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4. Justification of Averaging Method for Oscillation Systems with ω = ω (x, τ )

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Consider a multifrequency system of the form

dτ

= a (x, ϕ, τ) + εA (x, ϕ, τ, ε),

dϕ

dτ

= ω(x, τ )

+ b (x, ϕ, τ, ε),

(4.1)

where (x, ϕ, τ, ε) ∈ D×Rm×R+×(0, ε0] ≡ G and the real vector functions a,

A, ω, and b belong to certain classes of smooth functions 2π-periodic in ϕν,

ν = 1,m, m ≥ 2. We also consider the corresponding averaged (with respect to

ϕ) system of equations of the first approximation for slow variables:

dτ

= a (x, τ ) ≡ (2π)−m

_2π

. . .

_2π

a (x, ϕ, τ ) dϕ1 . . . dϕm. (4.2)

The main result on this section is the following: we establish an estimate for

_x−x_ on a finite time interval and prove an analog of the Banfi–Filatov theorem

[Fil, Ban] for systems of the standard form in the case of an infinite time interval.

For a two-frequency system, the estimate

_x − x_ ≤ c

√

ε ln2 1

∀τ ∈ [0, L]

52 Averaging Method in Systems with Variable Frequencies Chapter 1

was first established by Arnol’d [Arn2]. Later, Neishtadt improved this estimate

as follows: _x−x_ ≤ c

√

ε [Arn4]. The main assumption was the following: the

rate of the variation of the ratio of frequencies

ω1

ω2

along integral curves of system

(4.1) is nonzero. This assumption becomes obvious if we represent the equation

(k, ω(x, τ)) = 0, k _= 0, of the resonance surface in the form

ω1(x, τ )

ω2(x, τ )

= −k2

In other words, the resonance surfaces in the two-frequency case are level surfaces.

If the number of frequencies m is greater than 3, then the structure of

such surfaces is often fairly complex, which significantly complicates the investigation

of multifrequency systems. Therefore, it is necessary to impose certain

restrictions on resonance surfaces [Bak1, Gre, GrR1–GrR3, Neis2, Sam5]. One

should also note Khapaev’s paper [Kha1], in which the restrictions considered are

related only to resonance harmonics of the function a(x, ϕ, τ ); however, in this

case, the estimate of the error of the averaging method is not expressed explicitly

in terms of ε. Below, we establish the estimate _x−x_ ≤ c

√

ε under analogous

assumptions for system (4.1).

Assume that x = x(τ ) is a certain solution of Eqs. (4.2) defined on the

semiaxis R+ and lying in D together with its ρ-neighborhood. Denote by P the

set of m-dimensional vectors k = (k1, . . . , km) with integer-valued coordinates

for which the Fourier coefficients ak(x, τ ) of the function a(x, ϕ, τ ) are not

identically equal to zero on the set Dρ1(x(τ )) × R+, where

Dρ1(x(τ )) = {x : x ∈ D, _x − x(τ )_ < ρ1 ∀ τ ∈ R+}

and ρ1 ∈ (0, ρ] is a fixed constant.

Assume that the functions ω(x, τ ),

∂

∂x

ω(x, τ ), and

∂

∂τ

ω(x, τ ) are continuous

on D × R+ and, for all k ∈ P and (x, ϕ, τ, ε) ∈ G, the following

inequality is true:

|(k, ω(x, τ ))| + |(k, Ω(x, ϕ, τ, ε))| ≥ σ1_k_−s, (4.3)

where σ1 > 0 and s ≥ −1 are constants, (k, ω) and (k, Ω) are the scalar

products of vectors,

Ω(x, ϕ, τ, ε) = ∂ω(x, τ )

∂τ

+ ∂ω(x, τ )

∂x

δ (x, ϕ, τ, ε),

δ(x, ϕ, τ, ε) = a(x, τ) +

k∈P

ak(x, τ )hεα((k, ω(x, τ ))) exp{i(k,ϕ)},

Section 4 Justification of Averaging Method for Systems with ω = ω (x, τ ) 53

hα(τ ) for d = εα is the function defined in Section 3, and α ∈

is a fixed

constant.

