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18. Decomposition of Equations in a Neighborhood of Asymptotically Stable Integral Manifold

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Consider the system of ordinary differential equations

dτ

= a(x, ϕ, τ, ε),

dϕ

dτ

= ω(τ )

+ b(x, ϕ, τ, ε), (18.1)

where x ∈ D ⊂ Rn, ϕ ∈ Rm, m≥ 2, τ ∈ R, ε ∈ (0, ε0] is a small parameter,

the vector functions a and b are defined on the set G = D×Rm×R×(0, ε0],

2π-periodic in ϕν, ν = 1,m, and thrice 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 independent of ε. Assume that

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

where the function _a(x, ϕ, τ ) averaged with respect to ϕ over the cube of periods

is identically equal to zero, and the Fourier coefficients ck(x, τ, ε) of the function

c(x, ϕ, τ, ε) = [_a(x, ϕ, τ ); b(x, ϕ, τ, ε)] satisfy the inequality

k_=0

_k_3 sup

_ck_ + _k_2

sup

___

∂ck

∂τ

___

+ sup

___

∂ck

∂x

___

≤ c1. (18.2)

Consider the system of equations of the first approximation for slow variables

averaged with respect to all angular variables ϕ:

∂x

∂τ

= a(x, τ ).

Section 18 Decomposition of Equations 207

Assume that there exists a solution x = x(τ ) of this system defined on the entire

axis that lies, together with a certain ρ-neighborhood of it, in the domain D and

for which the normal fundamental matrix Q(τ, t) of solutions of the variational

equation

dτ

= ∂a(x(τ ), τ)

∂x

satisfies the estimate

_Q(τ, t)_ ≤ Ke

−γ(τ−t) ∀τ ≥ t ∈ R, (18.3)

where γ > 0 and K ≥ 1 are certain constants. Let

σ0 =

K sup

ϕ,τ

___

∂

∂x

_a(x(τ ), ϕ, τ)

___

< 1. (18.4)

We also impose certain restrictions on the components ων(τ ), ν = 1,m, of

the frequency vector ω(τ ). Assume that the functions

ω(μ)

ν (τ ) ≡ dμ

dτμων(τ ), ν= 1,m, μ = 0, p − 1, p≥ m,

are uniformly continuous on the entire axis, and

_(WT

p (τ )Wp(τ ))−1WT

p (τ )_ ≤ c1 ∀τ ∈ R, (18.5)

where

Wp(τ) = (ω(μ−1)

ν (τ ))m,p

ν,μ=1

and WT

p (τ ) is the transposed matrix. As proved above, under these conditions

there exists an asymptotically stable integral manifold x = X(ψ, τ, ε) = x(τ) +

Y (ψ, τ, ε) of system (18.1) for which the function Y (ψ, τ, ε) is 2π-periodic

in ψν, ν = 1,m, twice continuously differentiable with respect to ψ and τ

for every value of ε, and such that its second derivatives satisfy the Lipschitz

condition and

_Y (ψ, τ, ε)_ +

___

∂

∂ψ

Y (ψ, τ, ε)

___

ν=1

___

∂2

∂ψ∂ψν

Y (ψ, τ, ε)

___

≤ c1εα,

ν=1

___

∂2

∂ψ∂ψν

Y (ψ, τ, ε) − ∂2

∂ψ∂ψν

Y (ψ, τ, ε)

___

≤ c1εα_ψ − ψ_ (18.6)

∀(ψ, τ, ε) ∈ G1, ψ ∈ Rm, G1 = Rm × R × (0, ε0], α=

208 Integral Manifolds Chapter 3

Performing the change of variables y = x − X(ϕ, τ, ε) in (18.1), we obtain

the system

dτ

= a(y + X(ϕ, τ, ε), ϕ, τ, ε) − a(X(ϕ, τ, ε), ϕ, τ, ε)

− ∂X(ϕ, τ, ε)

∂ϕ

[b(y + X(ϕ, τ, ε), ϕ, τ, ε) − b(X(ϕ, τ, ε), ϕ, τ, ε)],

dϕ

dτ

= ω(τ )

+ b(y + X(ϕ, τ, ε), ϕ, τ, ε). (18.7)

For every value of the small parameter ε, the right-hand side of this system has

continuous partial derivatives of the first order with respect to y, ϕ, and τ, and

the derivatives with respect to y and ϕ satisfy the Lipschitz condition with respect

to y and ϕ with a Lipschitz constant independent of ε.

