About this Journal Submit a Manuscript Table of Contents
Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 239357, 32 pages
http://dx.doi.org/10.1155/2012/239357
Research Article

Application of the Hori Method in the Theory of Nonlinear Oscillations

Departamento de Matemática, Instituto Tecnológico de Aeronáutica, 12228-900, São José dos Campos, SP, Brazil

Received 12 November 2011; Accepted 25 January 2012

Academic Editor: Antonio F. Bertachini A. Prado

Copyright © 2012 Sandro da Silva Fernandes. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Some remarks on the application of the Hori method in the theory of nonlinear oscillations are presented. Two simplified algorithms for determining the generating function and the new system of differential equations are derived from a general algorithm proposed by Sessin. The vector functions which define the generating function and the new system of differential equations are not uniquely determined, since the algorithms involve arbitrary functions of the constants of integration of the general solution of the new undisturbed system. Different choices of these arbitrary functions can be made in order to simplify the new system of differential equations and define appropriate near-identity transformations. These simplified algorithms are applied in determining second-order asymptotic solutions of two well-known equations in the theory of nonlinear oscillations: van der Pol equation and Duffing equation.

1. Introduction

In da Silva Fernandes [1], the general algorithm proposed by Sessin [2] for determining the generating function and the new system of differential equations of the Hori method for noncanonical systems has been revised considering a new approach for the integration theory which does not depend on the auxiliary parameter 𝑡 introduced by Hori [3, 4].

In this paper, this new approach is applied to the theory of nonlinear oscillations for a second-order differential equation and two simplified versions of the general algorithm are derived. The first algorithm is applied to systems of two first-order differential equations corresponding to the second-order differential equation, and the second algorithm is applied to the equations of variation of parameters associated with the original equation. According to these simplified algorithms, the determination of the unknown functions 𝑇𝑗(𝑚) and 𝑍𝑗(𝑚), defined in the 𝑚th-order equation of the algorithm of the Hori method, is not unique, since these algorithms involve at each order arbitrary functions of the constants of integration of the general solution of the new undisturbed system. Different choices of the arbitrary functions can be made in order to simplify the new system of differential equations and define appropriate near-identity transformations. The problem of determining second-order asymptotic solutions of two well-known equations in the theory of nonlinear oscillations—van der Pol and Duffing equations—is taken as example of application of the simplified algorithms. For van der Pol equation, two generating functions are determined: one of these generating functions is the same function obtained by Hori [4], and, the other function provides the well-known averaged equations of variation of parameters in the theory of nonlinear oscillations. For Duffing equation, only one generating function is determined, and the second-order asymptotic solution is the same solution obtained through Krylov-Bogoliubov method [5], through the canonical version of Hori method [6] or through a different integration theory for the noncanonical version of Hori method [7]. For completeness, brief descriptions of the noncanonical version of the Hori method [4] and the general algorithm proposed by Sessin [2] are presented in the next two sections.

2. Hori Method for Noncanonical Systems

The noncanonical version of the Hori method [4] can be briefly described as follows.

Consider the differential equations:𝑑𝑧𝑗𝑑𝑡=𝑍𝑗(𝑧,𝜀),𝑗=1,,𝑛,(2.1) where 𝑍𝑗(𝑧,𝜀), 𝑗=1,,𝑛, are the elements of the vector function 𝑍(𝑧,𝜀). It is assumed that 𝑍(𝑧,𝜀) is expressed in power series of a small parameter 𝜀: 𝑍(𝑧,𝜀)=𝑍(0)(𝑧)+𝑚=1𝜀𝑚𝑍(𝑚)(𝑧).(2.2) The system of differential equations described by 𝑍(0)(𝑧) is supposed to be solvable.

Let the transformation of variables (𝑧1,,𝑧𝑛)(𝜁1,,𝜁𝑛) be generated by the vector function 𝑇(𝜁,𝜀). This transformation of variables is such that the new system:𝑑𝜁𝑗𝑑𝑡=𝑍𝑗(𝜁,𝜀),𝑗=1,,𝑛,(2.3) is easier to solve or captures essential features of the system. 𝑍𝑗(𝜁,𝜀), 𝑗=1,,𝑛, are the elements of the vector function 𝑍(𝜁,𝜀), also expressed in power series of 𝜀:𝑍(𝜁,𝜀)=𝑍(0)(𝜁)+𝑚=1𝜀𝑚𝑍(𝑚)(𝜁).(2.4) It is assumed that the vector function 𝑇(𝜁,𝜀), that defines a near-identity transformation, is also expressed in powers series of 𝜀:𝑇(𝜁,𝜀)=𝑚=1𝜀𝑚𝑇(𝑚)(𝜁).(2.5)

Following Hori [4], the transformation of variables (𝑧1,,𝑧𝑛)(𝜁1,,𝜁𝑛) generated by 𝑇(𝜁,𝜀) is given by𝑧𝑗=𝜁𝑗+𝑘=11𝐷𝑘!𝑘𝑇𝜁𝑗=𝑒𝐷𝑇𝜁𝑗,𝑗=1,,𝑛.(2.6) For an arbitrary function 𝑓(𝑧), the expansion formula is given by𝑓(𝑧)=𝑓(𝜁)+𝑘=11𝐷𝑘!𝑘𝑇𝑓(𝜁)=𝑒𝐷𝑇𝑓(𝜁).(2.7) The operator 𝐷𝑇 is defined by𝐷𝑇𝑓(𝜁)=𝑛𝑗=1𝑇𝑗𝜕𝑓𝜕𝜁𝑗,𝐷𝑛𝑇𝑓(𝜁)=𝐷𝑇𝑛1𝑛𝑗=1𝑇𝑗𝜕𝑓𝜕𝜁𝑗.(2.8)

According to the algorithm of the perturbation method proposed by Hori [4], the vector functions 𝑍 and 𝑇 are obtained, at each order in the small parameter 𝜀, from the following equations:order0:𝑍𝑗(0)=𝑍𝑗(0),𝑍(2.9)order1:(0),𝑇(1)𝑗+𝑍𝑗(1)=𝑍𝑗(1),𝑍(2.10)order2:(0),𝑇(2)𝑗+12𝑍(1)+𝑍(1),𝑇(1)𝑗+𝑍𝑗(2)=𝑍𝑗(2),(2.11)𝑗=1,,𝑛, where []𝑗 stands for the generalized Poisson brackets []𝑍,𝑇𝑗=𝑛𝑘=1𝑇𝑘𝜕𝑍𝑗𝜕𝜁𝑘𝑍𝑘𝜕𝑇𝑗𝜕𝜁𝑘.(2.12)𝑍(0), 𝑍(𝑚), 𝑍(𝑚), and 𝑇(𝑚) are written in terms of the new variables 𝜁1,,𝜁𝑛.

The 𝑚th-order equation of the algorithm can be put in the general form:𝑍(0),𝑇(𝑚)𝑗+Ψ𝑗(𝑚)=𝑍𝑗(𝑚),𝑗=1,,𝑛,(2.13) where the functions Ψ𝑗(𝑚) are obtained from the preceding orders.