For s = −1, condition (4.3) is an analog of condition (23) in [Sam5], which

allows one to obtain a uniform estimate for the oscillation integral. Also note that,

by virtue of the finiteness of the function hεα((k, ω)), restriction (4.3) deals only

with the resonance harmonics of the function a. This restriction was analyzed in

[Kha2].

Assume that the following conditions are satisfied:

[ω(x, τ ); a(x, ϕ, τ );A(x, ϕ, τ, ε); b(x, ϕ, τ, ε)] ∈ C1

x,ϕ,τ (G, σ2),

a ∈ Cl1

ϕ (G, σ2),

∂a

∂τ

∈ Cl2

ϕ (G, σ2),

∂a

∂x

∈ Cl3

ϕ (G, σ2), (4.4)

l1 > m+1+max

0; 2s; s + 1

1 − 2α

− 1

, min{l2; l3} ≥ m + max{0; s},

where σ2 is a certain constant. We also assume that there exists a solution

(x(τ, ε); ϕ(τ, ε)) of system (4.1) defined for any (τ, ε) ∈ R+ ×(0, ε0] and lying

in D1

2 ρ1

(x(τ )) × Rm.

Lemma 4.1. If f(τ) = (f1(τ ), . . . , fn(τ )) ∈ C1R

, L > 0 is a constant,

and conditions (4.3) and (4.4) are satisfied, then, for s > −1, one can find a

sufficiently large number N ∼ ε

2α−1

2(s+1) such that, for all (τ, τ, t, ε) ∈ R+ ×

R+ × [0, L] × (0, ε1] and k ∈ PN = {k : k ∈ P, _k_ ≤ N}, the oscillation

integral

Ik(τ, τ, t, ε) =

_τ+t

f(t) exp

_ i

(k, ω(x(z, ε), z)) dz

dt (4.5)

satisfies the inequality

_Ik_ ≤ σ3

√

(1 + _k_s)_k_s+1 max

[τ,τ+L]

_f(t)_

+ _k_s max

[τ,τ+L]

___

f (t)

___

, (4.6)

where σ3 and ε1 ∈ (0, ε0] are constants independent of τ, τ, t, ε, and k. If

s = −1, then (4.6) holds for all k ∈ P.

54 Averaging Method in Systems with Variable Frequencies Chapter 1

Proof. For (τ, ε) ∈ R+ × (0, ε0], we consider the functions

y (τ, ε) = x (τ, ε) + U (τ, ε),

U(τ, ε) = ε

k∈P

ak(x(τ, ε), τ)

i(k, ω(x(τ, ε), τ))

× [1 − hμ((k, ω(x(τ, ε), τ)))] exp{i(k,ϕ(τ, ε))}. (4.7)

The smoothness conditions (4.4) and the estimates for the Fourier coefficients

∀(τ, ε) ∈ R+ × (0, ε0] presented in [BMS] yield

_U(τ, ε)_ ≤ ε1−α

k_=0

sup

_ak_ ≤ ε1−α

k_=0

ml1σ2_k_−l1

≤ ε1−αml12mσ2

1 +

l1 − m

≡ σ4ε1−α. (4.8)

We set

ε2 = min

__1

2 ρ1σ

−1

_ 1

1−α ; ε0

Then, for τ ∈ R+ and ε ∈ (0, ε2], the curve y = y(τ, ε) lies in Dρ1(x(τ )). By

direct differentiation, one can easily verify that

dy(τ, ε)

dτ

= δ(x(τ, ε), ϕ(τ, ε), τ, ε) + B(τ, ε),

where the function B(τ, ε) is continuous in τ ∈ R+ for every fixed ε and

satisfies the inequality

_B(τ, ε)_ ≤ ε1−2α

k_=0

sup _ak__k_

sup _b_ + 17

sup _a + εA_ sup

___

∂ω

∂x

___

+ sup

___

∂ω

∂τ

___

+ sup

___

∂ak

∂τ

___+ sup

___

∂ak

∂x

___

sup _a + εA_

+ ε sup _A_

≤ σ5ε1−2α, (4.9)

