Abstract
We highlight some results from normal form theory for symmetric bifurcations that give a rational way to organize higher-order interactions between phase oscillators in networks with fully symmetric coupling. For systems near Hopf bifurcation the lowest order (pairwise) interactions correspond to the system of Kuramoto and Sakaguchi. At next asymptotic order one must generically include higher-order interactions of up to four oscillators. We discuss some dynamical consequences of these interactions in terms of heteroclinic attractors, chaos, and chimeras for related systems.
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