16. Smoothness of Integral Manifold

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In Sections 12–15, we have proved the existence of the integral manifold x =

X(ψ, τ, ε) of the system of n + m differential equations

dx

dτ

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

dϕ

dτ

= ω(τ )

ε

+ b(x, ϕ, τ, ε). (16.1)

In the present section, we study the problem of the smoothness of the function

X(ψ, τ, ε). Assume that the following conditions are satisfied:

(a) the functions a, _a, A, ω, and b are l ≥ 2 times continuously differentiable

with respect to (x, ϕ, τ ) ∈ D×Rm×R ≡ G3 for every ε ∈ (0, ε0],

and all their partial derivatives are uniformly bounded in G = G3 × (0, ε0]

by a constant c1 independent of ε;

(b) the following relation is true:

_

k_=0

_

_k_l sup

G

_ck_ + _k_l−1

_

sup

G

___

∂ck

∂τ

___

+ sup

G

___

∂ck

∂x

___

__

≤ c1, (16.2)

where ck = ck(x, τ, ε) are the Fourier coefficients of the function

[_a(x, ϕ, τ ); b(x, ϕ, τ, ε)].

Theorem 16.1. Suppose that conditions (a) and (b) are satisfied and relations

(12.1), (13.2), and (13.3) are true. Then there exist constants ε1 > 0 and c2 > 0

such that, for all (ψ, τ, ε) ∈ G1 = Rm × R × (0, ε0], ε0 ≤ ε1, the function

X(ψ, τ, ε) is l−1 times continuously differentiable with respect to ψ and τ for

every fixed ε,

__ _

Dsψ

∂q

∂τqX(ψ, τ, ε)

___

≤ c2εα−q ∀(ψ, τ, ε) ∈ G1, 1 ≤ s + q ≤ l − 1, (16.3)

and the derivatives of the (l − 1)th order satisfy the Lipschitz condition with

respect to the variables ψ and τ. Here, Dsψ

is an arbitrary partial derivative of

order s with respect to ψ.

It follows from Theorem 16.1 that the smoothness of the function X(ψ, τ, ε)

decreases as compared with the smoothness of the right-hand side of (16.1). UnSection

16 Smoothness of Integral Manifold 175

der the conditions imposed on system (16.1), this situation is typical of the theory

of integral manifolds, which is confirmed, e.g., by the analysis carried out in

[Sam4].

Prior to the proof of Theorem 16.1, we prove the lemma presented below,

in which Yj(ψ, τ, ε) are the functions defined by (13.7), and ϕt

τ,j+1(ψ, ε) is a

solution of the Cauchy problem

d

dt

ϕt

τ,j+1 = ω(t)

ε

+b(x(t)+Yj(ϕt

τ,j+1, t, ε), ϕt

τ,j+1, t, ε), ϕτ

τ,j+1 = ψ. (16.4)

Lemma 16.1. If, for certain j ≥ 0, the function Yj(ψ, τ, ε) is l ≥ 2 times

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

and such that

__ _

Dsψ

∂q

∂τq Yj(ψ, τ, ε)

___

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

then one can find sufficiently large constants ds,q and a sufficiently small constant

ε0 = ε0(ds,q) > 0 such that the function Yj+1(ψ, τ, ε) is l times continuously

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

__ _

Dsψ

∂q

∂τq Yj+1(ψ, τ, ε)

___

≤ ds,qεα−q (16.5)

for all (ψ, τ, ε) ∈ G1 and 0 ≤ s + q ≤ l.

Proof. For l = 2, the statement of the lemma follows from Theorem 13.1.

Therefore, we assume that l > 2. According to the theorems on the existence of

a solution of the Cauchy problem and its differentiability with respect to initial

data, for all t ∈ R the function ϕt

τ,j+1(ψ, ε) has l continuous partial derivatives

with respect to (ψ, τ) ∈ Rm × R for every fixed ε ∈ (0, ε0]. On the basis of

problem (16.4), we consider the derivatives of the function ϕt

τ,j+1 with respect

to ψ. According to Lemmas 12.1 and 12.3, we have

___

∂

∂ψ

(ϕt

τ,j+1

− ψ)

___

≤ c(1)

0 εαeγ|t−τ|

,

___

∂

∂ψ

ϕt

τ,j+1

___

≤ c(1)

0 eγ|t−τ|

,

__ _

D2ψ

ϕt

τ,j+1

___

≤ c(2)

0 εαe2γ|t−τ|

, (16.6)

176 Integral Manifolds Chapter 3

where

γ = γ

2l

, c(1)

0 = c1(1 + md1,0) max

_

1;

2

γ

_

, c(1)

0 = m + c(1)

0 ,

c(2)

0 = c10(1 + m2d2,0) max

_

1;

2

γ

_

, εα0

≤ min

_ γ

c11

; γ

2c2(1 + md1,0)

_

,

and c1, c2, c10, and c11 are the constants defined in Lemmas 12.1 and 12.3.

Assume that, for all p = 2, s − 1, s ≤ l, the following inequalities are true:

_Dp

ψϕt

τ,j+1

_ ≤ c(p)

0 εαepγ|t−τ|

, (ψ, τ, ε) ∈ G1, t ∈ R, (16.7)

where the constants c(p)

0 depend on d0,0, d1,0, . . . , dp,0. Then the functions Yj =

Yj(ϕt

τ,j+1, t, ε) satisfy the estimate

_Dp

ψYj_ ≤

_p

ν=1

_Dν

ϕt

τ,j+1

Yj_

_

β

cνβ_Dψϕt

τ,j+1

_β1 . . . _Dp

ψϕt

τ,j+1

_βp

≤ εα

_p

ν=1

dν,0

_

β

cνβ(c(1)

0 )β1 . . . (c(p)

0 )βpeγp|t−τ| ≡ εαMpeγp|t−τ|

.