σ5 = 2mσ2(2 + 18σ2 + 17nσ2)mmax{l1;l2;l3}

3 +

l1 − m − 1

l2 − m

l3 − m

Section 4 Justification of Averaging Method for Systems with ω = ω (x, τ ) 55

In the last inequality, the supremum is taken over all (x, ϕ, τ, ε) ∈ G, and this

inequality can be established with the use of (4.4) by analogy with (4.8).

We now consider an arbitrary vector k ∈ P. Assuming that x = x(τ, ε),

y = y(τ, ε), and ϕ = ϕ(τ, ε), we obtain

|(k, ω(y, τ))| +

___

dτ

(k, ω(y, τ))

___

≥ |(k, ω(y, τ))| + |(k, Ω(y, ϕ, τ, ε))| − _k_ sup

___

∂ω

∂x

___

sup

τ∈R+

_B_ + sup

τ∈R+

_δ(x, ϕ, τ, ε) − δ(y, ϕ, τ, ε)_

. (4.10)

Using inequalities (4.8) and (4.9) and conditions (4.4), we get

_B(τ, ε)_ + _δ(x, ϕ, τ, ε) − δ(y, ϕ, τ, ε)_ ≤ σ6ε1−2α,

σ6 = σ5 + σ2σ4[n + (16nσ2 + 1)]2m

_ ml1

l1 − m − 1

+ ml3

l3 − m

+ 2

which, together with (4.3) and (4.10), yields

|(k, ω(y(τ, ε), τ))| +

___

dτ

(k, ω(y(τ, ε), τ))

___

≥ σ1

_k_−s (4.11)

for s > −1, ε ∈ (0, ε2], τ ∈ R+, and k ∈ PN1 ; here,

N1 = E

2α−1

s+1

_ σ1

2nσ2σ6

_ 1

s+1

and E{t} is the integer part of the number t. If s = −1, then inequality (4.11)

is satisfied for all k ∈ P and

ε ≤ ε3 = min

ε2; (2nσ

−1

1 σ2σ6) 1

2α−1

It follows from (4.11) that, for all k ∈ PN1 and s > −1 or for k ∈ P,

s = −1, ε ∈ (0, ε3], and τ0 ∈ [τ, τ+t], at least one of the following inequalities

is satisfied:

|(k, ω(y0, τ0))| ≥ 1

4σ1_k_−s,

___

dτ

(k, ω(y0, τ0))

___

≥ 1

4σ1_k_−s, (4.12)

56 Averaging Method in Systems with Variable Frequencies Chapter 1

where y0 = y(τ0, ε). Let |(k, ω(y0, τ0))| ≥ 1

4σ1_k_−s. Then, according to the

Lagrange mean-value theorem and the condition of the boundedness of

___

dτ

___

on the segment

τ ∈ [τ0, τ0 + δk], δk = σ1

8σ2

[1 + n(σ4 + σ5)]−1_k_−s−1,

we have

|(k, ω(y(τ, ε), τ))| ≥ 1

8σ1_k_−s.

In view of (4.7) and (4.8), this estimate yields

|(k, ω(x(τ, ε), τ))| ≥ 1

16σ1_k_−s (4.13)

for all τ ∈ [τ0, τ0 + δk], ε ∈ (0, ε3], s>−1, and k ∈ PN2 , where

N2 = min

N1;E

s+1

7 ε

α−1

s+1

, σ7 =

_16

σ1

nσ2σ4

_−1

For s = −1 and ε ≤ ε4 = min{ε3; σ

α−1

}, estimate (4.13) holds for all k ∈ P.

