A fast Galerkin method to obtain the periodic solutions of a nonlinear oscillator
The authors propose an algorithm to determine periodic solutions of nonlinear second-order equations of type \[ x''+ kx'+ (a_1x+ a_2x^2+ a_3x^3)= g(t),\tag{1} \] where \(k\) and \(a_j\) are given constants and \(g(t)\) is a \(T\)-periodic forcing term. Assuming the existence of a periodic solution of (1) with some period \(T'= pT\) and taking the \(m\)th order trigonometrical series approximation to this solution the coefficients of this expansion are determined by a Galerkin type method where the coefficients are approximated iteratively by means of a fast Fourier transform on equally distributed points on a period \([0,T']\). No proofs of the convergence of the proposed iterative schemes and the global errors of the overall process are given. Further, a stability analysis of periodic solutions for equations of this type is also presented. The original equation is linearized around the assumed periodic solution and by Floquet theory some stability conditions on the problem parameters can be derived. Finally, the authors present some numerical results for a Duffing nonlinear oscillator given by the equation. \(x''+ 0.4x'+ x- x^3= b\sin(0.65t)\) that show several families of stable and unstable periodic orbits for different ranges of the \(b\)-parameter.
scientific article; zbMATH DE number 1183153
scientific article