Nonlinear Dynamics

, Volume 70, Issue 2, pp 1079–1094 | Cite as

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

  • Yo Horikawa
  • Hiroyuki Kitajima
Original Paper


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.


Parametric oscillator Quasiperiodic oscillation Traveling phase wave Exponential transient Noise-sustained propagation 


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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Faculty of EngineeringKagawa UniversityTakamatsuJapan

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