Stability of a chain of phase oscillators

J. Sieber, T. Kalmar-Nagy

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Abstract

We study a chain of N+1 phase oscillators with asymmetric but uniform coupling. This type of chain possesses 2N 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.
Original languageEnglish
Pages (from-to)016227
Number of pages1
JournalPhysical Review E
Volume84
Issue number1
DOIs
Publication statusPublished - 29 Jul 2011

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