# Application of the Hori Method in the Theory of Nonlinear Oscillations

Authors: Sandro da Silva Fernandes
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 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 functionsTj(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:(2.1)dzjdt=Zj(z,ε),j=1,…,n,
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:
(2.2)Z(z,ε)=Z(0)(z)+∑m=1εmZ(m)(z).
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:(2.3)dζjdt=Zj*(ζ,ε),j=1,…,n,
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:(2.4)Z*(ζ,ε)=Z*(0)(ζ)+∑m=1εmZ*(m)(ζ).
It is assumed that the vector function T(ζ,ε), that defines a near-identity transformation, is also expressed in powers series of ε:(2.5)T(ζ,ε)=∑m=1εmT(m)(ζ).Following Hori [4], the transformation of variables (z1,…,zn)→(ζ1,…,ζn) generated by T(ζ,ε) is given by(2.6)zj=ζj+∑k=11k!DTkζj=eDTζj,j=1,…,n.
For an arbitrary function f(z), the expansion formula is given by(2.7)f(z)=f(ζ)+∑k=11k!DTkf(ζ)=eDTf(ζ).
The operator DT is defined by(2.8)DTf(ζ)=∑j=1nTj∂f∂ζj,DTnf(ζ)=DTn-1(∑j=1nTj∂f∂ζj).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:(2.9)order0:Zj(0)=Zj*(0),(2.10)order1:[Z(0),T(1)]j+Zj(1)=Zj*(1),(2.11)order2:[Z(0),T(2)]j+12[Z(1)+Z*(1),T(1)]j+Zj(2)=Zj*(2),⋮j=1,…,n, where []j stands for the generalized Poisson brackets
(2.12)[Z,T]j=∑k=1n[Tk∂Zj∂ζk-Zk∂Tj∂ζk].Z*(0), Z(m), Z*(m), and T(m) are written in terms of the new variables ζ1,…,ζn.Themth-order equation of the algorithm can be put in the general form:(2.13)[Z*(0),T(m)]j+Ψj(m)=Zj*(m),j=1,…,n,
where the functions Ψj(m) are obtained from the preceding orders.


## 3. The General Algorithm

The determination of the functionsZj*(m) and Tj(m) from (2.13) is based on the following proposition presented in da Silva Fernandes [1].Proposition 3.1.
LetF be a n×1 vector function of the variables ζ1,…,ζn, which satisfy the system of differential equations:
(3.1)dζjdt=Zj*(0)(ζ)+Rj*(ζ;ε),j=1,…,n,
where Z*(0) describes an integrable system of differential equations:
(3.2)dζjdt=Zj*(0)(ζ),j=1,…,n,
a general solution of which is given by
(3.3)ζj=ζj(c1,…,cn,t),j=1,…,n,
being c1,…,cn arbitrary constants of integration; then
(3.4)[F,Z*(0)]j=∂Fj∂t-∑k=1n∂Zj*(0)∂ζkFk,j=1,…,n.A corollary of this proposition can be stated.Corollary 3.2.
Consider the same conditions of Proposition3.1 with the general solution of (3.2) given by
(3.5)ζj=ζj(c1,…,cn-1,M),j=1,…,n,
being c1,…,cn-1 arbitrary constants of integration and M=t+τ, where τ  is an additive constant; then
(3.6)[F,Z*(0)]j=∂Fj∂M-∑k=1n∂Zj*(0)∂ζkFk,j=1,…,n.Now, consider (2.13). According to Proposition 3.1, this equation can be put in the form:
(3.7)∂Tj(m)∂t-∑k=1n∂Zj*(0)∂ζkTk(m)=Ψj(m)-Zj*(m),j=1,…,n,
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,
(3.8)dTj(m)dt*-∑k=1n∂Zj*(0)∂ζkTk(m)=Ψj(m)-Zj*(m),j=1,…,n.To determineZj*(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 imposed(3.9)[Z(0),Z*]j=0,j=1,…,n.
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 determiningZj*(m) and Tj(m), j=1,…,n, proposed by Sessin [2] and revised in da Silva Fernandes [1] is briefly presented.Introducing then×1 matrices:
(3.10)T(m)=(Tj(m)),Ψ(m)=(Ψj(m)),Z*(m)=(Zj*(m)),j=1,…,n,
and the n×n Jacobian matrix(3.11)J(t)=(∂Zj*(0)∂ζk),j,k=1,…,n,the system of partial differential equations (3.7) can be put in the following matrix form:(3.12)∂T(m)∂t-J(t)T(m)=Ψ(m)-Z*(m).The vector functionsZ*(m) and T(m) are determined from the following equations:
(3.13)Z*(m)=Δζ∂∂t{Δζ-1[ΔζD(m)+Δζ∫Δζ-1Ψ(m)dt]s},(3.14)T(m)=[ΔζD(m)+Δζ∫Δζ-1Ψ(m)dt]p,
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 thatD(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:(4.1)ẍ+x=εf(x,ẋ).The first algorithm is applied to the system of first-order differential equations withx and ẋ 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 θ′are 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 variablesz1=x and z2=ẋ, (4.1) can be put in the form:(4.2)ż1=z2,ż2=-z1+εf(z1,z2).
According to the notation introduced in (2.1) and (2.2):(4.3)Z(0)=[z2-z1],Z(1)=[0f(z1,z2)].Following the algorithm of the Hori method for noncanonical systems, one finds from zero-th-order equation, (2.9), that(4.4)Z*(0)=[ζ2-ζ1].Applying Proposition3.1, it follows that the undisturbed system (3.2) is given by(4.5)ζ̇1=ζ2,ζ̇2=-ζ1,
general solution of which can be written in terms of the exponential matrix as [2]:(4.6)[ζ1ζ2]=eEt[c1c2],
where(4.7)eEt=[cos⁡tsin⁡t-sin⁡tcos⁡t],
and E is the symplectic matrix:
(4.8)E=[01-10],
and ci, i=1,2, are constants of integration. The Jacobian matrix Δζ associated with the solution (4.6) is then given by(4.9)Δζ=eEt,
with inverse Δζ-1=e-Et=ΔζT, 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 as(4.10)ΨI(m)=∑k=0∞[ak,I(m)cos⁡kt+bk,I(m)sin⁡ktck,I(m)cos⁡kt+dk,I(m)sin⁡kt],
where the coefficients ak,I(m), bk,I(m), ck,I(m), and dk,I(m) are functions of the constants ci, i=1,2. The Fourier series can also be put in matrix form:(4.11)ΨI(m)=∑k=0∞(eEktAk,I(m)+e-EktBk,I(m)),
with(4.12)Ak,I(m)=12[ak,I(m)-dk,I(m)bk,I(m)+ck,I(m)],Bk,I(m)=12[ak,I(m)+dk,I(m)ck,I(m)-bk,I(m)].
The subscript I is introduced to denote the first simplified algorithm.Substituting (4.9) and (4.11) into (3.13), one finds(4.13)ZI*(m)=eEt∂∂t{e-Et[teEtA1,I(m)+periodicterms]s}=eEtA1,I(m).On the other hand,(4.14)〈e-EtΨI(m)〉=A1,I(m),
where 〈〉 stands for the mean value of the function.Therefore, from (4.9), (4.13), and (4.14), it follows that(4.15)ZI*(m)=eEt〈e-EtΨI(m)〉=Δζ〈Δζ-1ΨI(m)〉.The second equation of the general algorithm, (3.14) can be simplified as described bellow.From (3.14), (4.14), and (4.15), one finds(4.16)teEtA1,I(m)=Δζ∫〈Δζ-1ΨI(m)〉dt=⌊ΔζDI(m)+Δζ∫Δζ-1ΨI(m)dt⌋s.On the other hand,(4.17)ΔζDI(m)+Δζ∫Δζ-1ΨI(m)dt=⌊ΔζDI(m)+Δζ∫Δζ-1ΨI(m)dt⌋p+teEtA1,I(m).Thus, introducing (4.16) and (4.17) into (3.14), one finds(4.18)TI(m)=⌊ΔζDI(m)+Δζ∫Δζ-1ΨI(m)dt⌋p=ΔζDI(m)+Δζ∫⌊Δζ-1ΨI(m)-〈Δζ-1ΨI(m)〉⌋dt.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 ẋ 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), 〈Δζ-1ΨI(m)〉 does not depend explicitly on the time t, it follows that(4.19)∂Z*(m)∂t=∂Δζ∂t〈Δζ-1ΨI(m)〉=JΔζ〈Δζ-1ΨI(m)〉=JZ*(m).
Using Proposition 3.1 and taking into account that J=∂Zj*(0)/∂ζk, this equation can be put in the following form:(4.20)[Z*(m),Z(0)]j=∂Zj*(m)∂t-∑k=1n∂Zj*(0)∂ζkZk*(m)=0,j=1,2,
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
(4.21)ζ1=ccos⁡(t+θ),ζ2=-csin⁡(t+θ),
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
(4.22)Δζ=[cos⁡⁡(t+θ)-csin⁡⁡(t+θ)-sin⁡⁡(t+θ)-ccos⁡⁡(t+θ)].
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:
(4.23)c2=c12+c22,tanθ=-c2c1.
In view of this transformation, the Jacobian matrix Δζ1 can be written in terms of the Jacobian matrix Δζ2 as
(4.24)Δζ1=Δζ2ΔC,
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
(4.25)ZI*(m)=Δζ1〈(Δζ1)-1ΨI1(m)〉=Δζ2ΔC〈ΔC-1(Δζ2)-1ΨI2(m)〉=Δζ2ΔCΔC-1〈(Δζ2)-1ΨI2(m)〉=Δζ2〈(Δζ2)-1ΨI2(m)〉,TI(m)=Δζ1DI1(m)+Δζ1∫[(Δζ1)-1ΨI1(m)-〈(Δζ1)-1ΨI1(m)〉]dt=Δζ2ΔCDI1(m)+Δζ2ΔC∫[ΔC-1(Δζ2)-1ΨI2(m)-〈ΔC-1(Δζ2)-1ΨI2(m)〉]dt=Δζ2ΔCDI1(m)+Δζ2ΔCΔC-1∫[(Δζ2)-1ΨI2(m)-〈(Δζ2)-1ΨI2(m)〉]dt=Δζ2DI2(m)+Δζ2∫[(Δζ2)-1ΨI2(m)-〈(Δζ2)-1ΨI2(m)〉]dt.
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,ẋ)→(c′,θ′) defined by the following equations(4.26)x=c′cos⁡(t+θ′),ẋ=-c′sin⁡(t+θ′),
equation (4.1) is transformed into(4.27)dc′dt=-εf(c′cos⁡⁡(t+θ′),-c′sin⁡⁡(t+θ′))sin⁡⁡(t+θ′),dθ′dt=-ε1c′f(c′cos⁡⁡(t+θ′),-c′sin⁡⁡(t+θ′))cos⁡⁡(t+θ′).
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,ẋ)→(c′,ϕ′), can also be defined. This second transformation is given by
(4.28)x=c′cos⁡ϕ′,ẋ=-c'sinϕ′.
In this case, (4.1) is transformed into
(4.29)dc′dt=-εf(c'cosϕ′,-c'sinϕ′)sin⁡ϕ′,dϕ′dt=1-ε1c′f(c'cosϕ′,-c'sinϕ′)cos⁡ϕ′.
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 variablesz1=c′ and z2=θ′, one gets from (4.27) that
(4.30)Z(0)=[00],Z(1)=[-f(z1cos⁡(t+z2),-z1sin⁡(t+z2))sin⁡(t+z2)-1z1f(z1cos⁡(t+z2),-z1sin⁡(t+z2))cos⁡(t+z2)].
Applying Proposition3.1, it follows that the undisturbed system (3.2) is given by
(4.31)ζ̇1=0,ζ̇2=0,
and its general solution is very simple,
(4.32)ζi=ci,i=1,2,
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
(4.33)Δζ=I,
where I is the identity matrix.
Substituting (4.33) into (3.13) and (3.14), it follows that
(4.34)ZII*(m)=∂∂t{[DII(m)+∫ΨII(m)dt]s},TII(m)=[DII(m)+∫ΨII(m)dt]p.
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 ΨII(m) can be written as
(4.35)ΨII(m)=∑k=0∞[ak,II(m)cos⁡k(t+c2)+bk,II(m)sin⁡k(t+c2)ck,II(m)cos⁡k(t+c2)+dk,II(m)sin⁡k(t+c2)],
where the coefficients ak,II(m), bk,II(m), ck,II(m), and dk,II(m) are functions of the constant c1. The vector function ΨII(m) can also be put in matrix form:
(4.36)ΨII(m)=∑k=0∞(eEk(t+c2)Ak,II(m)+e-Ek(t+c2)Bk,II(m)),
with
(4.37)Ak,II(m)=12[ak,II(m)-dk,II(m)bk,II(m)+ck,II(m)],Bk,II(m)=12[ak,II(m)+dk,II(m)ck,II(m)-bk,II(m)].
Note that ΨII(m) 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
(4.38)DII(m)+∫ΨII(m)dt=DII(m)+(A0,II(m)+B0,II(m))t+periodic terms,
with the periodic terms given by
(4.39)∑k=1∞((Ek)-1eEk(t+c2)Ak,II(m)+(-Ek)-1e-Ek(t+c2)Bk,II(m)).
Therefore,
(4.40)[DII(m)+∫ΨII(m)dt]s=DII(m)+(A0,II(m)+B0,II(m))t,[DII(m)+∫ΨII(m)dt]p=∑k=1∞((Ek)-1eEk(t+c2)Ak,II(m)+(-Ek)-1e-Ek(t+c2)Bk,II(m)).
Substituting (4.40) into (4.34), one finds
(4.41)ZII*(m)=A0,II(m)+B0,II(m)=〈ΨII(m)〉,(4.42)TII(m)=DII(m)+∫(ΨII(m)-〈ΨII(m)〉)dt.
Note that DII(m) 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 DII(m) is assumed to be zero, since in this second approach:
(4.43)J(t)=(∂Zj*(0)∂ζk)=O,
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
(4.44)ζ̇1=0,ζ̇2=1,
with general solution defined by
(4.45)ζ1=c1,ζ1=t+c2,
and Jacobian matrix Δζ=I.
Finally, we note that (4.41) is the tractability condition (3.9) up to order m. Since 〈ΨII(m)〉 does not depend explicitly on the time t, it follows that
(4.46)∂Zj*(m)∂t=[Z*(m),Z(0)]j=0,j=1,2,
which is the tractability condition (3.9) up to order m.


