15. Conditional Asymptotic Stability of Integral Manifold

Back

In this section, we establish the conditional asymptotic stability of the integral

manifold x = X(ψ, τ, ε) of system (12.1) with respect to a certain set

of initial data for slow variables. In the theorem presented below, we denote

by (xττ

0(y, ψ, ε); ϕττ

0(y, ψ, ε)) the solution of system (12.1) that takes the value

(y; ψ) for τ = τ0 and by n0 the integer number defined in Section 13.

Theorem 15.1. Suppose that the conditions of Theorem 14.1 are satisfied.

Then, for sufficiently small ε0 > 0 and any (ψ, τ0, ε) ∈ G1, in a certain neighborhood

of the point x(τ0) there exist an (n−n0)-dimensional manifold S+ and

an n0-dimensional manifold S− such that, for τ ∈ [τ0,∞) (τ ∈ (−∞, τ0]),

the solution (xττ

0(y, ψ, ε); ϕττ

0(y, ψ, ε)) of system (12.1) is defined for all y ∈ S+

(y ∈ S−), and the slow variables xττ

0(y, ψ, ε) tend exponentially to the integral

manifold x = X(ψ, τ, ε) as τ → +∞ (τ →−∞) for y ∈ S+ (y ∈ S−).

Proof. We construct a sequence {Zj(ψ, τ, ε, τ0, d)} using the recurrence

formula

Zj+1(ψ, τ, ε, τ0, d)

= Q(τ, τ0)d +

∞ _

τ0

Q(τ, t)[F(Zj, t) + _a(x(t) + Zj , ϕt

τ,j+1, t)

+ εA(x(t) + Zj , ϕt

τ,j+1, t, ε)]dt, Z0 ≡ 0, (15.1)

where

Zj = Zj(ϕt

τ,j+1, t, ε, τ0, d), Q(τ, τ0) = diag (0,Q−(τ, τ0)),

d is a constant n-dimensional vector whose first n0 coordinates are equal to zero,

ε ∈ (0, ε0], τ ≥ τ0, and ϕt

τ,j+1 = ϕt

τ,j+1(ψ, ε, d) is a solution of the Cauchy

problem

d

dt

ϕt

τ,j+1 = ω(t)

ε

+ b(x(t) + Zj , ϕt

τ,j+1, t, ε), ϕτ

τ,j+1 = ψ. (15.2)

Taking into account that Q(τ, τ0) = Q(τ, τ0) for τ > τ0, we deduce from (15.1)

that

166 Integral Manifolds Chapter 3

_Zj+1(ψ, τ, ε, τ0, d)_

≤ K_d_ + σ0 sup

ψ,τ

_Zj(ψ, τ, ε, τ0, d)_

+

2

γ

K[εσ1 + n2σ1 sup

ψ,τ

_Zj(ψ, τ, ε, τ0, d)_2]

+

_

k_=0

_ ∞_

s=−q

___

τ+_s+1

τ+s

Q(τ, t)ak(x(t), t) exp{i(k, ϕt

τ,j+1)}dt

___

+

___

_τ−q

τ0

Q(τ, t)ak(x(t), t) exp{i(k, ϕt

τ,j+1)}dt

___

_

. (15.3)

Here, q is the integer part of the number τ − τ0. Using conditions (12.2) and

(13.5) and relation (1.20), we estimate the last term on the right-hand side of

inequality (15.3) from above by the value σ4εα, where σ4 is the constant defined

in Section 13. Thus, inequality (15.3) yields

sup

ψ,τ

_Zj+1_

≤ K_d_ + σ0 sup

ψ,τ

_Zj_ +

2

γ

Kn2σ1 sup

ψ,τ

_Zj_2 +

_

σ4 +

2

γ

Kσ1

_

εα0

,

which, for ε0 ≤ (nd1)− 2

α and _d_ ≤

_

σ4K

−1 +

4

γ

σ1

_

εα0

, leads to the estimate

_Zj(ψ, τ, ε, τ0, d)_ ≤ 2

4Kσ1 + γσ4

γ(1 − σ0) εα0

= d1εα0

(15.4)

for all

j ≥ 0, (ψ, τ, ε, d) ∈ Rm × [τ0,∞) × (0, ε0] × L ≡ G2,

L =

_

d: d ∈ Rn, _d_ ≤

_

σ4K

−1 +

4

γ

σ1

_

εα0

_

,

where the first n0 coordinates of the vector d are equal to zero.

