Noise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle
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Updated: Jul 6, 2026

Fabrication and Testing of Microfluidic Optomechanical Oscillators
Published on: May 29, 2014
Cyrill B Muratov1, Eric Vanden-Eijnden
1Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA.
Abstract:
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.
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