Bifurcation Analysis of Twisted States on Nearest-Neighbor Networks

The Kuramoto model provides a prototypical framework to study the dynamics interacting particle systems. The classical heterogeneous Kuramoto model exhibits two main dynamically important states - desynchronization and partial synchronization. Depending on the parameters of the system, the long term behavior always tends to either of these states. However, when considering identical oscillators on a nearest-neighbor graph, the Kuramoto model exhibits more interesting states such as uniformly twisted states. It was discovered by Wiley, Strogatz and Girvan in 2006 that the stability of these twisted stated depends on the coupling range of the nearest-neighbor graph. Since this original analysis was published, many generalizations and variants were developed. In this talk, we will analyze the bifurcation in which these twisted states loose their stability upon varying parameters, such as the coupling range, of the system. We investigate the existence and shape of bifurcating equilibria in the infinite particle limit. Moreover, we add higher-order interactions and show how these change the stability of twisted states and influence the bifurcation at which stability is lost or gained.