In the present section, we decompose Eqs. (18.7) in a neighborhood of the

integral manifold y ≡ 0 by introducing new variables according to the formula

ϕ = ψ +Φ(y, ψ, τ, ε) (Φ(0, ψ, τ, ε) ≡ 0), (18.8)

which reduces system (18.7) to the form

dτ

= a(y + X(ψ +Φ, τ, ε), ψ +Φ, τ, ε) − a(X(ψ +Φ, τ, ε), ψ +Φ, τ, ε)

− ∂X(ψ +Φ, τ, ε)

∂ϕ

[b(y + X(ψ +Φ, τ, ε), ψ +Φ, τ, ε)

− b(X(ψ +Φ, τ, ε), ψ +Φ, τ, ε)],

dψ

dτ

= ω(τ )

+ b(X(ψ, τ, ε), ψ, τ, ε), (18.9)

For ω = const, results concerning the canonical form of a dynamical system

in a neighborhood of an invariant torus are presented in Chapter 4. One

should also note the work [SaS], where the decomposition of equations was carried

out for systems with slowly varying phase, and the monograph [StS], where

the decomposition of singularly perturbed equations was carried out. We write

the following partial differential equation for the determination of the function

Φ = Φ(y, ψ, τ, ε):

Section 18 Decomposition of Equations 209

∂Φ

∂τ

+ ∂Φ

∂ψ

_ω(τ )

+ b(X(ψ, τ, ε), ψ, τ, ε)

+ ∂Φ

∂y

a(y + X(ψ +Φ, τ, ε), ψ +Φ, τ, ε)

− a(X(ψ +Φ, τ, ε), ψ +Φ, τ, ε)

− ∂X(ψ +Φ, τ, ε)

∂ϕ

(b(yX(ψ +Φ, τ, ε), ψ +Φ, τ, ε)

− b(X(ψ +Φ, τ, ε), ψ +Φ, τ, ε)

= b(y + X(ψ +Φ, τ, ε), ψ +Φ, τ, ε) − b(X(ψ, τ, ε), ψ, τ, ε). (18.10)

We construct a solution of Eq. (18.10) by the method of successive approximations,

defining these approximations by the formula

Φj+1(y, ψ, τ, ε) = −

∞ _

[bj −_b]dξ, Φ0 ≡ 0, (18.11)

where

bj = b(yξ,j

τ + X(ψξ

τ +Φj(yξ,j

τ , ψξ

τ, ξ, ε), ξ, ε), ψξ

τ +Φj(yξ,j

τ , ψξ

τ, ξ, ε), ξ, ε),

= b(X(ψξ

τ, ξ, ε), ψξ

τ, ξ, ε).

Here, (yξ,j

τ , ψξ

τ) = (yξ,j

τ (y, ψ, ε), ψξ

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

dξ

yξ,j

τ = aj − _aj − ∂Xj

∂ϕ

(bj −_bj), yτ,j

τ = y, (18.12)

dξ

ψξ

τ = ω(ξ)

+_b, ψτ

τ = ψ, (18.13)

where aj = a(yξ,j

τ + Xj, ψξ

τ + Φj, ξ, ε), _aj = a(Xj, ψξ

τ + Φj, ξ, ε), Xj =

X(ψξ

τ +Φj, ξ, ε), _bj = b(Xj, ψξ

τ +Φj, ξ, ε), and Φj = Φj(yξ,j

τ , ψξ

τ, ξ, ε).

