Synchronization of chaos

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.

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 ${\displaystyle F}$ is the vector field modeling the isolated chaotic dynamics and ${\displaystyle \alpha }$ is the coupling parameter. The regime ${\displaystyle x(t)=y(t)}$ defines an invariant subspace of the coupled system, if this subspace ${\displaystyle x(t)=y(t)}$ 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 ${\displaystyle v=x-y}$. Assuming that ${\displaystyle v}$ 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 ${\displaystyle DF(x(t))}$ denotes the Jacobian of the vector field along the solution. If ${\displaystyle \alpha =0}$ then we obtain