13. Construction of Successive Approximations

Back

Consider a solution x = x(τ ) of the averaged equations (12.4) that lies in

D together with its ρ-neighborhood ∀τ ∈ R and assume that the variational

system of equations

dz

dτ

= ∂

∂x

a(x(τ ), τ)z is hyperbolic [Pli2]. Without loss of

generality, we can rewrite the variational system in the form

dz+

dτ

= H+(τ )z+,

dz−

dτ

= H−(τ )z−, (13.1)

where z = (z+, z−), z+ and z− are, respectively, n0-dimensional and (n −

n0)-dimensional vectors, and

H(τ ) = diag [H+(τ ),H−(τ )] = ∂a(x(τ ), τ)

∂x

.

In this case, the normal fundamental matrices Q+(τ, t) and Q−(τ, t) of solutions

of the first and the second equations in (13.1) satisfy the inequalities

_Q+(τ, t)_ ≤ Keγ(τ−t) ∀τ ≤ t,

_Q−(τ, t)_ ≤ Ke

−γ(τ−t) ∀τ ≥ t, (13.2)

where K ≥ 1 and γ > 0 are certain constants. Furthermore, in what follows,

we assume that

σ0 =

2

γ

K sup

ϕ,τ

___

∂

∂x

_a(x(τ ), ϕ, τ)

___

< 1. (13.3)

146 Integral Manifolds Chapter 3

Performing the change of variables y = x − x(τ ), y = (y+, y−), we transform

Eqs. (12.1) as follows:

dy+

dτ

= H+(τ )y+ + F+(y, τ) + _a+(y + x(τ ), ϕ, τ) + εA+(y + x(τ ), ϕ, τ, ε),

dy−

dτ

= H−(τ )y− + F−(y, τ) + _a−(y + x(τ ), ϕ, τ)

+ εA−(y + x(τ ), ϕ, τ, ε),

(13.4)

dϕ

dτ

= ω(τ )

ε

+ b(y + x(τ ), ϕ, τ, ε),

where (_a+, _a−) = _a, (A+,A−) = A,

(F+, F−) = F = a(y + x(τ ), τ) − a(x(τ ), τ) − H(τ )y

≡

_1

0

_ ∂

∂y

a(ly + x(τ ), τ) − H(τ )

_

dly,

_F(y, τ)_ ≤ 1

2n2σ1_y_2,

___

∂

∂y

F(y, τ)

___

≤ n2σ1_y_.

Let Q(τ, t) denote the quadratic n-dimensional matrix

Q(τ, t) =

⎧⎨

⎩

−diag (Q+(τ, t), 0), τ<t,

diag (0,Q−(τ, t)), τ>t.

For τ _= t, we obviously have

dQ(τ, t)

dτ

= H(τ )Q(τ, t),

dQ(τ, t)

dt

= −Q(τ, t)H(t),

_Q(τ, t)_ ≤ Ke

−γ|τ−t|

. (13.5)

We determine the integral manifold of Eqs. (13.4) by the method of successive

approximations as the limit (as j → ∞) of the integral manifolds y =

Yj(ψ, τ, ε), (ψ, τ, ε) ∈ G1, of the equations

Section 13 Construction of Successive Approximations 147

dy

dτ

= H(τ )y + F(Yj−1(ϕ, τ, ε), τ) + _a(Xj−1(ϕ, τ, ε), ϕ, τ)

+ εA(Xj−1(ϕ, τ, ε), ϕ, τ, ε),

dϕ

dτ

= ω(τ )

ε

+ b(Xj−1(ϕ, τ, ε), ϕ, τ, ε), (13.6)

where Y0 ≡ 0 and Xj−1(ϕ, τ, ε) = x(τ) + Yj−1(ϕ, τ, ε).

