Synchronization of chaos is a phenomenon that may occur when two or more dissipative chaotic systems are coupled.
However, synchronization of coupled or driven chaotic oscillators is a phenomenon well established experimentally and reasonably well-understood theoretically.
Synchronization of chaos is a rich phenomenon and a multi-disciplinary subject with a broad range of applications.
In the simplest case of two diffusively coupled dynamics is described by where
is the vector field modeling the isolated chaotic dynamics and
is locally attractive then the coupled system exhibit identical synchronization.
If the coupling vanishes the oscillators are decoupled, and the chaotic behavior leads to a divergence of nearby trajectories.
Complete synchronization occurs due to the interaction, if the coupling parameter is large enough so that the divergence of trajectories of interacting systems due to chaos is suppressed by the diffusive coupling.
To find the critical coupling strength we study the behavior of the difference
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 where
denotes the Jacobian of the vector field along the solution.
denotes the maximum Lyapunov exponent of the isolated system.
Therefore, we obtain yield a critical coupling strength
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.
[4] 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.
that determine the state of the oscillators, generalized synchronization occurs when there is a functional,
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.
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 these cases, the synchronized state is characterized by a time interval τ such that the dynamical variables of the oscillators,
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.
This is a mild form of synchronization that may appear between two weakly coupled chaotic oscillators.
All these forms of synchronization share the property of asymptotic stability.
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.