The inequality _Q(τ, t)_ ≤ Ke−γ|t−τ| and condition (12.2) guarantee that

the integral on the right-hand side of (15.1) converges uniformly for any ψ ∈ Rm,

Section 15 Conditional Asymptotic Stability of Integral Manifold 167

τ ∈ [τ0, T], ε ∈ (0, ε0], and d ∈ L (T > τ0

is arbitrary). Since Z0 ≡ 0 and

ϕt

τ,1 = ϕt

τ,1, using Lemmas 12.1–12.4 we establish the estimates

sup

G2

___

∂

∂ψ

Z1

___

≤ d2εα0

,

_m

ν=1

sup

G2

___

∂2

∂ψ∂ψν

Z1

___

≤ d3εα0

and the uniform

_

for all ψ ∈ Rm, τ ∈ [τ0, T], ε ∈ [ε0, ε0], and d ∈ L, where

T > τ0

and ε0

∈ (0, ε0) are arbitrary

_

convergence of the integrals obtained

from (15.1) for j = 0 by differentiation with respect to ψ and τ under the

integral sign. Moreover, by direct differentiation, one can verify that

∂Z1

∂τ

+ ∂Z1

∂ψ

_ω(τ )

ε

+ b(x(τ ), ψ, τ, ε)

_

= H(τ )Z1 + _a(x(τ ), ψ, τ) + εA(x(τ ), ψ, τ, ε),

Z1 = Z1(ψ, τ, ε, τ0, d),

and prove that Z1 is 2π-periodic in ψν, ν = 1,m, and twice continuously

differentiable with respect to ψ and τ for fixed ε, τ0, and d.

Using the method of mathematical induction, by analogy with Section 13 we

obtain the inequalities

___

∂

∂ψ

Zj

___

≤ d2εα0

,

_m

ν=1

___

∂2

∂ψ∂ψν

Zj

___≤

d3εα0

(15.5)

and the identity

∂Zj+1

∂τ

+ ∂Zj+1

∂τ

_ω(τ )

ε

+ b(x(τ) + Zj, ψ, τ, ε)

_

= H(τ )Zj+1 + F(Zj, τ) + _a(x(τ) + Zj, ψ, τ)

+ εA(x(τ) + Zj, ψ, τ, ε) (15.6)

for all (ψ, τ, ε, d) ∈ G2 and j ≥ 0. Furthermore, the functions Zj =

Zj(ψ, τ, ε, τ0, d) are periodic in ψν, ν = 1,m, with period 2π and twice

continuously differentiable with respect to ψ and τ for fixed ε, τ0, and d.

Moreover, according to Lemma 12.5, they satisfy estimates (14.9) and (14.10)

with Yj and G1 replaced by Zj and G2, respectively. This implies that the

168 Integral Manifolds Chapter 3

sequences {Zj} and

_ ∂

∂ψ

Zj

_

converge uniformly on the set G2, and equality

(15.6) yields the uniform convergence of the sequence

_ ∂

∂τ

Zj

_

on the set

ψ ∈ Rm, τ ∈ [τ0, T], ε∈ [ε0, ε0], d∈ L

for arbitrary T > τ0 and ε0 ∈ (0, ε0). Passing to the limit as j →∞ in (15.6),

for any (ψ, τ, ε, d) ∈ G2 we get

∂Z

∂τ

+ ∂Z

∂ψ

_ω(τ )

ε

+ b(x(τ) + Z, ψ, τ, ε)

_

= H(τ )Z + F(Z, τ) + _a(x(τ) + Z, ψ, τ) + εA(x(τ) + Z, ψ, τ, ε), (15.7)

where

Z = Z(ψ, τ, ε, τ0, d) = lim

j→∞

Zj(ψ, τ, ε, τ0, d).

Let ϕττ

0 = ϕττ

0(ψ, ε, d) denote a solution of the Cauchy problem

dϕττ

0

dτ

= ω(τ )

ε

+ b(x(τ) + Z(ϕττ

0, τ, ε, τ0, d), ϕττ

0, τ, ε),

ϕτ0

τ0 = ψ ∈ Rm.

