17. Asymptotic Expansion of Integral Manifold

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Consider a system of ordinary differential equations of the form

dx

dτ

= a(x, τ) + _a(x, ϕ, τ) + εA(x, ϕ, τ, ε),

dϕ

dτ

= ω(τ )

ε

+ b(x, ϕ, τ, ε),

(17.1)

where the functions a, _a, A, ω, and b are defined on the set (x, ϕ, τ, ε) ∈

D×Rm×R×(0, ε0] ≡ G, 2π-periodic in ϕν, ν = 1,m, and l ≥ 2 times continuously

differentiable with respect to x, ϕ, and τ for every fixed ε ∈ (0, ε0],

and all their partial derivatives are uniformly bounded in G by a constant c1.

We also assume that the function _a(x, ϕ, τ ) averaged with respect to ϕ over the

cube of periods is identically equal to zero and conditions (12.3), (13.2), (13.3),

and (16.2) are satisfied. Under these restrictions, in Sections 12–16 we have established

the existence of the integral manifold x = X(ψ, τ, ε) = x(τ)+Y (ψ, τ, ε)

Section 17 Asymptotic Expansion of Integral Manifold 195

l−1 times continuously differentiable with respect to (ψ, τ) ∈ Rm×R for every

ε ∈ (0, ε0] and such that the function Y (ψ, τ, ε) satisfies inequalities (16.45).

In the present section, we study the problem of the asymptotic expansion of

Y (ψ, τ, ε) as a function of the parameter ε in the form of a functional sum,

namely

Y (ψ, τ, ε) =

_r−1

ν=0

uν(ψ, τ, ε) + v(ψ, τ, ε), (17.2)

where uν and v are defined on the set G1 = Rm × R × (0, ε0] and satisfy the

estimates

_uν(ψ, τ, ε)_ ≤ σνε

ν

p, ν= 0, r − 1, _v(ψ, τ, ε)_ ≤ σrε

r

p (17.3)

for all (ψ, τ, ε) ∈ G1 and 2 ≤ r ≤ l−2. Here, the integer p =

1

α

is determined

by condition (12.3), and σμ = const, μ = 0, r.

Lemma 17.1. Suppose that the conditions formulated above are satisfied and

a function f(ϕ, τ, ε) is 2π-periodic in ϕν, ν = 1,m, r times continuously

differentiable with respect to (ϕ, τ ) ∈ Rm ×R for every ε ∈ (0, ε0], 1 ≤ r ≤ l,

and such that

__ _

Dsϕ

∂q

∂τq f(ϕ, τ, ε)

___

≤ σεα−q ∀(ϕ, τ, ε) ∈ G1, 0 ≤ s + q ≤ r. (17.4)

Then, for sufficiently small ε0 > 0, there exists the integral manifold y =

Y (ψ, τ, ε) of the system

dy

dτ

= H(τ )y + ∂_a(x(τ ), ϕ, τ)

∂x

y + f(ϕ, τ, ε),

dϕ

dτ

= ω(τ )

ε

+ b(x(τ ), ϕ, τ, ε)

(17.5)

for which the function Y (ψ, τ, ε) is 2π-periodic in ψν, ν = 1,m, r times

continuously differentiable with respect to ψ and τ for every fixed ε ∈ (0, ε0],

and such that

__ _

Dsψ

∂q

∂τq Y (ψ, τ, ε)

___

≤ σεα−q ∀(ψ, τ, ε) ∈ G1, 0 ≤ s + q ≤ r. (17.6)

196 Integral Manifolds Chapter 3

Proof. To construct the function Y (ψ, τ, ε), we consider the iterations

Yj+1(ψ, τ, ε)

=

∞ _

−∞

Q(τ, t)

_ ∂

∂x

_a(x(t), ϕt

τ , t)Yj(ϕt

τ , t, ε) + f(ϕt

τ , t, ε)

_

dt, (17.7)

where ϕτt

= ϕτt

(ψ, ε) is a solution of the second equation of system (17.5) that

takes the value ψ for τ = t, and Y0 ≡ 0. It follows from Theorem 16.2 that

each function Yj(ψ, τ, ε), j ≥ 1, is r times continuously differentiable with

respect to (ψ, τ) ∈ Rm × R for every fixed value of ε ∈ (0, ε0], and

____

Dsψ

∂q

∂τq Yj(ψ, τ, ε)

____

≤ ds,qεα−q ∀(ψ, τ, ε) ∈ G1, (17.8)

j ≥ 0, 0 ≤ s + q ≤ r.

