One-Cluster States in Adaptive Networks of Coupled Phase Oscillators

Abstract

Numeric studies suggested that one-cluster states may serve as building blocks for multicluster states. In this chapter, we exhaustively analyze the properties of one-cluster states in a network of adaptively coupled phase oscillators. In particular, we find that there are only three types of one-cluster states: splay, antipodal, and double antipodal. It is shown that all one-cluster states of splay type form a high dimensional family and thus give rise to infinitely many states the system can achieve. In order to understand the stability of these cluster states, we perform a linear stability analysis. We provide analytic results for the stability of antipodal, double antipodal as well as of special types of splay states, namely, rotating-wave states. In particular, we show that double antipodal states are always of saddle type but may appear as physically important transient states. Their role for the global phase space structure is discussed.