If the order of differentiation and integration can be changed, then one can

easily verify that the function Φj+1=Φj+1(y, ψ, τ, ε) defined by equality (18.11)

satisfies the partial differential equation

210 Integral Manifolds Chapter 3

∂Φj+1

∂τ

+ ∂Φj+1

∂ψ

_ω(τ )

+ b(X(ψ, τ, ε), ψ, τ, ε)

+ ∂Φj+1

∂y

a(y + X(ψ +Φj, τ, ε), ψ +Φj, τ, ε)

− a(X(ψ +Φj, τ, ε), ψ +Φj, τ, ε)

− ∂X(ψ +Φj, τ, ε)

∂ϕ

(b(y + X(ψ +Φj, τ, ε), ψ +Φj, τ, ε)

− b(X(ψ +Φj, τ, ε), ψ +Φj, τ, ε)

= b(y + X(ψ +Φj, τ, ε), ψ +Φj, τ, ε) − b(X(ψ, τ, ε), ψ, τ, ε). (18.14)

Assuming that the norm of the matrix

∂b

∂ϕ

is sufficiently small, i.e.,

sup

___

∂b(x, ϕ, τ, ε)

∂ϕ

___

min

;

, (18.15)

we prove that the sequence {Φj(y, ψ, τ, ε)} converges uniformly to a certain

function Φ(y, ψ, τ, ε) on the set y ∈ Ph ≡ {y : y ∈ Rn, _y_ ≤ h}, ψ ∈ Rm,

τ ∈ R, ε ∈ (0, ε0], provided that ε0 > 0 and h > 0 are sufficiently small.

Moreover, we establish the convergence of the sequences

_ ∂

∂y

Φj

_ ∂

∂ψ

Φj

and

_ ∂

∂τ

Φj

∂

∂y

Φ,

∂

∂ψ

Φ, and

∂

∂τ

Φ, respectively. Then, passing to the

limit as j → ∞ in Eq. (18.14), we obtain equality (18.10) for the function

Φ(y, ψ, τ, ε) constructed above.

Theorem 18.1. Suppose that the conditions imposed on system (18.1) and

conditions (18.2)–(18.5) and (18.15) are satisfied. Then, for sufficiently small

h > 0 and ε0 > 0, there exists a change of variables (18.8) that reduces system

(18.1) to the decomposed form (18.9), where the function Φ(y, ψ, τ, ε) is 2π-

periodic in ψν, ν = 1,m, continuously differentiable with respect to y, ψ, and

τ for every fixed ε ∈ (0, ε0], and such that

_Φ_ ≤ d1_y_,

___

∂Φ

∂y

___

≤ d2,

___

∂Φ

∂ψ

___

≤ d3_y_ (18.16)

Section 18 Decomposition of Equations 211

∀(y, ψ, τ, ε) ∈ Ph × Rm × R × (0, ε0] ≡ G, and its partial derivatives with

respect to y and ψ satisfy the Lipschitz condition:

___

∂

∂y

Φ(y, ψ, τ, ε) − ∂

∂y

Φ(y, ψ, τ, ε)

___

≤ μ(_y − y_ + _ψ − ψ_),

___

∂

∂ψ

Φ(y, ψ, τ, ε) − ∂

∂ψ

Φ(y, ψ, τ, ε)

___

≤ ν_y − y_, (18.17)

___

∂

∂ψ

Φ(y, ψ, τ, ε) − ∂

∂ψ

Φ(y, ψ, τ, ε)

___

≤ ν_y_ ・ _ψ − ψ_.

Here, d1 − d3, μ, and ν are constants independent of ε and h.

We prove Theorem 18.1 in the next section. Here, we establish certain properties

of a solution of the Cauchy problem (18.12).