By using the matrix Q(τ, t), one can determine the integral manifold of

Eqs. (13.6) as follows:

Yj(ψ, τ, ε) =

∞ _

−∞

Q(τ, t)[F(Yj−1(ϕt

τ,j(ψ, ε), t, ε), t)

+ _a(Xj−1(ϕt

τ,j(ψ, ε), t, ε), ϕt

τ,j(ψ, ε), t)

+ εA(Xj−1(ϕt

τ,j(ψ, ε), t, ε), ϕt

τ,j(ψ, ε), t, ε)] dt, (13.7)

where ϕ = ϕτ

t,j(ψ, ε) is a solution of the second equation of system (13.6) that

takes the value ψ for τ = t. Indeed, assuming that the order of differentiation

and integration on the right-hand side of (13.7) may be changed, we get

∂Yj

∂τ

+ ∂Yj

∂ψ

_ω(τ )

ε

+_b(ψ, τ, ε)

_

= H(τ )Yj + F(Yj−1(ψ, τ, ε), τ)

+ _a(Xj−1(ψ, τ, ε), ψ, τ) + εA(Xj−1(ψ, τ, ε), ψ, τ, ε)

+

∞ _

−∞

∂B

∂ϕ

_∂ϕt

τ,j

∂τ

+

∂ϕt

τ,j

∂ψ

_ω(τ )

ε

+_b(ψ, τ, ε)

__

dt, (13.8)

where Yj = Yj(ψ, τ, ε), _b(ψ, τ, ε) = b(Xj−1(ψ, τ, ε), ψ, τ, ε), and B is the

integrand of integral (13.7). The second equation of system (13.6) yields

∂ϕt

τ,j

∂τ

+

∂ϕt

τ,j

∂ψ

_ω(τ )

ε

+_b(ψ, τ, ε)

_

=

_t

τ

∂_b(ϕl

τ,j, l, ε)

∂ϕ

_∂ϕl

τ,j

∂τ

+

∂ϕl

τ,j

∂ψ

_ω(τ )

ε

+_b(ψ, τ, ε)

__

dl.

148 Integral Manifolds Chapter 3

Setting

f(t) =

∂ϕt

τ,j

∂τ

+

∂ϕt

τ,j

∂ψ

_ω(τ )

ε

+ b(ψ, τ, ε)

_

,

we deduce from the last equality that

_f(t)_ ≤

___

_t

τ

_f(l)_ sup

___

∂_b(ψ, τ, ε)

∂ψ

___

dl

___

.

Solving this integral inequality, we get f(t) ≡ 0. Therefore, for all (ψ, τ, ε) ∈

G1, the following identity is true:

∂Yj

∂τ

+ ∂Yj

∂ψ

_ω(τ )

ε

+ b(Xj−1, ψ, τ, ε)

_

= H(τ )Yj + F(Yj−1, τ) + _a(Xj−1, ψ, τ) + εA(Xj−1, ψ, τ, ε), (13.9)

where the values of the functions Yj−1, Xj−1, and Yj are taken at a point

(ψ, τ, ε). Hence, y = Yj(ψ, τ, ε) is indeed the integral manifold of system

(13.6).

Theorem 13.1. Suppose that conditions (12.2), (12.3), (13.2), and (13.3) are

satisfied. Then, for sufficiently small ε0 > 0, the functions Yj = Yj(ψ, τ, ε),

j = 0,∞, defined by equality (13.7) are 2π-periodic in ψν, ν = 1,m, twice

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

ε ∈ (0, ε0], and such that, for any (ψ, τ, ε) ∈ G1, the following inequalities are

satisfied:

_Yj_ ≤ d1εα,

___

∂

∂ψ

Yj

___

≤ d2εα,

_m

ν=1

___

∂2

∂ψ∂ψν

Yj

___

≤ d3εα. (13.10)

If, in addition, the norms _ω(τ )_,

___

dω(τ )

dτ

__ _

,

and

___

∂

∂τ

A(x, ϕ, τ, ε)

___

are

uniformly bounded on the set G, then the following inequalities are also true:

___

∂

∂τ

Yj

___

≤ d4εα−1,

___

∂2

∂ψ∂τ

Yj

___

≤ d5εα−1,

___

∂2

∂τ2 Yj

___

≤ d6εα−2. (13.11)

Here, d1, . . . , d6 are constants independent of ε and j.