It now follows from (15.7) that (xττ

0(ψ, ε, d); ϕττ

0(ψ, ε, d)), where xττ

0(ψ, ε, d) =

x(τ)+Z(ϕττ

0(ψ, ε, d), τ, ε, τ0, d), is a solution of system (12.1) for τ ≥ τ0, and

_xττ

0(ψ, ε, d) − x(τ )_ ≤ d1εα0

∀(ψ, τ, ε, d) ∈ G2. (15.8)

Thus, x(τ0) + Z(ψ, τ0, ε, τ0, d) ≡ S+ (for fixed ψ, τ0, and ε) is an (n −

n0)-dimensional manifold that possesses the following property: every solution

of system (12.1) with initial data from the set S+×Rm is defined for all τ ≥ τ0,

and, according to (15.8), its slow variables are uniformly bounded.

To construct the manifold S−, it is necessary to consider the following sequence

instead of the sequence {Zj} defined by (15.1) and (15.2):

_ Zj+1(ψ,τ, ε, τ0, d)

= _Q(τ, τ0)d +

_τ

−∞

Q(τ, t)[F( _ Zj, t) + _a(x(t) + _ Zj , _ϕt

τ,j+1, t)

+ εA(x(t) + _ Zj , _ϕt

τ,j+1, t, ε)]dt, _ Z0 ≡ 0,

Section 15 Conditional Asymptotic Stability of Integral Manifold 169

where τ ≤ τ0,

_ Zj = _ Zj(_ϕt

τ,j+1, t, ε, τ0, d), _Q(τ, τ0) = diag (−Q+(τ, τ0), 0),

d is a constant n-dimensional vector the last n − n0 coordinates of which are

equal to zero, and _ϕt

τ,j+1 = _ϕt

τ,j+1(ψ, ε, d) is a solution of the Cauchy problem

d_ϕt

τ,j+1

dt

= ω(t)

ε

+ b(x(t) + _ Zj , _ϕt

τ,j+1, t, ε), _ϕτ

τ,j+1 = ψ.

Then

S− = x(τ0) + lim

j→∞

_ Zj(ψ, τ0, ε, τ0, d).

We now prove the second part of the theorem. Taking into account relations

(13.7) and the equality

Q−(τ, τ0)Q−(τ0, t) = Q−(τ, t), t≤ τ0 ≤ τ,

one can represent the function Yj+1(ψ, τ, ε) for τ ≥ τ0 in the form

Yj+1(ψ, τ, ε) = Q(τ, τ0)yj+1 +

∞ _

τ0

Q(τ, t)[F(Yj, t) + _a(x(t) + Yj, ϕt

τ,j+1, t)

+ εA(x(t) + Yj, ϕt

τ,j+1, t, ε)]dt, (15.9)

where Yj = Yj(ϕt

τ,j+1, t, ε), yj+1 = yj+1(ψ, τ, ε) is the n-dimensional vector

the first n0 coordinates of which are equal to zero and the other coordinates

coincide with the vector

_τ0

−∞

Q−(τ0, t)[F−(Yj, t) + _a−(x(t) + Yj, ϕt

τ,j+1, t)

+ εA−(x(t) + Yj, ϕt

τ,j+1, t, ε)]dt.

Here, we preserve the notation of Section 13. According to inequalities (1.20),

(12.2), and (13.3), for any (ψ, τ, ε) ∈ G1 we have

_yj+1(ψ, τ, ε)_ ≤

_ 1

γ

(n2d21

σ1εα0

+ σ1)K +

1

2σ0d1 + σ4

_

εα < d1εα.

170 Integral Manifolds Chapter 3

Now consider the inequality

_Zj+1(ψ, τ, ε, τ0, d) − Yj(ψ, τ, ε)_

≤ Ke

−γ(τ−τ0)_d − yj+1(ψ, τ, ε)_

+

____

_−∞

τ0

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

____

, (15.10)

which follows from (15.1) and (15.9). In this inequality, Fj , _aj , and Aj have

the same meaning as in (14.1), and

Fj = F(Zj, t), _aj = _a(x(t) + Zj , ϕt

τ,j+1, t),

Aj = A(x(t) + Zj , ϕt

τ,j+1, t, ε), Zj = Zj(ϕt

τ,j+1, t, ε, τ0, d).

