## 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. ẍ = (α+σ_{1}ξ_{1})x+βẋ+ax ^{3}+bx_{2}ẋ+ 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 extrem a 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 Itô 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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Pages (from-to) | 193-205 |

Number of pages | 13 |

Journal | American Society of Mechanical Engineers, Applied Mechanics Division, AMD |

Volume | 223 |

State | Published - Dec 1 1997 |

## Keywords

- Co-dimension two bifurcation
- Hopf bifurcation
- Stochastic Averaging
- Stochastic bifurcation

## ASJC Scopus subject areas

- Mechanical Engineering