Note that condition (17.8) is also satisfied for r = l because the function

∂

∂x

_a(x(τ ), ϕ, τ)y is l − 1 times continuously differentiable with respect to y,

ϕ, and τ and, according to (16.2), has l continuous derivatives with respect to

y and ϕ. Denote

Zj+1(ψ, τ, ε) = Yj+1(ψ, τ, ε) − Yj(ψ, τ, ε).

Then it follows from (17.7) that

sup

G1

_Zj+1(ψ, τ, ε)_

≤

∞ _

−∞

Ke

−γ|t−τ| sup

ϕ,τ

___∂

∂x

_a(x(τ ), ϕ, τ)

__ _

dt sup

G1

_Zj(ψ, τ, ε)_

= σ0 sup

G1

_Zj(ψ, τ, ε)_.

According to condition (13.3), the constant σ0 is less than 1; therefore, the last

relation guarantees the convergence of the numerical series

∞_

j=1

sup

G1

_Zj(ψ, τ, ε)_,

Section 17 Asymptotic Expansion of Integral Manifold 197

and, hence, the uniform convergence of the sequence {Yj(ψ, τ, ε)} on the set

G1. Further, we assume that each numerical series

∞_

j=1

sup

G1

_Dνψ

Zj(ψ, τ, ε)_, ν= 1, s − 1, s ≤ r, (17.9)

is also convergent. Consider the equality

Dsψ

Zj+1(ψ, τ, ε) =

∞ _

−∞

Q(τ, t)Dsψ

_ ∂

∂x

_a(x(t), ϕt

τ , t)Zj(ϕt

τ , t, ε)

_

dt.

To estimate the last integral, we differentiate the product in the square brackets

and use the following inequalities of the form (16.6) and (16.23):

___

∂

∂ψ

(ϕt

τ (ψ, ε) − ψ)

___

≤ c(1)

0 εαeγ|t−τ|

,

___

∂

∂ψ

ϕt

τ (ψ, ε)

___

≤ c(1)

0 eγ|t−τ|

,

_Dsψ

ϕt

τ (ψ, ε)_ ≤ c(s)

0 εαeγs|t−τ|

, s≥ 2, γ = γ

2l

. (17.10)

Taking into account that

∂

∂x

_a(x(τ ), ϕ, t) has r bounded derivatives with respect

to ϕ, we get

_Dsψ

Zj+1(ψ, τ, ε)_ ≤ K

∞ _

−∞

e

−γ|t−τ|

_

σ(s)

_s−1

ν=0

sup

G1

_Dνψ

Zj(ψ, τ, ε)_eγs|t−τ|

+ sup

ϕ,τ

___

∂

∂x

_a(x(τ ), ϕ, τ)

___

・_Dsψ

Zj(ϕt

τ , t, ε)_

_

dt. (17.11)

Here, σ(s) is a constant independent of ε and j. Applying the scheme of the

proof of Lemma 16.1, we obtain

_Dsψ

Zj(ϕt

τ , t, ε)_ ≤

_s−1

ν=0

sup

G1

_Dνψ

Zj(ψ, τ, ε)_

_

β

cνβ_Dψϕt

τ

_β1_Dsψ

ϕt

τ

_βs

+

___

_

p1+...+pm=s

∂sZj(ϕt

τ , t, ε)

∂ϕp1

1 . . . ∂ϕpm

m

×

_ m-

μ=1

m-

ν=1

(δνμ + ∂

∂ψν

(ϕt,μ

τ

− ψμ))β(μ)

ν

__ _

,

198 Integral Manifolds Chapter 3

where ϕt

τ = (ϕt,1

τ , . . . , ϕt,m

τ ), δν,μ is the Kronecker symbol, the symbol

_

means summation over all β(μ)

ν satisfying the conditions

_m

μ=1

β(μ)

ν = sν,

_m

ν=1

β(μ)

ν = pν, ν,μ= 1,m,

and

Dsψ

= ∂s

∂ψs1

1 . . . ∂ψsm

m

.