For p ≥ 2, an analogous estimate is also true for u = (u1, . . . , un+m) = (x(t)+

Yj, ϕt

τ,j+1), namely

_Dp

ψu_ ≤ _Dp

ψYj_ + _Dp

ψϕt

τ,j+1

_

≤ εα[Mp + c(p)

0 ]eγp|t−τ| ≡ εαc(p)

0 eγp|t−τ|

. (16.8)

If p = 1, then

_Dψu_ ≤ (_Dϕt

τ,j+1

Yj_ + 1)_Dψϕt

τ,j+1

_

≤ (md1,0 + 1)c(1)

0 eγ|t−τ| ≡ c(1)

0 eγ|t−τ|

. (16.9)

Further, differentiating equality (16.4) s = s1 + . . . + sm times with respect to

the variables ψ, we obtain

Section 16 Smoothness of Integral Manifold 177

d

dt

∂sϕt

τ,j+1

∂ψs1

1 . . . ∂ψsm

m

= ∂b

∂u

∂su

∂ψs1

1 . . . ∂ψsm

m

+ Fs,j

+

_

p1+...+pn+m=s

∂sb

∂up1

1 . . . ∂upn+m

n+m

_ m-

ν=1

m-+n

μ=1

_∂uμ

∂ψν

_β(μ)

ν

. (16.10)

Here, the symbol

,

in the third term on the right-hand side denotes summation

over all β(μ)

ν that satisfy the conditions

n_+m

μ=1

β(μ)

ν = sν, ν= 1,m,

_m

ν=1

β(μ)

ν = pμ, μ= 1,m + n,

and Fs,j satisfies the inequality

_Fs,j_ ≤

_s−1

p=2

_Dpu

b_

_

β

cpβ_Dψu_β1 . . . _Ds−1

ψ u_βs−1 ,

where at least one of the numbers β2, . . . , βs−1 is not equal to zero. Since the

partial derivatives of the function b(x, ϕ, τ, ε) with respect to all variables xk

and ϕν, k = 1, n, ν = 1,m, up to the order l inclusive are bounded by a

constant c1 and inequalities (16.8) and (16.9) are satisfied, we have

_Fs,j_ ≤ εα

_s−1

p=2

c1

_

β

cpβ(c(1)

0 )β1 . . . (c(s−1)

0 )βs−1 × eγs|t−τ|

≡ εασ(s)

1 eγs|t−τ|

. (16.11)

We represent the first term on the right-hand side of (16.10) in the form

∂b

∂u

∂su

∂ψs1

1 . . . ∂ψsm

m

= ∂b

∂x

∂sYj

∂ψs1

1 . . . ∂ψsm

m

+ ∂_____________b

∂ϕ Lt

τ

=

_ ∂b

∂x

∂Yj

∂ϕ

+ ∂b

∂φ

_

Lt

τ +Φs,j , (16.12)

where

Yj = Yj(ϕt

τ,j+1, t, ε), Lt

τ =

∂sϕt

τ,j+1

∂ψs1

1 . . . ∂ψsm

m

,

178 Integral Manifolds Chapter 3

_Φs,j_ ≤

___

∂b

∂x

___

_s

p=2

_Dp

ϕt

τ,j+1

Yj_

_

β

cpβ_Dψϕt

τ,j+1

_β1 . . . _Ds−1

ψ ϕt

τ,j+1

_βs−1

≤ nc1

_s

p=2

dp,0εα

_

ν

cpβ(c(1)

0 )β1 . . . (c(s−1)

0 )βs−1eγs|t−τ|

≡ εασ(s)

2 eγs|t−τ|

. (16.13)

Moreover, taking into account that

___

∂uμ

∂ψν

___

≤

___

∂

∂ϕ

Y (μ)

j

___

___

∂

∂ψν

ϕt

τ,j+1

___

≤ md1,0c(1)

0 εαeγ|t−τ| for μ = 1, n

and

∂un+μ

∂ψν

= ∂

∂ψν

(ϕt,μ

τ,j+1

− ψμ) + δνμ for μ = 1,m,

where δνμ is the Kronecker symbol, ϕt

τ,j+1 = (ϕt,1

τ,j+1, . . . , ϕt,m

τ,j+1), and Yj =

(Y (1)

j , . . . , Y (n)

j ), we deduce from condition (a) and inequality (16.6) that

_

p1+...+pn+m=s

∂sb

∂up1

1 . . . ∂upn+m

n+m

_ m-

ν=1

m-+n

μ=1

_∂uμ

∂ψν

_β(μ)

ν

= ∂sb

∂ϕs1

1 . . . ∂ϕsm

m

+ Rs,j . (16.14)

Here,

_Rs,j_

≤

_

p1+...+pn+m=s

c1

_ m-

ν=1

m-+n

μ=1

_

max{1 + c(1)

0 ;md1,0c(1)

0

}

_β(μ)

ν εαeγs|t−τ|

≡ εασ(s)

3 eγs|t−τ|

. (16.15)

Thus, combining (16.12) and (16.14), we can rewrite Eq. (16.10) in the form

d

dt

Lt

τ =

_ ∂b

∂x

∂Yj

∂ϕ

+ ∂b

∂ϕ

_

Lt

τ + ∂sb

∂ϕs1

1 . . . ∂ϕsm

m

+ Fs,j +Φs,j + Rs,j , (16.16)

Section 16 Smoothness of Integral Manifold 179

where the functions Fs,j , Φs,j , and Rs,j satisfy inequalities (16.11), (16.13),

and (16.15), respectively. For s ≥ 2, equation (16.16) yields

_Lt

τ

_ ≤ nmc1d1,0εα

___

_t

τ

_Lξ

τ

_dξ

___

+

___

_t

τ

∂sb

∂ϕs1

1 . . . ∂ϕsm

m

dξ

___

+

___

_t

τ

∂b

∂ϕ

Lξ

τ dξ

___

+

1

γs

(σ(s)

1 + σ(s)

2 + σ(s)

3 )eγs|t−τ|

εα. (16.17)

Since

∂sb

∂ϕs1

1 . . . ∂ϕsm

m

=

_

k_=0

bk(x(ξ) + Yj(ϕξ

τ,j+1, ξ, ε), ξ, ε)isks1

1 . . . ksm

m exp{i(k,ϕξ

τ,j+1)},

sup

G

_bkks1

1 . . . ksm

m

_ ≤ _k_s sup

G

_bk_,

it follows from the condition for Fourier coefficients (16.2) and the estimate for

oscillation integrals (1.20) that

___

_t

τ

∂sb

∂ϕs1

1 . . . ∂ϕsm

m

dξ

___

≤ εασ3c1[1 + 3c1 + md1,0c1(1 + 2n)](1 + |t − τ |)

≤ εασ4eγs|t−τ|

, (16.18)

σ4 = σ3c1[1 + 3c1 + md1,0c1(1 + 2n)] max

_

1;

1

γs

_

.

