Dynamics of Coupled Oscillator Systems - Abstract
Globally coupled Stuart-Landau oscillators are a generic model system for the study of collective behavior in oscillatory systems beyond the weak coupling limit. In this talk we will address two fundamental questions concerning emergent behavior in globally coupled oscillatory networks: 1) How is clustering behavior in minimal networks linked to clustering dynamics in large ensembles of oscillators. 2) Which symmetries are realized in states with partially broken symmetry, also called chimera states. In both cases, we start out by considering a minimal system of four globally coupled Stuart-Landau oscillators. To answer the first question, we elaborate how 2-cluster states crowd when increasing the number of oscillators. Furthermore, using persistence, we discuss how this crowding leads to a continuous transition from balanced cluster states to synchronized solutions via the intermediate unbalanced 2-cluster states. These cascade-like transitions emerge from what we call cluster singularities. As for the second question, we demonstrate that when the four mean-coupled Stuart-Landau oscillators form states with partially broken symmetry, states with different set-wise symmetries in the incoherent oscillators may arise, some of which are and some are not invariant under a permutation symmetry on average. We conclude our report with a discussion of how the results apply to spatially extended systems.