Chinese Journal of Theoretical and Applied Mechani ›› 2013, Vol. 45 ›› Issue (2): 257-264.DOI: 10.6052/0459-1879-12-169

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Hao Ying, Wu Zhiqiang   

  1. Department of Mechanics, Tianjin University, Tianjin 300072, China
  • Received:2012-06-04 Revised:2012-09-29 Online:2013-03-23 Published:2013-03-25
  • Supported by:
    The project was supported by the National Natural Science Foundation of China (11172198), Ministry of Education Found for the Dotroal (2009003211005) and the Program for New Century Excellent Talents in University (NCET-15-0247).

Abstract: This paper aims to investigate the stochastic P-bifurcations in the tri-stable Van der Pol-Duffing oscillator with additive and multiplicative Gauss noise. By using the stochastic averaging method, the stationary probability density function of amplitude is obtained. Then based on the singularity theory of the deterministic system, the explicit parameter conditions for P-bifurcation are deduced, and eleven types of qualitatively different probability density curves are founded. Finally the effects of three coe cients, one for linear damping and two for random excitation strength, are discussed. The results are verified by Monte-Carlo numerical simulations. The method used here is also suitable for other systems' P-bifurcation analysis.

Key words: stochastic P-bifurcation|probability density function|singularity theory

CLC Number: