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High-order synchronization, transitions, and competition among Arnold tongues in a rotator under harmonic forcing

David Garcia-Alvarez·Aneta Stefanovska·Peter V. E. McClintock
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Abstract

We consider a rotator whose equation of motion for the angle $θ$ consists of the zeroth and first Fourier modes. Numerical analysis based on the trailing of saddle-node bifurcations is used to locate the n:1 Arnold tongues where synchronization occurs. Several of them are wide enough for high-order synchronization to be seen in passive observations. By sweeping the system parameters within a certain range, we find that the stronger the dependence of $\dotθ$ on $θ$, the wider the regions of synchronization. Use of a synchronization index reveals a vast number of very narrow n:m Arnold tongues. A competition phenomenon among the tongues is observed, in that they "push" and "squeeze" one another: as some tongues widen, others narrow. Two mechanisms for transitions between different n:m synchronization states are considered: slow variation of the driving frequency, and the influence of low-frequency noise on the rotator.

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David Garcia-Alvarez, Aneta Stefanovska, Peter V. E. McClintock. 2008-11-14. High-order synchronization, transitions, and competition among Arnold tongues in a rotator under harmonic forcing. https://doi.org/10.1103/physreve.77.056203

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