14. Existence of Integral Manifold

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Below, we show that the sequence {Xj(ψ, τ, ε)}, Xj(ψ, τ, ε) = x(τ) +

Yj(ψ, τ, ε), constructed in the previous section converges to the integral manifold

x = X(ψ, τ, ε) of system (12.1).

Theorem 14.1. If conditions (12.2), (12.3), (13.2), and (13.3) are satisfied,

then, for sufficiently small ε0 > 0, the following assertions are true:

(a) there exists an integral manifold x = X(ψ, τ, ε) of system (12.1) that lies

in a d1εα-neighborhood of the curve x = x(τ ) ∀(ψ, τ, ε) ∈ G1 ;

(b) the function X(ψ, τ, ε) is 2π-periodic in ψν, ν = 1,m, and continuously

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

matrix of partial derivatives with respect to ψ satisfies the inequality

___

∂

∂ψ

X(ψ, τ, ε)

___

≤ d2εα

for all (ψ, τ, ε) ∈ G1 and the Lipschitz condition with respect to the variables

ψ:

___

∂X(ψ, τ, ε)

∂ψ

− ∂X(ψ, τ, ε)

∂ψ

___

≤d3εα_ψ − ψ_

∀(ψ, τ, ε) ∈ G1, ψ ∈ Rm;

(c) on the integral manifold, system (12.1) takes the form

dϕ

dτ

= ω(τ )

ε

+ b(X(ϕ, τ, ε), ϕ, τ, ε).

158 Integral Manifolds Chapter 3

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

uniformly on the set G1 to a certain function Y (ψ, τ, ε). For this purpose, we

establish an estimate of the norm _Yj+1 − Yj_. It follows from (13.7) that

Yj+1(ψ, τ, ε) − Yj(ψ, τ, ε)

=

_+∞

−∞

Q(τ, t){[Fj − Fj−1] + [_aj − _aj−1] + ε[Aj − Aj−1]}dt, (14.1)

where

Fl = F(Yl, t), _al = _a(x(t) + Yl, ϕt

τ,l+1, t),

Al = A(x(t) + Yl, ϕt

τ,l+1, t, ε), Yl = Yl(ϕt

τ,l+1, t, ε), l= j, j − 1.

Further, we represent the difference _aj − _aj−1 in the form

_aj − _aj−1

= [_a(x(t), ϕt

τ,j+1, t) − _a(x(t), ϕt

τ,j, t)] + ∂

∂x

_a(x(t), ϕt

τ,j, t)[Yj − Yj−1]

+

_1

0

_ ∂

∂x

_a(x(t) + rYj, ϕt

τ,j+1, t) − ∂

∂x

_a(x(t) + rYj−1, ϕt

τ,j, t)

_

drYj

+

_1

0

_ ∂

∂x

_a(x(t) + rYj−1, ϕt

τ,j, t) − ∂

∂x

_a(x(t), ϕt

τ,j, t)

_

dr[Yj − Yj−1]. (14.2)

Using the smoothness conditions (12.2) and estimate (13.13), we obtain

_Fj − Fj−1_ ≤ n2σ1d1εα_Yj − Yj−1_,

ε_Aj − Aj−1_ ≤ εσ1(n + m)(_Yj − Yj−1_ + _ϕt

τ,j+1

− ϕt

τ,j

_),

___

_1

0

_ ∂

∂x

_a(x(t) + rYj, ϕt

τ,j+1, t) − ∂

∂x

_a(x(t) + rYj−1, ϕt

τ,j, t)

_

drYj

___

≤ nσ1(n + m)d1εα(_Yj − Yj−1_ + _ϕt

τ,j+1

− ϕt

τ,j

_),

Section 14 Existence of Integral Manifold 159

___

_1

0

_ ∂

∂x

_a(x(t) + rYj−1, ϕt

τ,j, t) − ∂

∂x

_a(x(t), ϕt

τ,j, t)

_

dr(Yj − Yj−1)

___

≤ n2σ1d1εα_Yj − Yj−1_. (14.3)

We now estimate the difference Yj − Yj−1 as follows:

_Yj − Yj−1_ ≤ _Yj(ϕt

τ,j+1, t, ε) − Yj(ϕt

τ,j, t, ε)_

+ _Yj(ϕt

τ,j, t, ε) − Yj−1(ϕt

τ,j, t, ε)_ ≤ d2εα_ϕt

τ,j+1

− ϕt

τ,j

_

+ sup

G1

_Yj(ψ, τ, ε) − Yj−1(ψ, τ, ε)_. (14.4)