3. The General Algorithm

The determination of the functions 𝑍𝑗(𝑚) and 𝑇𝑗(𝑚) from (2.13) is based on the following proposition presented in da Silva Fernandes [1].

Proposition 3.1. Let 𝐹 be a 𝑛×1 vector function of the variables 𝜁1,,𝜁𝑛, which satisfy the system of differential equations: 𝑑𝜁𝑗𝑑𝑡=𝑍𝑗(0)(𝜁)+𝑅𝑗(𝜁;𝜀),𝑗=1,,𝑛,(3.1) where 𝑍(0) describes an integrable system of differential equations: 𝑑𝜁𝑗𝑑𝑡=𝑍𝑗(0)(𝜁),𝑗=1,,𝑛,(3.2) a general solution of which is given by 𝜁𝑗=𝜁𝑗𝑐1,,𝑐𝑛,𝑡,𝑗=1,,𝑛,(3.3) being 𝑐1,,𝑐𝑛 arbitrary constants of integration; then 𝐹,𝑍(0)𝑗=𝜕𝐹𝑗𝜕𝑡𝑛𝑘=1𝜕𝑍𝑗(0)𝜕𝜁𝑘𝐹𝑘,𝑗=1,,𝑛.(3.4)

A corollary of this proposition can be stated.

Corollary 3.2. Consider the same conditions of Proposition 3.1 with the general solution of (3.2) given by 𝜁𝑗=𝜁𝑗𝑐1,,𝑐𝑛1,𝑀,𝑗=1,,𝑛,(3.5) being 𝑐1,,𝑐𝑛1 arbitrary constants of integration and 𝑀=𝑡+𝜏, where τ is an additive constant; then 𝐹,𝑍(0)𝑗=𝜕𝐹𝑗𝜕𝑀𝑛𝑘=1𝜕𝑍𝑗(0)𝜕𝜁𝑘𝐹𝑘,𝑗=1,,𝑛.(3.6)

Now, consider (2.13). According to Proposition 3.1, this equation can be put in the form: 𝜕𝑇𝑗(𝑚)𝜕𝑡𝑛𝑘=1𝜕𝑍𝑗(0)𝜕𝜁𝑘𝑇𝑘(𝑚)=Ψ𝑗(𝑚)𝑍𝑗(𝑚),𝑗=1,,𝑛,(3.7) with Ψ𝑗(𝑚) written in terms of the general solution (3.3) of the undisturbed system (3.2), involving 𝑛 arbitrary constants of integration—𝑐1,,𝑐𝑛. 𝑍𝑗(𝑚) and 𝑇𝑗(𝑚) are unknown functions.

Equation (3.7) is very similar to the one presented by Hori [4], which is written in terms of an auxiliary parameter 𝑡 through an ordinary differential equation, that is, 𝑑𝑇𝑗(𝑚)𝑑𝑡𝑛𝑘=1𝜕𝑍𝑗(0)𝜕𝜁𝑘𝑇𝑘(𝑚)=Ψ𝑗(𝑚)𝑍𝑗(𝑚),𝑗=1,,𝑛.(3.8)

To determine 𝑍𝑗(𝑚) and 𝑇𝑗(𝑚), 𝑗=1,,𝑛, Hori [4] extends the averaging principle applied in the canonical version: 𝑍𝑗(𝑚) are determined so that the 𝑇𝑗(𝑚) are free from secular or mixed secular terms. However, this procedure is not sufficient to determine 𝑍 such that the new system of differential equations (2.3) becomes more tractable, and, a tractability condition is imposed𝑍(0),𝑍𝑗=0,𝑗=1,,𝑛.(3.9) This condition is analogous to the condition {𝐹(0),𝐹}=0 in the canonical case, which provides the first integral 𝐹0(𝜉,𝜂)=const, where 𝜉,𝜂 denotes the new set of canonical variables, and, 𝐹(0) and 𝐹 are the undisturbed Hamiltonian and the new Hamiltonian, respectively, and {} stands for Poisson brackets [3, 4].

In the next paragraphs, the general algorithm for determining 𝑍𝑗(𝑚) and 𝑇𝑗(𝑚), 𝑗=1,,𝑛, proposed by Sessin [2] and revised in da Silva Fernandes [1] is briefly presented.

Introducing the 𝑛×1 matrices: 𝑇(𝑚)=𝑇𝑗(𝑚),Ψ(𝑚)=Ψ𝑗(𝑚),𝑍(𝑚)=𝑍𝑗(𝑚),𝑗=1,,𝑛,(3.10) and the 𝑛×𝑛 Jacobian matrix𝐽(𝑡)=𝜕𝑍𝑗(0)𝜕𝜁𝑘,𝑗,𝑘=1,,𝑛,(3.11)

the system of partial differential equations (3.7) can be put in the following matrix form:𝜕𝑇(𝑚)𝜕𝑡𝐽(𝑡)𝑇(𝑚)=Ψ(𝑚)𝑍(𝑚).(3.12)

The vector functions 𝑍(𝑚) and 𝑇(𝑚) are determined from the following equations: 𝑍(𝑚)=Δ𝜁𝜕Δ𝜕𝑡𝜁1Δ𝜁𝐷(𝑚)+Δ𝜁Δ𝜁1Ψ(𝑚)𝑑𝑡𝑠,𝑇(3.13)(𝑚)=Δ𝜁𝐷(𝑚)+Δ𝜁Δ𝜁1Ψ(𝑚)𝑑𝑡𝑝,(3.14) where Δ𝜁=[𝜕𝜁𝑗(𝑐1,,𝑐𝑛,𝑡)/𝜕𝑐𝑘] is the Jacobian matrix associated to the general solution (3.3) of the undisturbed system (3.2), 𝑠 denotes the secular or mixed secular terms, and 𝑝 denotes the remaining part. 𝐷(𝑚) is the 𝑛×1 vector, 𝐷(𝑚)=(𝐷𝑗(𝑚)), which depends only on the arbitrary constants of integration 𝑐1,,𝑐𝑛 of the general solution (3.3). The choice of 𝐷(𝑚) is arbitrary. Recall that in the integration process, the arbitrary constants of integration of the general solution (3.3) are taken as parameters.

Equations (3.13) and (3.14) assure that 𝑇(𝑚) is free from secular or mixed secular terms. Moreover, these equations provide the tractability condition (3.9) as it will be shown in the case of nonlinear oscillation problems presented in the next section.

Finally, it should be noted that 𝐷(𝑚) can be chosen at each order to simplify the generating function 𝑇 and the new system of differential equations (2.3). This aspect is discussed thoroughly in the examples of Section 5.

4. Simplified Algorithms in the Theory of Nonlinear Oscillations

In this section two simplified algorithms will be derived from the general algorithm in the case of nonlinear oscillations described by a second-order differential equation of the general form:̈𝑥+𝑥=𝜀𝑓(𝑥,̇𝑥).(4.1)

The first algorithm is applied to the system of first-order differential equations with 𝑥 and ̇𝑥 as elements of the vector 𝑧, and, the second algorithm is applied to the system of equations of variation of parameters associated to the differential equation with 𝑐 and 𝜃 as elements of the vector 𝑧; 𝑐 and 𝜃are defined in (4.26).

