Analysis of equilibria in a ring of phase oscillators with nearest-neighbor and next-nearest-neighbor interactions

Coupled phase oscillators with both attractive and repulsive interactions offer a valuable framework for exploring collective dynamics in nonlinear dynamics, serving as an analog to the frustrated Ising model in statistical physics. In this work, we study a ring of phase oscillators with nearest-neighbor and next-nearest-neighbor interactions. The coupling strengths, denoted as J_{1} and J_{2}, dictate the behavior of these interactions. For an infinite number of oscillators, the model allows for a continuum of equilibria. For identical oscillators, linear stability analysis shows the prevalence of multistability. To complement the linear stability analysis, we delve into the basin stabilities of these equilibria. We categorize them based on basin stability into in-phase equilibria with nearly zero phase difference between adjacent oscillators, antiphase equilibria with nearly π phase difference, and heterogeneous equilibria characterized by the phase difference, concentrating on the most unstable modes of the uniform equilibrium. The resulting phase diagram in the J_{1}-J_{2} plane mirrors that of the zero-temperature J_{1}-J_{2} Ising model. For nonidentical phase oscillators, similar qualitative outcomes are observed.