Patterns of synchronized clusters in adaptive networks

Adaptive networks with time-varying connectivity, often called plasticity, provide a fundamental paradigm to model complex dynamical systems. In these systems, different groups of elements frequently exhibit different yet synchronized dynamics within each group. Here we propose a framework to study patterns of synchronous solutions in a large class of plastic networks and derive a general approach to analyze the stability of these solutions. This approach decouples the role of the network topology from that of the dynamic, thus leading to a dimensionality reduction of the stability problem and allowing us to investigate how adaptation affects the emergence of multi-stable patterns of synchronized activity. To illustrate its potentialities, we apply our method to three networks of oscillators, with distinct topology, dynamics, and adaptation rules. Our working framework encompasses a large class of heterogeneous multi-layer dynamical networks, connected (even with delays) via different plastic links, and can have a broad impact on the analysis of complex plastic networks.