4.1. Simplified Algorithm I

For completeness, we present now the first simplified algorithm [1]. Additional remarks are included at the end of section.

Introducing the variables 𝑧1=𝑥 and 𝑧2=̇𝑥, (4.1) can be put in the form:̇𝑧1=𝑧2,̇𝑧2=𝑧1𝑧+𝜀𝑓1,𝑧2.(4.2) According to the notation introduced in (2.1) and (2.2):𝑍(0)=𝑧2𝑧1,𝑍(1)=0𝑓𝑧1,𝑧2.(4.3)

Following the algorithm of the Hori method for noncanonical systems, one finds from zero-th-order equation, (2.9), that𝑍(0)=𝜁2𝜁1.(4.4)

Applying Proposition 3.1, it follows that the undisturbed system (3.2) is given bẏ𝜁1=𝜁2,̇𝜁2=𝜁1,(4.5) general solution of which can be written in terms of the exponential matrix as [2]:𝜁1𝜁2=𝑒𝐸𝑡𝑐1𝑐2,(4.6) where𝑒𝐸𝑡=cos𝑡sin𝑡sin𝑡cos𝑡,(4.7) and 𝐸 is the symplectic matrix: 𝐸=0110,(4.8) and 𝑐𝑖, 𝑖=1,2, are constants of integration. The Jacobian matrix Δ𝜁 associated with the solution (4.6) is then given byΔ𝜁=𝑒𝐸𝑡,(4.9) with inverse Δ𝜁1=𝑒𝐸𝑡=Δ𝑇𝜁, since Δ𝜁 is an orthogonal matrix.

In view of (4.6), the functions Ψ𝑗(𝑚) defined at each order of the algorithm (see (2.13) or (3.7)) are expressed by Fourier series with multiples of 𝑡 as arguments such that the vector function Ψ𝐼(𝑚) can be written asΨ𝐼(𝑚)=𝑘=0𝑎(𝑚)𝑘,𝐼cos𝑘𝑡+𝑏(𝑚)𝑘,𝐼𝑐sin𝑘𝑡(𝑚)𝑘,𝐼cos𝑘𝑡+𝑑(𝑚)𝑘,𝐼sin𝑘𝑡,(4.10) where the coefficients 𝑎(𝑚)𝑘,𝐼, 𝑏(𝑚)𝑘,𝐼, 𝑐(𝑚)𝑘,𝐼, and 𝑑(𝑚)𝑘,𝐼 are functions of the constants 𝑐𝑖, 𝑖=1,2. The Fourier series can also be put in matrix form:Ψ𝐼(𝑚)=𝑘=0𝑒𝐸𝑘𝑡𝐴(𝑚)𝑘,𝐼+𝑒𝐸𝑘𝑡𝐵(𝑚)𝑘,𝐼,(4.11) with𝐴(𝑚)𝑘,𝐼=12𝑎(𝑚)𝑘,𝐼𝑑(𝑚)𝑘,𝐼𝑏(𝑚)𝑘,𝐼+𝑐(𝑚)𝑘,𝐼,𝐵(𝑚)𝑘,𝐼=12𝑎(𝑚)𝑘,𝐼+𝑑(𝑚)𝑘,𝐼𝑐(𝑚)𝑘,𝐼𝑏(𝑚)𝑘,𝐼.(4.12) The subscript 𝐼 is introduced to denote the first simplified algorithm.

Substituting (4.9) and (4.11) into (3.13), one finds𝑍𝐼(𝑚)=𝑒𝐸𝑡𝜕𝑒𝜕𝑡𝐸𝑡𝑡𝑒𝐸𝑡𝐴(𝑚)1,𝐼+periodicterms𝑠=𝑒𝐸𝑡𝐴(𝑚)1,𝐼.(4.13)

On the other hand,𝑒𝐸𝑡Ψ𝐼(𝑚)=𝐴(𝑚)1,𝐼,(4.14) where stands for the mean value of the function.

Therefore, from (4.9), (4.13), and (4.14), it follows that𝑍𝐼(𝑚)=𝑒𝐸𝑡𝑒𝐸𝑡Ψ𝐼(𝑚)=Δ𝜁Δ𝜁1Ψ𝐼(𝑚).(4.15)

The second equation of the general algorithm, (3.14) can be simplified as described bellow.

From (3.14), (4.14), and (4.15), one finds𝑡𝑒𝐸𝑡𝐴(𝑚)1,𝐼=Δ𝜁Δ𝜁1Ψ𝐼(𝑚)Δ𝑑𝑡=𝜁𝐷𝐼(𝑚)+Δ𝜁Δ𝜁1Ψ𝐼(𝑚)𝑑𝑡𝑠.(4.16)

On the other hand,Δ𝜁𝐷𝐼(𝑚)+Δ𝜁Δ𝜁1Ψ𝐼(𝑚)Δ𝑑𝑡=𝜁𝐷𝐼(𝑚)+Δ𝜁Δ𝜁1Ψ𝐼(𝑚)𝑑𝑡𝑝+𝑡𝑒𝐸𝑡𝐴(𝑚)1,𝐼.(4.17)

Thus, introducing (4.16) and (4.17) into (3.14), one finds𝑇𝐼(𝑚)=Δ𝜁𝐷𝐼(𝑚)+Δ𝜁Δ𝜁1Ψ𝐼(𝑚)𝑑𝑡𝑝=Δ𝜁𝐷𝐼(𝑚)+Δ𝜁Δ𝜁1Ψ𝐼(𝑚)Δ𝜁1Ψ𝐼(𝑚)𝑑𝑡.(4.18)

Equations (4.15) and (4.18) define the first simplified form of the general algorithm applicable to the nonlinear oscillations problems described by (4.1) with 𝑥 and ̇𝑥 as elements of vector 𝑧.

Finally, we note that (4.15) satisfies the tractability condition (3.9) up to order 𝑚. In order to show this equivalence, one proceeds as follows. Since, from (4.14), Δ𝜁1Ψ𝐼(𝑚) does not depend explicitly on the time 𝑡, it follows that𝜕𝑍(𝑚)=𝜕𝑡𝜕Δ𝜁Δ𝜕𝑡𝜁1Ψ𝐼(𝑚)=𝐽Δ𝜁Δ𝜁1Ψ𝐼(𝑚)=𝐽𝑍(𝑚).(4.19) Using Proposition 3.1 and taking into account that 𝐽=𝜕𝑍𝑗(0)/𝜕𝜁𝑘, this equation can be put in the following form:𝑍(𝑚),𝑍(0)𝑗=𝜕𝑍𝑗(𝑚)𝜕𝑡𝑛𝑘=1𝜕𝑍𝑗(0)𝜕𝜁𝑘𝑍𝑘(𝑚)=0,𝑗=1,2,(4.20) which is the tractability condition (3.9) up to order 𝑚.

