C-oscillators and stability of stationary cluster structures in lattices of diffusively coupled oscillators

This paper studies the conditions for existence and stability of stationary cluster structures in lattices of diffusively coupled dynamical systems within the framework of a new interpretation of cluster synchronization as classical synchronization of cluster oscillators (C-oscillators). The study of existence of cluster attractors is based on the linear chains of cluster oscillators, defining possible types of cluster structures in chains. First, we present interval estimates for the range of coupling strengths in which cluster attractors can exist. Then we formulate and prove the basic theorems about the local stability of the various cluster structures. The presented methodology can be extended to study cluster structures on lattices of different geometry and forms such as linear cluster structures in two-dimensional lattices, layered cluster structures in three-dimensional lattices and cluster structures in ring-shaped systems.