Then, for t ∈ [τ, τ + 2), inequality (16.17) yields

_Lt

τ

_ ≤ σ(s)

4 εα ≤ εασ(s)

4 eγs|t−τ|

, (16.19)

σ(s)

4 =

_ 1

γs

(σ(s)

1 + σ(s)

2 + σ(s)

3 ) + σ4

_

exp{2(γs + (mnd1,0 + m)c1)}.

180 Integral Manifolds Chapter 3

If t ≥ τ +2, then we represent the third term on the right-hand side of inequality

(16.17) in the form

___

_t

τ

∂b

∂ϕ

Lξ

τ dξ

___

≤

_

k_=0

__q−1

q=0

___

τ+_q+1

τ+q

BkLξ

τ exp{i(k, θξ

τ,j+1)} exp

_ i

ε

_ξ

τ

(k, ω(r))dr

_

dξ

___

+

___

_t

τ+q

BkLξ

τ exp{i(k, θξ

τ,j+1)}dξ

___

_

, (16.20)

where q is the integer part of the number t − τ − 1, 1 ≤ t − (τ + q) < 2, and

Bk = Bk(x(ξ)+Yj(ϕξ

τ,j+1, ξ, ε), ξ, ε) are the Fourier coefficients of the function

∂b

∂ϕ

. By analogy with the proof of Lemma 12.1, we estimate each of the integrals

over the segments [τ + q, τ + q + 1] and [τ + q, t] with regard for inequality

(1.20). As a result, we establish that the integral over the segment [τ+q, τ+q+1]

does not exceed the value

_

sup

G

_Bk_ +

1

_k_

_

sup

G

___

∂

∂τ

Bk

___

+sup

G

___

∂

∂x

Bk

___

__

σ(s)

5 εα

×

_

max

[τ+q,τ+q+1]

_Lξ

τ

_ +

τ+_q+1

τ+q

esγ(ξ−τ)dξ

_

, (16.21)

where

σ(s)

5 = σ3

_

1+(m+1)c1+c1md1,0(2+n)+2nc1d0,0+

1

γs

(σ(s)

1 +σ(s)

2 +σ(s)

3 )

_

e2γs.

The integral over the segment [τ + q, t] can also be estimated by a value

of the form (16.21) with the only difference that the maximum of _Lξ

τ _ over

ξ ∈ [τ + q, τ + q + 1] must be replaced by the corresponding maximum over

ξ ∈ [τ +q, t], and the integral of the exponent over the segment [τ +q, τ +q+1]

must be replaced by the corresponding integral over ξ ∈ [τ + q, t].

Section 16 Smoothness of Integral Manifold 181

We estimate the maximum of the function _Lξ

τ _ on the segments [τ +q, τ +

q + 1] and [τ + q, t] by analogy with the estimation of the maximum of the

function _zlτ

_ in the proof of Lemma 12.1, namely

max

[τ+q,τ+q+1]

_Lξ

τ

_ ≤ σ(s)

6

τ+_q+1

τ+q

[_Lξ

τ

_ + esγ(ξ−τ)]dξ,

max

[τ+q,t]

_Lξ

τ

_ ≤ σ(s)

6

_t

τ+q

[_Lξ

τ

_ + esγ(ξ−τ)]dξ,

σ(s)

6 = m

_

1 + mc1(1 + nd1,0) +

1

γs

(σ(s)

1 + σ(s)

2 + σ(s)

3 )

_

.

Thus, using (16.2), (16.18), (16.20), and (16.21), we can rewrite inequality (16.17)

for t ≥ τ + 2 in the form

_Lt

τ

_ ≤ σ(s)

7 εα

__t

τ

_Lξ

τ

_dξ + eγs(t−τ)

_

,

where

σ(s)

7 = max

_

(mnd1,0 + σ(s)

5 σ(s)

6 )c1;

σ4 +

1

γs

_

σ(s)

1 + σ(s)

2 + σ(s)

3 + c1σ(s)

5 (1 + σ(s)

6 )

__

.

The last inequality, together with inequality (16.19), yields

___

∂sϕt

τ,j+1

∂ψs1

1 . . . ∂ψsm

m

___

≤ c(s)

0 esγ|τ−t|

εα, c(s)

0 = max

_

σ(s)

4 ;

2sσ(s)

7

2s − 1

_

(16.22)

for all t ≥ τ and s ≥ 2. By analogy, we establish estimate (16.22) for t < τ.

Hence, by induction, for all (ψ, τ, ε) ∈ G1, t ∈ R, and s = 2, l we get

_Dsψ

ϕt

τ,j+1(ψ, ε)_ ≤ c(s)

0 εαeγs|τ−t|

, (16.23)

where the constants c(s)

0 depend on d0,0, . . . , ds,0.

We now prove that inequalities (16.6) and (16.23) yield estimate (16.5) for

q = 0. If s = 0, 1, 2, then relation (16.5) follows from Theorem 13.1. Assume

182 Integral Manifolds Chapter 3

that estimate (16.5) holds for p = 0, s − 1, s ≥ 3, s ≤ l, and q = 0. For

s = s1 + . . . + sm, we consider

Dsψ

Yj+1(ψ, τ, ε)

=

∞ _

−∞

Q(τ, t)[Dsψ

F(Yj, t) + Dsψ

_a(x(t) + Yj, ϕt

τ,j+1, t)

+ εDsψ

A(x(t) + Yj, ϕt

τ,j+1, t, ε)]dt, (16.24)

where

Dsψ

= ∂s

∂ψs1

1 . . . ∂ψsm

m

, Yj = Yj(ϕt

τ,j+1, t, ε).

