Noise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle

Cyrill B. Muratov, Eric Vanden-Eijnden

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65 Scopus citations


A detailed asymptotic study of the effect of small Gaussian white noise on a relaxation oscillator undergoing a supercritical Hopf bifurcation is presented. The analysis reveals an intricate stochastic bifurcation leading to several kinds of noise-driven mixed-mode oscillations at different levels of amplitude of the noise. In the limit of strong time-scale separation, five different scaling regimes for the noise amplitude are identified. As the noise amplitude is decreased, the dynamics of the system goes from the limit cycle due to self-induced stochastic resonance to the coherence resonance limit cycle, then to bursting relaxation oscillations, followed by rare clusters of several relaxation cycles (spikes), and finally to small-amplitude oscillations (or stable fixed point) with sporadic single spikes. These scenarios are corroborated by numerical simulations.

Original languageEnglish (US)
Article number015111
Issue number1
StatePublished - 2008

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • General Physics and Astronomy
  • Applied Mathematics


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