## 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 variablesz1=x and z2=ẋ, (4.1) can be put in the form:(4.2)ż1=z2,ż2=-z1+εf(z1,z2).
According to the notation introduced in (2.1) and (2.2):(4.3)Z(0)=[z2-z1],Z(1)=[0f(z1,z2)].Following the algorithm of the Hori method for noncanonical systems, one finds from zero-th-order equation, (2.9), that(4.4)Z*(0)=[ζ2-ζ1].Applying Proposition3.1, it follows that the undisturbed system (3.2) is given by(4.5)ζ̇1=ζ2,ζ̇2=-ζ1,
general solution of which can be written in terms of the exponential matrix as [2]:(4.6)[ζ1ζ2]=eEt[c1c2],
where(4.7)eEt=[cos⁡tsin⁡t-sin⁡tcos⁡t],
and E is the symplectic matrix:
(4.8)E=[01-10],
and ci, i=1,2, are constants of integration. The Jacobian matrix Δζ associated with the solution (4.6) is then given by(4.9)Δζ=eEt,
with inverse Δζ-1=e-Et=ΔζT, 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 as(4.10)ΨI(m)=∑k=0∞[ak,I(m)cos⁡kt+bk,I(m)sin⁡ktck,I(m)cos⁡kt+dk,I(m)sin⁡kt],
where the coefficients ak,I(m), bk,I(m), ck,I(m), and dk,I(m) are functions of the constants ci, i=1,2. The Fourier series can also be put in matrix form:(4.11)ΨI(m)=∑k=0∞(eEktAk,I(m)+e-EktBk,I(m)),
with(4.12)Ak,I(m)=12[ak,I(m)-dk,I(m)bk,I(m)+ck,I(m)],Bk,I(m)=12[ak,I(m)+dk,I(m)ck,I(m)-bk,I(m)].
The subscript I is introduced to denote the first simplified algorithm.Substituting (4.9) and (4.11) into (3.13), one finds(4.13)ZI*(m)=eEt∂∂t{e-Et[teEtA1,I(m)+periodicterms]s}=eEtA1,I(m).On the other hand,(4.14)〈e-EtΨI(m)〉=A1,I(m),
where 〈〉 stands for the mean value of the function.Therefore, from (4.9), (4.13), and (4.14), it follows that(4.15)ZI*(m)=eEt〈e-EtΨI(m)〉=Δζ〈Δζ-1ΨI(m)〉.The second equation of the general algorithm, (3.14) can be simplified as described bellow.From (3.14), (4.14), and (4.15), one finds(4.16)teEtA1,I(m)=Δζ∫〈Δζ-1ΨI(m)〉dt=⌊ΔζDI(m)+Δζ∫Δζ-1ΨI(m)dt⌋s.On the other hand,(4.17)ΔζDI(m)+Δζ∫Δζ-1ΨI(m)dt=⌊ΔζDI(m)+Δζ∫Δζ-1ΨI(m)dt⌋p+teEtA1,I(m).Thus, introducing (4.16) and (4.17) into (3.14), one finds(4.18)TI(m)=⌊ΔζDI(m)+Δζ∫Δζ-1ΨI(m)dt⌋p=ΔζDI(m)+Δζ∫⌊Δζ-1ΨI(m)-〈Δζ-1ΨI(m)〉⌋dt.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 ẋ 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), 〈Δζ-1ΨI(m)〉 does not depend explicitly on the time t, it follows that(4.19)∂Z*(m)∂t=∂Δζ∂t〈Δζ-1ΨI(m)〉=JΔζ〈Δζ-1ΨI(m)〉=JZ*(m).
Using Proposition 3.1 and taking into account that J=∂Zj*(0)/∂ζk, this equation can be put in the following form:(4.20)[Z*(m),Z(0)]j=∂Zj*(m)∂t-∑k=1n∂Zj*(0)∂ζkZk*(m)=0,j=1,2,
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
(4.21)ζ1=ccos⁡(t+θ),ζ2=-csin⁡(t+θ),
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
(4.22)Δζ=[cos⁡⁡(t+θ)-csin⁡⁡(t+θ)-sin⁡⁡(t+θ)-ccos⁡⁡(t+θ)].
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:
(4.23)c2=c12+c22,tanθ=-c2c1.
In view of this transformation, the Jacobian matrix Δζ1 can be written in terms of the Jacobian matrix Δζ2 as
(4.24)Δζ1=Δζ2ΔC,
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
(4.25)ZI*(m)=Δζ1〈(Δζ1)-1ΨI1(m)〉=Δζ2ΔC〈ΔC-1(Δζ2)-1ΨI2(m)〉=Δζ2ΔCΔC-1〈(Δζ2)-1ΨI2(m)〉=Δζ2〈(Δζ2)-1ΨI2(m)〉,TI(m)=Δζ1DI1(m)+Δζ1∫[(Δζ1)-1ΨI1(m)-〈(Δζ1)-1ΨI1(m)〉]dt=Δζ2ΔCDI1(m)+Δζ2ΔC∫[ΔC-1(Δζ2)-1ΨI2(m)-〈ΔC-1(Δζ2)-1ΨI2(m)〉]dt=Δζ2ΔCDI1(m)+Δζ2ΔCΔC-1∫[(Δζ2)-1ΨI2(m)-〈(Δζ2)-1ΨI2(m)〉]dt=Δζ2DI2(m)+Δζ2∫[(Δζ2)-1ΨI2(m)-〈(Δζ2)-1ΨI2(m)〉]dt.
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,ẋ)→(c′,θ′) defined by the following equations(4.26)x=c′cos⁡(t+θ′),ẋ=-c′sin⁡(t+θ′),
equation (4.1) is transformed into(4.27)dc′dt=-εf(c′cos⁡⁡(t+θ′),-c′sin⁡⁡(t+θ′))sin⁡⁡(t+θ′),dθ′dt=-ε1c′f(c′cos⁡⁡(t+θ′),-c′sin⁡⁡(t+θ′))cos⁡⁡(t+θ′).
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,ẋ)→(c′,ϕ′), can also be defined. This second transformation is given by
(4.28)x=c′cos⁡ϕ′,ẋ=-c'sinϕ′.
In this case, (4.1) is transformed into
(4.29)dc′dt=-εf(c'cosϕ′,-c'sinϕ′)sin⁡ϕ′,dϕ′dt=1-ε1c′f(c'cosϕ′,-c'sinϕ′)cos⁡ϕ′.
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 variablesz1=c′ and z2=θ′, one gets from (4.27) that
(4.30)Z(0)=[00],Z(1)=[-f(z1cos⁡(t+z2),-z1sin⁡(t+z2))sin⁡(t+z2)-1z1f(z1cos⁡(t+z2),-z1sin⁡(t+z2))cos⁡(t+z2)].
Applying Proposition3.1, it follows that the undisturbed system (3.2) is given by
(4.31)ζ̇1=0,ζ̇2=0,
and its general solution is very simple,
(4.32)ζi=ci,i=1,2,
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
(4.33)Δζ=I,
where I is the identity matrix.
Substituting (4.33) into (3.13) and (3.14), it follows that
(4.34)ZII*(m)=∂∂t{[DII(m)+∫ΨII(m)dt]s},TII(m)=[DII(m)+∫ΨII(m)dt]p.
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 ΨII(m) can be written as
(4.35)ΨII(m)=∑k=0∞[ak,II(m)cos⁡k(t+c2)+bk,II(m)sin⁡k(t+c2)ck,II(m)cos⁡k(t+c2)+dk,II(m)sin⁡k(t+c2)],
where the coefficients ak,II(m), bk,II(m), ck,II(m), and dk,II(m) are functions of the constant c1. The vector function ΨII(m) can also be put in matrix form:
(4.36)ΨII(m)=∑k=0∞(eEk(t+c2)Ak,II(m)+e-Ek(t+c2)Bk,II(m)),
with
(4.37)Ak,II(m)=12[ak,II(m)-dk,II(m)bk,II(m)+ck,II(m)],Bk,II(m)=12[ak,II(m)+dk,II(m)ck,II(m)-bk,II(m)].
Note that ΨII(m) 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
(4.38)DII(m)+∫ΨII(m)dt=DII(m)+(A0,II(m)+B0,II(m))t+periodic terms,
with the periodic terms given by
(4.39)∑k=1∞((Ek)-1eEk(t+c2)Ak,II(m)+(-Ek)-1e-Ek(t+c2)Bk,II(m)).
Therefore,
(4.40)[DII(m)+∫ΨII(m)dt]s=DII(m)+(A0,II(m)+B0,II(m))t,[DII(m)+∫ΨII(m)dt]p=∑k=1∞((Ek)-1eEk(t+c2)Ak,II(m)+(-Ek)-1e-Ek(t+c2)Bk,II(m)).
Substituting (4.40) into (4.34), one finds
(4.41)ZII*(m)=A0,II(m)+B0,II(m)=〈ΨII(m)〉,(4.42)TII(m)=DII(m)+∫(ΨII(m)-〈ΨII(m)〉)dt.
Note that DII(m) 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 DII(m) is assumed to be zero, since in this second approach:
(4.43)J(t)=(∂Zj*(0)∂ζk)=O,
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
(4.44)ζ̇1=0,ζ̇2=1,
with general solution defined by
(4.45)ζ1=c1,ζ1=t+c2,
and Jacobian matrix Δζ=I.
Finally, we note that (4.41) is the tractability condition (3.9) up to order m. Since 〈ΨII(m)〉 does not depend explicitly on the time t, it follows that
(4.46)∂Zj*(m)∂t=[Z*(m),Z(0)]j=0,j=1,2,
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 vectorD(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:(5.1)ẍ+ε(x2-1)ẋ+x=0.
Introducing the variables z1=x and z2=ẋ, this equation can be written in the form:(5.2)dz1dt=z2,dz1dt=-z1-ε(z12-1)z2.
Thus(5.3)Z(0)=[z2-z1],(5.4)Z(1)=[0-(z12-1)z2].As described in preceding paragraphs, two different choices ofD(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
(5.5)Z(1)=[0(-c+14c3)sin⁡(t+θ)+14c3sin⁡3(t+θ)].
ComputingΔζ-1Z(1),
(5.6)Δζ-1Z(1)=[12c(1-14c2)-12ccos⁡2(t+θ)+18c3cos⁡4(t+θ)12(1-12c2)sin⁡2(t+θ)-18c2sin⁡4(t+θ)],
and taking its secular part, one finds
(5.7)〈Δζ-1Z(1)〉=[12c(1-14c2)0].
From (4.15) and (4.22), it follows that Z*(1) is given by
(5.8)Z*(1)=[12c(1-14c2)cos⁡(t+θ)-12c(1-14c2)sin⁡(t+θ)].
In view of (4.21), Z*(1) can be written explicitly in terms of the new variables ζ1 and ζ2 as follows:
(5.9)Z*(1)=[12ζ1(1-14(ζ12+ζ22))12ζ2(1-14(ζ12+ζ22))].
To determineT(1), the indefinite integral ∫⌊Δζ-1Z(1)-〈Δζ-1Z(1)〉⌋dt is calculated:
(5.10)∫[Δζ-1Z(1)-〈Δζ-1Z(1)〉]dt=[-14csin⁡2(t+θ)+132c3sin⁡4(t+θ)-(14-18c2)cos⁡2(t+θ)+132c2cos⁡4(t+θ)].
Thus, from (4.18), it follows that T(1) is given by
(5.11)T(1)=[-14c(1-14c2)sin⁡(t+θ)-132c3sin⁡3(t+θ)14c(1-14c2)cos⁡(t+θ)-332c3cos⁡3(t+θ)]+ΔζD(1).
In view of (4.21), T(1) can be written explicitly in terms of the new variables ζ1 and ζ2 as follows:
(5.12)T(1)=[14ζ2(1+18(ζ12+ζ22))-18ζ2314ζ1(1+78(ζ12+ζ22))-38ζ13]+ΔζD(1),
with ΔζD(1) put in the form:
(5.13)ΔζD(1)=[d1(1)ζ1+d2(1)ζ2d1(1)ζ2-d2(1)ζ1],
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 Section2, the second-order equation, (2.11), involves the term Ψ(2) given by
(5.14)Ψ(2)=12[Z(1)+Z*(1),T(1)].
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=ζ12+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
(5.15)(Δζ-1Ψ(2))1=(764c3-3128c5)sin⁡2(t+θ)-132c3sin⁡4(t+θ)-1128c5sin⁡6(t+θ)+d1(1)(-14c3+18c3cos⁡4(t+θ))+d2(1)(12csin⁡2(t+θ)-14c3sin⁡4(t+θ))+dd1(1)dζ1(-14(c2-14c4)cos⁡(t+θ))+dd1(1)dζ2(34(c2-14c4)sin⁡(t+θ)-18c4sin⁡3(t+θ)),(Δζ-1Ψ(2))2=-18+316c2-11256c4-(132c2+164c4)cos⁡2(t+θ)+(-132c2+1128c4)cos⁡4(t+θ)-1128c4cos⁡6(t+θ)+d1(1)(-14c2sin⁡2(t+θ)-18c2sin⁡4(t+θ))+d2(1)((12-14c2)cos⁡2(t+θ)-14c2cos⁡4(t+θ))+dd2(1)dζ1(-14(c-14c3)cos⁡(t+θ))+dd2(1)dζ2(34(c-14c3)sin⁡(t+θ)-18c3sin⁡3(t+θ)).
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
(5.16)d(1)=[0-14+116c2],
it follows from (5.15) that
(5.17)〈Δζ-1Ψ(2)〉=[0-18+18c2-7256c4].
From (4.15), (4.21), and (5.17), one finds
(5.18)Z*(2)=[-18ζ2(1-(ζ12+ζ22)+732(ζ12+ζ22)2)18ζ1(1-(ζ12+ζ22)+732(ζ12+ζ22)2)].
In view of the choice the auxiliary vectord(1), (5.12) can be simplified, and T(1) is then given by
(5.19)T(1)=[132ζ2(3ζ12-ζ22)132ζ1(16-7ζ12+5ζ22)].
Computing the indefinite integral∫⌊Δζ-1Ψ(2)-〈Δζ-1Ψ(2)〉⌋dt and substituting the general solution of the new undisturbed system, it follows that T(2) is given by
(5.20)T(2)=[116ζ1-564ζ13+13768ζ15+196ζ13ζ22+11768ζ1ζ24-116ζ2+364ζ12ζ2+116ζ23-29768ζ2ζ14-5192ζ12ζ23-7768ζ25]+ΔζD(2),
with ΔζD(2) put in the form:
(5.21)ΔζD(2)=[d1(2)ζ1+d2(2)ζ2d1(2)ζ2-d2(2)ζ1],
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:
(5.22)d(2)=[-116+15256c2-7512c40].
Accordingly, T(2) is given by
(5.23)T(2)=[51536ζ15-13768ζ13ζ22+11536ζ1ζ24-5256ζ13+15256ζ1ζ22-351536ζ25-41768ζ12ζ23-791536ζ2ζ14+31256ζ23+27256ζ12ζ2-18ζ2].
The new system of differential equations and the generating function are given, up to the second-order of the small parameter, by(5.24)dζ1dt=ζ2+ε12ζ1(1-14(ζ12+ζ22))-ε218ζ2(1-(ζ12+ζ22)+732(ζ12+ζ22)2),dζ2dt=-ζ1+ε12ζ2(1-14(ζ12+ζ22))+ε218ζ1(1-(ζ12+ζ22)+732(ζ12+ζ22)2),(5.25)T1=ε132ζ2(3ζ12-ζ22)+ε2(51536ζ15-13768ζ13ζ22+11536ζ1ζ24-5256ζ13+15256ζ1ζ22),T2=ε132ζ1(16-7ζ12+5ζ22)+ε2(-351536ζ25-41768ζ12ζ23-791536ζ2ζ14+31256ζ23+27256ζ12ζ2-18ζ2).
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
(5.26)dCdt=Δζ-1R*,
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:
(5.27)dCdt=∑m=1εm〈Δζ-1ΨI(m)〉.
Accordingly, the Lagrange variational equations for the new system of differential equations, (5.24), are given by(5.28a)dcdt=εc2(1-14c2),(5.28b)dθdt=ε2(-18+18c2-7256c4).
