled.

Because of the exponential divergence of the nearby trajectories of chaotic systems, having two chaotic systems evolving in synchrony might appear surprising. However, synchronization of coupled or driven chaotic oscillators is a phenomenon well established experimentally and reasonably well-understood theoretically.

The stability of synchronization for coupled systems can be analyzed using master stability. Synchronization of chaos is a rich phenomenon and a multi-disciplinary subject with a broad range of applications.

Synchronization may present a variety of forms depending on the nature of the interacting systems and the type of coupling, and the proximity between the systems.

Identical synchronization

This type of synchronization is also known as complete synchronization. It can be observed for identical chaotic systems. The systems are said to be completely synchronized when there is a set of initial conditions so that the systems eventually evolve identically in time. In the simplest case of two diffusively coupled dynamics is described by

is locally attractive then the coupled system exhibit identical synchronization.

is small we can expand the vector field in series and obtain a linear differential equation - by neglecting the Taylor remainder - governing the behavior of the difference

then we obtain

. Therefore, we obtain

the system exhibit complete synchronization. The existence of a critical coupling strength is related to the chaotic nature of the isolated dynamics.

In general, this reasoning leads to the correct critical coupling value for synchronization. However, in some cases one might observe loss of synchronization for coupling strengths larger than the critical value. This occurs because the nonlinear terms neglected in the derivation of the critical coupling value can play an important role and destroy the exponential bound for the behavior of the difference. It is however, possible to give a rigorous treatment to this problem and obtain a critical value so that the nonlinearities will not affect the stability.

Generalized synchronization

is the identity.

Phase synchronization

Phase synchronization occurs when the coupled chaotic oscillators keep their phase difference bounded while their amplitudes remain uncorrelated. This phenomenon occurs even if the oscillators are not identical. 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

; 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 the 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.

Some chaotic systems allow even stronger control of chaos, and both synchronization of chaos and control of chaos constitute parts of what's known as "cybernetical physics".

External links

Notes

References

See also

• Kuramoto model