If the first inequality in (4.12) is not satisfied, then, by virtue of the continuity

of the function

dτ

(k, ω(y(τ, ε), τ)) in τ, the inequality

___

dτ

(k, ω(y(τ, ε), τ))

___

≥ 1

8 σ1_k_−s (4.14)

holds on a certain segment [τ0, αk] of maximum length that does not exceed δk.

Let τ ∗

k denote the minimum point of the function |(k, ω(y(τ, ε), τ))| on this

segment. It follows from (4.14) that

|(k, ω(y(τ, ε), τ))| ≥ 1

8σ1_k_−s|τ − τ

∗

| ∀τ ∈ [τ0, αk].

Therefore,

|(k, ω(x(τ, ε), τ))| ≥ 1

16 σ1_k_−s

√

ε (4.15)

∀τ ∈ [τ0, αk]\[τ

∗

−

√

ε, τ

∗

k +

√

ε]

for

s > −1, ε ∈ (0, ε4], k ∈ PN, N = min

E{σ

s+1

7 ε

2α−1

2(s+1) };N2

Section 4 Justification of Averaging Method for Systems with ω = ω (x, τ ) 57

If s = −1, then estimate (4.15) holds for every k ∈ P and ε ≤ ε1 =

min{ε4; σ

1−2α

We now represent Ik(τ, τ, t, ε) in the form of the sum of the integrals

Ik =

q_k−1

r=0

τ+δ_k(r+1)

τ+δkr

Fdt +

_τ+t

τ+δkqk

Fdt, (4.16)

where F is the integrand of integral (4.5) and qk is the integer part of the number

tδ

−1

k ,

qk ≤ 1

σ1

8σ2L[1 + n(σ4 + σ5)]_k_s+1 ≡ σ8_k_s+1.

Let us estimate each integral on the right-hand side of (4.16). If the first inequality

in (4.12) is satisfied at the point τ0 = τ +δkr, then, integrating by parts and using

(4.13), we get

_______

τ+δ_k(r+1)

τ+δkr

Fdt

_______

≤ ε

σ1

δk_k_s max

[τ,τ+L]

____

f(t)

____

(nσ2 + 1)σ2

_16

σ1

δk_k_1+2s

σ1

_k_s

max

[τ,τ+L]

_f(t)_

. (4.17)

Now assume that, at the point τ0 = τ + δkr, the first inequality in (4.12) is

not satisfied, but the second inequality in (4.12) is true. Then, on a segment

of length 2

√

ε, the integral under investigation can be estimated by the value

√

ε max _f(t)_, and, outside this segment, inequality (4.15) holds. Therefore,

the following estimate holds for αk ≤ τ + δk(r + 1) :

_____

_αk

τ+δkr

Fdt

_____ ≤

2√

max

[τ,τ+L]

_f(t)_

1 +

σ1

_k_s

+ max

[τ,τ+L]

___

f(t)

___

σ1

δk_k_s

. (4.18)

58 Averaging Method in Systems with Variable Frequencies Chapter 1

Note that if αk < τ +δk(r+1), then the definition of the number αk yields

|(k, ω(y(αk, ε), αk))| ≥ σ1

_k_−s.

In view of this inequality, the integral of the function F over the segment [αk, τ+

δk(r + 1)] can be estimated from above by the value presented on the right-hand

side of (4.17). Combining (4.16)–(4.18), we obtain estimate (4.6) ∀k ∈ PN for

s > −1 or ∀k ∈ P for s = −1 with the constant

σ3 =

σ1

L + 2 (1 + σ8)

1 +

σ1

_16

σ1

(2 + 2nσ2) σ2L.

Lemma 4.1 is proved.

Theorem 4.1. Suppose that there exists a solution x = x(τ ) of the averaged

system (4.2) that lies in D together with its ρ-neighborhood ∀τ ∈ [0, L] and

conditions (4.3) and (4.4) are satisfied for τ ∈ [0, L] . Then one can find constants

ε0 ∈ (0, ε0] and σ9 > 0 such that

_x (τ, x (0), ψ, ε) − x (τ )_ ≤ σ9

√

ε (4.19)

for all τ ∈ [0, L], ψ ∈ Rm, and ε ∈ (0, ε0].