Remark 4.1. It should be noted that (4.15) and (4.18) for determining the vector functions 𝑍𝐼(𝑚) and 𝑇𝐼(𝑚), respectively, are invariant with respect to the form of the general solution of the undisturbed system described by 𝑍(0). This means that if the general solution of the undisturbed system is written in terms of a second set of constants of integration, for instance, if this solution is given by 𝜁1𝜁=𝑐cos(𝑡+𝜃),2=𝑐sin(𝑡+𝜃),(4.21) where 𝑐 and 𝜃 denote new constants of integration, then 𝑍𝐼(𝑚) and 𝑇𝐼(𝑚) are determined through (4.15) and (4.18), with the Jacobian matrix Δ𝜁 given by Δ𝜁=cos(𝑡+𝜃)𝑐sin(𝑡+𝜃)sin(𝑡+𝜃)𝑐cos(𝑡+𝜃).(4.22) This result can be proved as follows. The two sets of constants of integration (𝑐1,𝑐2) and (𝑐,𝜃) are related through the following transformation: 𝑐2=𝑐21+𝑐22,𝑐tan𝜃=2𝑐1.(4.23) In view of this transformation, the Jacobian matrix Δ1𝜁 can be written in terms of the Jacobian matrix Δ2𝜁 as Δ1𝜁=Δ2𝜁Δ𝐶,(4.24) where the superscripts 1 and 2 are introduced to denote the form of the general solution of the undisturbed system described by 𝑍(0) with respect to the set of constants of integration (𝑐1,𝑐2) and (𝑐,𝜃), respectively. Δ𝐶 is the Jacobian matrix of the transformation. Since Δ𝐶 does not depend on the time 𝑡, it follows from (4.15) and (4.18) that 𝑍𝐼(𝑚)=Δ1𝜁Δ1𝜁1Ψ1𝐼(𝑚)=Δ2𝜁Δ𝐶Δ𝐶1Δ2𝜁1Ψ2𝐼(𝑚)=Δ2𝜁Δ𝐶Δ𝐶1Δ2𝜁1Ψ2𝐼(𝑚)=Δ2𝜁Δ2𝜁1Ψ2𝐼(𝑚),𝑇𝐼(𝑚)=Δ1𝜁𝐷1𝐼(𝑚)+Δ1𝜁Δ1𝜁1Ψ1𝐼(𝑚)Δ1𝜁1Ψ1𝐼(𝑚)𝑑𝑡=Δ2𝜁Δ𝐶𝐷1𝐼(𝑚)+Δ2𝜁Δ𝐶Δ𝐶1Δ2𝜁1Ψ2𝐼(𝑚)Δ𝐶1Δ2𝜁1Ψ2𝐼(𝑚)𝑑𝑡=Δ2𝜁Δ𝐶𝐷1𝐼(𝑚)+Δ2𝜁Δ𝐶Δ𝐶1Δ2𝜁1Ψ2𝐼(𝑚)Δ2𝜁1Ψ2𝐼(𝑚)𝑑𝑡=Δ2𝜁𝐷2𝐼(𝑚)+Δ2𝜁Δ2𝜁1Ψ2𝐼(𝑚)Δ2𝜁1Ψ2𝐼(𝑚)𝑑𝑡.(4.25)
Finally, we note that the general solution given by (4.21) is more suitable in practical applications than the general solution given by (4.6), that, in turn, is more suitable for theoretical purposes.

4.2. Simplified Algorithm II

In this section, a second simplified algorithm is derived from the general one. Introducing the transformation of variables (𝑥,̇𝑥)(𝑐,𝜃) defined by the following equations𝑥=𝑐cos𝑡+𝜃,̇𝑥=𝑐sin𝑡+𝜃,(4.26) equation (4.1) is transformed into𝑑𝑐𝑐𝑑𝑡=𝜀𝑓cos(𝑡+𝜃),𝑐sin(𝑡+𝜃)sin(𝑡+𝜃),𝑑𝜃1𝑑𝑡=𝜀𝑓𝑐𝑐cos(𝑡+𝜃),𝑐sin(𝑡+𝜃)cos(𝑡+𝜃).(4.27) These differential equations are the well-known variation of parameters equations associated to the second-order differential equation (4.1). Equation (4.27) define a nonautonomous system of differential equations.

The sets (𝑐,𝜃) and (𝑐,𝜃), defined, respectively in (4.21) and (4.26), have different meanings in the theory: in (4.21), 𝑐 and 𝜃 are constants of integration of the general solution of the new undisturbed system described by 𝑍(0)(𝜁1,𝜁2); in (4.26), 𝑐 and 𝜃 are new variables which represent the constants of integration of the general solution of the original undisturbed system described by 𝑍(0)(𝑧1,𝑧2) in the variation of parameter method. These sets, (𝑐,𝜃) and (𝑐,𝜃), are connected through a near identity transformation.

Remark 4.2. It should be noted that a second transformation of variables involving a fast phase, (𝑥,̇𝑥)(𝑐,𝜙), can also be defined. This second transformation is given by 𝑥=𝑐cos𝜙,̇𝑥=𝑐'sin𝜙.(4.28) In this case, (4.1) is transformed into 𝑑𝑐𝑑𝑡=𝜀𝑓(𝑐'cos𝜙,𝑐'sin𝜙)sin𝜙,𝑑𝜙1𝑑𝑡=1𝜀𝑐𝑓(𝑐'cos𝜙,𝑐'sin𝜙)cos𝜙.(4.29) These equations define an autonomous system of differential equations. In what follows, the first set of variation of parameters equations, (4.27), will be considered.
Now, introducing the variables 𝑧1=𝑐 and 𝑧2=𝜃, one gets from (4.27) that 𝑍(0)=00,𝑍(1)=𝑧𝑓1cos𝑡+𝑧2,𝑧1sin𝑡+𝑧2sin𝑡+𝑧21𝑧1𝑓𝑧1cos𝑡+𝑧2,𝑧1sin𝑡+𝑧2cos𝑡+𝑧2.(4.30)
Applying Proposition 3.1, it follows that the undisturbed system (3.2) is given by ̇𝜁1̇𝜁=0,2=0,(4.31) and its general solution is very simple, 𝜁𝑖=𝑐𝑖,𝑖=1,2,(4.32) where 𝑐𝑖, 𝑖=1,2, are constants of integration. The Jacobian matrix Δ𝜁 associated with this general solution is also very simple, and it is given by Δ𝜁=𝐼,(4.33) where 𝐼 is the identity matrix.
Substituting (4.33) into (3.13) and (3.14), it follows that 𝑍(𝑚)𝐼𝐼=𝜕𝐷𝜕𝑡(𝑚)𝐼𝐼+Ψ(𝑚)𝐼𝐼𝑑𝑡𝑠,𝑇(𝑚)𝐼𝐼=𝐷(𝑚)𝐼𝐼+Ψ(𝑚)𝐼𝐼𝑑𝑡𝑝.(4.34) The subscript 𝐼𝐼 is introduced to denote the second simplified algorithm.
Equation (4.34) can be put in a more suitable form as follows. In view of (4.30), the functions Ψ𝑗(𝑚) defined at each order of the algorithm (see (2.13) or (3.7)) are expressed by Fourier series with multiples of 𝑡+𝑧2 as arguments such that the vector function Ψ(𝑚)𝐼𝐼 can be written as Ψ(𝑚)𝐼𝐼=𝑘=0𝑎(𝑚)𝑘,𝐼𝐼cos𝑘𝑡+𝑐2+𝑏(𝑚)𝑘,𝐼𝐼sin𝑘𝑡+𝑐2𝑐(𝑚)𝑘,𝐼𝐼cos𝑘𝑡+𝑐2+𝑑(𝑚)𝑘,𝐼𝐼sin𝑘𝑡+𝑐2,(4.35) where the coefficients 𝑎(𝑚)𝑘,𝐼𝐼, 𝑏(𝑚)𝑘,𝐼𝐼, 𝑐(𝑚)𝑘,𝐼𝐼, and 𝑑(𝑚)𝑘,𝐼𝐼 are functions of the constant 𝑐1. The vector function Ψ(𝑚)𝐼𝐼 can also be put in matrix form: Ψ(𝑚)𝐼𝐼=𝑘=0𝑒𝐸𝑘(𝑡+𝑐2)𝐴(𝑚)𝑘,𝐼𝐼+𝑒𝐸𝑘(𝑡+𝑐2)𝐵(𝑚)𝑘,𝐼𝐼,(4.36) with 𝐴(𝑚)𝑘,𝐼𝐼=12𝑎(𝑚)𝑘,𝐼𝐼𝑑(𝑚)𝑘,𝐼𝐼𝑏(𝑚)𝑘,𝐼𝐼+𝑐(𝑚)𝑘,𝐼𝐼,𝐵(𝑚)𝑘,𝐼𝐼=12𝑎(𝑚)𝑘,𝐼𝐼+𝑑(𝑚)𝑘,𝐼𝐼𝑐(𝑚)𝑘,𝐼𝐼𝑏(𝑚)𝑘,𝐼𝐼.(4.37) Note that Ψ(𝑚)𝐼𝐼 is very similar to Ψ𝐼(𝑚), defined by (4.11). They represent different forms of Fourier series of Ψ(𝑚), but they are not the same, since they involve different sets of arbitrary constants of integration.
Thus, it follows from (4.36) that 𝐷(𝑚)𝐼𝐼+Ψ(𝑚)𝐼𝐼𝑑𝑡=𝐷(𝑚)𝐼𝐼+𝐴(𝑚)0,𝐼𝐼+𝐵(𝑚)0,𝐼𝐼𝑡+periodicterms,(4.38) with the periodic terms given by 𝑘=1(𝐸𝑘)1𝑒𝐸𝑘(𝑡+𝑐2)𝐴(𝑚)𝑘,𝐼𝐼+(𝐸𝑘)1𝑒𝐸𝑘(𝑡+𝑐2)𝐵(𝑚)𝑘,𝐼𝐼.(4.39) Therefore, 𝐷(𝑚)𝐼𝐼+Ψ(𝑚)𝐼𝐼𝑑𝑡𝑠=𝐷(𝑚)𝐼𝐼+𝐴(𝑚)0,𝐼𝐼+𝐵(𝑚)0,𝐼𝐼𝐷𝑡,(𝑚)𝐼𝐼+Ψ(𝑚)𝐼𝐼𝑑𝑡𝑝=𝑘=1(𝐸𝑘)1𝑒𝐸𝑘(𝑡+𝑐2)𝐴(𝑚)𝑘,𝐼𝐼+(𝐸𝑘)1𝑒𝐸𝑘(𝑡+𝑐2)𝐵(𝑚)𝑘,𝐼𝐼.(4.40)
Substituting (4.40) into (4.34), one finds 𝑍(𝑚)𝐼𝐼=𝐴(𝑚)0,𝐼𝐼+𝐵(𝑚)0,𝐼𝐼=Ψ(𝑚)𝐼𝐼𝑇,(4.41)(𝑚)𝐼𝐼=𝐷(𝑚)𝐼𝐼+Ψ(𝑚)𝐼𝐼Ψ(𝑚)𝐼𝐼𝑑𝑡.(4.42) Note that 𝐷(𝑚)𝐼𝐼 depends only on 𝑐1=𝜁1.
It should be noted that (4.41) and (4.42) can be straightforwardly obtained from (3.12) by applying the averaging principle if 𝐷(𝑚)𝐼𝐼 is assumed to be zero, since in this second approach: 𝐽(𝑡)=𝜕𝑍𝑗(0)𝜕𝜁𝑘=𝑂,(4.43) where 𝑂 denotes the null matrix. Thus, the general algorithm defined by (3.13) and (3.14) is equivalent to the averaging principle usually applied in the theory of nonlinear oscillations [5, 7].

Remark 4.3. Equations (4.41) and (4.42) are also obtained, if the second set of variation of parameters equations is considered (see Remark 4.2). In this case, the undisturbed system (3.2) is given by ̇𝜁1̇𝜁=0,2=1,(4.44) with general solution defined by 𝜁1=𝑐1,𝜁1=𝑡+𝑐2,(4.45) and Jacobian matrix Δ𝜁=𝐼.
Finally, we note that (4.41) is the tractability condition (3.9) up to order 𝑚. Since Ψ(𝑚)𝐼𝐼 does not depend explicitly on the time 𝑡, it follows that 𝜕𝑍𝑗(𝑚)=𝑍𝜕𝑡(𝑚),𝑍(0)𝑗=0,𝑗=1,2,(4.46) which is the tractability condition (3.9) up to order 𝑚.

5. Application to Nonlinear Oscillations Problems

In order to illustrate the application of the simplified algorithms, two examples are presented. The noncanonical version of the Hori method will be applied in determining second-order asymptotic solutions for van der Pol and Duffing equations. For the van der Pol equation, two different choices of the vector 𝐷(𝑚) will be made, and two generating functions 𝑇(𝑚) will be determined, one of these generating functions is the same function obtained by Hori [4] through a different approach, and, the other function gives the well-known averaged variation of parameters equations in the theory of nonlinear oscillations obtained through Krylov-Bogoliubov method [5]. It should be noted that the solution presented by Hori defines a new system of differential equations with a different frequency for the phase in comparison with the solution obtained by Ahmed and Tapley [7] and by Nayfeh [5], using different perturbation methods. For the Duffing equation, only one generating function is determined, and the second simplified algorithm gives the same generating function obtained through Krylov-Bogoliubov method.

The section is organized in two subsections: in the first subsection, the asymptotic solutions are determined through the first simplified algorithm, and, in the second subsection, they are determined through the second simplified algorithm.

5.1. Determination of Asymptotic Solutions through Simplified Algorithm I
5.1.1. Van der Pol Equation

Consider the well-known van der Pol equation:𝑥̈𝑥+𝜀21̇𝑥+𝑥=0.(5.1) Introducing the variables 𝑧1=𝑥 and 𝑧2=̇𝑥, this equation can be written in the form:𝑑𝑧1𝑑𝑡=𝑧2,𝑑𝑧1𝑑𝑡=𝑧1𝑧𝜀21𝑧12.(5.2) Thus𝑍(0)=𝑧2𝑧1𝑍,(5.3)(1)=0𝑧21𝑧12.(5.4)

As described in preceding paragraphs, two different choices of 𝐷(𝑚) will be made, and two generating functions 𝑇(𝑚) will be determined. Firstly, we present the solution obtained by Hori [4].

