High-order phase reduction using the parameterization methodOnline: attend (945856)

Networks of coupled oscillators are important models for a wide variety of complex systems. In the absence of coupling, such a system has an invariant torus given by the product of the individual periodic orbits. This invariant torus persists when the coupling strength is small, and many questions deal with the behavior on or near this torus. As it turns out, a major challenge lies in obtaining expressions for the higher-order terms of the restricted vector field in the (small) coupling parameter. We present a novel way of performing this so-called high-order phase reduction, which uses the parameterization method. Our techniques only require the uncoupled periodic orbits to be hyperbolic (as opposed to the usual condition of stability), and directly allow us to obtain phase-dynamics that are in normal form. This means we obtain a conjugate system that has only resonant terms, up to arbitrary finite order. We finish with an example that shows remote synchronisation in a network of three coupled Stuart-Landau oscillators. This is joint work with Sören von der Gracht and Bob Rink.