Predicting and controlling spatiotemporal patterns in networks of Stuart-Landau oscillators

A long-standing question in nonlinear physics is to understand the precise link between the pattern of connections in the system and the resulting dynamics. An approach that addresses this problem would have direct applications across physics, engineering, and biology. Here, we focus on the Stuart-Landau (SL) model, which can display rich dynamics both in phase and amplitude. These dynamics result from the interplay of interactions with other oscillators in a network and single-node properties. It remains difficult, however, to describe the result of this interaction with a microscopic mathematical theory. Recent work has shown that a complex-valued formulation of the Kuramoto model allows an analytical understanding of the collective behavior through analyzing the eigenvalues and eigenvectors of the network adjacency matrix [Budzinski et al., Phys Rev Res, 2023]. We now generalize this approach to SL networks. This analytical approach allows predicting the precise spatiotemporal patterns that occur in SL networks, including phase synchronization, traveling waves, and chimera states. These predictions are valid for individual realizations of the systems: for an individual network with specific sets of connections and an individual initial state, our approach offers an analytical prediction for the precise spatiotemporal dynamics that emerges. Taken together, these results show that this complex-valued approach can offer new insights into the nonlinear dynamics of oscillator networks.