For the difference _aj−_aj , we use a representation of the form (14.2) with Zj

instead of Yj−1. Then, taking into account conditions (12.2), (13.13), and (15.5),

we deduce from (15.10) the following inequality:

_Zj+1(ψ, τ, ε, τ0, d) − Yj(ψ, τ, ε)_

≤ 2d1εα0

Ke

−γ(τ−τ0) + K

∞ _

τ0

e

−γ|τ−t|

__

(md1 + 3nd1 + 1)nσ1εα0

+ sup

ψ,τ

___

∂_a(x(τ ), ψ, τ)

∂x

___

_

_Zj − Yj_

+ εα0

(m + n(m + n)d1)σ1_ϕt

τ,j+1

− ϕt

τ,j+1

_

_

dt

+

_

k_=0

____

_τ−q

τ0

Q(τ, t)ak(x(t), t)(exp{i(k, θ

t

τ,j+1)}

− exp{i(k, θt

τ,j+1)}) exp

_ i

ε

(k,

_t

τ

ω(r)dr)

_

dt

___

Section 15 Conditional Asymptotic Stability of Integral Manifold 171

+

∞_

s=−q

___

τ+_s+1

τ+s

Q(τ, t)ak(x(t), t)(exp{i(k, θ

t

τ,j+1)}

− exp{i(k, θt

τ,j+1)}) exp

_ i

ε

_

k,

_t

τ

ω(r)dr

__

dt

___

_

, (15.11)

where q is the integer part of the number τ − τ0 and

θ

t

τ,j+1 = ϕt

τ,j+1

− 1

ε

_t

τ

ω(r)dr.

Denote

Mj = sup

τ∈[τ0,∞)

[e

γ

l (τ−τ0)pj(τ, ε)], l=

⎧⎪⎨

⎪⎩

2, σ0 = 0,

_

1 +

2σ0

1 − σ0

_1

2, σ0 > 0,

pj(τ, ε) = sup

ψ∈Rm

_Zj(ψ, τ, ε, τ0, d) − Yj(ψ, τ, ε)_.

It is clear that

pj(τ, ε) ≤ Mje

γ

l (τ−τ0) ∀τ ≥ τ0,

∞ _

τ0

e

−γ_τ−t_

pj(t, ε)dt ≤ 2l2

γ(l2 − 1)Mje

−γ

l (τ−τ0). (15.12)

To estimate _ϕt

τ,j+1

− ϕt

τ,j+1

_ ≡ Δϕj+1, we use Lemma 12.5. As a result, we

get

Δϕj+1 ≤ σ8e

γ

2

l−1

l

|t−τ| max

ξ∈N(τ,t)

pj(ξ, ε)

for c19εα0

≤ γ(l − 1)

4l

and σ8 = c18 max

_

1;

4l

γ(l − 1)

_

. If t < τ, then

max

ξ∈[t,τ ]

pj(ξ, ε) ≤ e

−γ

l (t−τ0) max

ξ∈[t,τ ]

[e

−γ

l (ξ−τ0)pj(ξ, ε)] ≤ Mje

−γ

l (t−τ0);

if t ≥ τ, then

max

ξ∈[τ,t]

pj(ξ, ε) ≤ Mje

−γ

l (τ−τ0).

172 Integral Manifolds Chapter 3

Taking this arguments into account, we obtain

_ϕt

τ,j+1

− ϕt

τ,j+1

_ ≤ σ8e

γ

2

l−1

l

|t−τ|− γ

l (min{τ;t}−τ0),

∞ _

τ0

e

−γ|τ−t|_ϕt

τ,j+1

− ϕt

τ,j+1

_dt ≤ σ8

4l2

γ(l2 − 1)Mje

−γ

l (τ−τ0). (15.13)

Then it follows from (15.12) and (15.13) that

∞ _

τ0

e

−γ|τ−t|_Zj(ϕt

τ,j+1, t, ε, τ0, d) − Yj(ϕt

τ,j+1, t, ε)_dt

≤ 2l2

γ(l2 − 1)