Section 13 Construction of Successive Approximations 149

Proof. Consider the sequence {Yj(ψ, τ, ε)}. Let us prove that it is bounded

∀(ψ, τ, ε) ∈ G1. Denote

θt

τ,j = ϕt

τ,j

− 1

ε

_t

τ

ω(r)dr.

Since

_a(x(τ) + Yj, ψ, τ)

= _a(x(τ ), ψ, τ) + ∂

∂x

_a(x(τ ), ψ, τ)Yj + _ A(x(τ ), Yj, ψ, τ),

where

_ _ A_ =

____

_1

0

_ ∂

∂x

_a(x(τ) + lYj, ψ, τ) − ∂

∂x

_a(x(τ ), ψ, τ)

_

dlYj

____ ≤

1

2n2σ1_Yj_2,

it follows from (13.7) that

_Yj+1(ψ, τ, ε)_ ≤ σ0 sup

ψ,τ

_Yj(ψ, τ, ε)_

+

2

γ

K

_

εσ1 + n2σ1 sup

ψ,τ

_Yj(ψ, τ, ε)_2

_

+

_

k_=0

∞_

s=−∞

___

τ+_s+1

τ+s

Q(τ, t)ak(x(t), t) exp{i(k, θt

τ,j+1)}

× exp

_ i

ε

_t

τ

(k, ω(r))dr

_

dt

__ _

.

(

13.12)

Here, ak(x, τ ) are the Fourier coefficients of the function _a(x, ϕ, τ ). Using conditions

(12.2) and (13.5) and the uniform estimate (1.20) of the oscillation integral,

we estimate the last term on the right-hand side of (13.12) from above by the value

150 Integral Manifolds Chapter 3

σ3K

_

k_=0

∞_

s=−∞

__

sup

G

_ak_ +

1

_k_

_

sup

G

___

∂ak

∂τ

___

+ sup

G

___

∂ak

∂x

___

__

× (1 + σ1 + nσ1) max

[τ+s,τ+s+1]

e

−γ|t−τ|

_

εα ≤ σ4εα,

σ4 = 2Kσ1σ3(1 + σ1 + nσ1) eγ

1 − e−γ .

Then relation (13.12) yields

sup

ψ,τ

_Yj+1_ ≤ σ0 sup

ψ,τ

_Yj_ +

2

γ

Kn2σ1 sup

ψ,τ

_Yj_2 +

_

σ4 +

2

γ

Kσ1

_

εα,

which leads to the following estimate in view of the fact that Y0 ≡ 0 and σ0 < 1:

sup

ψ,τ

_Yj(ψ, τ, ε)_ ≤ 1

2d1εα < d1εα ∀j ≥ 0, ε∈ (0, ε0], (13.13)

where ε0 ≤ (nd1)− 2

α and d1 =

1

γ

(4Kσ1 + γσ4)(1 − σ0)−1. Note that it

is necessary to impose the restriction d1εα0

≤ ρ. If this condition is satisfied,

then Xj(ψ, τ, ε) = x(τ) + Yj(ψ, τ, ε) lies in the

1

2ρ-neighborhood of the curve

x = x(τ ) ∀(ψ, τ, ε) ∈ G1, i.e., in the course of the construction of successive

approximations, we do not leave the domain of definition of the right-hand side of

system (12.1).

Let us prove that the sequence

_ ∂

∂ψ

Yj(ψ, τ, ε)

_

is also uniformly bounded

in G1 by the value d2εα, where d2 is a constant independent of ε and j ≥ 0.