Lemma 18.1. Suppose that the conditions of Theorem 18.1 are satisfied and,

for every ε ∈ (0, ε0] and certain j ≥ 0, the function Φj(y, ψ, τ, ε) is continuously

differentiable with respect to y, ψ, and τ. Then one can find constants

ε0 > 0, h1 > 0, and γ1 ∈

_γ

2 , γ

such that, for all (y, ψ, τ, ε) ∈ G, h ≤ h1,

ε0 ≤ ε0, and ξ ≥ τ, the solution yξ,j

τ = yξ,j

τ (y, ψ, ε) of the Cauchy problem

(18.12) satisfies the inequality

_yξ,j

_ ≤ K_y_e

−γ1(ξ−τ).

Proof. It follows from (18.12) and the smoothness conditions for the righthand

side of system (18.1) that

dyξ,j

dξ

= H(ξ)yξ,j

τ + ∂_a(x(ξ), ψξ

τ +Φj, ξ)

∂x

yξ,j

τ + F1(yξ,j

τ , ψξ

τ ,Φj, ξ, ε), (18.18)

where

_F1_ ≤

ν=1

sup

___

∂2

∂x∂xν

a(x, ϕ, τ, ε)

___

(_yξ,j

_ + c1εα)_yξ,j

+ ε sup

___

∂

∂x

A(x, ϕ, τ, ε)

___

_yξ,j

_ + c1εα sup

___

∂

∂x

b(x, ϕ, τ, ε)

___

_yξ,j

≤ c2(_yξ,j

_ + εα0

), c2 = nc1(1 + n + c1), H(ξ) = ∂

∂x

a(x(ξ), ξ).

212 Integral Manifolds Chapter 3

For the function yξ,j

τ , we can write the representation

yξ,j

τ = Q(ξ, τ)y +

_ξ

Q(ξ, l)

_ ∂

∂x

_a(x(l), ψlτ

+Φj(yl,j

τ , ψlτ

, l, ε), l)yl,j

τ + F1(yl,j

τ , ψlτ

,Φj(yl,j

τ , ψlτ

, l, ε), l, ε)

dl,

whence

_yξ,j

_ ≤ K_y_e

−γ(ξ−τ) +

_ξ

−γ(ξ−l)

K sup

ϕ,τ

___

∂

∂x

_a(x(τ ), ϕ, τ)

___

+ Kc2(_yl,j

_ + εα0

)

_yl,j

_dl.

Assume that _yξ,j

τ _ < 2K_y_ on the maximum half-interval ξ ∈ [τ, T). Then,

denoting

zξ

τ = _yξ,j

_eγ(ξ−τ),

we obtain the inequality

zξ

≤ K_y_ +

_ξ

K sup

ϕ,τ

___

∂

∂x

_a(x(t), ϕ, τ)

___

+ Kc2(εα0

+ 2K_y_)

zlτ

dl,

whose solution (according to the Gronwall–Bellman lemma) satisfies the estimate

zξ

≤ K_y_ exp

K sup

ϕ,τ

___

∂

∂x

_a(x(t), ϕ, τ)

___

+ Kc2(εα0

+ 2K_y_)

(ξ − τ )

Taking into account that, according to (18.4), we have σ0 < 1. and assuming that

2K2c2h ≤ 1

8γ(1 − σ0), Kc2

εα0

≤ 1

8γ(1 − σ0),

we deduce from the last estimate that

_yξ,j

_ ≤ K_y_e

−γ1(ξ−τ), γ1 =

3 − σ0

4 , (18.19)

for all y ∈ Ph, ψ∈ Rm, ξ ∈ [τ, T), and ε ∈ (0, ε0]. It is obvious that

_yξ,j

_ ≤ K_y_ < 2K_y_ ∀ξ ∈ [τ, T).

Therefore, relation (18.19) holds for all ξ ∈ [τ,∞). Lemma 18.1 is proved.