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 originalvariablesx and ẋ are calculated through (2.6), and the second-order asymptotic solution is
(5.29)x=ζ1+ε132ζ2(3ζ12-ζ22)+ε2(-436144ζ15+293072ζ13ζ22-596144ζ1ζ24+1256ζ13+9256ζ1ζ22),ẋ=ζ2+ε132ζ1(16-7ζ12+5ζ22)+ε2(-1556144ζ25-353072ζ23ζ12-7156144ζ2ζ14+29256ζ23+53256ζ2ζ12-18ζ2).
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 vectord(1). Taking d(1) as a null vector, it follows straightforwardly from (5.15) that
(5.30)〈Δζ-1Ψ(2)〉=[0-18+316c2-11256c4].
Thus, from (4.15), (4.21), (5.28a), and (5.28b), one finds
(5.31)Z*(2)=[-18ζ2(1-32(ζ12+ζ22)+1132(ζ12+ζ22)2)18ζ1(1-32(ζ12+ζ22)+1132(ζ12+ζ22)2)].
Sinced(1) is a null vector, (5.12) simplifies, and T(1) is given by
(5.32)T(1)=[14ζ2(1+18(ζ12+ζ22))-18ζ2314ζ1(1+78(ζ12+ζ22))-38ζ13].
Now, repeating the procedure described in the previous section, that is, computing the indefinite integral∫⌊Δζ-1Ψ(2)-〈Δζ-1Ψ(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
(5.33)T(2)=[-364ζ13+116ζ1ζ22+5384ζ15-1192ζ13ζ22+7384ζ1ζ24-764ζ12ζ2+116ζ23-1384ζ2ζ14-196ζ12ζ23-5384ζ25].
So, the new system of differential equations and the generating function are given, up to the second-order of the small parameterε, by
(5.34)dζ1dt=ζ2+ε12ζ1(1-14(ζ12+ζ22))-ε218ζ2(1-32(ζ12+ζ22)+1132(ζ12+ζ22)2),dζ2dt=-ζ1+ε12ζ2(1-14(ζ12+ζ22))+ε218ζ1(1-32(ζ12+ζ22)+1132(ζ12+ζ22)2),(5.35)T1=ε(14ζ2(1+18(ζ12+ζ22))-18ζ23)+ε2(-364ζ13+116ζ1ζ22+5384ζ15-1192ζ13ζ22+7384ζ1ζ24),T2=ε(14ζ1(1+78(ζ12+ζ22))-38ζ13)+ε2(-764ζ12ζ2+116ζ23-1384ζ2ζ14-196ζ12ζ23-5384ζ25).
The Lagrange variational equations for the new system of differential equations, defined by (5.34), are given by(5.36a)dcdt=εc2(1-14c2),(5.36b)dθdt=ε2(-18+316c2-11256c4).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 variablesx and ẋ are calculated through (2.6), which provides the following second-order asymptotic solution,
(5.37)x=ζ1+ε14ζ2(1+18(ζ12-3ζ22))+ε2(656144ζ15+653072ζ13ζ22-956144ζ1ζ24-116ζ13+116ζ1ζ22+132ζ1),ẋ=ζ2+ε14ζ1(1-18(5ζ12-7ζ22))+ε2(-1436144ζ25+1933072ζ23ζ12-2716144ζ2ζ14+564ζ23-764ζ2ζ12+132ζ2).
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:(5.38)ẍ+εx3+x=0.Introducing the variables z1=x and z2=ẋ, this equation can be written in the form:(5.39)dz1dt=z2,dz2dt=-z1-εz13.Thus,(5.40)Z(1)=[0-z13].Following the simplified algorithm I and repeating the procedure described in Section5.1.1, 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 gets(5.41)〈Δζ-1Z(1)〉=[038c2].
Thus, from (4.15) and (5.41), it follows that Z*(1) is given by(5.42)Z*(1)=[-38c3sin⁡(t+θ)-38c3cos⁡(t+θ)].In view of (4.21), Z*(1) can be written explicitly in terms of the new variables ζ1 and ζ2:(5.43)Z*(1)=[38ζ2(ζ12+ζ22)-38ζ1(ζ12+ζ22)].Calculating the indefinite integral∫⌊Δζ-1Z(1)-〈Δζ-1Z(1)〉⌋dt, one finds(5.44)∫[Δζ-1Z(1)-〈Δζ-1Z(1)〉]dt=[-132c3(4cos⁡2(t+θ)+cos⁡4(t+θ))132c2(8sin⁡2(t+θ)+sin⁡4(t+θ))].Multiplying this result byΔζ, it follows, according to (4.18), that T(1) is given by(5.45)T(1)=[-316c3cos⁡(t+θ)+132c3cos⁡3(t+θ)-316c3sin⁡(t+θ)-332c3sin⁡3(t+θ)]+ΔζD(1).TakingD(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:(5.46)T(1)=[-532ζ13-932ζ1ζ221532ζ12ζ2+332ζ23].In the second-order approximation, one finds after lengthy calculations using MAPLE software:(5.47)Ψ(2)=[-69256ζ14ζ2+27128ζ12ζ23+27256ζ25165256ζ15+69128ζ13ζ22-27256ζ24ζ1].Repeating the procedure described in the above paragraphs, one finds(5.48)〈Δζ-1Ψ(2)〉=[0-51256c4],Z*(2)=[-51256ζ2(ζ12+ζ22)251256ζ1(ζ12+ζ22)2].
Taking D(2) as a null vector, it follows that(5.49)T(2)=[19256ζ15+1332ζ13ζ22+65256ζ1ζ24-95256ζ2ζ14-1332ζ12ζ23-13256ζ25].The new system of differential equations and the generating function are given, up to the second-order of the small parameterε, by
(5.50)dζ1dt=ζ2+ε38ζ2(ζ12+ζ22)-ε251256ζ2(ζ12+ζ22)2,dζ2dt=-ζ1-ε38ζ1(ζ12+ζ22)+ε251256ζ1(ζ12+ζ22)2,(5.51)T1=-ε(532ζ13+932ζ1ζ22)+ε2(19256ζ15+1332ζ13ζ22+65256ζ1ζ24),T2=ε(1532ζ12ζ2+332ζ23)+ε2(-95256ζ2ζ14-1332ζ12ζ23-13256ζ25).The Lagrange variational equations for the new system of differential equations, defined by (5.50), are given by
(5.52)dcdt=0,dθdt=ε38c2-ε251256c4.
These differential equations are the well-known equations obtained through Krylov-Bogoliubov method [5].As described in Section5.1.1, 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 variablesx and ẋ are calculated through (2.6), which provides the following second-order asymptotic solution:
(5.53)x=ζ1-ε132ζ1(5ζ12+9ζ22)+ε212048(227ζ15+742ζ13ζ22+547ζ1ζ24),ẋ=ζ2+ε332ζ2(5ζ12+ζ22)-ε212048(77ζ25+922ζ23ζ12+685ζ2ζ14).
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 functionf(x,ẋ) is written in terms of the variables z1=c′ and z2=θ′ by(5.54)f(x,ẋ)=f(z1cos⁡(t+z2),-z1sin⁡(t+z2))=-(z12co⁡s2(t+z2)-1)z1sin⁡(t+z2).Thus, it follows from (4.30) that(5.55)Z(1)=[12z1(1-14z12-cos⁡2(t+z2)+14z12cos⁡4(t+z2))12(1-12z12)sin⁡2(t+z2)-18z12sin⁡4(t+z2)].
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 ofZ(1), with ζ replacing z, one finds
(5.56)Z*(1)=[12ζ1(1-14ζ12)0],
and, integrating the remaining part,
(5.57)T(1)=[-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2)-14(1-12ζ12)cos⁡2(t+ζ2)+132ζ12cos⁡4(t+ζ2)]+D(1),
with Di(1)=Di(1)(ζ1), i=1,2.
Following the algorithm of the Hori method described in Section2, the second-order equation, (2.11), involves the term Ψ(2) given by
(5.58)Ψ(2)=12[Z(1)+Z*(1),T(1)].
After tedious lengthy calculations using MAPLE software, one finds
(5.59)Ψ1(2)=(764ζ13-3128ζ15)sin⁡2(t+ζ2)-132ζ13sin⁡4(t+ζ2)-1128ζ15sin⁡6(t+ζ2)+D1(1)(12-38ζ12-14cos⁡2(t+ζ2)+316ζ12cos⁡4(t+ζ2))+D2(1)(12ζ1sin⁡2(t+ζ2)-14ζ13sin⁡4(t+ζ2))+dD1(1)dζ1(-12ζ1+18ζ13+14ζ1cos⁡2(t+ζ2)-116ζ13cos⁡4(t+ζ2)),Ψ2(2)=-18+316ζ12-11256ζ14-(132ζ12+164ζ14)cos⁡2(t+ζ2)+(-132ζ12+1128ζ14)cos⁡4(t+ζ2)-1128ζ14cos⁡6(t+ζ2)+D1(1)(-14ζ1sin⁡2(t+ζ2)-18ζ1sin⁡4(t+ζ2))+D2(1)((12-14ζ12)cos⁡2(t+ζ2)-14ζ12cos⁡4(t+ζ2))+dD2(1)dζ1(-12ζ1+18ζ13+14ζ1cos⁡2(t+ζ2)-116ζ13cos⁡4(t+ζ2)).
In order to obtain the same averaged Lagrange variational equations given by (5.28a) and (5.28b), D(1) must be taken as
(5.60)D(1)=[0-14+116ζ12].
Thus, it follows that
(5.61)Z*(2)=[0-18+18ζ12-7256ζ14].
In view of the choice ofD(1), T(1) is then given by
(5.62)T(1)=[-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2)-14+116ζ12-14(1-12ζ12)cos⁡2(t+ζ2)+132ζ12cos⁡4(t+ζ2)].
Repeating the procedure for the third-order approximation, and, taking
(5.63)D(2)=[-116ζ1+15256ζ13-7512ζ150],
one finds
(5.64)T1(2)=11536(-96ζ1+90ζ13-21ζ15)+(116ζ1-9128ζ13+9768ζ15)cos⁡2(t+ζ2)+(-1128ζ13+1256ζ15)cos⁡4(t+ζ2)+1768ζ15cos⁡6(t+ζ2),T2(2)=(-116+364ζ12-164ζ14)sin⁡2(t+ζ2)+(1128ζ12-1256ζ14)sin⁡4(t+ζ2)-1768ζ14sin⁡6(t+ζ2).
The new system of differential equations is given, up to the second-order of the small parameterε, by(5.65a)dζ1dt=ε12ζ1(1-14ζ12),(5.65b)dζ2dt=ε2(-18+18ζ12-7256ζ14).These differential equations are exactly the same equations given by (5.28a) and (5.28b).
The generating function is obtained from (5.62), (5.64), and it is given, up to the second-order of the small parameter ε, by
(5.66)T1=ε(-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2))+ε2(11536(-96ζ1+90ζ13-21ζ15)+(116ζ1-9128ζ13+9768ζ15)cos⁡2(t+ζ2)+(-1128ζ13+1256ζ15)cos⁡4(t+ζ2)+1768ζ15cos⁡6(t+ζ2)),T2=ε(116ζ12-14(1-12ζ12)cos⁡2(t+ζ2)+132ζ12cos⁡4(t+ζ2))+ε2((-116+364ζ12-164ζ14)sin⁡2(t+ζ2)+(1128ζ12-1256ζ14)×sin⁡4(t+ζ2)-1768ζ14sin⁡6(t+ζ2)).
The original variablesx and ẋ are calculated through (2.7), which provides, up to the second-order of the small parameter, the following solution:
(5.67)x=ζ1cos⁡(t+ζ2)-ε132ζ13sin⁡3(t+ζ2)+ε2{[3256ζ13-92048ζ15]cos⁡(t+ζ2)-[1128ζ13+11024ζ15]cos⁡3(t+ζ2)-53072ζ15cos⁡5(t+ζ2)},ẋ=-ζ1sin⁡⁡(t+ζ2)+ε{[12ζ1-18ζ13]cos⁡⁡(t+ζ2)-332ζ13cos⁡⁡3(t+ζ2)}+ε2{[18ζ1-35256ζ13+652048ζ15]sin⁡⁡(t+ζ2)-[3128ζ13-151024ζ15]×sin⁡⁡3(t+ζ2)+253072ζ15sin⁡5(t+ζ2)},
with ζ1 and ζ2 given by the solution of (5.65a) and (5.65b).Note that (5.29) and (5.67) give the same second-order asymptotic solution for the van der Pol equation. Recall that ζ1 and ζ2 have different meaning in these equations, but they are related through an equation similar to (4.21).(2) Second Asymptotic Solution
Now, let us to takeD(1) and D(2) as null vectors. Equations (5.59) simplifies, and Z*(2) is given by(5.68)Z*(2)=[0-18+316ζ12-11256ζ14].
The functionsT(1) and T(2) are then given by
(5.69)T(1)=[-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2)-14(1-12ζ12)cos⁡2(t+ζ2)+132ζ12cos⁡4(t+ζ2)],T1(2)=-(7128ζ13-3256ζ15)cos⁡2(t+ζ2)+1128ζ13cos⁡4(t+ζ2)+1768ζ15cos⁡6(t+ζ2),T2(2)=-(164ζ12+1128ζ14)sin⁡2(t+ζ2)+(-1128ζ12+1512ζ14)sin⁡4(t+ζ2)-1768ζ14sin⁡6(t+ζ2).
The new system of differential equations is obtained from (5.56) and (5.68), and it is given, up to the second-order of the small parameter, by
(5.70)dζ1dt=ε12ζ1(1-14ζ12),dζ2dt=ε2(-18+316ζ12-11256ζ14).
These differential equations are exactly the same equations given by (5.36a) and (5.36b).
The generating function is obtained from (5.69), and it is given, up to the second-order of the small parameter ε, by
(5.71)T1=ε(-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2))+ε2(-(7128ζ13-3256ζ15)cos⁡2(t+ζ2)+1128ζ13cos⁡4(t+ζ2)+1768ζ15cos⁡⁡6(t+ζ2)),T2=ε(-14(1-12ζ12)cos⁡⁡2(t+ζ2)+132ζ12cos⁡⁡4(t+ζ2))+ε2(-(164ζ12+1128ζ14)sin⁡2(t+ζ2)+(-1128ζ12+1512ζ14)sin⁡⁡4(t+ζ2)-1768ζ14sin⁡⁡6(t+ζ2)).
Equations (5.71) are in agreement with the solution obtained by Ahmed and Tapley [7] through a different integration theory for the Hori method.
A second-order asymptotic solution for the original variablesx and ẋ is calculated through (2.7), and it is given by
(5.72)x=ζ1cos⁡(t+ζ2)+ε{[-14ζ1+116ζ13]sin⁡(t+ζ2)-132ζ13sin⁡3(t+ζ2)}+ε2{[132ζ1-132ζ13+152048ζ15]cos⁡(t+ζ2)+[-132ζ13+51024ζ15]×cos⁡3(t+ζ2)-53072ζ15cos⁡5(t+ζ2)},ẋ=-ζ1sin⁡(t+ζ2)+ε{[14ζ1-116ζ13]cos⁡(t+ζ2)-332ζ13cos⁡3(t+ζ2)}+ε2{[-132ζ1-132ζ13+252048ζ15]sin⁡(t+ζ2)+[364ζ13-31024ζ15]×sin⁡3(t+ζ2)+253072ζ15sin⁡5(t+ζ2)},
with ζ1 and ζ2 given by the solution of (5.70).
As before, note that (5.37) and (5.72) give the same second-order asymptotic solution for the van der Pol equation. Recall that ζ1 and ζ2 have different meaning in these equations, but they are related through an equation similar to (4.21).