Taking inequalities (17.10) into account, we get

_Dsψ

Zj(ϕt

τ , t, ε)_

≤ sup

G1

_Dsψ

Zj(ψ, τ, ε)_

+ σ(s)

__s−1

ν=1

sup

G1

_Dνψ

Zj(ψ, τ, ε)_ + εαLj(s)

_

eγs|t−τ|

, (17.12)

Lj(s) =

_

p1+...+pm=s

sup

G1

___

∂sZj(ψ, τ, ε)

∂ψp1

1 . . . ∂ψpm

m

__ _

.

Inequalities (17.11) and (17.12) yield

Lj+1(s) ≤

_

σ0 +

2

γ − sγ

Kσ(s)smεα0

_

Lj(s)

+

2

γ − sγ

Ksm

_

σ(s) + σ(s) sup

ϕ,τ

___

∂

∂x

_a(x(τ ), ϕ, τ)

___

_

×

_s−1

ν=1

sup

G1

_Dνψ

Zj(ψ, τ, ε)_. (17.13)

Since, for sufficiently small ε0 > 0, the constant in the square brackets on the

right-hand side of (17.13) is less than 1 and series (17.9) are convergent, it follows

from (17.13) that each series

∞_

j=1

sup

G1

_Dsψ

Zj(ψ, τ, ε)_ (17.14)

Section 17 Asymptotic Expansion of Integral Manifold 199

is convergent. Thus, according to the principle of mathematical induction, each

numerical series (17.14) is convergent for 0 ≤ s ≤ r. We now write a partial

differential equation for the function Yj+1 = Yj+1(ψ, τ, ε):

∂Yj+1

∂τ

= −∂Yj+1

∂ψ

_ω(τ )

ε

+ b(x(τ ), ψ, τ, ε)

_

+H(τ )Yj+1

+ ∂

∂x

_a(x(τ ), ψ, τ)Yj + f(ψ, τ, ε).

This yields

∂Zj+1

∂τ

= −∂Zj+1

∂ψ

_ω(τ )

ε

+ b(x(τ ), ψ, τ, ε)

_

+H(τ )Zj+1

+ ∂

∂x

_a(x(τ ), ψ, τ)Zj , (17.15)

where Zν = Zν(ψ, τ, ε) for ν = j, j + 1. Let us fix an arbitrary ε ∈ (0, ε0].

It follows from Eq. (17.15) and the condition of the boundedness of the functions

ω, b, and

∂_a

∂x

and their derivatives that the series

∞_

j=1

sup

ψ,τ

___

∂

∂τ

Zj(ψ, τ, ε)

___

is convergent. Differentiating equality (17.15) ν times, 1 ≤ ν ≤ r − 1, with respect

to ψ and using the convergence of series (17.14) for s = 0, r, we establish

the convergence of each series

∞_

j=1

sup

ψ,τ

__ _

Dsψ

∂

∂τ

Zj(ψ, τ, ε)

__ _

,

0

≤

s

≤

r

−

1.

Further, differentiating equality (17.15) with respect to τ and ν times with respect

to ψ, 0 ≤ ν ≤ r − 2, we establish the convergence of the series

∞_

j=1

sup

ψ,τ

__ _

Dsψ∂2

∂τ2Zj(ψ, τ, ε)

__ _

,

0

≤

s

≤

r

−

2,

and so on. Thus, all numerical series

∞_

j=1

sup

ψ,τ

__ _

Dsψ

∂q

∂τq Zj(ψ, τ, ε)

__ _

,

0

≤

s

+

q

≤

r,

200 Integral Manifolds Chapter 3

are convergent for any value of the small parameter ε ∈ (0, ε0]. This is sufficient

for the limit function

Y (ψ, τ, ε) = lim

j→∞

Yj(ψ, τ, ε)

to have r continuous derivatives with respect to (ψ, τ) ∈ Rm × R for every

fixed ε ∈ (0, ε0]. Passing to the limit as j →∞ in inequalities (17.8), we obtain

estimates (17.6). Lemma 17.1 is proved.

Remark 4. System (17.5) satisfies all conditions of Theorem 16.2, which

guarantees that the function Y (ψ, τ, ε) is smooth with respect to ψ and τ up

to the order r − 1, r ≥ 2, and the derivatives of the (r − 1)th order satisfy

the Lipschitz condition. Since system (17.5) is linear with respect to y and the

equations for ϕ are independent of y, Lemma 17.1 establishes the smoothness

of the function Y (ψ, τ, ε) with respect to ψ and τ up to the order r, which can

be equal to 1.

