# American Institute of Mathematical Sciences

August  2012, 32(8): 2971-2995. doi: 10.3934/dcds.2012.32.2971

## Stochastic phase dynamics of noise driven synchronization of two conditional coherent oscillators

 1 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, V6T 1Z2, Canada, Canada 2 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2

Received  April 2011 Revised  September 2011 Published  March 2012

We consider a pair of uncoupled conditional oscillators near a subcritical Hopf bifurcation that are driven by two weak white noise sources, one intrinsic and one common. In this context the noise drives oscillations in a setting where the underlying deterministic dynamics are quiescent. Synchronization of these noise-driven oscillations is considered, where the noise is also driving synchronization. We first derive the envelope equations of the noise-driven oscillations using a stochastic multiple scales method, providing access to phase and amplitude information. Using both a linearized approximation and an asymptotic analysis of the nonlinear system, we obtain approximations for the probability density of the phase difference of the oscillators. It is found that common noise increases the degree of synchrony in the pair of oscillators, which can be characterized by the ratio of intrinsic to common noise. Asymptotic expressions for the phase difference density provide explicit parametric expressions for the probability of observing different phase dynamics: in-phase synchronization, phase shifted oscillations, and non-synchronized states. Computational results are provided to support analytical conclusions.
Citation: William F. Thompson, Rachel Kuske, Yue-Xian Li. Stochastic phase dynamics of noise driven synchronization of two conditional coherent oscillators. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2971-2995. doi: 10.3934/dcds.2012.32.2971
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