Hierarchy and stability of partially synchronous oscillations of diffusively coupled dynamical systems

The paper presents a qualitative analysis of an array of diffusively coupled identical continuous time dynamical systems.The effects of full, partial, anti-phase and in-phase-anti-phase chaotic synchronization are investigated via the linear invariant manifolds of the corresponding differential equations. Existence of various invariant manifolds, a self-similar behavior, a hierarchy and embedding of the manifolds of the coupled system are discovered. Sufficient conditions for the stability of the invariant manifolds are obtained via the method of Lyapunov functions. Conditions under which full global synchronization can not be achieved even for the largest coupling constant are defined. The general rigorous results are illustrated through examples of coupled Lorenz-like and coupled Rössler systems.