(1) First Asymptotic Solution: Hori’s [4] Solution
Following the simplified algorithm I defined by (4.15) and (4.18), the first-order terms 𝑍(1) and 𝑇(1) are calculated as follows.
Introducing the general solution given by (4.21) of the undisturbed system described by 𝑍(0)(𝜁1,𝜁2) into (5.4), with 𝜁 replacing 𝑧, one gets 𝑍(1)=01𝑐+4𝑐31sin(𝑡+𝜃)+4𝑐3sin3(𝑡+𝜃).(5.5)
Computing Δ𝜁1𝑍(1), Δ𝜁1𝑍(1)=12𝑐114𝑐2121𝑐cos2(𝑡+𝜃)+8𝑐31cos4(𝑡+𝜃)2112𝑐21sin2(𝑡+𝜃)8𝑐2sin4(𝑡+𝜃),(5.6) and taking its secular part, one finds Δ𝜁1𝑍(1)=12𝑐114𝑐20.(5.7)
From (4.15) and (4.22), it follows that 𝑍(1) is given by 𝑍(1)=12𝑐114𝑐21cos(𝑡+𝜃)2𝑐114𝑐2sin(𝑡+𝜃).(5.8) In view of (4.21), 𝑍(1) can be written explicitly in terms of the new variables 𝜁1 and 𝜁2 as follows: 𝑍(1)=12𝜁1114𝜁21+𝜁2212𝜁2114𝜁21+𝜁22.(5.9)
To determine 𝑇(1), the indefinite integral Δ𝜁1𝑍(1)Δ𝜁1𝑍(1)𝑑𝑡 is calculated: Δ𝜁1𝑍(1)Δ𝜁1𝑍(1)1𝑑𝑡=41𝑐sin2(𝑡+𝜃)+𝑐3231sin4(𝑡+𝜃)418𝑐21cos2(𝑡+𝜃)+𝑐322cos4(𝑡+𝜃).(5.10) Thus, from (4.18), it follows that 𝑇(1) is given by 𝑇(1)=14𝑐114𝑐21sin(𝑡+𝜃)𝑐3231sin3(𝑡+𝜃)4𝑐114𝑐23cos(𝑡+𝜃)𝑐323cos3(𝑡+𝜃)+Δ𝜁𝐷(1).(5.11) In view of (4.21), 𝑇(1) can be written explicitly in terms of the new variables 𝜁1 and 𝜁2 as follows: 𝑇(1)=14𝜁211+8𝜁21+𝜁2218𝜁3214𝜁171+8𝜁21+𝜁2238𝜁31+Δ𝜁𝐷(1),(5.12) with Δ𝜁𝐷(1) put in the form: Δ𝜁𝐷(1)=𝑑1(1)𝜁1+𝑑2(1)𝜁2𝑑1(1)𝜁2𝑑2(1)𝜁1,(5.13) being 𝑑𝑖(1)=𝑑𝑖(1)(𝑐), 𝑖=1,2, 𝐷1(1)=𝑐𝑑1(1), and 𝐷2(1)=𝑑2(1). The auxiliary vector 𝑑(1) is introduced in order to simplify the calculations, and, it is calculated in the second-order approximation as described below.
Following the algorithm of the Hori method described in Section 2, the second-order equation, (2.11), involves the term Ψ(2) given by Ψ(2)=12𝑍(1)+𝑍(1),𝑇(1).(5.14) The determination of Ψ(2) is very arduous. The generalized Poisson brackets must be calculated in terms of 𝜁1 and 𝜁2 through (2.12), and, the general solution of the undisturbed system, defined by (3.1), must be introduced. It should be noted that 𝑑𝑖(1),𝑖=1,2, in (5.12) are functions of the new variables 𝜁1 and 𝜁2 through 𝑐2=𝜁21+𝜁22. So, their partial derivatives must be considered in the calculation of the generalized Poisson brackets. After lengthy calculations performed using MAPLE software, one finds Δ𝜁1Ψ(2)1=7𝑐6433𝑐12851sin2(𝑡+𝜃)𝑐3231sin4(𝑡+𝜃)𝑐1285sin6(𝑡+𝜃)+𝑑1(1)14𝑐3+18𝑐3cos4(𝑡+𝜃)+𝑑2(1)121𝑐sin2(𝑡+𝜃)4𝑐3+sin4(𝑡+𝜃)𝑑𝑑1(1)𝑑𝜁114𝑐214𝑐4+cos(𝑡+𝜃)𝑑𝑑1(1)𝑑𝜁234𝑐214𝑐41sin(𝑡+𝜃)8𝑐4,Δsin3(𝑡+𝜃)𝜁1Ψ(2)21=8+3𝑐16211𝑐25641𝑐322+1𝑐644+1cos2(𝑡+𝜃)𝑐322+1𝑐12841cos4(𝑡+𝜃)𝑐1284cos6(𝑡+𝜃)+𝑑1(1)14𝑐21sin2(𝑡+𝜃)8𝑐2sin4(𝑡+𝜃)+𝑑2(1)1214𝑐21cos2(𝑡+𝜃)4𝑐2+cos4(𝑡+𝜃)𝑑𝑑2(1)𝑑𝜁1141𝑐4𝑐3+cos(𝑡+𝜃)𝑑𝑑2(1)𝑑𝜁2341𝑐4𝑐31sin(𝑡+𝜃)8𝑐3.sin3(𝑡+𝜃)(5.15)
In order to obtain the same result presented by Hori [4] for the new system of differential equations and the near-identity transformation, the following choice is made for the auxiliary vector 𝑑(1). Taking 𝑑(1)=014+1𝑐162,(5.16) it follows from (5.15) that Δ𝜁1Ψ(2)=018+18𝑐27𝑐2564.(5.17)
From (4.15), (4.21), and (5.17), one finds 𝑍(2)=18𝜁2𝜁121+𝜁22+7𝜁3221+𝜁22218𝜁1𝜁121+𝜁22+7𝜁3221+𝜁222.(5.18)
In view of the choice the auxiliary vector 𝑑(1), (5.12) can be simplified, and 𝑇(1) is then given by 𝑇(1)=1𝜁3223𝜁21𝜁221𝜁321167𝜁21+5𝜁22.(5.19)
Computing the indefinite integral Δ𝜁1Ψ(2)Δ𝜁1Ψ(2)𝑑𝑡 and substituting the general solution of the new undisturbed system, it follows that 𝑇(2) is given by 𝑇(2)=1𝜁1615𝜁6431+13𝜁76851+1𝜁9631𝜁22+11𝜁7681𝜁421𝜁162+3𝜁6421𝜁2+1𝜁163229𝜁7682𝜁415𝜁19221𝜁327𝜁76852+Δ𝜁𝐷(2),(5.20) with Δ𝜁𝐷(2) put in the form: Δ𝜁𝐷(2)=𝑑1(2)𝜁1+𝑑2(2)𝜁2𝑑1(2)𝜁2𝑑2(2)𝜁1,(5.21) being 𝑑𝑖(2)=𝑑𝑖(2)(𝑐), 𝑖=1,2, 𝐷1(2)=𝑐𝑑1(2), and 𝐷2(2)=𝑑2(2). 𝐷(2) is obtained from the third-order approximation.
In order to get the same result presented by Hori [4], one finds, repeating the procedure described in the preceding paragraphs, that the auxiliary vector 𝑑(2) must be taken as follows: 𝑑(2)=1+1615𝑐25627𝑐51240.(5.22) Accordingly, 𝑇(2) is given by 𝑇(2)=5𝜁15365113𝜁76831𝜁22+1𝜁15361𝜁425𝜁25631+15𝜁2561𝜁2235𝜁15365241𝜁76821𝜁3279𝜁15362𝜁41+31𝜁25632+27𝜁25621𝜁218𝜁2.(5.23)
The new system of differential equations and the generating function are given, up to the second-order of the small parameter, by 𝑑𝜁1𝑑𝑡=𝜁21+𝜀2𝜁1114𝜁21+𝜁22𝜀218𝜁2𝜁121+𝜁22+7𝜁3221+𝜁222,𝑑𝜁2𝑑𝑡=𝜁11+𝜀2𝜁2114𝜁21+𝜁22+𝜀218𝜁1𝜁121+𝜁22+7𝜁3221+𝜁222,𝑇(5.24)11=𝜀𝜁3223𝜁21𝜁22+𝜀25𝜁15365113𝜁76831𝜁22+1𝜁15361𝜁425𝜁25631+15𝜁2561𝜁22,𝑇21=𝜀𝜁321167𝜁21+5𝜁22+𝜀235𝜁15365241𝜁76821𝜁3279𝜁15362𝜁41+31𝜁25632+27𝜁25621𝜁218𝜁2.(5.25) These results are in agreement with the ones obtained by Hori [4] using a different approach.
Following da Silva Fernandes [1], the Lagrange variational equations—equations of variation of parameters—for the noncanonical version of the Hori method are given by 𝑑𝐶𝑑𝑡=Δ𝜁1𝑅,(5.26) where 𝑅=𝑚=1𝜀𝑚𝑍(𝑚), and 𝐶 is the 𝑛×1 vector of constants of integration of the general solution of the new undisturbed system (3.3). In view of (4.15), Lagrange variational equations can be put in the form: 𝑑𝐶=𝑑𝑡𝑚=1𝜀𝑚Δ𝜁1Ψ𝐼(𝑚).(5.27) Accordingly, the Lagrange variational equations for the new system of differential equations, (5.24), are given by𝑑𝑐=𝑑𝑡𝜀𝑐2114𝑐2,(5.28a)𝑑𝜃𝑑𝑡=𝜀218+18𝑐27𝑐2564.(5.28b)
The solution of the new system of differential equations can be obtained by introducing the solution of the above variational equations into (4.21).
The originalvariables 𝑥 and ̇𝑥 are calculated through (2.6), and the second-order asymptotic solution is 𝑥=𝜁11+𝜀𝜁3223𝜁21𝜁22+𝜀243𝜁614451+29𝜁307231𝜁2259𝜁61441𝜁42+1𝜁25631+9𝜁2561𝜁22,̇𝑥=𝜁21+𝜀𝜁321167𝜁21+5𝜁22+𝜀2155𝜁61445235𝜁307232𝜁21715𝜁61442𝜁41+29𝜁25632+53𝜁2562𝜁2118𝜁2.(5.29) Equations (5.29) define exactly the same solution presented by Hori.

