Symmetry-breaking bifurcation analysis of stochastic van der Pol system via Chebyshev polynomial approximation
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.
scientific article; zbMATH DE number 5080137