Proof. It follows from the conditions of the smoothness of the right-hand

side of system (4.1) that the slow variables x(τ, x(0), ψ, ε) of every solution

(x(τ, x(0), ψ, ε); ϕ(τ, x(0), ψ, ε)), ψ ∈ Rm, ε ≤ ε0, lie in the

2ρ1-neighborhood

of the curve x = x(τ ) for all τ from a certain segment [0, L1] ⊂ [0, L],

L1 = L1(ψ, ε). Here, ρ1 ∈ (0, ρ] is the constant determined by condition (4.3).

Then it follows from Eqs. (4.1) and (4.2) and the Gronwall–Bellman lemma that,

for any τ ∈ [0, L1],

_x(τ, x(0), ψ, ε) − x(τ )_ ≤ eLnσ2

εσ2L + Lsup

_RNa(x, ϕ, τ )_

k∈PN

sup

___

_τ

ak(x(t, x(0), ψ, ε), t)

___

exp{i(k, θ)}

× exp

_ i

(k, ω(x(z, x(0), ψ, ε), z))dz

dt, (4.20)

Section 4 Justification of Averaging Method for Systems with ω = ω (x, τ ) 59

where

θ = ϕ (t, x(0), ψ, ε) − 1

ω (x(z, x(0), ψ, ε), z)dz

and

RNa (x, ϕ, τ) =

_k_>N

ak(x, τ ) exp {i(k,ϕ)}

for s > −1. If s = −1, then we set PN = P and RNa(x, ϕ, τ ) ≡ 0. Here, N

is the number defined in Lemma 4.1. It is obvious that

N >

2 σ10 ε

2α−1

2(s+1) ∀ε ∈ (0, ε0], ε0 ≤ σ

2(s+1)

1−2α

10 ,

σ10 = min

s+1

7 ;

_ 1

σ1

2nσ2σ6

_− 1

s+1

The smoothness conditions (4.4) for the function a(x, ϕ, τ ) guarantee that

sup

_RNa(x, ϕ, τ )_ ≤

_k_>N

sup

_ak(x, τ )_ ≤

_k_>N

σ2ml1_k_−l1

≤ 2mml1σ2

∞_

r=N+1

rm−1−l1 ≤ 2mml1σ2

∞ _

rm−1−l1dr

= 2mml1σ2

l1 − m

Nm−l1 ≤ σ11ε

(l1

−m)(1−2α)

2(s+1) , (4.21)

σ11 =

2mml1σ2

l1 − m

_ 2

σ10

_l1−m

for s > −1. Using Lemma 4.1, conditions (4.4), and inequality (4.21), we obtain

the following estimate for s > −1:

_x(τ, x(0), ψ, ε) − x(τ )_ ≤ σ12(

√

ε + ε

(l1

−m)(1−2α)

2(s+1) ) ≤ 2σ12

√

ε ≡ σ9

√

ε,

where τ ∈ [0, L1], ψ ∈ Rm, ε ≤ min{ε0; σ

2(s+1)

1−2α

10 ; ε1} ≡ ε0, and

60 Averaging Method in Systems with Variable Frequencies Chapter 1

σ12 = eLnσ2

σ2L + σ11L + 2mσ2σ3(1 + σ2)mmax{l1;l2;l3}

4 +

l1 − 2s − 1 − m

l1 − m − s − 1

l2 − s − m

l3 − s − m

Here, ε1 is the constant defined in Lemma 4.1. It is easy to see that the last

estimate remains true if s = −1. We set

ε0 = min

ε0;

_ 1

ρ1

4σ9

_−2_

which guarantees the validity of the inequality

_x(τ, x(0), ψ, ε) − x(τ )_ ≤ 1

4ρ1 ∀τ ∈ [0, L1].