By analogy, using estimate (1.20) for oscillation integrals, one can prove the

following statement:

Lemma 17.2. Under the conditions imposed on system (17.1), there exists the

integral manifold y = Y (ψ, τ, ε) of the equations

dy

dτ

= H(τ )y + _a(x(τ ), ϕ, τ) + ∂

∂x

_a(x(τ ), ϕ, τ)y,

dϕ

dτ

= ω(τ )

ε

+ b(x(τ ), ϕ, τ, ε),

where Y (ψ, τ, ε) is 2π-periodic in ψν, ν = 1,m, and l times continuously

differentiable with respect to ψ and τ for every value of ε ∈ (0, ε0], ε0 > 0 is

sufficiently small, and

__ _

Dsψ

∂q

∂τq Y (ψ, τ, ε)

___

≤ cεα−q ∀(ψ, τ, ε) ∈ G1, 0 ≤ s + q ≤ l.

Section 17 Asymptotic Expansion of Integral Manifold 201

To establish relations (17.2) and (17.3), we rewrite Eq. (14.13) for the function

X(ψ, τ, ε) = x(τ) + Y (ψ, τ, ε) in the form

∂Y

∂τ

+ ∂Y

∂ψ

_ω(τ )

ε

+ b(x(τ ), ψ, τ, ε)

_

= H(τ )Y + F(Y, τ) + _a(x(τ) + Y,ψ, τ) + εA(x(τ) + Y,ψ, τ, ε)

+ ∂Y

∂ψ

_

b(x(τ ), ψ, τ, ε) − b(x(τ) + Y,ψ, τ, ε)

_

, (17.16)

where

H(τ) = ∂

∂x

a(x(τ ), τ), F(Y, τ) = a(x(τ) + Y, τ) − a(x(τ ), τ) − H(τ )Y.

We now substitute the value of Y from (17.2) into (17.16) and then expand

the right-hand side into the sum over values of the same order ε

ν

p , assuming that

uν and its derivatives

∂uν

∂ψμ

, μ = 1,m, are values of order ε

ν

p . Equating the

expression on the left-hand side of (17.16) for Y = uν to the term of order ε

ν

p

of the indicated expansion of the right-hand side of (17.16), we obtain a partial

differential equation for the determination of the function uν = uν(ψ, τ, ε). It

follows from estimate (16.45) for s = q = 0 that u0(ψ, τ, ε) ≡ 0 for any

(ψ, τ, ε) ∈ G1.

Now consider the following equation for the determination of u1:

∂u1

∂τ

+ ∂u1

∂ψ

_ω(τ )

ε

+ b(x(τ ), ψ, τ, ε)

_

= H(τ )u1 + _a(x(τ ), ψ, τ) + ∂_a(x(τ ), ψ, τ)

∂x

u1.

It is clear that the solution of this equation is the integral manifold y = u1(ψ, τ, ε)

of the system

dy

dτ

= H(τ )y + _a(x(τ ), ϕ, τ) + ∂_a(x(τ ), ϕ, τ)

∂x

y,

dϕ

dτ

= ω(τ )

ε

+ b(x(τ ), ϕ, τ, ε).

According to Lemma 17.2, the integral manifold y = u1(ψ, τ, ε) of this system

exists; moreover, the function u1(ψ, τ, ε) is 2π-periodic in ψν, ν = 1,m, and

202 Integral Manifolds Chapter 3

l times continuously differentiable with respect to (ψ, τ) ∈ Rm × R for every

fixed ε, and its derivatives satisfy the estimates

____

Dsψ

∂q

∂τq u1(ψ, τ, ε)

____

≤ ds,q,1ε

1

p

−q (17.17)

∀(ψ, τ, ε) ∈ G1, 0 ≤ s + q ≤ l.

Further, we write an equation for the determination of uν(ψ, τ, ε) for ν ≥ 2:

∂uν

∂τ

+ ∂uν

∂ψ

_ω(τ )

ε

+ b(x(τ ), ψ, τ, ε)

_

= H(τ )uν + ∂_a(x(τ ), ψ, τ)

∂x

uν

+ fν

_

ψ, τ, ε, u1(ψ, τ, ε), . . . , uν−1(ψ, τ, ε),

∂

∂ψ

u1(ψ, τ, ε), . . . ,

∂

∂ψ

uν−1(ψ, τ, ε)

_

.

