Cooperative Behavior In Networks Of Coupled Oscillators
Martens, Erik Andreas
This thesis comprises three problems related to the dynamics of coupled phase oscillators, described by variants of the Kuramoto model. Kuramoto originally made strong assumptions to simplify the analysis of the oscillator behavior: the phases obey a sinusoidal response curve, their natural frequencies are unimodally distributed, and the oscillators are globally coupled, i.e. all oscillators are coupled with equal strength. Investigating three problems we study what behavior may emerge as we relax the last two of these assumptions. In the ?rst problem, we study the impact of replacing the unimodal with a bimodal frequency distribution on the oscillator dynamics. Based on a recent breakthrough in the ?eld, we are able to determine the complete stability diagram, and determine all types of cooperative behavior that may occur. In the next two problems we break with the assumption of global coupling; a similar simpli?cation that is frequently used is to consider the limit of local, i.e. nearest neighbor coupling. We investigate what types of new behavior emerge in the intermediate regime that we call nonlocal coupling, and study under which conditions it persists. For nonlocal coupling, a new kind of state has been observed, where synchronized and desynchronized oscillators coexist side by side in a stable fashion. This state is referred to as a chimera state. In the second problem we discuss a triangular network of oscillator populations with nonlocal coupling. For this network topology, we discover that bistable chimera attractors are possible. For the third problem, we study a generalization of this system, and break the ro- tational symmetry inherent to the triangle by introducing an additional parameter. This parameter allows us to change the topology of the network continuously, such that the network attains a more chain-like character; this enables us to study the effect of network topology on the existence of chimera attractors.
dissertation or thesis