Abstract

This paper aims to directly extend the Hamiltonian approach and coupled homotopy-variational formulation to study the periodic solutions of the Helmholtz-Duffing oscillator.The results of numerical example are presented and only a few terms are required to obtain accurate solutions. Results derived from this method are shown graphically. The behaviors of the solutions in the positive and negative directions are quite different, the asymmetric equation is separated into two auxiliary equations and the auxiliary equations are solved by two different method. Finally, the phase plane to show the stability of systems is plotted and discussed. An excellent agreement between the approximate periods with the exact period is achieved. The numerical results presented show that thesemethods are very accurate.

Key words: Helmholtz-Duffing oscillator; nonlinear oscillation, analytical methods, periodic solution.