It follows from the smoothness conditions for the right-hand side of system (17.1)

that fν is a polynomial of at most νth degree with respect to u1, . . . , uν−1,

∂

∂ψ

u1, . . . ,

∂

∂ψ

uν−1 whose coefficients are l − ν times continuously differentiable

with respect to ψ and τ for fixed ε ∈ (0, ε0], and all their partial derivatives

are uniformly bounded in G1. Moreover, if the functions uμ(ψ, τ, ε), μ =

1, ν − 1, are 2π-periodic in ψk, k = 1,m, and l − μ times continuously differentiable

with respect to ψ and τ, and their derivatives satisfy the inequalities

____

Dsψ

∂q

∂τq uμ(ψ, τ, ε)

____

≤ ds,q,με

μ

p

−q ∀(ψ, τ, ε) ∈ G1 (17.19)

for 0 ≤ s+q ≤ l−μ, then fν, as a function of ψ, τ, and ε, is obviously l−ν

times continuously differentiable with respect to ψ and τ for every ε ∈ (0, ε0],

2π-periodic in ψk, k = 1,m, and such that

____

Dsψ

∂q

∂τq fν

____

≤ ds,q,νε

ν

p

−q ∀(ψ, τ, ε) ∈ G1, 0 ≤ s + q ≤ l − ν,

where ds,q,ν is a certain constant independent of ε. We set

uν = ε

ν−1

p uν(ψ, τ, ε).

Section 17 Asymptotic Expansion of Integral Manifold 203

Then it follows from (17.18) that y = uν(ψ, τ, ε) is the integral manifold of the

system

dy

dτ

= H(τ )y + ∂_a(x(τ ), ϕ, τ)

∂x

y + fν(ϕ, τ, ε),

dϕ

dτ

= ω(τ )

ε

+ b(x(τ ), ϕ, τ, ε),

where

fν(ϕ, τ, ε) = ε

1−ν

p fν

_

ϕ, τ, ε, u1(ϕ, τ, ε), . . . ,

∂

∂ϕ

uν−1(ϕ, τ, ε)

_

.

Taking into account that

____

Dsϕ

∂q

∂τq fν(ϕ, τ, ε)

____

≤ ds,q,νεα−q, α=

1

p

,

∀(ϕ, τ, ε) ∈ G1, 0 ≤ s + q ≤ l − ν,

we conclude that, according to Lemma 17.1, the function uν(ψ, τ, ε) is 2π-

periodic in ψk, k = 1,m, has continuous partial derivatives with respect to

ψ and τ for every ε ∈ (0, ε0] up to the order l − ν, and satisfies the estimates

____

Dsψ

∂q

∂τq uν(ψ, τ, ε)

____

≤ ds,q,νεα−q ∀(ψ, τ, ε) ∈ G1, 0 ≤ s + q ≤ l − ν.

This yields estimate (17.19) with μ = ν for the function uν = uνε

ν−1

p . Thus,

according to the principle of mathematical induction, every function uμ(ψ, τ, ε),

μ = 1, r − 1, is 2π-periodic in ψν, ν = 1,m, has continuous

_

in (ψ, τ) ∈

Rm × R for every ε ∈ (0, ε0]

_

partial derivatives to within the order l − μ, and

satisfies estimates (17.19).

Let us determine the asymptotic character of expansion (17.2). For this purpose,

we denote

u(ψ, τ, ε) =

_r−1

ν=1

uν(ψ, τ, ε) (17.20)

and change the variables in Eq. (17.16) as follows:

Y = u(ψ, τ, ε) + ε

r−1

p z(ψ, τ, ε). (17.21)

204 Integral Manifolds Chapter 3

For z, we obtain the following equation:

∂z

∂τ

+ ∂z

∂ψ

_ω(τ )

ε

+ b(x(τ ), ψ, τ, ε) + εαB2(z,ψ, τ, ε)

_

= H(τ )z + ∂_a(x(τ ), ψ, τ)

∂x

z + εαB1(z,ψ, τ, ε), (17.22)

where

B1(z,ψ, τ, ε) = ε

−r

p

_

_a(x(τ) + u + ε

r−1

p z,ψ, τ)

− _a(x(τ ), ψ, τ) − ∂_a(x(τ ), ψ, τ)

∂x

(u + ε

r−1

p z)

+ ∂u

∂ψ

_

b(x(τ ), ψ, τ, ε) − b(x(τ) + u + ε

r−1

p z,ψ, τ, ε)

_

+ F(u + ε

r−1

p z, τ) + εA(x(τ) + u + ε

r−1

p z,ψ, τ, ε)

−

_r−1

ν=1

fν(ψ, τ, ε, u1(ψ, τ, ε), . . . ,

∂

∂ψ

uν−1(ψ, τ, ε)

_

, f1 ≡ 0,

B2(z,ψ, τ, ε) = ε

−1

p

_

b(x(τ) + u + ε

r−1

p z,ψ, τ, ε) − b(x(τ ), ψ, τ, ε)

_

.

