Symmetry-breaking bifurcation analysis of stochastic van der Pol system via Chebyshev polynomial approximation
type: Item
instance of: scholarly article
title: Symmetry-breaking bifurcation analysis of stochastic van der Pol system via Chebyshev polynomial approximation
zbMATH Open document ID: 1120.34041
author: Wei Xu
author: Yanfei Jin
author: W. Li
author: T. Fang
author: Shao-juan Ma
published in: Communications in Nonlinear Science and Numerical Simulation
publication date: 2006-12-14T00:00:00Z
review text: The authors study the bifurcation phenomena of the stochastic van der Pol equation \[ \ddot{x} + b x^2 \dot{x} + c \dot{x} + x = F \cos \omega t, \] where \(b = \bar{b} + \delta u \) is a bounded random parameter with mean \(\bar{b}\), standard deviation \(\delta\) and standardization \(u\) distributed on \([-1, 1]\) according to the law \(d \mu = p(u) \, du = \frac{\pi}{2} \sqrt{1 - u^2}\, du\). The authors expand the solution \(x(t,u)\) in \(L^2( [-1, 1], \, {\mathcal B}([-1,1]), \, \mu)\) with help of the orthonormal system of the Chebychev polynomials \(U_n(u)\), \(n= 0,1,2,\dots\), that is \[ x(t,u) = \sum_{n=0}^{\infty} x_n(t) U_n(u), \quad x_n(t) = \int_{-1}^1 x(t,u) U_n(u) p(u) \, d \mu \] and derive the equivalent system of countably many nonlinear deterministic equations for the coefficients \(x_n(t)\), approximated by \(N\) such equations. They show numerically that this approximated deterministic nonlinear system exhibits the same bifurcation phenomena that appear with the original stochastic van der Pol equation, if the the frequency \(\omega\) and the amplitude \(F\) of the harmonic excitation are taken as bifurcation parameters.
reviewed by: Volker Wihstutz
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H35
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 5080137
zbMATH Keywords: bifurcation of the van der Pol equation
MaRDI profile type: Publication
full work available at URL: j.cnsns.2005.03.004
OpenAlex ID: W2094174318
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DOI: 10.1016/J.CNSNS.2005.03.004
description: scientific article; zbMATH DE number 5080137