Since

_Dsψ

F(Yj, t)_ ≤

_s

ν=2

_DνY

jF(Yj, t)_

_

β

cνβ_DψYj_β1 . . . _Ds−1

ψ Yj_βs−1

+

___

∂

∂Yj

F(Yj, t)

___

・_Dsψ

Yj_

and

_Dp

ψYj_ ≤ εαMpeγp|t−τ|

, Mp = Mp(d0,0, . . . , dp,0),

___

∂

∂Yj

F(Yj, t)

___

≤ n2c1d0,0εα,

the following estimate holds for all (ψ, τ, ε) ∈ G1 and t ∈ R :

_Dsψ

F(Yj, t)_ ≤ ε2α

__s

ν=2

c1

_

β

cνβMβ1

1 . . . Mβs−1

s−1 + n2c1d0,0Ms

_

eγs|t−τ|

≡ ε2αc(s)

1 eγs|t−τ|

, (16.25)

where the constant c(s)

1 depends on d0,0, . . . , ds,0.

By analogy, using inequalities (16.8) and (16.9), we obtain

_Dsψ

A_ ≤

_s

ν=1

c1

_

β

cνβ(c(1)

0 )β1 . . . (c(s)

0 )βseγs|t−τ| ≡ c(s)

2 eγs|t−τ|

, (16.26)

Section 16 Smoothness of Integral Manifold 183

where c(s)

2 = c(s)

2 (d0,0, . . . , ds,0). Further, we consider the second term in the

square brackets on the right-hand side of equality (16.24) and represent it in the

form

Dsψ

_a = ∂_a

∂u

Dsψ

u +

_

p1+...+pn+m=s

∂s_a

∂up1

1 . . . ∂upn+m

n+m

_ m-

ν=1

m-+n

μ=1

_∂uμ

∂ψν

_β(μ)

ν

+ _ Fs,j . (16.27)

Here, the symbol

,

denotes summation over all β(μ)

ν that satisfy the conditions

m_+n

μ=1

β(μ)

ν = sν, ν= 1,m,

_m

ν=1

β(μ)

ν = pμ, μ= 1,m + n,

and _ Fs,j satisfies the inequality

_ _ Fs,j_ ≤

_s−1

p=2

_Dpu

_a_

_

β

cpβ_Dψu_β1 . . . _Ds−1

ψ u_βs−1

≤ εα

_s−1

p=2

c1

_

β

cpβ(c(1)

0 )β1 . . . (c(s−1)

0 )βs−1eγs|t−τ|

≡ εασ(s)

1 eγs|t−τ|

, (16.28)

where at least one of the numbers β2, . . . , βs−1 is not equal to zero and σ(s)

1 =

σ(s)

1 (d0,0, . . . , ds−1,0).

The second term on the right-hand side of (16.27) (denote it by v) admits a

representation of the form (16.14), namely

v = Dsϕ

_a + _Rs,j , (16.29)

where

__Rs,j_ ≤ εασ(s)

3 eγs|t−τ|

, σ(s)

3 = σ(s)

3 (d0,0, . . . , ds−1,0). (16.30)

It remains to transform the first term on the right-hand side of (16.27). It is

obvious that

184 Integral Manifolds Chapter 3

∂_a

∂u

Dsψ

u =

_∂_a

∂x

∂Yj

∂ϕ

+ ∂_a

∂ϕ

_

Dsψ

ϕt

τ,j+1 +_Φs,j

+ ∂a

∂x

_

p1+...+pm=s

∂sYj

∂ϕp1

1 . . . ∂ϕpm

m

_ m-

μ=1

m-

ν=1

_∂ϕμ

∂ψν

_β(μ)

ν

where the symbol

,

denotes summation over all β(μ)

ν that satisfy the conditions

_m

μ=1

β(μ)

ν = sν, ν= 1,m,

_m

ν=1

β(μ)

ν = pμ, μ= 1,m,

and _Φs,j satisfies the inequality

__Φs,j_ ≤ εαc(s)

3 eγs|t−τ|

, (16.31)

c(s)

3 = c(s)

3 (d0,0, . . . , ds−1,0) = nc1

_s−1

p=2

dp,0

_

β

cpβ(c(1)

0 )β1 . . . (c(s−1)

0 )βs−1 .

Since

∂_a

∂x

_

p1+...+pm=s

∂sYj

∂ϕp1

1 . . . ∂ϕpm

m

_ m-

μ=1

m-

ν=1

_ ∂

∂ψν

(ϕμ − ψμ) + δνμ

_β(μ)

ν

= ∂_a

∂x

Dsψ

Yj + _ Ns,j ,

where

_ _ Ns,j_ ≤ εαc(s)

4 eγs|t−τ| _

p1+...+pm=s

sup

ψ,τ

___

∂sYj(ψ, τ, ε)

∂ψp1

1 . . . ∂ψpm

m

__ _

,

(

16.32)

c(s)

4 = nc1 max

p1+...+pm=s

_ m-

μ=1

m-

ν=1

(max{1 + c(1)

0 ;md1,0c(1)

0

})β(μ)

ν ,

c(s)

4 = c(s)

4 (d0,0, d1,0),

we get

∂_a

∂u

Dsψ

u =

_∂_a

∂x

∂Yj

∂ϕ

+ ∂_a

∂ϕ

_

Dsψ

ϕt

τ,j+1 + ∂_a

∂x

Dsϕ

Yj +_Φs,j + _ Ns,j . (16.33)

Section 16 Smoothness of Integral Manifold 185

Hence, taking into account equalities (16.27), (16.29), and (16.33) and estimates

(16.25), (16.26), (16.28), and (16.30)–(16.32), for s ≥ 3 we deduce the

following inequality from (16.24):

_Dsψ

Yj+1(ψ, τ, ε)_

≤ εαK

_

c(s)

4

_

p1+...+pm=s

sup

ψ,τ

___

∂sYj(ψ, τ, ε)

∂ψp1

1 . . . ∂ψpm

m

___

+c(s)