Section 18 Decomposition of Equations 213

Lemma 18.2. If the conditions of Lemma 18.1 are satisfied and the function

Φj(y, ψ, τ, ε) satisfies the inequality

___

∂

∂y

Φj

___

≤ d2, then, for sufficiently small

h > 0 and ε0 > 0, the following estimate is true:

___

∂

∂y

yξ,j

τ (y, ψ, ε)

___

≤ Ke

−γ1(ξ−τ) ∀(y, ψ, τ, ε) ∈ G, ξ≥ τ. (18.20)

Proof. Differentiating Eq. (18.12) with respect to y and using the equality

∂

∂y

yτ,j

τ (y, ψ, ε) = En, where En is the n-dimensional identity matrix, we get

∂

∂y

yξ,j

τ = Q(ξ, τ) +

_ξ

Q(ξ, l)

_ ∂

∂x

_a(x(l), ψlτ

+Φj(yl,j

τ , ψlτ

, l, ε), l)

+ F2

yl,j

τ , ψlτ

,Φj(yl,j

τ , ψlτ

, l, ε),

∂

∂y

Φj(yl,j

τ , ψlτ

, l, ε), l, ε

__ ∂

∂y

yl,j

τ dl, (18.21)

where

_F2_ ≤ c3(εα0

+ _y_),

c3 = nc1K[n + nc1 + c1 + d2(nc1 + nc21

+ m + 2mc1)].

Equality (18.21) yields

___

∂

∂y

yξ,j

___

eγ(ξ−τ) ≤ K +

_ξ

eγ(l−τ)

_γ

2 σ0 + Kc3(εα0

+ h)

____ ∂

∂y

yl,j

___

dl,

Solving this inequality, we obtain estimate (18.20) under the conditions Kc3εα0

≤

8γ(1 − σ0) and Kc3h ≤ 1

8γ(1 − σ0). Lemma 18.2 is proved.

Lemma 12.1–12.3 yield the following estimates for a solution ψξ

τ = ψξ

τ (ψ, ε)

of the Cauchy problem (18.13):

___

∂

∂ψ

(ϕξ

− ψ)

___

≤ c4εα(1 + |ξ − τ |)ec4εα|ξ−τ|

214 Integral Manifolds Chapter 3

___

∂

∂τ

ψξ

___

≤ c4

_ω(τ )_ + 1

ec4εα|ξ−τ|

, (18.22)

ν=1

___

∂2

∂ψ∂ψν

ϕξ

___

≤ c4εα(1 + |ξ − τ |2)ec4εα|ξ−τ|

Here, ψ ∈ Rm, ε ∈ (0, ε0], τ ∈ R, ξ ∈ R, , and c4 is the constant equal to

the greatest constant in the corresponding estimates in Lemmas 12.1–12.3. Using

(18.22) and repeating the scheme of the proof of Lemmas 18.1 and 18.2, we

establish the following statements:

Lemma 18.3. Let

___

∂

∂y

Φj(y, ψ, τ, ε)

___

≤ d2,

___

∂

∂ψ

Φj(y, ψ, τ, ε)

___

≤ d3_y_

∀(y, ψ, τ, ε) ∈ G.

Then, for sufficiently small h > 0 and ε0 > 0 and all (y, ψ, τ, ε) ∈ G and

ξ ≥ τ, the following inequalities are true:

___

∂

∂ψ

yξ,j

___

≤ c5_y_(1 + _y_d3)e

−γ2(ξ−τ), γ2 =

5 − σ0

8 γ, (18.23)

___

∂

∂τ

yξ,j

___

≤ c5_y_

_ω(τ )_ + d3_y_

−γ2(ξ−τ), (18.24)

where the constant c5 depends on d2 and does not depend on d3 and ε.

Lemma 18.4. If the conditions of Lemma 18.3 are satisfied and the function

∂

∂y

Φj(y, ψ, τ, ε) satisfies the Lipschitz condition

___

∂

∂y

Φj(y, ψ, τ, ε) − ∂

∂y

Φj(y, ψ, τ, ε)

___

≤ μ(_y − y_ + _ψ − ψ_) (18.25)

∀y, y ∈ Ph, ψ,ψ ∈ Rm, τ ∈ R, ε ∈ (0, ε0],

then the following estimate is true:

___

∂

∂y

yξ,j

τ (y, ψ, ε)− ∂

∂y

yξ,j

τ (y, ψ, ε)

___

≤ c6(1+hμ)e

−γ3(ξ−τ)(_y−y_+_ψ−ψ_),

where ξ ≥ τ, γ3 = 1

16(9 − σ0)γ, ε0 > 0 and h > 0 are sufficiently small, and

c6 is a constant independent of μ, h, and ε.

