The extended homotopy analysis method (EHAM) is presented to establish the analytical approximate solutions for two-degree-of-freedom (2-DOF) coupled van der Pol-Duffing oscillator. Meanwhile, the comparisons between the results of the EHAM and standard Runge-Kutta numerical method are also presented. The results demonstrate that the analytical approximate solutions of the EHAM agree well with the numerical integration solutions. For EHAM as an analytical approximation method, we are not sure whether it can apply to all of the nonlinear systems; we can only verify its effectiveness through specific cases. As a result of the existence of nonlinear terms, we must study different types of systems, no matter from the complication of calculation and physical significance.

Mathematical methods for the natural and engineering sciences problems have drawn considerable attention in recent years. In normal circumstances, most of the nonlinear dynamical models can be governed by a set of differential equations and auxiliary conditions from modeling processes [

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 [

In the present work, the exact analytical series solutions of the two-degree-of-freedom (2-DOF) coupled van der Pol-Duffing system are obtained by using the EHAM, and we also establish the comparisons between the results of the EHAM and standard Runge-Kutta numerical method. It is shown that the periodic solutions of the EHAM are in excellent agreement with the numerical integration ones, even if time

The MDOF dynamical system is considered by the following equation:

From (

In (

According to the fundamental concepts and working procedures of the HAM [

The operator

Setting

If the auxiliary linear operator, initial guess solution, auxiliary parameters

Differentiating the zeroth-order deformation equation (

The

In this section, we apply the EHAM for analysis of the two coupled van der Pol-Duffing oscillators:

We introduce a new variable

For the initial approximation,

We can define a nonlinear operator as the following by EHAM:

In terms of the principle of solution expression, we select the auxiliary functions as

For

While

Obviously, as the embedding parameter

With the help of the Taylor series expansion and (

If the auxiliary parameters

For simplicity, the following vectors are defined as

By differentiating the zeroth-order deformation equation (

Because of the principle of solution expression and the linear operator

The solutions of

In this section, numerical experiment is conducted to verify the accuracy of the present approach.

Taking

For simplicity and accuracy, we set

Comparison of the phase portrait curves of the forth-order approximation with the numerical integration method.

The initial conditions of the numerical integration method are

Moreover, the fifth-order analytical solutions (

The above results demonstrate that the system has an in-phase solution. While giving the initial approximations of

In this case, the comparison between the phase curves of the fifth-order approximation and the numerical integration solution is portrayed in Figure

Comparison between the phase curves of the fifth-order approximation and the numerical integration out-of-phase solution.

The initial conditions of the numerical integration method are

In the present paper, the EHAM approach is applied to get asymptotic analytical series solutions of 2-DOF van der Pol-Duffing oscillators with a nonlinear coupling. The basic idea described in this paper is expected to be more employed in solving other dynamical systems in engineering and physical sciences. Comparisons with the numerical results are presented to demonstrate the validity of this method. In summary, compared with some other methods, the EHAM has the following advantages.

The EHAM provides an ingenious avenue for controlling the convergences of approximation series. Numerical comparisons demonstrate that the EHAM is an effective and robust analytical method of 2-DOF van der Pol-Duffing oscillators.

Because of its flexibility, the present techniques can also be further generalized to analyze more complicated nonlinear MDOF dynamical systems that can only be analyzed by numerical methods.

The authors declare that there is no conflict of interests regarding the publication of this paper.

All the authors contributed equally and significantly to the writing of this paper. All the authors read and approved the final paper.

The author Y. H. Qian gratefully acknowledges the support of the National Natural Science Foundations of China (NNSFC) through Grants nos. 11202189 and 11304286 and the Natural Science Foundation of Zhejiang Province of China through Grant no. LY12A02002. The author S. M. Chen gratefully appreciates the financial support from the NNSFC through Grants no. 11371326. The author L. Shen gratefully acknowledges the support from open experiment project of Zhejiang Normal University. The authors are also grateful to the anonymous reviewers for their constructive comments and suggestions.