1 εα

+ c(s)

2 ε1−α + σ(s)

1 + σ(s)

3 + c(s)

3

_ ∞ _

−∞

e(−γ+γs)|t−τ|

dt

+ K

∞ _

−∞

e

−γ|t−τ|

___

∂_a

∂x

___

___

∂Yj

∂ϕ

___

_Dsψ

ϕt

τ,j+1

_dt

+ K

∞ _

−∞

e

−γ|t−τ|

_

sup

ϕ,τ

___

∂_a(x(τ ), ϕ, τ)

∂x

___

+

___

∂

∂x

_a(x(τ) + Yj, ϕt

τ,j+1, t) − ∂

∂x

_a(x(t), ϕt

τ,j+1, t, ε)

___

_

dt

+

___

∞ _

−∞

Q(τ, t)Dsϕ

_a(x(t) + Yj, ϕt

τ,j+1, t)dt

___

+

___

∞ _

−∞

Q(τ, t) ∂

∂ϕ

_a(x(t) + Yj, ϕt

τ,j+1, t)Dsψ

ϕt

τ,j+1dt

__ _

,

(

16.34)

where Yj = Yj(ϕt

τ,j+1, t, ε), γ = γ

2l

, and the constants σ(s)

1 , σ(s)

3 , c(s)

3 , and

c(s)

4 are independent of ds,0. We choose ε0 > 0 so small that c(s)

1 εα0

≤ 1 and

c(s)

2 εα0

≤ 1 and denote

2 + σ(s)

1 + σ(s)

3 + c(s)

3 + c(s)

4

2

γ − γs

K = c(s)

5 ,

c(s)

5 = c(s)

5 (d0,0, d0,1, . . . , ds−1,0).

To estimate the last two terms on the right-hand side of inequality (16.34), we

represent the corresponding integrals in the form of the infinite sum of integrals

186 Integral Manifolds Chapter 3

over segments of unit length. Then, using inequalities (1.20), (16.2), and (16.23),

we establish that the next to the last term on the right-hand side of (16.34) is

estimated from above by the value

2

1 − e−γ Kc1σ3

_

1 + c1

_

n+2+md1,0 + nd0,0

_

1 +

1

2nd0,0

___

εα ≡ c6εα,

and the last term is estimated from above by the value

2

1 − e−γ+γsKc1σ3eγs

_

2 + (m + n + 2)c1 + (m + n)c1d0,0

+ nc1d0,0

_

1 +

1

2nd0,0

__

εα ≡ c(s)

6 εα

for c(s)

0 εα0

≤ 1 and (σ(s)

1 + σ(s)

2 + σ(s)

3 )εα0

≤ 1.

Since σ0 < 1 and the constants c(s)

5 , c6, and c(s)

6 are independent of ds,0,

s ≥ 3, we conclude that, for

εα0

≤ min

_ 1 − σ0

2n2c1d0,0

;

1

ds,0

_

relation (16.34) yields

sup

ψ,τ

_Dsψ

Yj+1(ψ, τ, ε)_

≤ 1 + σ0

2 ds,0εα +

_

c(s)

5 (1 + ms) + c6 + c(s)

6 ++

2

γ − sγ

Knc1d1,0

_

εα

≤ ds,0εα,

ds,0 =

2

1 − σ0

_

c(s)

5 (1 + ms) + c6 + c(s)

6 +

2

γ − sγ

Knc1d1,0

_

, (16.35)

for all (ψ, τ, ε) ∈ G1 and s = 3, l.

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

(16.1) and the functions Yj(ψ, τ, ε) and ϕt

τ,j+1(ψ, ε) that estimate (16.35) remains

true if we change the order of the differentiation of the function Yj(ψ, τ, ε)

with respect to the variables ψ1, . . . , ψm. Also note that inequalities (16.23) and

(13.2) guarantee the uniform convergence of the improper integral (16.24) on the

set (ψ, τ, ε) ∈ Rm × [−T,T] × (0, ε0] (T > 0 is arbitrary). Therefore, the

functions Dsψ

Yj+1(ψ, τ, ε) are continuous in (ψ, τ) ∈ Rm × [−T,T]. Taking

Section 16 Smoothness of Integral Manifold 187

into account that T is arbitrary, we conclude that, for every fixed ε ∈ (0, ε0], the

functions Dsψ

Yj+1(ψ, τ, ε), s = 0, l, are continuous for all (ψ, τ) ∈ Rm × R.

Thus, Lemma 16.1 is proved for q = 0 and s = 0, l.

Let us prove the statement of the lemma for q ≥ 1. Using Lemmas 12.2 and

12.4, we get

___

∂

∂τ

ϕt

τ,j+1

___

≤ c(0)

7 ε

−1eγ|t−τ|

,

___

∂

∂τ

∂

∂ψ

ϕt

τ,j+1

___

≤ c(1)

7 εα−1e2γ|t−τ|

, (16.36)

where

c(0)

7 = c7(1 + c1), c(1)

7 = c10(1 + c1) max

_

1;

1

γ

_

, εα0

≤ γ max

_ 1

c8

,

1

c11

_

,

and c7, c8, c10, and c11 are the constants defined by Lemmas 12.2 and 12.4.

