## Abstract

In this paper, we shall re-examine the stochastic version of the Duffing-van der Pol equation. As in [2], [9], [19], [22], we shall introduce a multiplicative and an additive stochastic excitation in our case, i.e. d^{2}x/dt^{2} = (α + σ_{1}ξ_{1})x + βdx/dt + ax^{3} + bx^{2}dx/dt + σ_{2}ξ_{2}, where, α and β are the bifurcation parameters, ξ_{1} and ξ_{2} are white noise processes with intensities σ_{1} and σ_{2}, respectively. The existence of the extrema of the probability density function is presented for the stochastic system. The method used in this paper is essentially the same as what has been used in [19]. We first reduce the above system to a weakly perturbed conservative system by introducing an appropriate scaling. The corresponding unperturbed system is then studied. Subsequently, by transforming the variables and performing stochastic averaging, we obtain a one-dimensional Ito equation of the Hamiltonian H. The probability density function is found by solving the Fokker-Planck equation. The extrema of the probability density function are then calculated in order to study the so-called P-Bifurcation. The bifurcation pictures for the stochastic version Duffing-van der Pol oscillator with a = -1.0, b = -1.0 over the whole (α, β)-plane are given. The related mean exit time problem has also been studied.

Original language | English (US) |
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Title of host publication | Active/Passive Vibration Control and Nonlinear Dynamics of Structures |

Publisher | ASME |

Pages | 193-205 |

Number of pages | 13 |

Volume | 95 |

State | Published - 1997 |

Event | Proceedings of the 1997 ASME International Mechanical Engineering Congress and Exposition - Dallas, TX, USA Duration: Nov 16 1997 → Nov 21 1997 |

### Other

Other | Proceedings of the 1997 ASME International Mechanical Engineering Congress and Exposition |
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City | Dallas, TX, USA |

Period | 11/16/97 → 11/21/97 |

## ASJC Scopus subject areas

- Engineering(all)