This inequality and the smoothness conditions for the right-hand side of system

(4.1) allow one to extend the solution (x(τ, x(0), ψ, ε); ϕ(τ, x(0), ψ, ε)) to the

entire segment [0, L]. In this case, inequality (4.19) does not change. Theorem

4.1 is proved.

As an example, we consider the Cauchy problem

dτ

= λ[f1(x, _ϕ, τ) + f2(x,ϕ∼

, τ) + cos ϕ + 2.5], x(0) = 0,

d_ϕ

dτ

= λ

dϕ∼

dτ

= x

dϕ

dτ

= τ + 2

, _ϕ(0) = ϕ∼(0) = 0, ϕ(0) = 0,

where x, _ϕ, and ϕ∼

are m-dimensional vectors, m ≥ 2, x ∈ D = {x: _x_ ≤

3_λ_}, ϕ is a scalar, τ ∈ [0, 1], λ = (λ1, . . . , λm) is a constant nonzero

vector, and f1 and f2 are scalar 2π-periodic (in _ϕ and ϕ∼

) functions satisfying

conditions (4.4) for α = 0 and s = m + 1. We also assume that the Fourier

coefficients f1,k and f2,k of the functions f1 and f2 satisfy the relations

f1,0(x, τ) = f2,0(x, τ ) ≡ 0,

k_=0

[|f1,k(x, τ )| + |f2(x, τ )|] ≤ 2,

and the vector λ and every nonzero vector k = (k1, . . . , km) with integer-valued

coordinates satisfy the inequality

|(k, λ)| =

___

ν=1

kνλν

___

≥ c

_k_m+1, c= const > 0.

Section 4 Justification of Averaging Method for Systems with ω = ω (x, τ ) 61

It is known [Arn4] that, in the ball _λ_ ≤ 1, the Lebesgue measure of the numbers

λ = (λ1, . . . , λm) for which the last inequality is not true tends to zero as

c → 0. In our case, the Cauchy problem for slow variables averaged with respect

to all angular variables has the solution x(τ) = 2.5λτ, which lies in D

together with its

_λ_-neighborhood for any τ ∈ [0, 1]. It is easy to verify that,

for the system under consideration, inequality (4.3) is satisfied for s = m + 1,

α = 0, and σ1 = min

; 2

. Therefore, according to Theorem 4.1, for any

(τ, ε) ∈ [0, 1] × (0, ε0] (ε0 is sufficiently small), the following estimate is true:

_x(τ, ε) − x(τ )_ ≤ c

√

ε, c = const.

Note that, in this special case, we cannot use the results of [Sam5, Kha1, Kha2]

for the justification of the averaging method.

We now study the problem of the qualitative relationship between solutions of

original and averaged equations on the infinite time interval [0,∞).

Theorem 4.2. Suppose that the following conditions are satisfied:

(i) conditions (4.3) and (4.4) are satisfied;

(ii) there exists an asymptotically stable solution x = x(τ ), τ ∈ R+, of

Eqs. (4.2) that lies in D together with its ρ-neighborhood.

Then the following assertions are true:

(a) there exist positive constants σ13, σ13, and β < ρ such that, for all

τ ∈ R+, ε ∈ (0, σ13], ϕ0 ∈ Rm, and x0 ∈ Dβ(x(0)), the slow variables

x(τ, x0, ϕ0, ε) of every solution (x(τ, x0, ϕ0, ε); ϕ(τ, x0, ϕ0, ε)) of system

(4.1) are uniformly bounded, i.e.,

_x (τ, x0, ϕ0, ε)_ ≤ σ13; (4.22)

(b) for arbitrary η ∈ (0, β), there exists ε(η) > 0 such that

_x (τ, x(0), ϕ0, ε) − x (τ )_ < η (4.23)

for all τ ∈ R+, ϕ0 ∈ Rm, and ε ∈ (0, ε(η)].

