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

and Applied Analysis 3 where χ m = { 0, m ≤ 1 1, m > 1, R m (u m−1 , r, t) = 1 (m − 1)! ∂ m−1 N(Φ (r, t; p)) ∂p m−1 󵄨󵄨󵄨󵄨󵄨󵄨󵄨p=0 , Δ m (r, t) = 1 m! ∂ m Π(Φ (r, t; p)) ∂p m 󵄨󵄨󵄨󵄨󵄨󵄨󵄨p=0 . (14) Themth-order deformation equation (13) is a linear equation, which can be readily solved by the symbolic software such as Mathematica. 3. Application of the EHAM In this section, we apply the EHAM for analysis of the two coupled van der Pol-Duffing oscillators: ?̈? 1 + εη 1 (x 2 1 − 1) ?̇? 1 + x 1 + εα 1 x 3 1 + εδ 1 x 1 x 2 2 = 0, (15a) ?̈? 2 + εη 2 (x 2 2 − 1) ?̇? 2 + x 2 + εα 2 x 3 2 + εδ 2 x 2 x 2 1 = 0, (15b) where the superscript denotes the differentiation with respect to time t, x 1 (t) and x 2 (t) are the unknown real functions, and ε, α 1 , α 2 , η 1 , η 2 , δ 1 , and δ 2 are parameters. We introduce a new variable τ and substitute τ = ω t, x 1 (t) = u 1 (τ), and x 2 (t) = u 2 (τ) into (15a) and (15b). Therefore, we have ω 2 u 󸀠󸀠 1 + ωεη 1 (u 2 1 − 1) u 󸀠 1 + u 1 + εα 1 u 3 1 + εδ 1 u 1 u 2 2 = 0, (16a) ω 2 u 󸀠󸀠 2 + ωεη 2 (u 2 2 − 1) u 󸀠 2 + u 2 + εα 2 u 3 2 + εδ 2 u 2 u 2 1 = 0, (16b) subject to the initial conditions u 1 (0) = a, u 󸀠 1 (0) = b, u 2 (0) = c, u 󸀠 2 (0) = 0, (17) where a prime denotes the derivative with respect to variable τ. Provided that the periodic solutions in (16a) and (16b) can be expressed by a set of base functions {cos (kτ) , sin (kτ) | k = 0, 1, 2, . . .} , (18) one obtains


Introduction
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 [1].Numerous analytical methods have been developed to deal with the nonlinear differential equations, such as the modified perturbation methods [2][3][4][5], improved harmonic balance methods [6,7], energy balance method [8,9], and the frequency-amplitude formulation [10,11].Enlightening from the basic concepts of the homotopy in topology [12,13], Liao developed the homotopy analysis method (HAM) [14][15][16] which does not require small parameters as one of the efficient analytical techniques in solving a variety of nonlinear vibration problems.
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].More recently, Qian et al. [20] extended the HAM to deal with strongly nonlinear coupled van der Pol oscillators.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.By solving such example, it is illustrated that the present techniques are not an adhoc approach; it can be generalized to investigate more complicated nonlinear multi-degree-of-freedom (MDOF) dynamical 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  progresses to a certain large domain.In what follows, Section 2 presents the EHAM of the MDOF dynamical system.Moreover, the EHAM is presented to establish the analytical approximate solutions for 2-DOF coupled van der Pol-Duffing oscillator in the next section.In Section 4, numerical comparisons are carried out to authenticate the

The Extended Homotopy Analysis Method
The MDOF dynamical system is considered by the following equation: where  is an -dimensional unknown vector, a dot denotes the derivative with respect to time , , , and  are, respectively,  ×  mass, damping, and stiffness matrixes, and  is the vector function of q , , and .Let ( q , , ) ≡ 0; then (1) is an autonomous dynamical system.
From (1), we define a nonlinear operator as where (, ) is an unknown vector value function and  and  are spatial and temporal variables, respectively.In (2), the unknown vector functions of (, ), (, )/, and  2 (, )/ According to the fundamental concepts and working procedures of the HAM [1,2], the zeroth-order deformation equation can be constructed as follows: where  ∈ [0, 1] is an embedding parameter,  0 (, ) is the solution of initial guess,  is an auxiliary linear operator, and ℎ  and   () are the auxiliary parameters and the functions, respectively.The operator Π[Φ(, ; )] has the following property: When  = 0 and  = 1, the zeroth-order deformation equation ( 4) is Φ(, ; 0) =  0 (, ) and Φ(, ; 1) = (, ), respectively.Hence, as  increases from 0 to 1, the solution Φ(, ; ) varies from the initial guess solution  0 (, ) to the exact solution (, ).In this paper, where with the initial conditions Setting and expanding Φ(, ; ) into the Taylor series expansion with respect to  in accordance with the theorem of vector-valued function, we obtain If the auxiliary linear operator, initial guess solution, auxiliary parameters ℎ  , and auxiliary functions   () are properly chosen, the series equation (10) converges at  = 1, and we arrive at For brevity, the vector of u  is defined as Differentiating the zeroth-order deformation equation ( 4)  times with respect to  then dividing the equation by ! and setting  = 0 yield where The th-order deformation equation ( 13) is a linear equation, which can be readily solved by the symbolic software such as Mathematica.
With the help of the Taylor series expansion and (13), we obtain where If the auxiliary parameters ℎ 1 and ℎ 2 are properly chosen, the power series solutions in (29a), (29b), and (29c) are converged at  = 1.Then from (28), we get For simplicity, the following vectors are defined as By differentiating the zeroth-order deformation equation (24)  times with respect to , then dividing the equation by !, and setting  = 0, the th-order deformation equation is formulated as follows: with the initial conditions in which Figure 1: Comparison of the phase portrait curves of the forth-order approximation with the numerical integration method.( Because of the principle of solution expression and the linear operator , the right side of (33) should not contain the terms of sin  and cos  or the secular terms  sin  and  cos .The coefficients are set to be zero to yield The solutions of   ,   ,   , and   ( = 0, 1, 2,...) from ( 33) and (36a), (36b), (36c), and (36d) can be computed successively.To achieve more accurate results, we modify the solution of  as follows: where  is a small parameter.

Numerical Simulation and Discussion
In this section, numerical experiment is conducted to verify the accuracy of the present approach.
For simplicity and accuracy, we set ℎ 1 = −0.1,ℎ 2 = 300, and  = −0.1;then the comparison of the phase curves of the fifth-order approximation with the numerical integration solution is shown in Figure 1. Figure 2: Comparison between the phase curves of the fifth-order approximation and the numerical integration out-of-phase solution.
In this case, the comparison between the phase curves of the fifth-order approximation and the numerical integration solution is portrayed in Figure 2.

Conclusions
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.
(1) 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.