# Synchronization of Chaos - Identical Synchronization

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

where is the vector field modeling the isolated chaotic dynamics and is the coupling parameter. The regime defines an invariant subspace of the coupled system, if this subspace 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 . Assuming that 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. If then we obtain

and since the dynamics of chaotic we have, where denotes the maximum Lyapunov exponent of the isolated system. Now using the ansatz we pass from the equation for to the equation for . Therefore, we obtain

yield a critical coupling strength, for all 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.