Asymptotic analytical solutions of the two-degree-of-freedom strongly nonlinear van der Pol oscillators with cubic couple terms using extended homotopy analysis method

Abstract

This paper adopts the extended homotopy analysis method (EHAM) to obtain the asymptotic analytical series solutions of the strongly nonlinear two-degree-of-freedom (2DOF) van der Pol oscillators with cubic couple terms. It turns out that the oscillators occur essentially in only two variations: If the system has periodic solutions, then it either has only one solution which is out-of-phase (i.e., x1(t) = −x2(t)) or has two solutions that are not only in-phase (i.e., x1(t) = x2(t)) but also out-of-phase. Two examples, as two types of the problem have been raised, correspondingly. Employing the EHAM for those two problems, the explicit analytical solutions of frequency ω and displacements x1(t) and x2(t) are well formulated, but the conventional homotopy analysis method (HAM) can hardly do it if the cubic couple terms are complex. Thus, the EHAM is rather general. Moreover, the fifth-order analytical solutions are then compared with those derived from the established Runge–Kutta method in order to verify the accuracy and validity of this approach. It is shown that there is excellent agreement between the two sets of results, even if the time variable t progresses to a comparatively large domain in the time-history responses. Finally, the convergence theorem for the present method is also presented and discussed. All these results confirm that the EHAM can solve the presented problem successfully and completely, and that the EHAM will be a powerful and efficient tool for solving other multi-degree-of-freedom (MDOF) dynamical systems in engineering and physical sciences.