Denote by Ak(x(τ ), τ) the m × n rectangular matrix

Ak(x(τ ), τ) = (a(μ)

k (x(τ ), τ)kν)n,m

μ,ν=1,

ak(x, τ) = (a(1)

k (x, τ ), . . . , a(n)

k (x, τ )).

It is obvious that

_Ak_ ≤ _ak_ ・ _k_,

___∂

∂τ

Ak

___

≤

___

∂ak

∂τ

___

・ _k_,

___

∂

∂x

Ak

___

≤ n

___

∂ak

∂x

___

・ _k_.

Section 13 Construction of Successive Approximations 151

Consider the inequality

___

∂

∂ψ

Y1(ψ, τ, ε)

___

≤ εnσ1K

∞ _

−∞

e

−γ|t−τ|

_

m +

___

∂

∂ψ

(ϕt

τ,1

− ψ)

___

_

dτ

+

_

k_=0

∞_

s=−∞

___

s+_τ+1

s+τ

Q(τ, t)Ak(x(t), t)

_

Em + ∂

∂ψ

(ϕt

τ,1

− ψ)

_

× exp{i(k, θt

τ,1)} exp

_ i

ε

_t

τ

(k, ω(r)) dr

_

dt

__ _

,

(

13.14)

which follows from (13.7) for j = 1. Estimating the last term on the right-hand

side of (13.14) [denote it by Δ] using inequalities (1.20) and (12.2), we get

Δ ≤ (1 + σ1 + nσ1)nσ1Kσ3εα

×

∞_

s=−∞

_

m + max

[τ+s,τ+s+1]

_ztτ

_ + max

[τ+s,τ+s+1]

___

d

dt

ztτ

___

_

× max

[τ+s,τ+s+1]

e

−γ|t−τ|

, (13.15)

where ztτ

= ∂

∂ψ

(ϕt

τ,1

− ψ). Further, we use Lemma 12.1 for the function Y =

Y0(ψ, τ, ε) ≡ 0. Since

___

∂

∂ψ

Y0(ψ, τ, ε)

___

≤ d2εα (the constant d2 > 0 is fixed

in what follows) and

___

∂Y0

∂τ

+ ∂Y0

∂ψ

ω(τ )

ε

___

≤ d1 = σ1

_

3 + nd1 +

1

2n2d21

_

,

(1 + |t − τ |)ec2(1+d2)εα|t−τ| ≤ σ5e

γ

2

|t−τ|

, σ5 = max

_

1;

4

γ

_

,

152 Integral Manifolds Chapter 3

for c2(1+d2)εα0

≤ 1

4γ and d2εα0

≤ 1, inequality (12.8) and Lemma 12.1 imply

that, for all s ≥ 0, the following relation is true:

μ ≡ m + max

[τ+s,τ+s+1]

_ztτ

_ + max

[τ+s,τ+s+1]

___

d

dt

ztτ

___

≤ σ(1)

5 + σ(2)

5 e

γ

2 (s+1),

σ(1)

5 = m + (m + n)σ1, σ(2)

5 = (1 + (m + n)σ1)2c1σ5.

For s < 0, we obviously have μ ≤ σ(1)

5 +σ(2)

5 e

−γ

2 s. Taking this into account,

we can rewrite inequality (13.15) in the form

Δ ≤ (1 + σ1 + nσ1)nσ1K2σ3

_

σ(1)

5

1

1 − e−γ + σ(2)

5

e

γ

2

1 − e

−γ

2

_

εα

≡ σ6εα. (13.16)

Inequalities (13.14) and (13.16) yield

___

∂

∂ψ

Y1(ψ, τ, ε)

___

≤ εnσ1K

_2m

γ

+

8

γ

c1σ5

_

+ σ6εα < d2εα

∀(ψ, τ, ε) ∈ G1.