#### 5.2.2. Duffing Equation

For the Duffing equation, the functionf(x,ẋ) is written in terms of the variables z1=c′and z2=θ′, by(5.73)f(x,ẋ)=f(z1cos⁡(t+z2),-z1sin⁡(t+z2))=-z13co⁡s3(t+z2).
Thus, it follows from (4.30) that(5.74)Z(1)=[14z13sin⁡2(t+z2)+18z13sin⁡4(t+z2)38z12+12z12cos⁡2(t+z2)+18z12cos⁡4(t+z2)].Following the simplified algorithm II and repeating the procedure described in Section5.2.1, the first-order terms Z*(1) and T(1) are obtained as follows. Taking the secular part of Z*(1), with ζ replacing z, and, integrating the remaining part, one finds(5.75)Z*(1)=[038ζ12],(5.76)T(1)=[-132ζ13(4cos⁡2(t+ζ2)+cos⁡4(t+ζ2))132ζ12(8sin⁡2(t+ζ2)+sin⁡4(t+ζ2))].In the second-order approximation, one finds(5.77)Ψ(2)=[1256ζ15(-33sin⁡2(t+ζ2)-12sin⁡4(t+ζ2)+3sin⁡6(t+ζ2))1256ζ14(-51-99cos⁡2(t+ζ2)-18cos⁡4(t+ζ2)+3cos⁡6(t+ζ2))].
Taking the secular part of Ψ(2), and, integrating the remaining part, one finds(5.78)Z*(2)=[0-51256ζ14],(5.79)T(2)=[1512ζ15(33cos⁡2(t+ζ2)+6cos⁡4(t+ζ2)-cos⁡6(t+ζ2))1512ζ14(-99sin⁡2(t+ζ2)-9sin⁡4(t+ζ2)+sin⁡6(t+ζ2))].The new system of differential equations is given, up to the second-order of the small parameterε, by(5.80)dζ1dt=0,dζ2dt=ε38ζ12-ε251256ζ14.
These differential equations are exactly the same equations given by (5.52).The generating function is obtained from (5.76) and (5.79), and it is given, up to the second-order of the small parameter ε, by
(5.81)T1=-ε132ζ13(4cos⁡2(t+ζ2)+cos⁡4(t+ζ2))+ε21512ζ15(33cos⁡2(t+ζ2)+6cos⁡4(t+ζ2)-cos⁡6(t+ζ2)),T2=ε132ζ12(8sin⁡2(t+ζ2)+sin⁡4(t+ζ2))+ε21512ζ14(-99sin⁡2(t+ζ2)-9sin⁡4(t+ζ2)+sin⁡6(t+ζ2)).A second-order asymptotic solution for the original variablesx and ẋ is calculated through (2.7), and it is given by
(5.82)x=ζ1cos⁡(t+ζ2)+ε132ζ13(-6cos⁡(t+ζ2)+cos⁡3(t+ζ2))+ε212048ζ15(303cos⁡(t+ζ2)-78cos⁡3(t+ζ2)+2cos⁡5(t+ζ2)),ẋ=-ζ1sin⁡(t+ζ2)-ε132ζ13(6sin⁡(t+ζ2)+3sin⁡3(t+ζ2))+ε212048ζ15(249sin⁡(t+ζ2)+162sin⁡3(t+ζ2)-10sin⁡5(t+ζ2)).
These equations are in agreement with the solution obtained through the canonical version of the Hori method [6]. Note that (5.53) and (5.82) give the same second-order asymptotic solution for the Duffing equation. Recall that ζ1 and ζ2 have different meaning in these equations, but they are related through an equation similar to (4.21).