Following the proof of inequality (16.23), one can easily show that

___

∂

∂τ

Dsψ

ϕt

τ,j+1

___

≤ c(s)

7 εα−1e(s+1)γ|t−τ| (16.37)

for all s = 1, l − 1, (ψ, τ, ε) ∈ G1, and t ∈ R. Inequalities (16.23), (16.36),

and (16.37) yield the uniform convergence of the integral obtained from (16.24)

by differentiation with respect to τ under the integral sign on the set

ψ ∈ Rm, τ ∈ [−T,T], ε∈ [ε0, ε0],

(T > 0 and ε0

∈ (0, ε0) are arbitrary). The smoothness conditions for the

right-hand side of system (16.1) and the functions Yj(ψ, τ, ε) and ϕt

τ,j+1(ψ, ε)

guarantee the continuity of the functions

∂

∂τ

DsψYj+1(ψ, τ, ε), s = 0, l − 1, in

(ψ, τ) ∈ Rm × [−T,T]. This yields

∂

∂τ

Dsψ

Yj+1(ψ, τ, ε) ∈ Cψ,τ , (16.38)

where Cψ,τ denotes the set of vector functions f(ψ, τ, ε) continuous in (ψ, τ) ∈

Rm × R for every fixed ε ∈ (0, ε0]. Let us write Eq. (13.9) for the function

Yj+1 = Yj+1(ψ, τ, ε). We have

∂Yj+1

∂τ

=

1

ε

_

−∂Yj+1

∂ψ

ω(τ ) − ε

∂Yj+1

∂ψ

b(x(τ) + Yj, ψ, τ, ε)

_

+ εH(τ )Yj+1 + εP (Yj, ψ, τ, ε). (16.39)

188 Integral Manifolds Chapter 3

Here,

P(Yj, ψ, τ, ε) = F(Yj, τ) + _a(x(τ) + Yj, ψ, τ) + εA(x(τ) + Yj, ψ, τ, ε);

furthermore, this function has l −1 continuous partial derivatives with respect to

(ψ, τ) ∈ Rm × R for every ε ∈ (0, ε0]. Since

Dsψ

Yj+1(ψ, τ, ε) ∈ Cψ,τ, s= 0, l, (16.40)

Eq. (16.39) yields

Dρ

ψ

∂

∂τ

Yj+1(ψ, τ, ε) ∈ Cψ,τ, ρ= 0, l − 1. (16.41)

Conditions (16.38) for s = 1 and (16.41) for ρ = 1 yield

∂2Yj+1(ψ, τ, ε)

∂ψν∂τ

= ∂2Yj+1(ψ, τ, ε)

∂τ∂ψν

∀(ψ, τ, ε) ∈ G1, ν= 1,m. (16.42)

Further, we consider the chain of equalities

∂3Yj+1

∂ψν∂ψμ∂τ

(16.41) = ∂3Yj+1

∂ψμ∂ψν∂τ

(16.42) = ∂3Yj+1

∂ψμ∂τ∂ψν

(16.38) =

(16.41)

∂3Yj+1

∂τ∂ψμ∂ψν

(16.40) = ∂3Yj+1

∂τ∂ψν∂ψμ

, μ,ν= 1,m.

Here, the marks above and below the equality signs indicate the relations used.

By analogy, one can establish the continuity and, hence, the equality of all partial

derivatives of the (ρ + 1)th order:

Dρ

ψ

∂

∂τ

Yj+1 = Dρ−1

ψ

∂

∂τ

∂

∂ψν

Yj+1 = . . . = ∂

∂τ

Dρ

ψYj+1.

To estimate

__ _

D

ρ

ψ

∂

∂τ

Yj+1

__ _

,

ρ= 0, l

−

1, we note that the right-hand side

of Eq. (16.39) and the right-hand sides of the equations obtained from (16.39)

by ρ-fold differentiation with respect to ψ are independent of Dρ

ψ

∂

∂τ

Yj . It

is also clear that the main contribution to the estimates is made by the term

Dρ

ψ

_ ∂

∂ψ

Yj+1ω(τ )

_

because the other terms in (16.39) and in the equations differentiated

ρ times with respect to ψ have a higher order of smallness with respect

to ε as ε → 0. Consequently,

__ _

D

ρψ

∂

∂τ

Yj+1(ψ, τ, ε)

___

≤ 1

ε

___ _Dρ

ψ

∂

∂ψ

Yj+1(ψ, τ, ε)

___

・_ω(τ )_ + εσρ,1

_

,

Section 16 Smoothness of Integral Manifold 189

where σρ,1 is a constant that depends on d0,0, d1,0, . . . , dρ+1,0 but does not depend

on dρ,1, ρ = 0, l − 1. For σρ,1εα0

≤ 1, the last inequality yields

__ _

D

ρ

ψ

∂

∂τ

Yj+1(ψ, τ, ε)

___

≤ (mc1dρ+1,0 + 1)εα−1 ≤ dρ,1εα−1

for all (ψ, τ, ε) ∈ G1 and ρ = 0, l − 1.

Now assume that all partial derivatives of the (ρ+μ)th order (ρth order with

respect to ψ and μth order with respect to τ ) of the function Yj+1(ψ, τ, ε) are

continuous in (ψ, τ) ∈ Rm × R for every fixed ε ∈ (0, ε0] and such that

____

Dρ

ψ

∂μ

∂τμ Yj+1(ψ, τ, ε)

____

≤ dρ,μεα−μ ∀(ψ, τ, ε) ∈ G1 (16.43)

for 0 ≤ μ ≤ q < l and 0 ≤ ρ ≤ l − μ. Differentiating equality (16.39) q times

with respect to τ and ρ times with respect to ψ, one can easily verify that all

partial derivatives of the (ρ + (q + 1))th order of the function Yj+1(ψ, τ, ε) are

continuous with respect to ψ, τ ∈ Rm × R for every fixed ε ∈ (0, ε0]. Moreover,

Dρ

ψ

∂q+1

∂τq+1 Yj+1 depends on Dρ

ψYj,Dρ

ψ

∂

∂τ

Yj, . . . , Dρ

ψ

∂q

∂τq Yj but does not

depend on Dρ

ψ

∂q+1

∂τq+1 Yj . The smoothness conditions for the right-hand side of

system (16.1), inequalities (16.43), and analogous inequalities for Yj imply that

the differentiation of the right-hand side of Eq. (16.39) with respect to ψ does

not worsen its order estimates with respect to ε, and each time it is differentiated

with respect to τ this order decreases by one. Thus,

__ _

D

ρ

ψ

∂q+1

∂τq+1 Yj+1(ψ, τ, ε)

___

≤ 1

ε

___ _Dρ

ψ

∂

∂ψ

∂q

∂τq Yj+1(ψ, τ, ε)

___

・_ω(τ )_ + ε1−qσρ,q+1

_

≤ [mc1dρ+1,q + 1]εα−q−1 ≤ dρ,q+1εα−q−1 ∀(ψ, τ, ε) ∈ G1

for σρ,q+1εα0

≤ 1 and 0 ≤ ρ ≤ l − (q + 1). Here, σρ,q+1 is a constant that

depends on ds,ν, s = 0, ρ + 1, ν = 0, q but does not depend on dρ,q+1.