Note that, for τ ∈ [−T,T] and N >T, we have

____

∞ _

N

Q(τ, t)

_

F(Y0, t) + _a(x(t) + Y0, ϕt

τ,1, t) + εA(x(t) + Y0, ϕt

τ,1, t, ε)

_

dt

____

≤ 1

γ

K

_1

2n2σ1d21

εα0

+ σ1 + ε0σ1

_

e

−γ(N−T),

___

∞ _

N

∂

∂ψ

_

Q(τ, t)[F(Y0, t) + _a(x(t) + Y0, ϕt

τ,1, t)

+ εA(x(t) + Y0, ϕt

τ,1, t, ε)]

_

dt

___

≤ Kσ1(n2d1d2ε2α

0 + nd2εα0

+ m

+ nd2ε1+α

0 + ε0m)

_m

γ

+

4

γ

c1σ5

_

e

−γ

2 (N−T),

Section 13 Construction of Successive Approximations 153

and the corresponding inequalities with the integration interval [N,∞) replaced

by (−∞,−N] are also true. Taking this into account, we conclude that integral

(13.7) for j = 1 and the integral obtained from (13.7) by differentiation

with respect to ψ under the integral sign are uniformly convergent on the set

(ψ, τ, ε) ∈ Rm×[−T,T]×(0, ε0]. By virtue of the smoothness of the right-hand

side of Eqs. (12.1) and the arbitrariness of T > 0, this implies that the functions

Y1(ψ, τ, ε) and

∂

∂ψ

Y1(ψ, τ, ε) are continuous in (ψ, τ) ∈ Rm × R for every

fixed ε ∈ (0, ε0]. Using (13.5) and Lemma 12.2, we can similarly establish the

uniform convergence (for (ψ, τ, ε) ∈ Rm × [−T,T] × [ε0, ε0], where T > 0

and ε0

∈ (0, ε0) are arbitrary) of the integral obtained from (13.7) by differentiation

with respect to τ under the integral sign. Therefore,

∂

∂τ

Y1(ψ, τ, ε) is also

continuous in (ψ, τ) ∈ Rm × R for every ε ∈ (0, ε0]. Moreover, the uniform

convergence of the corresponding integrals enables us to change the order of integration

and differentiation with respect to ψ and τ. As a result, we establish

that the function Y1(ψ, τ, ε) satisfies identity (13.9) ∀(ψ, τ, ε) ∈ G1 and the

inequality

___

∂Y1

∂τ

+ ∂Y1

∂ψ

ω(τ )

ε

___

≤ σ1

_

3 + nd1 +

1

2n2d21

_

= d1.

We now assume that, for all j = 2, l − 1, l > 2, the functions Yj =

Yj(ψ, τ, ε) are continuously differentiable with respect to (ψ, τ) ∈ Rm × R

for every ε ∈ (0, ε0] and satisfy identity (13.9) and the inequalities

___

∂Yj

∂ψ

___

≤ d2εα,

___

∂Yj

∂τ

+ ∂Yj

∂ψ

ω(τ )

ε

___

≤ d1 ∀(ψ, τ, ε) ∈ G1. (13.17)

Let us prove that Yl(ψ, τ, ε) is also continuously differentiable with respect to ψ

and τ for every fixed ε and satisfies (13.9) and (13.17) for j = l. It follows

from (13.7) that

___

∂

∂ψ

Yl(ψ, τ, ε)

___

≤ K

∞ _

−∞

e

−γ|t−τ|

_

εmσ1 + (n + m)nσ1d1εα

+

_

n2σ1d1εα + εnσ1 + sup

ϕ,τ

___

∂_a(x(τ ), ϕ, τ)

∂x

___

_

× sup

ψ,τ

___

∂

∂ψ

Yl−1(ψ, τ, ε)

___

__

1 +

___

∂

∂ψ

(ϕt

τ,l

− ψ)

___

_

dt

154 Integral Manifolds Chapter 3

+

_

k_=0

∞_

s=−∞

___

τ+_s+1

τ+s

Q(τ, t)Ak(x(t), t)

×

_

Em + ∂

∂ψ

(ϕt

τ,l

− ψ)

_

exp{i(k,ϕt

τ,l)} dt

__ _

.