## 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:(5.1)ẍ+ε(x2-1)ẋ+x=0.
Introducing the variables z1=x and z2=ẋ, this equation can be written in the form:(5.2)dz1dt=z2,dz1dt=-z1-ε(z12-1)z2.
Thus(5.3)Z(0)=[z2-z1],(5.4)Z(1)=[0-(z12-1)z2].As described in preceding paragraphs, two different choices ofD(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
(5.5)Z(1)=[0(-c+14c3)sin⁡(t+θ)+14c3sin⁡3(t+θ)].
ComputingΔζ-1Z(1),
(5.6)Δζ-1Z(1)=[12c(1-14c2)-12ccos⁡2(t+θ)+18c3cos⁡4(t+θ)12(1-12c2)sin⁡2(t+θ)-18c2sin⁡4(t+θ)],
and taking its secular part, one finds
(5.7)〈Δζ-1Z(1)〉=[12c(1-14c2)0].
From (4.15) and (4.22), it follows that Z*(1) is given by
(5.8)Z*(1)=[12c(1-14c2)cos⁡(t+θ)-12c(1-14c2)sin⁡(t+θ)].
In view of (4.21), Z*(1) can be written explicitly in terms of the new variables ζ1 and ζ2 as follows:
(5.9)Z*(1)=[12ζ1(1-14(ζ12+ζ22))12ζ2(1-14(ζ12+ζ22))].
To determineT(1), the indefinite integral ∫⌊Δζ-1Z(1)-〈Δζ-1Z(1)〉⌋dt is calculated:
(5.10)∫[Δζ-1Z(1)-〈Δζ-1Z(1)〉]dt=[-14csin⁡2(t+θ)+132c3sin⁡4(t+θ)-(14-18c2)cos⁡2(t+θ)+132c2cos⁡4(t+θ)].
Thus, from (4.18), it follows that T(1) is given by
(5.11)T(1)=[-14c(1-14c2)sin⁡(t+θ)-132c3sin⁡3(t+θ)14c(1-14c2)cos⁡(t+θ)-332c3cos⁡3(t+θ)]+ΔζD(1).
In view of (4.21), T(1) can be written explicitly in terms of the new variables ζ1 and ζ2 as follows:
(5.12)T(1)=[14ζ2(1+18(ζ12+ζ22))-18ζ2314ζ1(1+78(ζ12+ζ22))-38ζ13]+ΔζD(1),
with ΔζD(1) put in the form:
(5.13)ΔζD(1)=[d1(1)ζ1+d2(1)ζ2d1(1)ζ2-d2(1)ζ1],
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 Section2, the second-order equation, (2.11), involves the term Ψ(2) given by
(5.14)Ψ(2)=12[Z(1)+Z*(1),T(1)].
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=ζ12+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
(5.15)(Δζ-1Ψ(2))1=(764c3-3128c5)sin⁡2(t+θ)-132c3sin⁡4(t+θ)-1128c5sin⁡6(t+θ)+d1(1)(-14c3+18c3cos⁡4(t+θ))+d2(1)(12csin⁡2(t+θ)-14c3sin⁡4(t+θ))+dd1(1)dζ1(-14(c2-14c4)cos⁡(t+θ))+dd1(1)dζ2(34(c2-14c4)sin⁡(t+θ)-18c4sin⁡3(t+θ)),(Δζ-1Ψ(2))2=-18+316c2-11256c4-(132c2+164c4)cos⁡2(t+θ)+(-132c2+1128c4)cos⁡4(t+θ)-1128c4cos⁡6(t+θ)+d1(1)(-14c2sin⁡2(t+θ)-18c2sin⁡4(t+θ))+d2(1)((12-14c2)cos⁡2(t+θ)-14c2cos⁡4(t+θ))+dd2(1)dζ1(-14(c-14c3)cos⁡(t+θ))+dd2(1)dζ2(34(c-14c3)sin⁡(t+θ)-18c3sin⁡3(t+θ)).
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
(5.16)d(1)=[0-14+116c2],
it follows from (5.15) that
(5.17)〈Δζ-1Ψ(2)〉=[0-18+18c2-7256c4].
From (4.15), (4.21), and (5.17), one finds
(5.18)Z*(2)=[-18ζ2(1-(ζ12+ζ22)+732(ζ12+ζ22)2)18ζ1(1-(ζ12+ζ22)+732(ζ12+ζ22)2)].
In view of the choice the auxiliary vectord(1), (5.12) can be simplified, and T(1) is then given by
(5.19)T(1)=[132ζ2(3ζ12-ζ22)132ζ1(16-7ζ12+5ζ22)].
Computing the indefinite integral∫⌊Δζ-1Ψ(2)-〈Δζ-1Ψ(2)〉⌋dt and substituting the general solution of the new undisturbed system, it follows that T(2) is given by
(5.20)T(2)=[116ζ1-564ζ13+13768ζ15+196ζ13ζ22+11768ζ1ζ24-116ζ2+364ζ12ζ2+116ζ23-29768ζ2ζ14-5192ζ12ζ23-7768ζ25]+ΔζD(2),
with ΔζD(2) put in the form:
(5.21)ΔζD(2)=[d1(2)ζ1+d2(2)ζ2d1(2)ζ2-d2(2)ζ1],
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:
(5.22)d(2)=[-116+15256c2-7512c40].
Accordingly, T(2) is given by
(5.23)T(2)=[51536ζ15-13768ζ13ζ22+11536ζ1ζ24-5256ζ13+15256ζ1ζ22-351536ζ25-41768ζ12ζ23-791536ζ2ζ14+31256ζ23+27256ζ12ζ2-18ζ2].
The new system of differential equations and the generating function are given, up to the second-order of the small parameter, by(5.24)dζ1dt=ζ2+ε12ζ1(1-14(ζ12+ζ22))-ε218ζ2(1-(ζ12+ζ22)+732(ζ12+ζ22)2),dζ2dt=-ζ1+ε12ζ2(1-14(ζ12+ζ22))+ε218ζ1(1-(ζ12+ζ22)+732(ζ12+ζ22)2),(5.25)T1=ε132ζ2(3ζ12-ζ22)+ε2(51536ζ15-13768ζ13ζ22+11536ζ1ζ24-5256ζ13+15256ζ1ζ22),T2=ε132ζ1(16-7ζ12+5ζ22)+ε2(-351536ζ25-41768ζ12ζ23-791536ζ2ζ14+31256ζ23+27256ζ12ζ2-18ζ2).
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
(5.26)dCdt=Δζ-1R*,
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:
(5.27)dCdt=∑m=1εm〈Δζ-1ΨI(m)〉.
Accordingly, the Lagrange variational equations for the new system of differential equations, (5.24), are given by(5.28a)dcdt=εc2(1-14c2),(5.28b)dθdt=ε2(-18+18c2-7256c4).
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 originalvariablesx and ẋ are calculated through (2.6), and the second-order asymptotic solution is
(5.29)x=ζ1+ε132ζ2(3ζ12-ζ22)+ε2(-436144ζ15+293072ζ13ζ22-596144ζ1ζ24+1256ζ13+9256ζ1ζ22),ẋ=ζ2+ε132ζ1(16-7ζ12+5ζ22)+ε2(-1556144ζ25-353072ζ23ζ12-7156144ζ2ζ14+29256ζ23+53256ζ2ζ12-18ζ2).
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 vectord(1). Taking d(1) as a null vector, it follows straightforwardly from (5.15) that
(5.30)〈Δζ-1Ψ(2)〉=[0-18+316c2-11256c4].
Thus, from (4.15), (4.21), (5.28a), and (5.28b), one finds
(5.31)Z*(2)=[-18ζ2(1-32(ζ12+ζ22)+1132(ζ12+ζ22)2)18ζ1(1-32(ζ12+ζ22)+1132(ζ12+ζ22)2)].
Sinced(1) is a null vector, (5.12) simplifies, and T(1) is given by
(5.32)T(1)=[14ζ2(1+18(ζ12+ζ22))-18ζ2314ζ1(1+78(ζ12+ζ22))-38ζ13].
Now, repeating the procedure described in the previous section, that is, computing the indefinite integral∫⌊Δζ-1Ψ(2)-〈Δζ-1Ψ(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
(5.33)T(2)=[-364ζ13+116ζ1ζ22+5384ζ15-1192ζ13ζ22+7384ζ1ζ24-764ζ12ζ2+116ζ23-1384ζ2ζ14-196ζ12ζ23-5384ζ25].
So, the new system of differential equations and the generating function are given, up to the second-order of the small parameterε, by
(5.34)dζ1dt=ζ2+ε12ζ1(1-14(ζ12+ζ22))-ε218ζ2(1-32(ζ12+ζ22)+1132(ζ12+ζ22)2),dζ2dt=-ζ1+ε12ζ2(1-14(ζ12+ζ22))+ε218ζ1(1-32(ζ12+ζ22)+1132(ζ12+ζ22)2),(5.35)T1=ε(14ζ2(1+18(ζ12+ζ22))-18ζ23)+ε2(-364ζ13+116ζ1ζ22+5384ζ15-1192ζ13ζ22+7384ζ1ζ24),T2=ε(14ζ1(1+78(ζ12+ζ22))-38ζ13)+ε2(-764ζ12ζ2+116ζ23-1384ζ2ζ14-196ζ12ζ23-5384ζ25).
The Lagrange variational equations for the new system of differential equations, defined by (5.34), are given by(5.36a)dcdt=εc2(1-14c2),(5.36b)dθdt=ε2(-18+316c2-11256c4).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 variablesx and ẋ are calculated through (2.6), which provides the following second-order asymptotic solution,
(5.37)x=ζ1+ε14ζ2(1+18(ζ12-3ζ22))+ε2(656144ζ15+653072ζ13ζ22-956144ζ1ζ24-116ζ13+116ζ1ζ22+132ζ1),ẋ=ζ2+ε14ζ1(1-18(5ζ12-7ζ22))+ε2(-1436144ζ25+1933072ζ23ζ12-2716144ζ2ζ14+564ζ23-764ζ2ζ12+132ζ2).
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:(5.38)ẍ+εx3+x=0.Introducing the variables z1=x and z2=ẋ, this equation can be written in the form:(5.39)dz1dt=z2,dz2dt=-z1-εz13.Thus,(5.40)Z(1)=[0-z13].Following the simplified algorithm I and repeating the procedure described in Section5.1.1, 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 gets(5.41)〈Δζ-1Z(1)〉=[038c2].
Thus, from (4.15) and (5.41), it follows that Z*(1) is given by(5.42)Z*(1)=[-38c3sin⁡(t+θ)-38c3cos⁡(t+θ)].In view of (4.21), Z*(1) can be written explicitly in terms of the new variables ζ1 and ζ2:(5.43)Z*(1)=[38ζ2(ζ12+ζ22)-38ζ1(ζ12+ζ22)].Calculating the indefinite integral∫⌊Δζ-1Z(1)-〈Δζ-1Z(1)〉⌋dt, one finds(5.44)∫[Δζ-1Z(1)-〈Δζ-1Z(1)〉]dt=[-132c3(4cos⁡2(t+θ)+cos⁡4(t+θ))132c2(8sin⁡2(t+θ)+sin⁡4(t+θ))].Multiplying this result byΔζ, it follows, according to (4.18), that T(1) is given by(5.45)T(1)=[-316c3cos⁡(t+θ)+132c3cos⁡3(t+θ)-316c3sin⁡(t+θ)-332c3sin⁡3(t+θ)]+ΔζD(1).TakingD(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:(5.46)T(1)=[-532ζ13-932ζ1ζ221532ζ12ζ2+332ζ23].In the second-order approximation, one finds after lengthy calculations using MAPLE software:(5.47)Ψ(2)=[-69256ζ14ζ2+27128ζ12ζ23+27256ζ25165256ζ15+69128ζ13ζ22-27256ζ24ζ1].Repeating the procedure described in the above paragraphs, one finds(5.48)〈Δζ-1Ψ(2)〉=[0-51256c4],Z*(2)=[-51256ζ2(ζ12+ζ22)251256ζ1(ζ12+ζ22)2].
Taking D(2) as a null vector, it follows that(5.49)T(2)=[19256ζ15+1332ζ13ζ22+65256ζ1ζ24-95256ζ2ζ14-1332ζ12ζ23-13256ζ25].The new system of differential equations and the generating function are given, up to the second-order of the small parameterε, by
(5.50)dζ1dt=ζ2+ε38ζ2(ζ12+ζ22)-ε251256ζ2(ζ12+ζ22)2,dζ2dt=-ζ1-ε38ζ1(ζ12+ζ22)+ε251256ζ1(ζ12+ζ22)2,(5.51)T1=-ε(532ζ13+932ζ1ζ22)+ε2(19256ζ15+1332ζ13ζ22+65256ζ1ζ24),T2=ε(1532ζ12ζ2+332ζ23)+ε2(-95256ζ2ζ14-1332ζ12ζ23-13256ζ25).The Lagrange variational equations for the new system of differential equations, defined by (5.50), are given by
(5.52)dcdt=0,dθdt=ε38c2-ε251256c4.
These differential equations are the well-known equations obtained through Krylov-Bogoliubov method [5].As described in Section5.1.1, 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 variablesx and ẋ are calculated through (2.6), which provides the following second-order asymptotic solution:
(5.53)x=ζ1-ε132ζ1(5ζ12+9ζ22)+ε212048(227ζ15+742ζ13ζ22+547ζ1ζ24),ẋ=ζ2+ε332ζ2(5ζ12+ζ22)-ε212048(77ζ25+922ζ23ζ12+685ζ2ζ14).
These equations are in agreement with the solution obtained through the canonical version of the Hori method [6].


