TY - JOUR
T1 - Bifurcation threshold of the delayed van der Pol oscillator under stochastic modulation
AU - Gaudreault, Mathieu
AU - Drolet, François
AU - Viñals, Jorge
PY - 2012/5/29
Y1 - 2012/5/29
N2 - We obtain the location of the Hopf bifurcation threshold for a modified van der Pol oscillator, parametrically driven by a stochastic source and including delayed feedback in both position and velocity. We introduce a multiple scale expansion near threshold, and we solve the resulting Fokker-Planck equation associated with the evolution at the slowest time scale. The analytical results are compared with a direct numerical integration of the model equation. Delay modifies the Hopf frequency at threshold and leads to a stochastic bifurcation that is shifted relative to the deterministic limit by an amount that depends on the delay time, the amplitude of the feedback terms, and the intensity of the noise. Interestingly, stochasticity generally increases the region of stability of the limit cycle.
AB - We obtain the location of the Hopf bifurcation threshold for a modified van der Pol oscillator, parametrically driven by a stochastic source and including delayed feedback in both position and velocity. We introduce a multiple scale expansion near threshold, and we solve the resulting Fokker-Planck equation associated with the evolution at the slowest time scale. The analytical results are compared with a direct numerical integration of the model equation. Delay modifies the Hopf frequency at threshold and leads to a stochastic bifurcation that is shifted relative to the deterministic limit by an amount that depends on the delay time, the amplitude of the feedback terms, and the intensity of the noise. Interestingly, stochasticity generally increases the region of stability of the limit cycle.
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U2 - 10.1103/PhysRevE.85.056214
DO - 10.1103/PhysRevE.85.056214
M3 - Article
AN - SCOPUS:84861925812
SN - 1539-3755
VL - 85
JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
IS - 5
M1 - 056214
ER -