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arXiv · 1101.4761

On the Stability of a Chain of Phase Oscillators

Jan Sieber·Tamas 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 $2^{N}$ ways to synchronize in so-called travelling 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 travelling 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 length 3 and 4 we locate the region in parameter space where rotations (corresponding to phase slipping) are present.

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Jan Sieber, Tamas Kalmar-Nagy. 2011-06-02. On the Stability of a Chain of Phase Oscillators. https://doi.org/10.1103/physreve.84.016227

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