Synchronization of nonlinearly coupled Stuart–Landau oscillators on networks

Abstract The dynamics of coupled Stuart–Landau (SL) oscillators plays a central role in the study of synchronization phenomena. Previous works have focused on linearly coupled oscillators in different configurations, such as all-to-all or generic complex networks, allowing for both reciprocal or non-reciprocal links. The emergence of synchronization can be deduced by proving the linear stability of the limit-cycle solution for the SL model; the linear coupling assumption allows for a complete analytical treatment of the problem, mostly because the linearized system turns out to be autonomous. In this work, we analyze SL oscillators coupled through nonlinear functions on both undirected and directed networks. Synchronization now depends on the study of a non-autonomous linear system and thus novel tools are required to tackle the problem. We provide a complete analytical description of the system behaviour for some choices of the nonlinear coupling, e.g. in the resonant case. Otherwise, we develop a semi-analytical framework based on Jacobi–Anger expansion and Floquet theory, which allows us to derive precise conditions for the emergence of complete synchronization. The obtained results extend the classical theory of coupled oscillators and pave the way for future studies of nonlinear interactions in networks of oscillators and beyond.