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Quasiperiodic and exponential transient phase waves and their bifurcations in a ring of unidirectionally coupled parametric oscillators

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Abstract

Phase waves rotating in a ring of unidirectionally coupled parametric oscillators are studied. The system has a pair of spatially uniform stable periodic solutions with a phase difference and an unstable quasiperiodic traveling phase wave solution. They are generated from the origin through a period doubling bifurcation and the Neimark–Sacker bifurcation, respectively. In transient states, phase waves rotating in a ring are generated, the duration of which increases exponentially with the number of oscillators (exponential transients). A power law distribution of the duration of randomly generated phase waves and the noise-sustained propagation of phase waves are also shown. These properties of transient phase waves are well described with a kinematical equation for the propagation of wave fronts. Further, the traveling phase wave is stabilized through a pitchfork bifurcation and changes into a standing wave through pinning. These bifurcations and exponential transient rotating waves are also shown in an autonomous system with averaging and a coupled map model, and they agree with each other.

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  1. Faculty of Engineering, Kagawa University, Takamatsu, 761-0396, Japan

    Yo Horikawa & Hiroyuki Kitajima

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  1. Yo Horikawa
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  2. Hiroyuki Kitajima
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Correspondence to Yo Horikawa.

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Horikawa, Y., Kitajima, H. Quasiperiodic and exponential transient phase waves and their bifurcations in a ring of unidirectionally coupled parametric oscillators. Nonlinear Dyn 70, 1079–1094 (2012). https://doi.org/10.1007/s11071-012-0514-z

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  • DOI: https://doi.org/10.1007/s11071-012-0514-z

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