Adaptation on Nonlocally Coupled Ring Networks

In this chapter, we analyze a nonlocal ring network of adaptively coupled phase oscillators. We observe a variety of frequency synchronized states such as phase-locked, multicluster and solitary states. For an important subclass of the phase-locked solutions, the rotating waves, we provide a rigorous stability analysis. This analysis shows a strong dependence of their stability on the coupling structure and the wavenumber which is a remarkable difference to an all-to-all coupled network. A special type of multicluster states are solitary states which have been observed in a plethora of dynamical systems. We show how solitary states emerge due to the adaptive feature of the network and classify several bifurcation scenarios in which these states are created and stabilized.