Centre for Complexity Science seminar. A frustrated Kuramoto model on simplicial complexes.

We formulate a generalisation of the Sakaguchi-Kuramoto model, a.k.a the frustrated Kuramoto model, on weighted simplicial complexes where phases oscillators are supported on simplices of any order k with linear and non-linear frustration terms that are independent of the orientation of the (k+1) simplices. We also introduce a generalized Kuramoto order parameter that explicitly uses the harmonic subspace of the Hodge decomposition of the dynamics of the oscillator. We study the properties of the dynamics of the simplicial Sakaguchi-Kuramoto model with oscillators on edges using a selection of simplicial complexes of increasingly complex structure, to highlight the complexity of dynamical behaviours emerging from even simple simplicial complexes. In particular, using the Hodge decomposition of the solution, we understand how the nonlinear frustration couples the dynamics in orthogonal subspaces. We discover a zoo of dynamical regimes that indicate an extremely rich model.