An Optimal Analytic Approximate Solution for the Limit Cycle of Duffing–van der Pol Equation

The present paper is concerned with the accurate analytic solution of the limit cycle of the Duffing–van der Pol equation. Instead of the traditional Taylor series or asymptotic methods, the homotopy analysis technique is employed, which does not require a small perturbation parameter or a large asymptotic parameter. It is known that such a method is extremely powerful in gaining the exact solution of the physical problem in terms of purely trigonometric functions, yet the computational cost of the method is considerably high. We propose here an approach that not only greatly reduces the computational efforts but also presents an easy to implement task of application of the homotopy analysis method to the Duffing–van der Pol equation. The explicit analytical expressions obtained using the proposed approach generates the displacement, amplitude, and frequency of the limit cycle that compare excellently with the numerically computed ones.