Coexistence of Hamiltonian-Like and Dissipative Dynamics in Rings of Coupled Phase Oscillators with Skew-Symmetric Coupling

We consider rings of coupled phase oscillators with anisotropic coupling. When the coupling is skew-symmetric, i.e., when the anisotropy is balanced in a specific way, the system shows robustly a coexistence of Hamiltonian-like and dissipative dynamics in the phase space. We relate this phenomenon to the time-reversibility of the system. The geometry of low-dimensional systems up to five oscillators is described in detail. In particular, we show that the boundary between the dissipative and Hamiltonian-like regions consists of families of heteroclinic connections. For larger rings with skew-symmetric coupling, some sufficient conditions for the coexistence are provided, and in the limit of N → ∞ oscillators, we formally derive an amplitude equation for solutions in the neighborhood of the synchronous solution. It has the form of a nonlinear Schrödinger equation and describes the Hamiltonian-like region existing around the synchronous state similarly to the case of finite rings.