(2) Second Asymptotic Solution
Now, let us to consider a different choice of the auxiliary vector 𝑑(1). Taking 𝑑(1) as a null vector, it follows straightforwardly from (5.15) that Δ𝜁1Ψ(2)=018+3𝑐16211𝑐2564.(5.30) Thus, from (4.15), (4.21), (5.28a), and (5.28b), one finds 𝑍(2)=18𝜁2312𝜁21+𝜁22+11𝜁3221+𝜁22218𝜁1312𝜁21+𝜁22+11𝜁3221+𝜁222.(5.31)
Since 𝑑(1) is a null vector, (5.12) simplifies, and 𝑇(1) is given by 𝑇(1)=14𝜁211+8𝜁21+𝜁2218𝜁3214𝜁171+8𝜁21+𝜁2238𝜁31.(5.32)
Now, repeating the procedure described in the previous section, that is, computing the indefinite integral Δ𝜁1Ψ(2)Δ𝜁1Ψ(2)𝑑𝑡, substituting the general solution of the new undisturbed system defined by (4.21), and taking 𝑑(2) is a null vector, it follows that 𝑇(2) is given by 𝑇(2)=3𝜁6431+1𝜁161𝜁22+5𝜁384511𝜁19231𝜁22+7𝜁3841𝜁427𝜁6421𝜁2+1𝜁16321𝜁3842𝜁411𝜁9621𝜁325𝜁38452.(5.33)
So, the new system of differential equations and the generating function are given, up to the second-order of the small parameter ε, by 𝑑𝜁1𝑑𝑡=𝜁21+𝜀2𝜁1114𝜁21+𝜁22𝜀218𝜁2312𝜁21+𝜁22+11𝜁3221+𝜁222,𝑑𝜁2𝑑𝑡=𝜁11+𝜀2𝜁2114𝜁21+𝜁22+𝜀218𝜁1312𝜁21+𝜁22+11𝜁3221+𝜁222,𝑇(5.34)11=𝜀4𝜁211+8𝜁21+𝜁2218𝜁32+𝜀23𝜁6431+1𝜁161𝜁22+5𝜁384511𝜁19231𝜁22+7𝜁3841𝜁42,𝑇21=𝜀4𝜁171+8𝜁21+𝜁2238𝜁31+𝜀27𝜁6421𝜁2+1𝜁16321𝜁3842𝜁411𝜁9621𝜁325𝜁38452.(5.35)
The Lagrange variational equations for the new system of differential equations, defined by (5.34), are given by𝑑𝑐=𝑑𝑡𝜀𝑐2114𝑐2,(5.36a)𝑑𝜃𝑑𝑡=𝜀218+3𝑐16211𝑐2564.(5.36b)These differential equations are the well-known averaged equations obtained through Krylov-Bogoliubov method [5]. Note that (5.28b) and (5.36b) define the phase 𝜃 with slightly different frequencies.
As described in the preceding subsection, the solution of the new system of differential equations, defined by (5.34), can be obtained by introducing the solution of the above variational equations into (4.21).
The original variables 𝑥 and ̇𝑥 are calculated through (2.6), which provides the following second-order asymptotic solution, 𝑥=𝜁11+𝜀4𝜁211+8𝜁213𝜁22+𝜀265𝜁614451+65𝜁307231𝜁2295𝜁61441𝜁421𝜁1631+1𝜁161𝜁22+1𝜁321,̇𝑥=𝜁21+𝜀4𝜁11185𝜁217𝜁22+𝜀2143𝜁614452+193𝜁307232𝜁21271𝜁61442𝜁41+5𝜁64327𝜁642𝜁21+1𝜁322.(5.37)
Finally, note that (5.29) and (5.37) give different second-order asymptotic solution for van der Pol equation.