## 5.1.1. Van der Pol Equation

Consider the well-known van der Pol equation:(5.1)ẍ+ε(x2-1)ẋ+x=0.
Introducing the variables z1=x and z2=ẋ, this equation can be written in the form:(5.2)dz1dt=z2,dz1dt=-z1-ε(z12-1)z2.
Thus(5.3)Z(0)=[z2-z1],(5.4)Z(1)=[0-(z12-1)z2].As described in preceding paragraphs, two different choices ofD(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
(5.5)Z(1)=[0(-c+14c3)sin⁡(t+θ)+14c3sin⁡3(t+θ)].
ComputingΔζ-1Z(1),
(5.6)Δζ-1Z(1)=[12c(1-14c2)-12ccos⁡2(t+θ)+18c3cos⁡4(t+θ)12(1-12c2)sin⁡2(t+θ)-18c2sin⁡4(t+θ)],
and taking its secular part, one finds
(5.7)〈Δζ-1Z(1)〉=[12c(1-14c2)0].
From (4.15) and (4.22), it follows that Z*(1) is given by
(5.8)Z*(1)=[12c(1-14c2)cos⁡(t+θ)-12c(1-14c2)sin⁡(t+θ)].
In view of (4.21), Z*(1) can be written explicitly in terms of the new variables ζ1 and ζ2 as follows:
(5.9)Z*(1)=[12ζ1(1-14(ζ12+ζ22))12ζ2(1-14(ζ12+ζ22))].
To determineT(1), the indefinite integral ∫⌊Δζ-1Z(1)-〈Δζ-1Z(1)〉⌋dt is calculated:
(5.10)∫[Δζ-1Z(1)-〈Δζ-1Z(1)〉]dt=[-14csin⁡2(t+θ)+132c3sin⁡4(t+θ)-(14-18c2)cos⁡2(t+θ)+132c2cos⁡4(t+θ)].
Thus, from (4.18), it follows that T(1) is given by
(5.11)T(1)=[-14c(1-14c2)sin⁡(t+θ)-132c3sin⁡3(t+θ)14c(1-14c2)cos⁡(t+θ)-332c3cos⁡3(t+θ)]+ΔζD(1).
In view of (4.21), T(1) can be written explicitly in terms of the new variables ζ1 and ζ2 as follows:
(5.12)T(1)=[14ζ2(1+18(ζ12+ζ22))-18ζ2314ζ1(1+78(ζ12+ζ22))-38ζ13]+ΔζD(1),
with ΔζD(1) put in the form:
(5.13)ΔζD(1)=[d1(1)ζ1+d2(1)ζ2d1(1)ζ2-d2(1)ζ1],
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 Section2, the second-order equation, (2.11), involves the term Ψ(2) given by
(5.14)Ψ(2)=12[Z(1)+Z*(1),T(1)].
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=ζ12+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
(5.15)(Δζ-1Ψ(2))1=(764c3-3128c5)sin⁡2(t+θ)-132c3sin⁡4(t+θ)-1128c5sin⁡6(t+θ)+d1(1)(-14c3+18c3cos⁡4(t+θ))+d2(1)(12csin⁡2(t+θ)-14c3sin⁡4(t+θ))+dd1(1)dζ1(-14(c2-14c4)cos⁡(t+θ))+dd1(1)dζ2(34(c2-14c4)sin⁡(t+θ)-18c4sin⁡3(t+θ)),(Δζ-1Ψ(2))2=-18+316c2-11256c4-(132c2+164c4)cos⁡2(t+θ)+(-132c2+1128c4)cos⁡4(t+θ)-1128c4cos⁡6(t+θ)+d1(1)(-14c2sin⁡2(t+θ)-18c2sin⁡4(t+θ))+d2(1)((12-14c2)cos⁡2(t+θ)-14c2cos⁡4(t+θ))+dd2(1)dζ1(-14(c-14c3)cos⁡(t+θ))+dd2(1)dζ2(34(c-14c3)sin⁡(t+θ)-18c3sin⁡3(t+θ)).
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
(5.16)d(1)=[0-14+116c2],
it follows from (5.15) that
(5.17)〈Δζ-1Ψ(2)〉=[0-18+18c2-7256c4].
From (4.15), (4.21), and (5.17), one finds
(5.18)Z*(2)=[-18ζ2(1-(ζ12+ζ22)+732(ζ12+ζ22)2)18ζ1(1-(ζ12+ζ22)+732(ζ12+ζ22)2)].
In view of the choice the auxiliary vectord(1), (5.12) can be simplified, and T(1) is then given by
(5.19)T(1)=[132ζ2(3ζ12-ζ22)132ζ1(16-7ζ12+5ζ22)].
Computing the indefinite integral∫⌊Δζ-1Ψ(2)-〈Δζ-1Ψ(2)〉⌋dt and substituting the general solution of the new undisturbed system, it follows that T(2) is given by
(5.20)T(2)=[116ζ1-564ζ13+13768ζ15+196ζ13ζ22+11768ζ1ζ24-116ζ2+364ζ12ζ2+116ζ23-29768ζ2ζ14-5192ζ12ζ23-7768ζ25]+ΔζD(2),
with ΔζD(2) put in the form:
(5.21)ΔζD(2)=[d1(2)ζ1+d2(2)ζ2d1(2)ζ2-d2(2)ζ1],
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:
(5.22)d(2)=[-116+15256c2-7512c40].
Accordingly, T(2) is given by
(5.23)T(2)=[51536ζ15-13768ζ13ζ22+11536ζ1ζ24-5256ζ13+15256ζ1ζ22-351536ζ25-41768ζ12ζ23-791536ζ2ζ14+31256ζ23+27256ζ12ζ2-18ζ2].
The new system of differential equations and the generating function are given, up to the second-order of the small parameter, by(5.24)dζ1dt=ζ2+ε12ζ1(1-14(ζ12+ζ22))-ε218ζ2(1-(ζ12+ζ22)+732(ζ12+ζ22)2),dζ2dt=-ζ1+ε12ζ2(1-14(ζ12+ζ22))+ε218ζ1(1-(ζ12+ζ22)+732(ζ12+ζ22)2),(5.25)T1=ε132ζ2(3ζ12-ζ22)+ε2(51536ζ15-13768ζ13ζ22+11536ζ1ζ24-5256ζ13+15256ζ1ζ22),T2=ε132ζ1(16-7ζ12+5ζ22)+ε2(-351536ζ25-41768ζ12ζ23-791536ζ2ζ14+31256ζ23+27256ζ12ζ2-18ζ2).
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
(5.26)dCdt=Δζ-1R*,
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:
(5.27)dCdt=∑m=1εm〈Δζ-1ΨI(m)〉.
Accordingly, the Lagrange variational equations for the new system of differential equations, (5.24), are given by(5.28a)dcdt=εc2(1-14c2),(5.28b)dθdt=ε2(-18+18c2-7256c4).
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 originalvariablesx and ẋ are calculated through (2.6), and the second-order asymptotic solution is
(5.29)x=ζ1+ε132ζ2(3ζ12-ζ22)+ε2(-436144ζ15+293072ζ13ζ22-596144ζ1ζ24+1256ζ13+9256ζ1ζ22),ẋ=ζ2+ε132ζ1(16-7ζ12+5ζ22)+ε2(-1556144ζ25-353072ζ23ζ12-7156144ζ2ζ14+29256ζ23+53256ζ2ζ12-18ζ2).
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 vectord(1). Taking d(1) as a null vector, it follows straightforwardly from (5.15) that
(5.30)〈Δζ-1Ψ(2)〉=[0-18+316c2-11256c4].
Thus, from (4.15), (4.21), (5.28a), and (5.28b), one finds
(5.31)Z*(2)=[-18ζ2(1-32(ζ12+ζ22)+1132(ζ12+ζ22)2)18ζ1(1-32(ζ12+ζ22)+1132(ζ12+ζ22)2)].
Sinced(1) is a null vector, (5.12) simplifies, and T(1) is given by
(5.32)T(1)=[14ζ2(1+18(ζ12+ζ22))-18ζ2314ζ1(1+78(ζ12+ζ22))-38ζ13].
Now, repeating the procedure described in the previous section, that is, computing the indefinite integral∫⌊Δζ-1Ψ(2)-〈Δζ-1Ψ(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
(5.33)T(2)=[-364ζ13+116ζ1ζ22+5384ζ15-1192ζ13ζ22+7384ζ1ζ24-764ζ12ζ2+116ζ23-1384ζ2ζ14-196ζ12ζ23-5384ζ25].
So, the new system of differential equations and the generating function are given, up to the second-order of the small parameterε, by
(5.34)dζ1dt=ζ2+ε12ζ1(1-14(ζ12+ζ22))-ε218ζ2(1-32(ζ12+ζ22)+1132(ζ12+ζ22)2),dζ2dt=-ζ1+ε12ζ2(1-14(ζ12+ζ22))+ε218ζ1(1-32(ζ12+ζ22)+1132(ζ12+ζ22)2),(5.35)T1=ε(14ζ2(1+18(ζ12+ζ22))-18ζ23)+ε2(-364ζ13+116ζ1ζ22+5384ζ15-1192ζ13ζ22+7384ζ1ζ24),T2=ε(14ζ1(1+78(ζ12+ζ22))-38ζ13)+ε2(-764ζ12ζ2+116ζ23-1384ζ2ζ14-196ζ12ζ23-5384ζ25).
The Lagrange variational equations for the new system of differential equations, defined by (5.34), are given by(5.36a)dcdt=εc2(1-14c2),(5.36b)dθdt=ε2(-18+316c2-11256c4).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 variablesx and ẋ are calculated through (2.6), which provides the following second-order asymptotic solution,
(5.37)x=ζ1+ε14ζ2(1+18(ζ12-3ζ22))+ε2(656144ζ15+653072ζ13ζ22-956144ζ1ζ24-116ζ13+116ζ1ζ22+132ζ1),ẋ=ζ2+ε14ζ1(1-18(5ζ12-7ζ22))+ε2(-1436144ζ25+1933072ζ23ζ12-2716144ζ2ζ14+564ζ23-764ζ2ζ12+132ζ2).
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:(5.38)ẍ+εx3+x=0.Introducing the variables z1=x and z2=ẋ, this equation can be written in the form:(5.39)dz1dt=z2,dz2dt=-z1-εz13.Thus,(5.40)Z(1)=[0-z13].Following the simplified algorithm I and repeating the procedure described in Section5.1.1, 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 gets(5.41)〈Δζ-1Z(1)〉=[038c2].
Thus, from (4.15) and (5.41), it follows that Z*(1) is given by(5.42)Z*(1)=[-38c3sin⁡(t+θ)-38c3cos⁡(t+θ)].In view of (4.21), Z*(1) can be written explicitly in terms of the new variables ζ1 and ζ2:(5.43)Z*(1)=[38ζ2(ζ12+ζ22)-38ζ1(ζ12+ζ22)].Calculating the indefinite integral∫⌊Δζ-1Z(1)-〈Δζ-1Z(1)〉⌋dt, one finds(5.44)∫[Δζ-1Z(1)-〈Δζ-1Z(1)〉]dt=[-132c3(4cos⁡2(t+θ)+cos⁡4(t+θ))132c2(8sin⁡2(t+θ)+sin⁡4(t+θ))].Multiplying this result byΔζ, it follows, according to (4.18), that T(1) is given by(5.45)T(1)=[-316c3cos⁡(t+θ)+132c3cos⁡3(t+θ)-316c3sin⁡(t+θ)-332c3sin⁡3(t+θ)]+ΔζD(1).TakingD(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:(5.46)T(1)=[-532ζ13-932ζ1ζ221532ζ12ζ2+332ζ23].In the second-order approximation, one finds after lengthy calculations using MAPLE software:(5.47)Ψ(2)=[-69256ζ14ζ2+27128ζ12ζ23+27256ζ25165256ζ15+69128ζ13ζ22-27256ζ24ζ1].Repeating the procedure described in the above paragraphs, one finds(5.48)〈Δζ-1Ψ(2)〉=[0-51256c4],Z*(2)=[-51256ζ2(ζ12+ζ22)251256ζ1(ζ12+ζ22)2].
Taking D(2) as a null vector, it follows that(5.49)T(2)=[19256ζ15+1332ζ13ζ22+65256ζ1ζ24-95256ζ2ζ14-1332ζ12ζ23-13256ζ25].The new system of differential equations and the generating function are given, up to the second-order of the small parameterε, by
(5.50)dζ1dt=ζ2+ε38ζ2(ζ12+ζ22)-ε251256ζ2(ζ12+ζ22)2,dζ2dt=-ζ1-ε38ζ1(ζ12+ζ22)+ε251256ζ1(ζ12+ζ22)2,(5.51)T1=-ε(532ζ13+932ζ1ζ22)+ε2(19256ζ15+1332ζ13ζ22+65256ζ1ζ24),T2=ε(1532ζ12ζ2+332ζ23)+ε2(-95256ζ2ζ14-1332ζ12ζ23-13256ζ25).The Lagrange variational equations for the new system of differential equations, defined by (5.50), are given by
(5.52)dcdt=0,dθdt=ε38c2-ε251256c4.
These differential equations are the well-known equations obtained through Krylov-Bogoliubov method [5].As described in Section5.1.1, 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 variablesx and ẋ are calculated through (2.6), which provides the following second-order asymptotic solution:
(5.53)x=ζ1-ε132ζ1(5ζ12+9ζ22)+ε212048(227ζ15+742ζ13ζ22+547ζ1ζ24),ẋ=ζ2+ε332ζ2(5ζ12+ζ22)-ε212048(77ζ25+922ζ23ζ12+685ζ2ζ14).
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 functionf(x,ẋ) is written in terms of the variables z1=c′ and z2=θ′ by(5.54)f(x,ẋ)=f(z1cos⁡(t+z2),-z1sin⁡(t+z2))=-(z12co⁡s2(t+z2)-1)z1sin⁡(t+z2).Thus, it follows from (4.30) that(5.55)Z(1)=[12z1(1-14z12-cos⁡2(t+z2)+14z12cos⁡4(t+z2))12(1-12z12)sin⁡2(t+z2)-18z12sin⁡4(t+z2)].
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 ofZ(1), with ζ replacing z, one finds
(5.56)Z*(1)=[12ζ1(1-14ζ12)0],
and, integrating the remaining part,
(5.57)T(1)=[-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2)-14(1-12ζ12)cos⁡2(t+ζ2)+132ζ12cos⁡4(t+ζ2)]+D(1),
with Di(1)=Di(1)(ζ1), i=1,2.
Following the algorithm of the Hori method described in Section2, the second-order equation, (2.11), involves the term Ψ(2) given by
(5.58)Ψ(2)=12[Z(1)+Z*(1),T(1)].
After tedious lengthy calculations using MAPLE software, one finds
(5.59)Ψ1(2)=(764ζ13-3128ζ15)sin⁡2(t+ζ2)-132ζ13sin⁡4(t+ζ2)-1128ζ15sin⁡6(t+ζ2)+D1(1)(12-38ζ12-14cos⁡2(t+ζ2)+316ζ12cos⁡4(t+ζ2))+D2(1)(12ζ1sin⁡2(t+ζ2)-14ζ13sin⁡4(t+ζ2))+dD1(1)dζ1(-12ζ1+18ζ13+14ζ1cos⁡2(t+ζ2)-116ζ13cos⁡4(t+ζ2)),Ψ2(2)=-18+316ζ12-11256ζ14-(132ζ12+164ζ14)cos⁡2(t+ζ2)+(-132ζ12+1128ζ14)cos⁡4(t+ζ2)-1128ζ14cos⁡6(t+ζ2)+D1(1)(-14ζ1sin⁡2(t+ζ2)-18ζ1sin⁡4(t+ζ2))+D2(1)((12-14ζ12)cos⁡2(t+ζ2)-14ζ12cos⁡4(t+ζ2))+dD2(1)dζ1(-12ζ1+18ζ13+14ζ1cos⁡2(t+ζ2)-116ζ13cos⁡4(t+ζ2)).
In order to obtain the same averaged Lagrange variational equations given by (5.28a) and (5.28b), D(1) must be taken as
(5.60)D(1)=[0-14+116ζ12].
Thus, it follows that
(5.61)Z*(2)=[0-18+18ζ12-7256ζ14].
In view of the choice ofD(1), T(1) is then given by
(5.62)T(1)=[-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2)-14+116ζ12-14(1-12ζ12)cos⁡2(t+ζ2)+132ζ12cos⁡4(t+ζ2)].
Repeating the procedure for the third-order approximation, and, taking
(5.63)D(2)=[-116ζ1+15256ζ13-7512ζ150],
one finds
(5.64)T1(2)=11536(-96ζ1+90ζ13-21ζ15)+(116ζ1-9128ζ13+9768ζ15)cos⁡2(t+ζ2)+(-1128ζ13+1256ζ15)cos⁡4(t+ζ2)+1768ζ15cos⁡6(t+ζ2),T2(2)=(-116+364ζ12-164ζ14)sin⁡2(t+ζ2)+(1128ζ12-1256ζ14)sin⁡4(t+ζ2)-1768ζ14sin⁡6(t+ζ2).
The new system of differential equations is given, up to the second-order of the small parameterε, by(5.65a)dζ1dt=ε12ζ1(1-14ζ12),(5.65b)dζ2dt=ε2(-18+18ζ12-7256ζ14).These differential equations are exactly the same equations given by (5.28a) and (5.28b).
The generating function is obtained from (5.62), (5.64), and it is given, up to the second-order of the small parameter ε, by
(5.66)T1=ε(-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2))+ε2(11536(-96ζ1+90ζ13-21ζ15)+(116ζ1-9128ζ13+9768ζ15)cos⁡2(t+ζ2)+(-1128ζ13+1256ζ15)cos⁡4(t+ζ2)+1768ζ15cos⁡6(t+ζ2)),T2=ε(116ζ12-14(1-12ζ12)cos⁡2(t+ζ2)+132ζ12cos⁡4(t+ζ2))+ε2((-116+364ζ12-164ζ14)sin⁡2(t+ζ2)+(1128ζ12-1256ζ14)×sin⁡4(t+ζ2)-1768ζ14sin⁡6(t+ζ2)).
The original variablesx and ẋ are calculated through (2.7), which provides, up to the second-order of the small parameter, the following solution:
(5.67)x=ζ1cos⁡(t+ζ2)-ε132ζ13sin⁡3(t+ζ2)+ε2{[3256ζ13-92048ζ15]cos⁡(t+ζ2)-[1128ζ13+11024ζ15]cos⁡3(t+ζ2)-53072ζ15cos⁡5(t+ζ2)},ẋ=-ζ1sin⁡⁡(t+ζ2)+ε{[12ζ1-18ζ13]cos⁡⁡(t+ζ2)-332ζ13cos⁡⁡3(t+ζ2)}+ε2{[18ζ1-35256ζ13+652048ζ15]sin⁡⁡(t+ζ2)-[3128ζ13-151024ζ15]×sin⁡⁡3(t+ζ2)+253072ζ15sin⁡5(t+ζ2)},
with ζ1 and ζ2 given by the solution of (5.65a) and (5.65b).Note that (5.29) and (5.67) give the same second-order asymptotic solution for the van der Pol equation. Recall that ζ1 and ζ2 have different meaning in these equations, but they are related through an equation similar to (4.21).(2) Second Asymptotic Solution
Now, let us to takeD(1) and D(2) as null vectors. Equations (5.59) simplifies, and Z*(2) is given by(5.68)Z*(2)=[0-18+316ζ12-11256ζ14].
The functionsT(1) and T(2) are then given by
(5.69)T(1)=[-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2)-14(1-12ζ12)cos⁡2(t+ζ2)+132ζ12cos⁡4(t+ζ2)],T1(2)=-(7128ζ13-3256ζ15)cos⁡2(t+ζ2)+1128ζ13cos⁡4(t+ζ2)+1768ζ15cos⁡6(t+ζ2),T2(2)=-(164ζ12+1128ζ14)sin⁡2(t+ζ2)+(-1128ζ12+1512ζ14)sin⁡4(t+ζ2)-1768ζ14sin⁡6(t+ζ2).
The new system of differential equations is obtained from (5.56) and (5.68), and it is given, up to the second-order of the small parameter, by
(5.70)dζ1dt=ε12ζ1(1-14ζ12),dζ2dt=ε2(-18+316ζ12-11256ζ14).
These differential equations are exactly the same equations given by (5.36a) and (5.36b).
The generating function is obtained from (5.69), and it is given, up to the second-order of the small parameter ε, by
(5.71)T1=ε(-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2))+ε2(-(7128ζ13-3256ζ15)cos⁡2(t+ζ2)+1128ζ13cos⁡4(t+ζ2)+1768ζ15cos⁡⁡6(t+ζ2)),T2=ε(-14(1-12ζ12)cos⁡⁡2(t+ζ2)+132ζ12cos⁡⁡4(t+ζ2))+ε2(-(164ζ12+1128ζ14)sin⁡2(t+ζ2)+(-1128ζ12+1512ζ14)sin⁡⁡4(t+ζ2)-1768ζ14sin⁡⁡6(t+ζ2)).
Equations (5.71) are in agreement with the solution obtained by Ahmed and Tapley [7] through a different integration theory for the Hori method.
A second-order asymptotic solution for the original variablesx and ẋ is calculated through (2.7), and it is given by
(5.72)x=ζ1cos⁡(t+ζ2)+ε{[-14ζ1+116ζ13]sin⁡(t+ζ2)-132ζ13sin⁡3(t+ζ2)}+ε2{[132ζ1-132ζ13+152048ζ15]cos⁡(t+ζ2)+[-132ζ13+51024ζ15]×cos⁡3(t+ζ2)-53072ζ15cos⁡5(t+ζ2)},ẋ=-ζ1sin⁡(t+ζ2)+ε{[14ζ1-116ζ13]cos⁡(t+ζ2)-332ζ13cos⁡3(t+ζ2)}+ε2{[-132ζ1-132ζ13+252048ζ15]sin⁡(t+ζ2)+[364ζ13-31024ζ15]×sin⁡3(t+ζ2)+253072ζ15sin⁡5(t+ζ2)},
with ζ1 and ζ2 given by the solution of (5.70).
As before, note that (5.37) and (5.72) give the same second-order asymptotic solution for the van der Pol equation. Recall that ζ1 and ζ2 have different meaning in these equations, but they are related through an equation similar to (4.21).