Thus, according to the principle of mathematical induction, estimate (16.5)

holds for all s and q that satisfy the condition 0 ≤ s + q ≤ l. Lemma 16.1 is

proved.

190 Integral Manifolds Chapter 3

Proof of Theorem 16.1. We consider iterations (13.7) and fix the constants

ds,q and ε0 for which inequalities (16.5) are satisfied. For ds,q and ε0 thus

chosen, the 2π-periodic (in ψν, ν = 1,m) function Y1(ψ, τ, ε) defined by

formula (13.7) for j = 1 satisfies the estimate

__ _

Dsψ

∂q

∂τq Y1(ψ, τ, ε)

___

≤ ds,qεα−q ∀(ψ, τ, ε) ∈ G1

for 0 ≤ s + q ≤ l. Using the function Y1(ψ, τ, ε) and formula (13.7) for j =

2, we obtain a 2π-periodic (in ψν, ν = 1,m) function Y2(ψ, τ, ε), which,

according to Lemma 16.1, satisfies inequality (16.5), and so on. Thus, relation

(13.7) defines a sequence of iterations Yj(ψ, τ, ε), j ≥ 1, 2π-periodic in ψν,

ν = 1,m, l times continuously differentiable with respect to (ψ, τ) ∈ Rm × R

for every fixed ε ∈ (0, ε0], and satisfying the inequalities

__ _

Dsψ

∂q

∂τq Yj(ψ, τ, ε)

___

≤ ds,qεα−q ∀(ψ, τ, ε) ∈ G1,

j ≥ 1, 0 ≤ s + q ≤ l. (16.44)

Inequality (16.44) implies that the functions Dρ

ψ

∂q

∂τq Yj(ψ, τ, ε), j ≥ 1, are uniformly

bounded in (ψ, τ) ∈ Rm ×R. According to the theorem on compactness

in the space of continuous functions [KoF], this is sufficient in order that, for every

ε ∈ (0, ε0], the function

Y (ψ, τ, ε) = lim

j→∞

Yj(ψ, τ, ε)

have continuous derivatives with respect to ψ and τ up to the order l−1 that satisfy

the Lipschitz condition with respect to ψ and τ from the set Rm × [−T,T]

(T > 0 is arbitrary). Moreover, inequalities (16.44) yield

__ _

Dsψ

∂q

∂τq Y (ψ, τ, ε)

___

≤ c2εα−q, 0 ≤ s + q ≤ l − 1, (16.45)

for all (ψ, τ, ε) ∈ Rm × [−T,T] × (0, ε0] and c2 = max

0≤s+q≤l

ds,q. Taking

into account that T is arbitrary, we conclude that estimates (16.45) hold for any

(ψ, τ, ε) ∈ G1. Since X(ψ, τ, ε) = x(τ)+Y (ψ, τ, ε), inequalities (16.3) follow

from (16.45) and condition (a). Theorem 16.1 is proved.

Corollary 3. If the condition of the boundedness of _ω(τ )_,

___

d

dτ

ω(τ )

__ _

,

.

.

.

,

___

dl

dτl ω(τ )

___is omitted from the conditions of Theorem 16.1, then the function

Section 16 Smoothness of Integral Manifold 191

X(ψ, τ, ε) satisfies inequalities (16.3) only for q = 0, i.e.,

_Dsψ

X(ψ, τ, ε)_ ≤ c2εα, (ψ, τ, ε) ∈ G1, 1 ≤ s ≤ l − 1.

The results of the present section remain true for a system of a more general

form, namely

dx

dτ

= a(x, τ) + _a(x, ϕ, τ) + εβB1(x, ϕ, τ, ε),

dϕ

dτ

= ω(τ )

ε

+ b(x, ϕ, τ, ε) + εδB2(x, ϕ, τ, ε).

(16.46)

Here, min{β; δ} ≥ α, a, _a, ω, and b satisfy the conditions of Theorem 16.1,

and B1 and B2 are 2π-periodic (in ϕν, ν = 1,m) functions that, for every

ε ∈ (0, ε0], have continuous and bounded (by a constant c1) partial derivatives

with respect to x and ϕ up to an order l ≥ 2, i.e.,

[B1;B2] ∈ Cl

x,ϕ(G, c1), (16.47)

and continuous partial derivatives with respect to x, ϕ, and τ up to the order

l − 1 that satisfy the inequalities

__ _

Ds

x,ϕ

∂q

∂τq [B1;B2]

___

≤ c1ε

−1−q ∀(x, ϕ, τ, ε) ∈ G,

q ≥ 1, s+ q ≤ l − 1. (16.48)

Indeed, in this case, iterations (13.7) for the construction of the integral manifold

of system (16.46) are determined by the relations

Yj(ψ, τ, ε) =

∞ _

−∞

Q(τ, t)[F(Yj−1, t) + _a(x(t) + Yj−1, ϕt

τ,j, t)

+ εβB1(x(t) + Yj−1, ϕt

τ,j, t, ε)]dt, Y0 ≡ 0,

dϕt

τ,j

dt

= ω(τ )

ε

+ b(x(t) + Yj−1, ϕt

τ,j, t, ε) + εγB2(x(t) + Yj−1, ϕt

τ,j, t, ε),

ϕτ

τ,j = ψ,

where Yj−1 = Yj−1(ϕt

τ,j, t, ε) and ϕt

τ,j = ϕt

τ,j(ψ, ε). Since β ≥ α, δ ≥ α,

and conditions (16.47) are satisfied, following the scheme of the proof of Lemmas

192 Integral Manifolds Chapter 3

12.1–12.5 and inequalities (16.23) one can easily verify that these statements and

inequalities are true for the functions Yj and ϕt

τ,j constructed above. In this case,

only the constants in the corresponding inequalities do change. Therefore, for the

functions Yj and their derivatives with respect to ψ, the following estimate of

the form (16.35) is true:

_Dsψ

Yj(ψ, τ, ε)_ ≤ ds,0εα ∀(ψ, τ, ε) ∈ G1, j ≥ 1, 0 ≤ s ≤ l. (16.49)

Using the equality

∂Yj+1

∂τ

=

1

ε

_

−∂Yj+1

∂ψ

(ω(τ) + εb(x(τ) + Yj, ψ, τ, ε)

+ ε1+δB2(x(t) + Yj, ψ, τ, ε)) + εH(τ )Yj+1 + εF (Yj, τ)

+ ε_a(x(τ) + Yj, ψ, τ) + ε1+βB1(x(τ) + Yj, ψ, τ, ε)

_

, (16.50)

where Yk = Yk(ψ, τ, ε), k = j, j + 1, we study the character of the estimates

for the derivatives of the functions Yj(ψ, τ, ε), j ≥ 1, with respect to τ. The

smoothness conditions for the right-hand side of (16.50) enable one to differentiate

this equality l − 1 times with respect to ψ, and condition (16.47) and

inequality (16.49) yield

__ _

Dsψ

∂

∂τ

Yj(ψ, τ, ε)

___

≤ ds,1εα−1 (16.51)

∀(ψ, τ, ε) ∈ G1, j≥ 1, 0 ≤ s ≤ l − 1

where ds,1 is a certain constant dependent on dν,0, ν = 0, s + 1. Let us differentiate

equality (16.50) with respect to τ and use inequalities (16.48) for q = 1

and (16.51). Then, taking into account that, on the right-hand side of equality

(16.50), the coefficients of the functions B1 and B2 and their derivatives contain,

respectively, the factors ε1+β and ε1+δ, min{β; δ} ≥ α, we get

____

∂2

∂τ2 Yj(ψ, τ, ε)

____

≤ d0,2εα−2.

According to (16.48) and (16.51), subsequent differentiation with respect to ψ

does not worsen the order estimates with respect to ε. Therefore,

____

Dsψ∂2

∂τ2 Yj(ψ, τ, ε)

____

≤ ds,2εα−2 ∀(ψ, τ, ε) ∈ G1,

j ≥ 1, 0 ≤ s ≤ l − 2.

Section 16 Smoothness of Integral Manifold 193

Here, ds,2 is a constant that depends on dν,1, ν = 0, s + 1 but does not depend

on j. By analogy, one can establish the estimates

____

Dsψ

∂q

∂τq Yj(ψ, τ, ε)

____

≤ ds,qεα−q

for all (ψ, τ, ε) ∈ G1, q = 3, l, and s = 0, l − q.

Thus, the following statement is true for Y (ψ, τ, ε) = lim

j→∞

Yj(ψ, τ, ε):

Theorem 16.2. Suppose that the conditions of Theorem 16.1 for A ≡ 0 and

conditions (16.47) and (16.48) are satisfied. Then, for sufficiently small ε0 > 0,

there exists the integral manifold x = X(ψ, τ, ε) = x(τ) + Y (ψ, τ, ε) of system

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

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

ε ∈ (0, ε0], and such that

____

Dsψ

∂q

∂τq Y (ψ, τ, ε)

____

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

and its partial derivatives of the (l − 1)th order satisfy the Lipschitz condition

with respect to the variables ψ and τ.

Corollary 4. The function Y (ψ, τ, ε) constructed in the proof of Theorem

16.2 defines the integral manifold y = Y (ψ, τ, ε) of the system

dy

dτ

= H(τ )y + F(y, τ) + _a(x(τ) + y, ϕ, τ) + εβB1(x(τ) + y, ϕ, τ, ε),

dϕ

dτ

= ω(τ )

ε

+ b(x(τ) + y, ϕ, τ, ε) + εδB2(x(τ) + y, ϕ, τ, ε). (16.52)

The statement below solves the problem of the smoothness of the integral

manifold of system (16.1) with respect to the parameter ε.

Theorem 16.3. If the conditions of Theorem 16.1 are satisfied and the functions

A(x, ϕ, τ, ε) and b(x, ϕ, τ, ε) have l ≥ 2 continuous and uniformly

bounded (by a certain constant) partial derivatives with respect to all variables

(x, ϕ, τ, ε) ∈ G, then the integral manifold x = X(ψ, τ, ε) = x(τ) + Y (ψ, τ, ε)

of system (16.1) is l − 1 times continuously differentiable with respect to

(ψ, τ, ε) ∈ G1,

194 Integral Manifolds Chapter 3

____

Dsψ

∂q

∂τq

∂r

∂εr Y (ψ, τ, ε)

____

≤ cεα−q−2r ∀(ψ, τ, ε) ∈ G1, (16.53)

0 ≤ s + q + r ≤ l − 1, and the derivatives of the (l − 1)th order satisfy the

Lipschitz condition with respect to the variables ψ, τ, and ε on the set ψ ∈ Rm,

τ ∈ R, ε ∈ [ε0, ε0], where ε0 is an arbitrary value from the interval (0, ε0).

In view of technical difficulties, we do not prove Theorem 16.3 here. We

only note that the iterations Yj(ψ, τ, ε), j ≥ 1, defined by equality (13.7) are l

times continuously differentiable with respect to (ψ, τ, ε) ∈ G1, and their partial

derivatives satisfy the inequalities [SPe6]

____

Dsψ

∂q

∂τq

∂r

∂εr Yj(ψ, τ, ε)

____

≤ ds,q,rεα−q−2r ∀(ψ, τ, ε) ∈ G1, j≥ 1, (16.54)

for 0 ≤ s+q+r ≤ l. To establish (16.54), one should use the methods proposed

in the proof of Lemma 16.1 and the estimates obtained in [Sam2] for oscillation

systems with constant frequency vector. In the case of multifrequency systems

(16.1) with ω = ω(τ ), it is necessary to carefully take into account the measure

of the set of points of a time interval of unit length for which the scalar product

(k, ω(τ )) is sufficiently small (k is an integer-valued vector), which substantially

affects the character of estimates of oscillation integrals.

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