Since, according to Lemma 12.5, we have

_ϕt

τ,j+1

− ϕt

τ,j

_ ≤ σ8e

1

2 γ|t−τ| sup

G1

_Yj(ψ, τ, ε) − Yj−1(ψ, τ, ε)_ (14.5)

for c19εα0

≤ 1

4γ and σ8 = c18 max

_

1;

4

γ

_

, combining (14.1)–(14.5) we get

_Yj+1(ψ,τ, ε) − Yj(ψ, τ, ε)_

≤ (σ0 + σ9εα) sup

G1

_Yj(ψ, τ, ε) − Yj−1(ψ, τ, ε)_

+

___

∞ _

−∞

Q(τ, t)[_a(x(t), ϕt

τ,j+1, t) − _a(x(t), ϕt

τ,j, t)]dt

__ _

,

(

14.6)

where

σ9 =

2

γ

K{2n2σ1d1 + σ1(n + m)(nd1 + 1)(1 + 2σ8)

+ 2σ1σ8d2[n + 2n2d1 + (n + m)(1 + nd1)]}.

To estimate the integral on the right-hand side of the last inequality (denote it

by Ij ), we represent it in the form of the sum of integrals over segments of unit

length and use estimate (1.20). As a result, we get

160 Integral Manifolds Chapter 3

_Ij_ ≤

_

k_=0

∞_

s=−∞

σ3Kεα max

[τ+s,τ+s+1]

e

−γ|t−τ|

__

(1 + nσ1) sup

G

_ak_

+

1 + σ1

_k_ sup

G

___

∂ak

∂x

___

+ sup

G

___

∂ak

∂τ

___

_

× max

[τ+s,τ+s+1]

| exp{i(k, θt

τ,j+1)} − exp{i(k, θt

τ,j)}|

+

1

_k_ sup

G

_ak_ max

[τ+s,τ+s+1]

___

_

k,

d

dt

θt

τ,j+1

_

exp{i(k, θt

τ,j+1)}

−

_

k,

d

dt

θt

τ,j

_

exp{i(k, θt

τ,j)}

___

_

, (14.7)

where ak = ak(x, τ ). Taking into account the inequalities

| exp{i(k, θt

τ,j+1)} − exp{i(k, θt

τ,j)}| ≤ _k__ϕt

τ,j+1

− ϕt

τ,j

_,

___

exp{i(k, θt

τ,j+1)} d

dt

(k, θt

τ,j+1) − exp{i(k, θt

τ,j)} d

dt

(k, θt

τ,j)

___

≤ _k_2 sup

G

_b__ϕt

τ,j+1

− ϕt

τ,j

_ + _k_

_

sup

G

___

∂b

∂x

___

_Yj(ϕt

τ,j+1, t, ε)

− Yj−1(ϕt

τ,j, t, ε)_ + sup

G

___

∂b

∂ϕ

___

_ϕt

τ,j+1

− ϕt

τ,j

_

_

(14.8)

and estimates (14.4), (14.5), and (14.7), we obtain

_Ij_ ≤ σ10εα sup

G1

_Yj(ψ, τ, ε) − Yj−1(ψ, τ, ε)_,

where

σ10 = 2Kσ1σ3[(1 + nσ1)(1 + σ8) + σ1(1 + m + nd2)] e

γ

2

1 − e

−γ

2

.

Combining the last inequality with (14.6), for j ≥ 1 and ε0 ≤ [2(1−σ0)−1(σ9+

σ10)]− 1

α we get

sup

G1

_Yj+1(ψ, τ, ε) − Yj(ψ, τ, ε)_

≤ 1 + σ0

2

sup

G1

_Yj(ψ, τ, ε) − Yj−1(ψ, τ, ε)_. (14.9)

Section 14 Existence of Integral Manifold 161

Since the constant 1

2(1+σ0) is less than 1 and _Y1(ψ, t, ε)_ ≤ d1εα0

, it follows

from (14.9) that the sequence {Yj(ψ, τ, ε)} is uniformly convergent on the set

G1. Therefore, the function

Y (ψ, τ, ε) = lim

j→∞

Yj(ψ, τ, ε)

is 2π-periodic in ψν, ν = 1,m, continuous in (ψ, τ) for every fixed ε, and

such that _Y (ψ, τ, ε)_ ≤ d1εα ∀(ψ, τ, ε) ∈ G1.

To prove the convergence of the sequence

_ ∂

∂ψ

Yj(ψ, τ, ε)

_

, we consider the

equality

∂

∂ψ

(Yj+1(ψ, τ, ε) − Yj(ψ, τ, ε))

=

∞ _

−∞

Q(τ, t)

__∂Fj

∂y

+ ∂_aj

∂x

+ ε

∂Aj

∂x

_∂Yj

∂ϕ

+ ε

∂Aj

∂ϕ

_ ∂

∂ψ

(ϕt

τ,j+1

− ϕt

τ,j)dt

+

_+∞

−∞

Q(τ, t)

___∂Fj

∂y

− ∂Fj−1

∂y

_

+

_∂_aj

∂x

− ∂_aj−1

∂x

_

+ ε

_∂Aj

∂x

− ∂Aj−1

∂x

__∂Yj

∂ϕ

+ ε

_∂Aj

∂ϕ

− ∂Aj−1

∂ϕ

_

+

_∂Fj−1

∂y

+ ∂_aj−1

∂x

+ ε

∂Aj−1

∂x

__∂Yj

∂ϕ

− ∂Yj−1

∂ϕ

__

×

_ ∂

∂ψ

(ϕt

τ,j

− ψ) + Em

_

dt

+

_+∞

−∞

Q(τ, t)

_∂_aj

∂ϕ

− ∂_aj−1

∂ϕ

__ ∂

∂ψ

(ϕt

τ,j

− ψ) + Em

_

dt

+

_+∞

−∞

Q(τ, t)∂_aj

∂ϕ

∂

∂ψ

(ϕt

τ,j+1

− ϕt

τ,j)dt,

which follows from (13.7). Using Lemmas 12.1 and 12.5 and following the proof

of inequality (14.9), we get

162 Integral Manifolds Chapter 3

sup

G1

___

∂

∂ψ

(Yj+1(ψ, τ, ε) − Yj(ψ, τ, ε))

___

≤ (σ0 + σ11εα0

) sup

G1

___

∂

∂ψ

(Yj(ψ, τ, ε) − Yj−1(ψ, τ, ε))

___

+ σ12 sup

G1

_Yj(ψ, τ, ε) − Yj−1(ψ, τ, ε)_, (14.10)

where the constants σ11 and σ12 are independent of ε and j. Since σ0 < 1,

we deduce from the last estimate (by choosing ε0 > 0 sufficiently small) that the

sequence

_ ∂

∂ψ

Yj(ψ, τ, ε)

_

converges uniformly on the set G1 to the function

∂

∂ψ

Y (ψ, τ, ε), and, according to (13.10), the following inequality is true:

___

∂

∂ψ

Y (ψ, τ, ε)

___

≤ d2εα ∀(ψ, τ, ε) ∈ G1.

Also note that the function

∂

∂ψ

Y (ψ, τ, ε) is continuous in ψ and τ for every

ε ∈ (0, ε0], and the Lipschitz condition with respect to ψ follows from the last

inequality in (13.10).

Now consider the sequence

_ ∂

∂τ

Yj(ψ, τ, ε)

_

. It follows from (13.9) that

∂

∂τ

(Yj+1 − Yj) = H(τ )(Yj+1 − Yj) + [F(Yj, τ) − F(Yj−1, τ)]

+ [_a(Xj, ψ, τ) − _a(Xj−1, ψ, τ)]

+ ε[A(Xj, ψ, τ, ε) − A(Xj−1, ψ, τ, ε)]

− ∂

∂ψ

(Yj+1 − Yj)

_ω(τ )

ε

+ b(Xj, ψ, τ, ε)

_

− ∂Yj

∂ψ

[b(Xj, ψ, τ, ε) − b(Xj−1, ψ, τ, ε)],

whence

Section 14 Existence of Integral Manifold 163

___

∂

∂τ

Yj+1(ψ, τ, ε) − ∂

∂τ

Yj(ψ, τ, ε)

___

≤ nσ1 sup

G1

_Yj+1(ψ, τ, ε) − Yj(ψ, τ, ε)_

+ (1 + ε0 + d2εα0

+ nd1εα0

)nσ1 sup

G1

_Yj(ψ, τ, ε) − Yj−1(ψ, τ, ε)_

+

__ω(τ )_

ε

+ σ1

_

sup

G1

___

∂

∂ψ

Yj+1(ψ, τ, ε) − ∂

∂ψ

Yj(ψ, τ, ε)

__ _

.