### 5.2.2. Duffing Equation

For the Duffing equation, the functionf(x,ẋ) is written in terms of the variables z1=c′and z2=θ′, by(5.73)f(x,ẋ)=f(z1cos⁡(t+z2),-z1sin⁡(t+z2))=-z13co⁡s3(t+z2).
Thus, it follows from (4.30) that(5.74)Z(1)=[14z13sin⁡2(t+z2)+18z13sin⁡4(t+z2)38z12+12z12cos⁡2(t+z2)+18z12cos⁡4(t+z2)].Following the simplified algorithm II and repeating the procedure described in Section5.2.1, the first-order terms Z*(1) and T(1) are obtained as follows. Taking the secular part of Z*(1), with ζ replacing z, and, integrating the remaining part, one finds(5.75)Z*(1)=[038ζ12],(5.76)T(1)=[-132ζ13(4cos⁡2(t+ζ2)+cos⁡4(t+ζ2))132ζ12(8sin⁡2(t+ζ2)+sin⁡4(t+ζ2))].In the second-order approximation, one finds(5.77)Ψ(2)=[1256ζ15(-33sin⁡2(t+ζ2)-12sin⁡4(t+ζ2)+3sin⁡6(t+ζ2))1256ζ14(-51-99cos⁡2(t+ζ2)-18cos⁡4(t+ζ2)+3cos⁡6(t+ζ2))].
Taking the secular part of Ψ(2), and, integrating the remaining part, one finds(5.78)Z*(2)=[0-51256ζ14],(5.79)T(2)=[1512ζ15(33cos⁡2(t+ζ2)+6cos⁡4(t+ζ2)-cos⁡6(t+ζ2))1512ζ14(-99sin⁡2(t+ζ2)-9sin⁡4(t+ζ2)+sin⁡6(t+ζ2))].The new system of differential equations is given, up to the second-order of the small parameterε, by(5.80)dζ1dt=0,dζ2dt=ε38ζ12-ε251256ζ14.
These differential equations are exactly the same equations given by (5.52).The generating function is obtained from (5.76) and (5.79), and it is given, up to the second-order of the small parameter ε, by
(5.81)T1=-ε132ζ13(4cos⁡2(t+ζ2)+cos⁡4(t+ζ2))+ε21512ζ15(33cos⁡2(t+ζ2)+6cos⁡4(t+ζ2)-cos⁡6(t+ζ2)),T2=ε132ζ12(8sin⁡2(t+ζ2)+sin⁡4(t+ζ2))+ε21512ζ14(-99sin⁡2(t+ζ2)-9sin⁡4(t+ζ2)+sin⁡6(t+ζ2)).A second-order asymptotic solution for the original variablesx and ẋ is calculated through (2.7), and it is given by
(5.82)x=ζ1cos⁡(t+ζ2)+ε132ζ13(-6cos⁡(t+ζ2)+cos⁡3(t+ζ2))+ε212048ζ15(303cos⁡(t+ζ2)-78cos⁡3(t+ζ2)+2cos⁡5(t+ζ2)),ẋ=-ζ1sin⁡(t+ζ2)-ε132ζ13(6sin⁡(t+ζ2)+3sin⁡3(t+ζ2))+ε212048ζ15(249sin⁡(t+ζ2)+162sin⁡3(t+ζ2)-10sin⁡5(t+ζ2)).
These equations are in agreement with the solution obtained through the canonical version of the Hori method [6]. Note that (5.53) and (5.82) give the same second-order asymptotic solution for the Duffing equation. Recall that ζ1 and ζ2 have different meaning in these equations, but they are related through an equation similar to (4.21).