Section 18 Decomposition of Equations 215

Lemma 18.5. Suppose that the conditions of Lemma 18.4 are satisfied and

the following estimates are true:

___

∂

∂ψ

Φj(y, ψ, τ, ε) − ∂

∂ψ

Φj(y, ψ, τ, ε)

___

≤ ν_y − y_, ν= const,

___

∂

∂ψ

Φj(y, ψ, τ, ε) − ∂

∂ψ

Φj(y, ψ, τ, ε)

___

≤ ν_y__ψ − ψ_. (18.26)

Then one can find a constant c7 independent of ν, h, and ε and such that the

following inequalities hold for all ξ ≥ τ :

___

∂

∂ψ

yξ,j

τ (y, ψ, ε) − ∂

∂ψ

yξ,j

τ (y, ψ, ε)

___

≤ c7(1 + hν)e

−γ4(ξ−τ)_y − y_,

___

∂

∂ψ

yξ,j

τ (y, ψ, ε) − ∂

∂ψ

yξ,j

τ (y, ψ, ε)

___

≤ c7(1 + hν)_y_e

−γ4(ξ−τ)_ψ − ψ_,

where γ4 =

(17 − σ0)γ.

Further, we denote

_Φj+1(y, ψ, τ, ε) − Φj(y, ψ, τ, ε)_ = _y_vj(y, ψ, τ, ε),

___

∂

∂ψ

(Φj+1(y, ψ, τ, ε) − Φj(y, ψ, τ, ε))

___

= _y_vj(y, ψ, τ, ε),

vj(0, ψ, τ, ε) ≡ 0, vj(0, ψ, τ, ε) ≡ 0.

We use the following lemma for the investigation of the convergence of the

iterations defined by (18.11):

Lemma 18.6. Suppose that, for all (y, ψ, τ, ε) ∈ G and j ≥ 0, the inequalities

___

∂

∂y

Φj(y, ψ, τ, ε)

___

≤ d2,

___

∂

∂ψ

Φj(y, ψ, τ, ε)

___

≤ d3_y_

are true and the Lipschitz conditions (18.25) and (18.26) are satisfied. Then there

exist constants ε0 > 0, h2 > 0, γ5 >

2 , and c8 such that, for j ≥ 0,

(y, ψ, τ, ε) ∈ G, ε0 ≤ ε0, h ≤ h2, and ξ ≥ τ, the following estimates are

true:

216 Integral Manifolds Chapter 3

_yξ,j+1

− yξ,j

_ ≤ c8_y_2 sup

vje

−γ5(ξ−τ), (18.27)

___

∂

∂y

(yξ,j+1

− yξ,j

τ )

___

≤ c8_y_

sup

+ sup

___

∂

∂y

(Φj+1(y, ψ, τ, ε) − Φj(y, ψ, τ, ε))

___

−γ5(ξ−τ), (18.28)

___

∂

∂ψ

(yξ,j+1

− yξ,j

τ )

___

≤ c8_y_

sup

___

∂

∂y

(Φj+1(y, ψ, τ, ε) − Φj(y, ψ, τ, ε))

___

+ _y_(sup

vj + sup

vj)

−γ5(ξ−τ), (18.29)

where yξ,s

τ = yξ,s

τ (y, ψ, ε) for s = j, j + 1.