(1 + 2σ8d2εα0

)Mje

−γ

l (τ−τ0). (15.14)

According to (1.20) and (14.8), each of the integrals over the segments [τ +s, τ +

s + 1] and [τ0, τ − q] on the right-hand side of (15.11) can be estimated from

above by the value

σ3K(1 + nσ1 + σ1 + mσ1)εα0

max

t

e

−γ|t−τ|

×

_

_k_ sup

G

_ak_ + sup

G

___

∂ak

∂τ

___

+ sup

G

___

∂ak

∂x

___

_

×

_

max

t

_ϕt

τ,j+1

− ϕt

τ,j+1

_(1 + d2εα0

) + max

t

p(t, ε)

_

, (15.15)

where ak = ak(x, τ ) and the maximum with respect to t is taken over all t ∈

[τ + s, τ + s + 1] or t ∈ [τ0, τ − q], depending on which integral is considered.

Therefore, taking into account conditions (12.2) for the Fourier coefficients of the

function _a(x, ϕ, τ ) and inequalities (15.12), (15.13) and (15.15), we can estimate

the last of the three terms on the right-hand side of (15.11) by the value

σ13εα0Mje

−γ

l (τ−τ0),

where

σ_____________13 = 4eγ

_

1 − e

γ(l−1)

2l

_−1

Kσ3(2 + d2εα0

)(1 + σ8)(1 + σ1(1 + n + m))σ1.

Thus, with regard for inequalities (15.13) and (15.14), inequality (15.11) takes the

form

Mj+1 ≤ 2Kd1εα0

+

_

σ14εα0

+ σ0

l2

l2 − 1

_

Mj, j≥ 0. (15.16)

Section 15 Conditional Asymptotic Stability of Integral Manifold 173

Here,

σ14 = σ13 +

2l2

γ(l2 − 1)K[2(m + (n + m)n)σ1σ8

+ nσ1(1 + md1 + 3nd1)(1 + 2σ8d2)] + γK

−1σ8d2σ0.

Since σ0

l2

l2 − 1

≤ 1 + σ0

2

and M0 = 0, for σ14εα0

≤ 1 − σ0

4

relation

(15.16) yields

Mj ≤ 8

1 − σ0

Kd1εα0

∀j ≥ 0,

or

_Zj(ψ, τ, ε, τ0, d) − Yj(ψ, τ, ε)_ ≤ 8

1 − σ0

Kd1εα0

e

−γ

l (τ−τ0) (15.17)

for all (ψ, τ, ε, d) ∈ G2 and j ≥ 0. Passing to the limit as j → ∞ in (15.17),

we get

_Z(ψ, τ, ε, τ0, d) − Y (ψ, τ, ε)_ ≤ 8

1 − σ0

Kd1εα0

e

−γ

l (τ−τ0)

for ψ ∈ Rm, τ ≥ τ0, ε ∈ (0, ε0], and d ∈ L. Hence, as τ → ∞, the

slow variables xττ

0(ψ, ε, d) = x(τ) + Z(ϕττ

0(ψ, ε, d), τ, ε, τ0, d) of the solution

(xττ

0(ψ, ε, d); ϕττ

0(ψ, ε, d)) of system (12.1) tend exponentially to the curve x =

x(τ)+Y (ϕττ

0(ψ, ε, d), τ, ε), which lies on the integral manifold x = X(ψ, τ, ε).

By analogy, one can establish that, as τ → −∞,

the slow variables of every

solution of system (12.1) with initial data from the set S− × Rm tend exponentially

to the integral manifold. Theorem 15.1 is proved.

Remark 2. Inequality (15.8) can be regarded as an error estimate of the averaging

method on the semiaxis [τ0,∞) under the condition xτ0

τ0(ψ, ε, d) ∈ S+.

Remark 3. Theorem 15.1 remains true for n0 = 0. In this case, the integral

manifold x = X(ψ, τ, ε) of system (12.1) is asymptotically stable for all initial

values of the slow variable x from a certain small neighborhood of the point

X(ψ, τ0, ε).

174 Integral Manifolds Chapter 3

Навигация