5.1.2. Duffing Equation

Consider the well-known Duffing equation:̈𝑥+𝜀𝑥3+𝑥=0.(5.38)Introducing the variables 𝑧1=𝑥 and 𝑧2=̇𝑥, this equation can be written in the form:𝑑𝑧1𝑑𝑡=𝑧2,𝑑𝑧2𝑑𝑡=𝑧1𝜀𝑧31.(5.39)

Thus,𝑍(1)=0𝑧31.(5.40)

Following the simplified algorithm I and repeating the procedure described in Section 5.1.1, the first-order terms 𝑍(1) and 𝑇(1) are obtained as follows. Introducing (4.21) into (5.40), and computing the secular part, one getsΔ𝜁1𝑍(1)=038𝑐2.(5.41) Thus, from (4.15) and (5.41), it follows that 𝑍(1) is given by𝑍(1)=38𝑐33sin(𝑡+𝜃)8𝑐3cos(𝑡+𝜃).(5.42)

In view of (4.21), 𝑍(1) can be written explicitly in terms of the new variables 𝜁1 and 𝜁2:𝑍(1)=38𝜁2𝜁21+𝜁2238𝜁1𝜁21+𝜁22.(5.43)

Calculating the indefinite integral Δ𝜁1𝑍(1)Δ𝜁1𝑍(1)𝑑𝑡, one findsΔ𝜁1𝑍(1)Δ𝜁1𝑍(1)1𝑑𝑡=𝑐3231(4cos2(𝑡+𝜃)+cos4(𝑡+𝜃))𝑐322(8sin2(𝑡+𝜃)+sin4(𝑡+𝜃)).(5.44)

Multiplying this result by Δ𝜁, it follows, according to (4.18), that 𝑇(1) is given by𝑇(1)=3𝑐1631cos(𝑡+𝜃)+𝑐3233cos3(𝑡+𝜃)𝑐1633sin(𝑡+𝜃)𝑐323sin3(𝑡+𝜃)+Δ𝜁𝐷(1).(5.45)

Taking 𝐷(1) as a null vector and using (4.21), 𝑇(1) can be written explicitly in terms of the new variables 𝜁1 and 𝜁2 as follows:𝑇(1)=5𝜁32319𝜁321𝜁2215𝜁3221𝜁2+3𝜁3232.(5.46)

In the second-order approximation, one finds after lengthy calculations using MAPLE software:Ψ(2)=69𝜁25641𝜁2+27𝜁12821𝜁32+27𝜁25652165𝜁25651+69𝜁12831𝜁2227𝜁25642𝜁1.(5.47)

Repeating the procedure described in the above paragraphs, one findsΔ𝜁1Ψ(2)=051𝑐2564,𝑍(2)=51𝜁2562𝜁21+𝜁22251𝜁2561𝜁21+𝜁222.(5.48) Taking 𝐷(2) as a null vector, it follows that𝑇(2)=19𝜁25651+13𝜁3231𝜁22+65𝜁2561𝜁4295𝜁2562𝜁4113𝜁3221𝜁3213𝜁25652.(5.49)

The new system of differential equations and the generating function are given, up to the second-order of the small parameter 𝜀, by 𝑑𝜁1𝑑𝑡=𝜁23+𝜀8𝜁2𝜁21+𝜁22𝜀251𝜁2562𝜁21+𝜁222,𝑑𝜁2𝑑𝑡=𝜁13𝜀8𝜁1𝜁21+𝜁22+𝜀251𝜁2561𝜁21+𝜁222,𝑇(5.50)15=𝜀𝜁3231+9𝜁321𝜁22+𝜀219𝜁25651+13𝜁3231𝜁22+65𝜁2561𝜁42,𝑇2=𝜀15𝜁3221𝜁2+3𝜁3232+𝜀295𝜁2562𝜁4113𝜁3221𝜁3213𝜁25652.(5.51)

The Lagrange variational equations for the new system of differential equations, defined by (5.50), are given by 𝑑𝑐𝑑𝑡=0,𝑑𝜃3𝑑𝑡=𝜀8𝑐2𝜀251𝑐2564.(5.52) These differential equations are the well-known equations obtained through Krylov-Bogoliubov method [5].

As described in Section 5.1.1, the solution of the new system of differential equations, defined by (5.50), can be obtained by introducing the solution of the above variational equations into (4.21).

The original variables 𝑥 and ̇𝑥 are calculated through (2.6), which provides the following second-order asymptotic solution: 𝑥=𝜁11𝜀𝜁3215𝜁21+9𝜁22+𝜀212048227𝜁51+742𝜁31𝜁22+547𝜁1𝜁42,̇𝑥=𝜁23+𝜀𝜁3225𝜁21+𝜁22𝜀21204877𝜁52+922𝜁32𝜁21+685𝜁2𝜁41.(5.53) These equations are in agreement with the solution obtained through the canonical version of the Hori method [6].

5.2. Determination of Asymptotic Solutions through Simplified Algorithm II
5.2.1. Van der Pol Equation

For the van der Pol equation, the function 𝑓(𝑥,̇𝑥) is written in terms of the variables 𝑧1=𝑐 and 𝑧2=𝜃 by𝑧𝑓(𝑥,̇𝑥)=𝑓1cos𝑡+𝑧2,𝑧1sin𝑡+𝑧2𝑧=21cos2𝑡+𝑧2𝑧11sin𝑡+𝑧2.(5.54)Thus, it follows from (4.30) that𝑍(1)=12𝑧1114𝑧21cos2𝑡+𝑧2+14𝑧21cos4𝑡+𝑧212112𝑧21sin2𝑡+𝑧218𝑧21sin4𝑡+𝑧2.(5.55) As mentioned before, two different choices of 𝐷(𝑚) will be made, and two generating functions will be determined.

(1) First Asymptotic Solution
Following the simplified algorithm II defined by (4.41) and (4.42), the first-order terms 𝑍(1) and 𝑇(1) are calculated as follows.
Taking the secular part of 𝑍(1), with 𝜁 replacing 𝑧, one finds 𝑍(1)=12𝜁1114𝜁210,(5.56) and, integrating the remaining part, 𝑇(1)=14𝜁1sin2𝑡+𝜁2+1𝜁3231sin4𝑡+𝜁214112𝜁21cos2𝑡+𝜁2+1𝜁3221cos4𝑡+𝜁2+𝐷(1),(5.57) with 𝐷𝑖(1)=𝐷𝑖(1)(𝜁1), 𝑖=1,2.
Following the algorithm of the Hori method described in Section 2, the second-order equation, (2.11), involves the term Ψ(2) given by Ψ(2)=12𝑍(1)+𝑍(1),𝑇(1).(5.58) After tedious lengthy calculations using MAPLE software, one finds Ψ1(2)=7𝜁64313𝜁12851sin2𝑡+𝜁21𝜁3231sin4𝑡+𝜁21𝜁12851sin6𝑡+𝜁2+𝐷1(1)1238𝜁2114cos2𝑡+𝜁2+3𝜁1621cos4𝑡+𝜁2+𝐷2(1)12𝜁1sin2𝑡+𝜁214𝜁31sin4𝑡+𝜁2+𝑑𝐷1(1)𝑑𝜁112𝜁1+18𝜁31+14𝜁1cos2𝑡+𝜁21𝜁1631cos4𝑡+𝜁2,Ψ2(2)1=8+3𝜁162111𝜁25