Dynamics of Coupled Oscillator Systems - Abstract
Dynamical systems on networks with adaptive couplings appear naturally in real-world systems such as power grid networks, social networks as well as neuronal networks. We investigate collective behaviour in a paradigmatic network of adaptively coupled phase oscillators. The coupling topology of the network changes slowly depending on the dynamics of the oscillators. We show that such a system gives rise to numerous complex dynamics, including relative equilibria and hierarchical multi-cluster states. An analytic treatment for equilibria and multi-cluster solutions as well as the existence of continuous families of these states is presented and parameter regimes of high multi-stability are found. In addition, we give an interpretation for equilibria as functional units which are building blocks in multi-cluster structures. Our results contribute to the understanding of mechanisms for pattern formation in adaptive networks, such as the emergence of multi-layer structure in neural systems.