(

13.18)

According to estimate (1.20) and Lemma 12.1, the last term on the right-hand side

of (13.18) satisfies inequalities (13.15) and (13.16), and, therefore, it is bounded

from above by the value σ6εα. Then (13.18) can be rewritten in the form

sup

ψ,τ

___

∂

∂ψ

Yl(ψ, τ, ε)

___

≤

_ 2

γ

K(m + (n + m)nd1)σ1(1 + 4c1σ5) + σ6

_

εα

+

_ 2

γ

Knσ1(1 + nd1)(1 + 4c1σ5)εα0

+ σ0

+ 2σ0c1(1 + d2)σ5εα0

_

sup

ψ,τ

___

∂

∂ψ

Yl−1(ψ, τ, ε)

__ _

.

Taking ε0 > 0 so small that

2

γ

Knσ1(1 + nd1)(1 + 4c1σ5)εα0

≤ 1 − σ0

4 ,

2σ0c1(1 + d2)σ5εα0

≤ 1 − σ0

4 ,

we get

sup

ψ,τ

___

∂

∂ψ

Yl(ψ, τ, ε)

___

≤ 1 + σ0

2

sup

ψ,τ

___

∂

∂ψ

Yl−1(ψ, τ, ε)

___

+ σ7εα,

where

σ7 =

2

γ

K(m + (n + m)nd1)σ1(1 + 4c1σ5) + σ6,

and the constant

1

2

(1+σ0) is less than 1 according to condition (13.3). The last

inequality yields

___

∂

∂ψ

Yl(ψ, τ, ε)

___

≤ 2σ7

1 − σ0

εα ≡ d2εα ∀(ψ, τ, ε) ∈ G1.

As in the case j = 1, one can easily verify that the improper integral on the righthand

side of (13.7) for j = l and the integrals obtained from it by differentiation

Section 13 Construction of Successive Approximations 155

with respect to ψ and τ under the integral sign are uniformly convergent on the

set

ψ ∈ Rm, τ ∈ [−T,T], ε∈ [ε0, ε0], (13.19)

where T > 0 and ε0

∈ (0, ε0] are arbitrary constants. Therefore, the function

Yl(ψ, τ, ε) is continuously differentiable with respect to ψ and τ for every fixed

ε on set (13.19) and satisfies identity (13.9) with j = l for all ψ, τ, and ε from

set (13.19). Since T and ε0 are arbitrary, we get relation (13.9) with j = l for

all (ψ, τ, ε) ∈ G1 and the inequality

___

∂Yl

∂τ

+ ∂Yl

∂ψ

ω(τ )

ε

___

≤ d1.

Thus, by induction, we establish that Yj(ψ, τ, ε), j = 0,∞, are continuously

differentiable with respect to (ψ, τ) ∈ Rm × R for every ε ∈ (0, ε0] and satisfy

the inequality

___

∂

∂ψ

Yj(ψ, τ, ε)

___

≤ d2εα ∀(ψ, τ, ε) ∈ G1, j ≥ 0.

By analogy, using the methods proposed above and Lemmas 12.3 and 12.4, we

prove the continuity of the functions

∂2

∂ψ∂ψν

Yj(ψ, τ, ε) and

∂2

∂τ∂ψ

Yj(ψ, τ, ε),

j ≥ 0, in (ψ, τ) ∈ Rm × R for every ε ∈ (0, ε0] and the estimate

_m

ν=1

___

∂2

∂ψ∂ψν

Yj(ψ, τ, ε)

___

≤ d3εα ∀(ψ, τ, ε) ∈ G1, j≥ 0,

where the constant d3 is independent of ε and j. By virtue of the smoothness

conditions (12.2) and the properties of the functions Yj(ψ, τ, ε) established

above, identity (13.9) yields the continuity of the functions

∂2

∂ψ∂τ

Yj(ψ, τ, ε) and

∂2

∂τ2 Yj(ψ, τ, ε) in ψ and τ for every ε. Hence, each of the functions Yj(ψ, τ, ε)

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

fixed ε ∈ (0, ε0].

