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

N. Sri Namachchivaya, Yan Liang

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 [3], [4], [5], [6], we shall introduce a multiplicative and an additive stochastic excitation in our case, i.e. ẍ = (α + σ1ξ1)x + βẋ + ax3 + bx2ẋ + σ2ξ2 (1) where, α and β are the bifurcation parameters, ξ1 and ξ2 are white noise processes with intensities σ1 and σ2 respectively. The method used in this paper is essentially the same as what has been used in [4]. We first reduce system (1) to a weakly perturbed conservative system by intruducing an appropriate rescaling. The corresponding unperturbed system is then studied. The problem of the existence of the extrema of the probability density function is presented for the stochastic system. Second, by transforming the variables and performing stochastic averaging, we obtain a one-dimensional Itô equation. The probability density function is found by solving the Fokker-Planck equation. The extrema of the probability density function are then culculated so we can study the so called P-bifurcation for the Duffing-van der Pol oscillator with a = -1.0, b = -1.0 over the whole (α, β)-plane by making use of the system Hamiltonian.

Original languageEnglish (US)
Pages (from-to)231-234
Number of pages4
JournalZAMM Zeitschrift fur Angewandte Mathematik und Mechanik
Issue numberSUPPL. 3
StatePublished - Dec 1 1996

ASJC Scopus subject areas

  • Computational Mechanics
  • Applied Mathematics


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