## 5.2.1. Van der Pol Equation

For the van der Pol equation, the functionf(x,ẋ) is written in terms of the variables z1=c′ and z2=θ′ by(5.54)f(x,ẋ)=f(z1cos⁡(t+z2),-z1sin⁡(t+z2))=-(z12co⁡s2(t+z2)-1)z1sin⁡(t+z2).Thus, it follows from (4.30) that(5.55)Z(1)=[12z1(1-14z12-cos⁡2(t+z2)+14z12cos⁡4(t+z2))12(1-12z12)sin⁡2(t+z2)-18z12sin⁡4(t+z2)].
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 ofZ(1), with ζ replacing z, one finds
(5.56)Z*(1)=[12ζ1(1-14ζ12)0],
and, integrating the remaining part,
(5.57)T(1)=[-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2)-14(1-12ζ12)cos⁡2(t+ζ2)+132ζ12cos⁡4(t+ζ2)]+D(1),
with Di(1)=Di(1)(ζ1), i=1,2.
Following the algorithm of the Hori method described in Section2, the second-order equation, (2.11), involves the term Ψ(2) given by
(5.58)Ψ(2)=12[Z(1)+Z*(1),T(1)].
After tedious lengthy calculations using MAPLE software, one finds
(5.59)Ψ1(2)=(764ζ13-3128ζ15)sin⁡2(t+ζ2)-132ζ13sin⁡4(t+ζ2)-1128ζ15sin⁡6(t+ζ2)+D1(1)(12-38ζ12-14cos⁡2(t+ζ2)+316ζ12cos⁡4(t+ζ2))+D2(1)(12ζ1sin⁡2(t+ζ2)-14ζ13sin⁡4(t+ζ2))+dD1(1)dζ1(-12ζ1+18ζ13+14ζ1cos⁡2(t+ζ2)-116ζ13cos⁡4(t+ζ2)),Ψ2(2)=-18+316ζ12-11256ζ14-(132ζ12+164ζ14)cos⁡2(t+ζ2)+(-132ζ12+1128ζ14)cos⁡4(t+ζ2)-1128ζ14cos⁡6(t+ζ2)+D1(1)(-14ζ1sin⁡2(t+ζ2)-18ζ1sin⁡4(t+ζ2))+D2(1)((12-14ζ12)cos⁡2(t+ζ2)-14ζ12cos⁡4(t+ζ2))+dD2(1)dζ1(-12ζ1+18ζ13+14ζ1cos⁡2(t+ζ2)-116ζ13cos⁡4(t+ζ2)).
In order to obtain the same averaged Lagrange variational equations given by (5.28a) and (5.28b), D(1) must be taken as
(5.60)D(1)=[0-14+116ζ12].
Thus, it follows that
(5.61)Z*(2)=[0-18+18ζ12-7256ζ14].
In view of the choice ofD(1), T(1) is then given by
(5.62)T(1)=[-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2)-14+116ζ12-14(1-12ζ12)cos⁡2(t+ζ2)+132ζ12cos⁡4(t+ζ2)].
Repeating the procedure for the third-order approximation, and, taking
(5.63)D(2)=[-116ζ1+15256ζ13-7512ζ150],
one finds
(5.64)T1(2)=11536(-96ζ1+90ζ13-21ζ15)+(116ζ1-9128ζ13+9768ζ15)cos⁡2(t+ζ2)+(-1128ζ13+1256ζ15)cos⁡4(t+ζ2)+1768ζ15cos⁡6(t+ζ2),T2(2)=(-116+364ζ12-164ζ14)sin⁡2(t+ζ2)+(1128ζ12-1256ζ14)sin⁡4(t+ζ2)-1768ζ14sin⁡6(t+ζ2).
The new system of differential equations is given, up to the second-order of the small parameterε, by(5.65a)dζ1dt=ε12ζ1(1-14ζ12),(5.65b)dζ2dt=ε2(-18+18ζ12-7256ζ14).These differential equations are exactly the same equations given by (5.28a) and (5.28b).
The generating function is obtained from (5.62), (5.64), and it is given, up to the second-order of the small parameter ε, by
(5.66)T1=ε(-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2))+ε2(11536(-96ζ1+90ζ13-21ζ15)+(116ζ1-9128ζ13+9768ζ15)cos⁡2(t+ζ2)+(-1128ζ13+1256ζ15)cos⁡4(t+ζ2)+1768ζ15cos⁡6(t+ζ2)),T2=ε(116ζ12-14(1-12ζ12)cos⁡2(t+ζ2)+132ζ12cos⁡4(t+ζ2))+ε2((-116+364ζ12-164ζ14)sin⁡2(t+ζ2)+(1128ζ12-1256ζ14)×sin⁡4(t+ζ2)-1768ζ14sin⁡6(t+ζ2)).
The original variablesx and ẋ are calculated through (2.7), which provides, up to the second-order of the small parameter, the following solution:
(5.67)x=ζ1cos⁡(t+ζ2)-ε132ζ13sin⁡3(t+ζ2)+ε2{[3256ζ13-92048ζ15]cos⁡(t+ζ2)-[1128ζ13+11024ζ15]cos⁡3(t+ζ2)-53072ζ15cos⁡5(t+ζ2)},ẋ=-ζ1sin⁡⁡(t+ζ2)+ε{[12ζ1-18ζ13]cos⁡⁡(t+ζ2)-332ζ13cos⁡⁡3(t+ζ2)}+ε2{[18ζ1-35256ζ13+652048ζ15]sin⁡⁡(t+ζ2)-[3128ζ13-151024ζ15]×sin⁡⁡3(t+ζ2)+253072ζ15sin⁡5(t+ζ2)},
with ζ1 and ζ2 given by the solution of (5.65a) and (5.65b).Note that (5.29) and (5.67) give the same second-order asymptotic solution for the van der Pol equation. Recall that ζ1 and ζ2 have different meaning in these equations, but they are related through an equation similar to (4.21).(2) Second Asymptotic Solution
Now, let us to takeD(1) and D(2) as null vectors. Equations (5.59) simplifies, and Z*(2) is given by(5.68)Z*(2)=[0-18+316ζ12-11256ζ14].
The functionsT(1) and T(2) are then given by
(5.69)T(1)=[-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2)-14(1-12ζ12)cos⁡2(t+ζ2)+132ζ12cos⁡4(t+ζ2)],T1(2)=-(7128ζ13-3256ζ15)cos⁡2(t+ζ2)+1128ζ13cos⁡4(t+ζ2)+1768ζ15cos⁡6(t+ζ2),T2(2)=-(164ζ12+1128ζ14)sin⁡2(t+ζ2)+(-1128ζ12+1512ζ14)sin⁡4(t+ζ2)-1768ζ14sin⁡6(t+ζ2).
The new system of differential equations is obtained from (5.56) and (5.68), and it is given, up to the second-order of the small parameter, by
(5.70)dζ1dt=ε12ζ1(1-14ζ12),dζ2dt=ε2(-18+316ζ12-11256ζ14).
These differential equations are exactly the same equations given by (5.36a) and (5.36b).
The generating function is obtained from (5.69), and it is given, up to the second-order of the small parameter ε, by
(5.71)T1=ε(-14ζ1sin⁡2(t+ζ2)+132ζ13sin⁡4(t+ζ2))+ε2(-(7128ζ13-3256ζ15)cos⁡2(t+ζ2)+1128ζ13cos⁡4(t+ζ2)+1768ζ15cos⁡⁡6(t+ζ2)),T2=ε(-14(1-12ζ12)cos⁡⁡2(t+ζ2)+132ζ12cos⁡⁡4(t+ζ2))+ε2(-(164ζ12+1128ζ14)sin⁡2(t+ζ2)+(-1128ζ12+1512ζ14)sin⁡⁡4(t+ζ2)-1768ζ14sin⁡⁡6(t+ζ2)).
Equations (5.71) are in agreement with the solution obtained by Ahmed and Tapley [7] through a different integration theory for the Hori method.
A second-order asymptotic solution for the original variablesx and ẋ is calculated through (2.7), and it is given by
(5.72)x=ζ1cos⁡(t+ζ2)+ε{[-14ζ1+116ζ13]sin⁡(t+ζ2)-132ζ13sin⁡3(t+ζ2)}+ε2{[132ζ1-132ζ13+152048ζ15]cos⁡(t+ζ2)+[-132ζ13+51024ζ15]×cos⁡3(t+ζ2)-53072ζ15cos⁡5(t+ζ2)},ẋ=-ζ1sin⁡(t+ζ2)+ε{[14ζ1-116ζ13]cos⁡(t+ζ2)-332ζ13cos⁡3(t+ζ2)}+ε2{[-132ζ1-132ζ13+252048ζ15]sin⁡(t+ζ2)+[364ζ13-31024ζ15]×sin⁡3(t+ζ2)+253072ζ15sin⁡5(t+ζ2)},
with ζ1 and ζ2 given by the solution of (5.70).
As before, note that (5.37) and (5.72) give the same second-order asymptotic solution for the van der Pol equation. Recall that ζ1 and ζ2 have different meaning in these equations, but they are related through an equation similar to (4.21).


## 5.2.2. Duffing Equation

For the Duffing equation, the functionf(x,ẋ) is written in terms of the variables z1=c′and z2=θ′, by(5.73)f(x,ẋ)=f(z1cos⁡(t+z2),-z1sin⁡(t+z2))=-z13co⁡s3(t+z2).
Thus, it follows from (4.30) that(5.74)Z(1)=[14z13sin⁡2(t+z2)+18z13sin⁡4(t+z2)38z12+12z12cos⁡2(t+z2)+18z12cos⁡4(t+z2)].Following the simplified algorithm II and repeating the procedure described in Section5.2.1, the first-order terms Z*(1) and T(1) are obtained as follows. Taking the secular part of Z*(1), with ζ replacing z, and, integrating the remaining part, one finds(5.75)Z*(1)=[038ζ12],(5.76)T(1)=[-132ζ13(4cos⁡2(t+ζ2)+cos⁡4(t+ζ2))132ζ12(8sin⁡2(t+ζ2)+sin⁡4(t+ζ2))].In the second-order approximation, one finds(5.77)Ψ(2)=[1256ζ15(-33sin⁡2(t+ζ2)-12sin⁡4(t+ζ2)+3sin⁡6(t+ζ2))1256ζ14(-51-99cos⁡2(t+ζ2)-18cos⁡4(t+ζ2)+3cos⁡6(t+ζ2))].
Taking the secular part of Ψ(2), and, integrating the remaining part, one finds(5.78)Z*(2)=[0-51256ζ14],(5.79)T(2)=[1512ζ15(33cos⁡2(t+ζ2)+6cos⁡4(t+ζ2)-cos⁡6(t+ζ2))1512ζ14(-99sin⁡2(t+ζ2)-9sin⁡4(t+ζ2)+sin⁡6(t+ζ2))].The new system of differential equations is given, up to the second-order of the small parameterε, by(5.80)dζ1dt=0,dζ2dt=ε38ζ12-ε251256ζ14.
These differential equations are exactly the same equations given by (5.52).The generating function is obtained from (5.76) and (5.79), and it is given, up to the second-order of the small parameter ε, by
(5.81)T1=-ε132ζ13(4cos⁡2(t+ζ2)+cos⁡4(t+ζ2))+ε21512ζ15(33cos⁡2(t+ζ2)+6cos⁡4(t+ζ2)-cos⁡6(t+ζ2)),T2=ε132ζ12(8sin⁡2(t+ζ2)+sin⁡4(t+ζ2))+ε21512ζ14(-99sin⁡2(t+ζ2)-9sin⁡4(t+ζ2)+sin⁡6(t+ζ2)).A second-order asymptotic solution for the original variablesx and ẋ is calculated through (2.7), and it is given by
(5.82)x=ζ1cos⁡(t+ζ2)+ε132ζ13(-6cos⁡(t+ζ2)+cos⁡3(t+ζ2))+ε212048ζ15(303cos⁡(t+ζ2)-78cos⁡3(t+ζ2)+2cos⁡5(t+ζ2)),ẋ=-ζ1sin⁡(t+ζ2)-ε132ζ13(6sin⁡(t+ζ2)+3sin⁡3(t+ζ2))+ε212048ζ15(249sin⁡(t+ζ2)+162sin⁡3(t+ζ2)-10sin⁡5(t+ζ2)).
These equations are in agreement with the solution obtained through the canonical version of the Hori method [6]. Note that (5.53) and (5.82) give the same second-order asymptotic solution for the Duffing equation. Recall that ζ1 and ζ2 have different meaning in these equations, but they are related through an equation similar to (4.21).


## 6. Conclusions

In this paper, the Hori method for noncanonical systems is applied to theory of nonlinear oscillations. Two different simplified algorithms are derived from the general algorithm proposed by Sessin. It has been shown that themth-order terms Tj(m) and Zj*(m) that define the near-identity transformation and the new system of differential equations, respectively, are not uniquely determined, since the algorithms involve at each order arbitrary functions of the constants of integration of the general solution of the undisturbed system. This arbitrariness is an intrinsic characteristic of perturbation methods, since some kind of averaging principle must be applied to determine these functions. The simplified algorithms are then applied in determining second-order asymptotic solutions of two well-known equations in the theory of nonlinear oscillations: van der Pol and Duffing equations. For van der Pol equation, the appropriate use of the arbitrary functions allows the determination of the solution presented by Hori. This solution defines a new system of differential equations with a different frequency for the phase in comparison with the solution obtained by Ahmed and Tapley, who used a different approach for determining the near-identity transformation and the new system of differential equations for the Hori method, and, with the solution obtained by Nayfeh through the method of averaging. 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.



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*Source: 239357-2012-04-24.xml*