Dynamics of the Kuramoto-Sakaguchi Oscillator Network with Asymmetric Order Parameter

Kuramoto oscillator networks are an important idealized class of oscillator models. We consider a generalized network in which the order parameter is the sum of the complex oscillator phases, but with non-identical coefficients. We analyze this model using a dimensional reduction from dynamics on an N-dimensional state space to a flow on the unit disk, where the natural hyperbolic metric facilitates the analysis. We give a fairly complete classification of the asymptotic dynamics with careful consideration of the subtleties of the flow near the disk's boundary, which includes both fully synchronized states and (N-1,1) states where all but one of the oscillators are synchronized. The geometric connection also allows us to identify conditions for the flows to be gradient, or Hamiltonian or even simultaneously gradient and Hamiltonian. Examples of new behavior in the asymmetric model include, (N-1,1) attractors with a basin of non-zero measure and homoclinic and heteroclinic non-periodic orbits to/from sync and (N−1,1) states in the Hamiltonian case.