Now let

_ω(τ )_ +

___

d

dτ

ω(τ )

___

+

___

∂

∂τ

A(x, ϕ, τ, ε)

___

≤ σ1 ∀(x, ϕ, τ, ε) ∈ G.

156 Integral Manifolds Chapter 3

Then identity (13.9) yields estimates (13.11) in which

d4 = σ1

_

d2 + (d2 + nd1)ε0 +

1

2n2d21

ε1+α

0 + ε1−α

0 + ε2−α

0

_

,

d5 = σ1(d3 + m(1 + ε0)ε1−α

0 + nd2ε20

+ (2nd2 + md2 + d3)ε0 + n2d2(d1 + d2)ε1+α

0 ),

d6 = σ1[d5 + (d5 + d2 + n(1 + nd1εα0

)d4)ε0

+ (2 + n2)ε2−α

0 + (nd1 + d2)ε2−α

0 + (nd1 + d2)ε20

+ n(ε1−α

0 + d4)(1 + d2εα0

+ ε0)ε0].

Finally, we prove that each function Yj(ψ, τ, ε), j ≥ 0, is 2π-periodic

in each component ψν, ν = 1,m, of the vector ψ. Indeed, the right-hand

side of (12.1) is 2π-periodic in ϕν, ν = 1,m. If we impose the condition

that Yl−1(ψ, τ, ε) is also periodic in ψν with period 2π, then the function

ϕ = ϕt

τ,l(ψ, ε) that is the solution of the Cauchy problem

dϕt

τ,l

dt

= ω(t)

ε

+ b(x(t) + Yl−1(ϕt

τ,l, t, ε), ϕt

τ,l, t, ε), ϕτ

τ,l = ψ (13.20)

can be represented in the form

ϕt

τ,l(ψ, ε) = ψ + _ϕt

τ,l(ψ, ε),

where _ϕt

τ,l(ψ, ε) is 2π-periodic in ψν, ν = 1,m. Let eν be the unit vector in

the space Rm. Then, taking into account that

ϕt

τ,l(ψ + 2πeν, ε) = ψ + 2πeν + _ϕt

τ,l(ψ, ε) = ϕt

τ,l(ψ, ε) + 2πeν,

we deduce from equality (13.7) for j = l that

Yl(ψ + 2πeν, τ, ε) = Yl(ψ, τ, ε), ν= 1,m.

Since Y0(ψ + 2πeν, τ, ε) = Y0(ψ, τ, ε) ≡ 0, this implies that Yj(ψ, τ, ε) are

2π-periodic functions with respect to ψν, ν = 1,m, for all j ≥ 0. Theorem

13.1 is proved.

Section 14 Existence of Integral Manifold 157

Remark 1. If we assume, in addition, that the right-hand side of system (12.1)

is continuous in all variables (x, ϕ, τ, ε) ∈ G, then the functions Yj(ψ, τ, ε) are

also continuous in (ψ, τ, ε) ∈ G1 for all j ≥ 0. Indeed, since Y0(ψ, τ, ε) ≡ 0

is continuous in G1, it follows from problem (13.20) that the function ϕt

τ,1(ψ, ε)

is continuous in (ψ, τ, ε). Then the uniform convergence of the improper integral

(13.7) guarantees the continuity of Y1(ψ, τ, ε) in G1. By analogy, one can

establish that Yj(ψ, τ, ε) is continuous for j > 1.

Навигация

• Главная
• Книги
• Статьи
• Обратная связь
• Поиск