Nonlinear waves on circle networks with excitable nodes (Q2854114)

The authors investigate networks of diffusively coupled phase oscillators NEWLINE\[NEWLINE\dot{x}_{i}=\omega-\epsilon\sin(x_{i})+d\sin(x_{i-1}-2x_{i}+x_{i+1}).NEWLINE\]NEWLINE They describe possible traveling wave solutions based on symmetry considerations and prove stability of regular waves in the case \(\epsilon=0\). Using the Poincaré-Lindstedt method, formulas for the case \(0<\epsilon\ll1\) are given via asymptotic expansions indicating the persistence of a stable regular wave. The stability of non-regular traveling waves is investigated numerically. For \(|\epsilon|>|\omega|\) the dynamics become excitable. A short section on the characterization of a circulating pulse in contrast to regular traveling waves is included. The relation between the investigated model and the Frenkel-Kontorova model is discussed.