Since the sequences {Yj(ψ, τ, ε)} and

_ ∂

∂ψ

Yj(ψ, τ, ε)

_

are uniformly convergent

on the set G1, the last inequality yields the uniform convergence of the

sequence

_ ∂

∂τ

Yj(ψ, τ, ε)

_

on the set

ψ ∈ Rm, τ ∈ [−T,T], ε∈ [ε0, ε0], (14.11)

where T > 0 and ε0

∈ (0, ε0) are arbitrary. Therefore,

lim

j→∞

∂Yj(ψ, τ, ε)

∂τ

= ∂Y (ψ, τ, ε)

∂τ

(14.12)

for all (ψ, τ, ε) from set (14.11). By virtue of the arbitrariness of T and ε0,

we obtain equality (14.12) for all (ψ, τ, ε) ∈ G1. It is clear that the function

∂

∂τ

Y (ψ, τ, ε) is continuous in (ψ, τ) ∈ Rm × R.

Passing to the limit as j →∞ in Eq. (13.9), we get

∂X

∂τ

+ ∂X

∂ψ

_ω(τ )

ε

+ b(X,ψ, τ, ε)

_

= a(X, τ) + _a(X,ψ, τ) + εA(X,ψ, τ, ε), (14.13)

where X = X(ψ, τ, ε) = x(τ) + Y (ψ, τ, ε).

Further, we consider the Cauchy problem

dϕ

dτ

= ω(τ )

ε

+ b(X(ϕ, τ, ε), ϕ, τ, ε), ϕ|τ=τ0 = ψ ∈ Rm, τ0 ∈ R.

The smoothness conditions enable one to extend the solution ϕ = ϕττ

0(ψ, ε) of

the Cauchy problem for all τ ∈ R. Using (14.13), one can easily verify that the

164 Integral Manifolds Chapter 3

function xττ

0(ψ, ε) = X(ϕττ

0(ψ, ε), τ, ε) satisfies the following equation for all

τ ∈ R:

dxττ

0

dτ

= a(xττ

0, τ) + _a(xττ

0, ϕττ

0, τ) + εA(xττ

0, ϕττ

0, τ, ε).

Therefore, by definition [MiLy], x = X(ψ, τ, ε) is the integral manifold of system

(12.1). The properties of the function X(ψ, τ, ε) follow from the properties

of x(τ ) and Y (ψ, τ, ε). Theorem 14.1 is proved.

Corollary 1. If the conditions of Theorem 14.1 are satisfied and the functions

A(x, ϕ, τ, ε) and b(x, ϕ, τ, ε) are continuous in the collection of variables on the

set G, then X(ψ, τ, ε) is continuous on G1.

Indeed, it follows from Remark 1 (Section 13) that each function Yj(ψ, τ, ε),

j ≥ 0, is continuous on the set G1. Since the sequence {Yj(ψ, τ, ε)} converges

uniformly on G1, the limit function Y (ψ, τ, ε) and, hence, X(ψ, τ, ε) are continuous

on G1.

Corollary 2. If the conditions of Theorem 14.1 are satisfied and _ω(τ )_, ___ d

dτ

ω(τ )

__ _

,

and

___

∂

∂τ

A(x, ϕ, τ, ε)

___

are uniformly bounded by a constant σ1 for

any (x, ϕ, τ, ε) ∈ G, then

___

∂

∂τ

X(ψ, τ, ε)

___

≤ (d4 + σ1)εα−1 ∀(ψ, τ, ε) ∈ G1,

and the matrices

∂

∂τ

X and

∂

∂ψ

X satisfy the Lipschitz conditions

___

∂

∂τ

X(ψ, τ, ε) − ∂

∂τ

X(ψ, τ, ε)

___

≤ d5εα−1_ψ − ψ_ + (d6 + σ1(1 + nσ1))εα−2|τ − τ |,

___

∂

∂ψ

X(ψ, τ, ε) − ∂

∂ψ

X(ψ, τ, ε)

___

≤ d3εα_ψ − ψ_ + d5εα−1|τ − τ |

for any τ, τ ∈ R, ψ, ψ ∈ Rm, and ε ∈ (0, ε0].

To prove this fact, it suffices to use inequalities (13.11) and the smoothness

condition for the function x(τ ).

Section 15 Conditional Asymptotic Stability of Integral Manifold 165

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