Sequent period-(2m − 1) motions to chaos in the van der Pol oscillator

Abstract

In this paper, independent, symmetric, periodic motions in a van der Pol oscillator are predicted through a semi-analytical method. This semi-analytic method is based on the discretization of the corresponding continuous nonlinear system for an implicit mapping. Through the implicit mapping structures, stable and unstable periodic motions are obtained analytically. A sequence of periodic motions to chaos via 1(S) ◁ 3(S) ◁ ··· ◁ (2m − 1)(S) ◁ ··· is discovered. The stability and bifurcations of periodic motions are determined through eigenvalue analysis. The frequency–amplitude characteristics of periodic motions are discussed. Numerical simulations of the periodic motions are carried out for comparison of numerical and analytical results. Such a periodic motion sequence is for a better understanding of dynamics of the van der Pol oscillator.