Series solutions of coupled Van der Pol equation by means of homotopy analysis method

Abstract

In this paper, the homotopy analysis method (HAM) is used to give series solutions of self-exited oscillation systems governed by two Van der Pol equations, which are coupled by a linear and a cubic term. The frequency and amplitude of all possible periodic solutions are investigated. It is found that there exist either in-phase or out-of-phase periodic solutions only. Besides, the in-phase periodic oscillations are decoupled, whose periods and amplitudes have nothing to do with the linear and cubic coupled terms. However, the out-of-phase periodic oscillations are strongly coupled, whose period and amplitude can be controlled by the linear and cubic coupled terms.

A single species population model in polluted environment solved by homotopy analysis method

“…Therefore, unlike the perturbation method, this method is independent of small parameters and can overcome the restrictions of the perturbation methods. The method has been used by many authors [5], [9] in a wide variety of scienti…c and engineering applications to solve di¤erent types of governing di¤erential equations. In this paper, the basic idea of the homotopy analysis method is introduced and then, the nonlinear equation of a model is solved through the homotopy analysis method.…”

Application of Extended Homotopy Analysis Method to the Two-Degree-of-Freedom Coupled van der Pol-Duffing Oscillator

“…Recently, great attention is paid to the discussion of coupled oscillators of nonlinear dynamical systems because most of practical engineering problems can be governed by such coupled systems [17][18][19]. The extended homotopy analysis method (EHAM) is one method based on the HAM envisioned first by Liao [16].…”

On the Homotopy Analysis Method for an Seir Tuberculosis Model

“…HAM is a general analytic technique developed for the purpose of obtaining approximate analytic series solutions to different types of nonlinear equations especially those with strong nonlinearity. This method has been successfully applied to solve many types of nonlinear problems arising in the field of science, engineering and finance [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. The HAM offers certain advantages over previous non-perturbative methods.…”