Using properties of the functions uν(ψ, τ, ε), ν = 1, r − 1, and the smoothness

conditions for the right-hand side of system (17.1), we establish that, for sufficiently

small ε0 > 0, the functions Bj(z,ψ, τ, ε), j = 1, 2, are defined on the

set

_z_ ≤ Δ, ψ∈ Rm, τ ∈ R, ε ∈ (0, ε0]

(Δ > 0 is fixed) and 2π-periodic in ψk, k = 1,m, and, for every fixed ε ∈

(0, ε0], they have continuous partial derivatives with respect to z, ψ, and τ up

to the order l − r inclusive that satisfy an inequality of the form

____

Ds

z,ψ

∂q

∂τqBj(z,ψ, τ, ε)

____

≤ cε

−q, 0 ≤ s + q ≤ l − r, j = 1, 2. (17.23)

A function z(ψ, τ, ε) that is a solution of Eq. (17.22) defines the integral manifold

y = z(ψ, τ, ε) of the system

Section 17 Asymptotic Expansion of Integral Manifold 205

dy

dτ

= H(τ )y + ∂_a(x(τ ), ϕ, τ)

∂x

y + εαB1(y, ϕ, τ, ε),

dϕ

dτ

= ω(τ )

ε

+ b(x(τ ), ϕ, τ, ε) + εαB2(y, ϕ, τ, ε),

(17.24)

which has the same form as system (16.52) for F ≡ 0, _a(x(τ) + y, ϕ, τ) =

∂

∂x

_a(x(τ ), ϕ, τ)y, and β = δ = α. Therefore, according to Theorem 16.2, for

l−r ≥ 2 there exists the integral manifold y = z(ψ, τ, ε) of system (17.24) that

satisfies the estimates

____

Dsψ

∂q

∂τq z(ψ, τ, ε)

____

≤ ds,qεα−q ∀(ψ, τ, ε) ∈ G1, 0 ≤ s + q ≤ l − r − 1.

It follows from the change of variables (17.21) that the function

X(ψ, τ, ε) = x(τ) + u(ψ, τ, ε) + v(ψ, τ, ε), (17.25)

where

v = ε

r−1

p z(ψ, τ, ε),

__ _

Dsψ

∂q

∂τq v(ψ, τ, ε)

___

≤ ds,qε

r

p

−q (17.26)

for all (ψ, τ, ε) ∈ G1, 0 ≤ s + q ≤ l − r − 1, determines the integral manifold

of system (17.1). Thus, the following statement is true:

Theorem 17.1. Suppose that the conditions imposed above on system (17.1)

are satisfied for l ≥ r + 2, r ≥ 2. Then, for sufficiently small ε0 > 0,

the function X(ψ, τ, ε) that defines the integral manifold of system (17.1) for

(ψ, τ, ε) ∈ G1 admits the asymptotic decomposition (17.25) in which the functions

u(ψ, τ, ε) and v(ψ, τ, ε) satisfy conditions (17.19), (17.20), and (17.26).

Corollary 5. If

∂_a(x(τ ), ϕ, τ)

∂x

≡ 0 ∀(ϕ, τ ) ∈ Rm × R,

206 Integral Manifolds Chapter 3

then, according to Lemmas 17.1 and 17.2, the functions uν are determined in

explicit form by the following formulas:

u1(ψ, τ, ε) =

∞ _

−∞

Q(τ, t)_a(x(t), ϕt

τ (ψ, ε), t)dt,

uν(ψ, τ, ε)

=

∞ _

−∞

Q(τ, t)fν(ϕt

τ (ψ, ε), t, ε, u1(ϕt

τ (ψ, ε), t, ε), . . . , uν−1(ϕt

τ (ψ, ε), t, ε),

∂

∂ϕ

u1(ϕt

τ (ψ, ε), t, ε), . . . ,

∂

∂ϕ

uν−1(ϕt

τ (ψ, ε), t, ε))dt, 2 ≤ ν ≤ r − 1.

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