title: Stability of a chain of phase oscillators.
abstract: We study a chain of N+1 phase oscillators with asymmetric but uniform coupling. This type of chain possesses 2(N) ways to synchronize in so-called traveling wave states, i.e., states where the phases of the single oscillators are in relative equilibrium. We show that the number of unstable dimensions of a traveling wave equals the number of oscillators with relative phase close to π. This implies that only the relative equilibrium corresponding to approximate in-phase synchronization is locally stable. Despite the presence of a Lyapunov-type functional, periodic or chaotic phase slipping occurs. For chains of lengths 3 and 4 we locate the region in parameter space where rotations (corresponding to phase slipping) are present.
authors.affiliation: Department of Mathematics, University of Portsmouth, Portsmouth, Hampshire, United Kingdom.
authors.forename: Jan
authors.initials: J
authors.lastname: Sieber
authors.forename: Tamás
authors.initials: T
authors.lastname: Kalmár-Nagy
mesh_terms: D008962:Models, Theoretical; D012399:Rotation