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Delay-induced stochastic bifurcations in a bistable system under white noise
Zhongkui Sun1, Jin Fu1, Yuzhu Xiao2
1Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China.
Chaos (Woodbury, N.Y.)
|September 3, 2015
Summary
This study investigates how noise and time delay affect stochastic bifurcations in a Duffing-Van der Pol oscillator. Results show time delay and noise intensity can induce bifurcations, impacting system dynamics.
Area of Science:
- Nonlinear Dynamics
- Stochastic Systems
- Control Theory
Background:
- The Duffing-Van der Pol oscillator is a fundamental model in nonlinear dynamics.
- Stochastic bifurcations are critical phenomena altering system behavior under random influences.
- Time delays can introduce complex dynamics and non-Markovian behavior in oscillators.
Purpose of the Study:
- To investigate the theoretical and numerical effects of noise and time delay on stochastic bifurcations.
- To analyze the influence of feedback intensity and noise intensity on system stability.
- To explore the relationship between time delay and stochastic bifurcation phenomena.
Main Methods:
- Theoretical analysis using approximate methods to derive the Fokker-Planck-Kolmogorov equation.
- Calculation of the stationary probability density function for the response amplitude.
- Numerical simulations to validate theoretical findings and confirm bifurcation behavior.
Main Results:
- Time delay and feedback intensity, along with noise intensity, were found to induce stochastic P-bifurcation.
- The system exhibits sensitive dependence on time delay, affecting stochastic bifurcation.
- Stochastic bifurcation is characterized by qualitative changes in the steady-state probability distribution.
Conclusions:
- Time delay significantly influences stochastic bifurcation phenomena in the Duffing-Van der Pol oscillator.
- The interplay between noise, time delay, and feedback intensity dictates system stability and bifurcation.
- Numerical simulations confirm the accuracy of the theoretical framework for analyzing these complex dynamics.
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