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Bifurcation threshold of the delayed van der Pol oscillator under stochastic modulation
Mathieu Gaudreault1, François Drolet, Jorge Viñals
1Department of Physics, McGill University, Montréal, Québec, Canada.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
Summary
We analyzed a modified van der Pol oscillator with stochastic forcing and delayed feedback. Stochasticity generally enhances the stability of the limit cycle, shifting the Hopf bifurcation threshold.
Area of Science:
- Nonlinear Dynamics
- Stochastic Systems
- Control Theory
Background:
- The van der Pol oscillator is a classic model for self-sustained oscillations.
- Delayed feedback and stochastic forcing are common in real-world systems, influencing system stability.
- Hopf bifurcations mark the onset of oscillations in dynamical systems.
Purpose of the Study:
- To determine the Hopf bifurcation threshold for a modified van der Pol oscillator with stochastic parametric driving and delayed feedback.
- To investigate the influence of delay and noise on the bifurcation dynamics.
- To analytically and numerically characterize the system's stability regions.
Main Methods:
- Multiple scale expansion near the Hopf bifurcation threshold.
- Solving the Fokker-Planck equation for the slowest time scale evolution.
- Direct numerical integration of the nonlinear delayed stochastic differential equation.
Main Results:
- The Hopf bifurcation threshold is analytically derived and numerically verified.
- Delayed feedback modifies the Hopf frequency and shifts the bifurcation point.
- The shift in bifurcation depends on delay time, feedback amplitude, and noise intensity.
- Stochasticity generally expands the stable limit cycle region.
Conclusions:
- Delayed feedback and stochastic noise significantly alter the bifurcation behavior of the van der Pol oscillator.
- The interplay between delay and noise determines the system's stability characteristics.
- The findings provide insights into the dynamics of complex oscillatory systems with feedback and noise.
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