Quasi-periodic solutions: analytical and numerical investigations

Abstract

First, an analytical asymptotic method to construct quasi-periodic solutions in autonomous dynamical systems governed by a nonlinear second order set of ordinary differential equations with delay is presented. The approach is based on the double asymptotic expansion of two independent perturbation parameters and is supported by symbolic computation using Mathematica package. Both resonance and non-resonance cases are successfully analyzed and the catastrophes of the torus solutions are classified and discussed. Second, a new method for numerical calculations of the quasi-periodic orbits, which is based on a concept of the general Poincaré map, is addressed. In both cases considered examples support the introduced theory.