P-bifurcations in the stochastic version of the Duffing-van der Pol equation

Yan Liang, N. Sri Namachchivaya

Research output: Contribution to journalArticlepeer-review


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+σ1ξ1)x+βẋ+ax 3+bx2ẋ+ σ2ξ2, where, α and β are the bifurcation parameters, ξ1and ξ2 are white noise processes with intensities σ1and σ2, 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 languageEnglish (US)
Pages (from-to)193-205
Number of pages13
JournalAmerican Society of Mechanical Engineers, Applied Mechanics Division, AMD
StatePublished - 1997


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

ASJC Scopus subject areas

  • Mechanical Engineering


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