Proof. Rewriting Eq. (18.12) in the form

dξ

yξ,j

τ = H(ξ)yξ,j

τ +

_ ∂

∂x

a(tyξ,j

τ + Xj, ψξ

τ +Φj, ξ, ε) − H(ξ)

dtyξ,j

− ∂Xj

∂ϕ

∂

∂x

b(tyξ,j

τ + Xj, ψξ

τ +Φj, ξ, ε)dtyξ,j

τ ,

one can easily obtain the estimate

_yξ,j+1

− yξ,j

_ ≤

_ξ

−γ(ξ−l)

____

_∂aj+1

∂x

− H(l)

___

_yl,j+1

− yl,j

___

_∂aj+1

∂x

− ∂aj

∂x

___

_yl,j

___∂

Xj+1

∂ϕ

− ∂Xj

∂ϕ

__ _

nc1_

yl,j+1

___

∂Xj

∂ϕ

___

_∂bj+1

∂x

yl,j+1

− ∂bj

∂x

yl,j

___

dl. (18.30)

Section 18 Decomposition of Equations 217

We now use Lemma 18.1 and inequalities (18.6). Since

___

∂aj+1

∂x

− H(l)

___

≤ sup

ϕ,τ

___

∂

∂x

_a(x(τ ), ϕ, τ)

___

+ nc1(1 + nc1 + Kn)(εα + _y_),

___

∂aj+1

∂x

− ∂aj

∂x

___

_yl,j

_ ≤ Knc1(n + m)(1 + c1)_y_e

−γ1(l−τ)(_yl,j+1

− yl,j

+ _Φj+1(yl,j+1

τ , ψlτ

, l, ε) − Φj(yl,j

τ , ψlτ

, l, ε)_,

___

∂Xj+1

∂ϕ

− ∂Xj

∂ϕ

__ _

nc1_

yl,j+1

_ +

___

∂Xj

∂ϕ

___

_∂bj+1

∂x

yl,j+1

− ∂bj

∂x

yl,j

___

≤ Knc1c1(1 + nc1 + n + m)[(εα + _y_)_yl,j+1

− yl,j

+ εα_y_e

−γ1(l−τ)_Φj+1(yl,j+1

τ , ψlτ

, l, ε) − Φj(yl,j

τ , ψlτ

, l, ε)_],

estimate (18.30) can be rewritten in the form

_yξ,j+1

− yξ,j

_ ≤

_ξ

−γ(ξ−l)

K sup

ϕ,τ

___

∂

∂x

_a(x(τ ), ϕ, τ)

___

+ (_y_ + εα)3c9K]_yl,j+1

− yl,j

_dl

+ 2c9K_y_

_ξ

−γ(ξ−l)−γ1(l−τ)_Φj+1 − Φj_dl,

where c9 = Knc1(1 + c1)(1 + nc1 + n + m). Taking into account that the

derivative of each function Φj(y, ψ, τ, ε), j ≥ 0, with respect to y is bounded

from above by the constant d2, we get

_Φj+1(yl,j+1

τ , ψlτ, l, ε) − Φj(yl,j

τ , ψlτ

, l, ε)_

≤ _yl,j+1

_ sup

vj + d2_yl,j+1

− yl,j

≤ K_y_e

−γ1(l−τ) sup

vj + d2_yl,j+1

− yl,j

_, (18.31)

218 Integral Manifolds Chapter 3

whence

_yξ,j+1

− yξ,j

_ ≤

_ξ

−γ(ξ−l)

_γ

2 σ0 + Kc9(3 + 2d2)(εα0

+ h)

_yl,j+1

− yl,j

_dl

2c9

2γ1 − γ

K2e

−γ(ξ−l)_y_2 sup

vj . (18.32)

We choose h > 0 and ε0 > 0 so small that

Kc9(3 + 2d2)εα0

≤ 1

8γ(1 − σ0), Kc9

(3 + 2d2)h ≤ 1

8γ(1 − σ0).

Then, solving inequality (18.32), we obtain the estimate

_yξ,j+1

− yξ,j

_ ≤ 2c9

2γ1 − γ

K2e

−γ

4 (3−σ0)(ξ−τ)_y_2 sup

vj ,

the form of which coincides with the form of estimate (18.27). Inequalities (18.28)

and (18.29) can be established by analogy.

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18. Decomposition of Equations in a Neighborhood of Asymptotically Stable Integral Manifold

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