Proof. According to the definition of uniform asymptotic stability [Fil], for

the number

2ρ one can find β > 0 such that _x(τ, t, x0) − x(τ )_ <

2ρ for all

62 Averaging Method in Systems with Variable Frequencies Chapter 1

τ ≥ t ∈ R+, provided that _x0 − x(t)_ < β. Here, x(τ, t, x0) is a solution of

system (4.2) that satisfies the condition x(t, t, x0) = x0. Moreover, one can find

a constant L = L(ρ) such that _x(τ, t, x0) − x(τ )_ ≤ 1

2β for τ ≥ t + L.

Using Theorem 4.1, for ε ≤ min

__ β

4σ9

; ε0

= σ13 we get

_xτ (0, x0, ϕ0, ε) − x(τ )_

≤ _xτ (0, x0, ϕ0, ε) − x(τ, 0, x0)_ + _x(τ, 0, x0) − x(τ )_

< σ9

√

ε +

2 ρ < ρ ∀τ ∈ [0, L), (4.24)

_xL(0, x0, ϕ0, ε) − x(L)_ < σ9

√

ε +

2 β < β,

i.e., the point _x0 = xL(0, x0, ϕ0, ε) is located in the β-neighborhood of the

point x(L). Here, ε0 and σ9 are the constants defined in Theorem 4.1, and

(xτ (t, x0, ϕ0, ε); ϕτ (t, x0, ϕ0, ε)) is a solution of system (4.1) that passes through

the point (x0; ϕ0) at τ = t.

We now consider the time interval [L, 2L]. By analogy with the above reasoning,

we can establish the inequalities

_xτ (L, _x0, _ϕ0, ε) − x(τ )_ < ρ ∀τ ∈ [L, 2L),

_x2L(L, _x0, _ϕ0, ε) − x(2L)_ < β, (4.25)

where _ϕ0 = ϕL(0, x0, ϕ0, ε). Inequalities (4.24) and (4.25) yield

_xτ (0, x0, ϕ0, ε) − x(τ )_ < ρ ∀τ ∈ [0, 2L),

_x2L(0, x0, ϕ0, ε) − x(2L)_ < β.

By induction, for an arbitrary natural p ≥ 3 we get

_xτ (0, x0, ϕ0, ε) − x(τ )_ < ρ ∀τ ∈ [0, pL),

_xpL(0, x0, ϕ0, ε) − x(pL)_ < β. (4.26)

This yields

_xτ (0, x0, ϕ0, ε)_ ≤ ρ + sup

R+

_x(τ )_ ≡ σ13

for all τ ∈ R+, ϕ0 ∈ Rm, x0 ∈ Dβ(x(0)), and ε ∈ (0, σ13]. Thus, assertion

(a) is proved.

Section 5 Averaging over All Fast Variables in Multifrequency Systems 63

We now fix an arbitrary η ∈ (0, β). According to the definition of uniform

asymptotic stability, for η there exist constants μ > 0 and L1 = L1(η) > 0

such that the inequality _x0 − x(t)_ < μ yields

_x(τ, t, x0) − x(τ )_ <

2 η ∀τ ≥ t,

_x(τ, t, x0) − x(τ )_ <

2 μ ∀τ ≥ τ + L1.

The further proof of assertion (b) is based on inequalities (4.24)–(4.26) with β, ρ,

L, and x0 replaced by μ, η, L1, and x(0), respectively. In this case, ε(η) =

min

ε0;

_ μ

4σ9

_2_

, and ε0 and σ9 are defined in Theorem 4.1 for L = L1.

Remark 5. If, in addition, we assume that a(x, τ ) ∈ C2

x(D ×R+, σ2) and

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

dτ

= ∂

∂x

a (x(τ ), τ) z satisfies the inequality

_Q(τ, t)_ ≤ Ke

−γ(τ−t) ∀τ ≥ t ∈ R+,

where K and γ are certain positive constants, then estimate (4.23) takes the form

_x (τ, x(0), ψ, ε) − x(τ )_ < σ14

√

ε, σ14 = const.

The proof of this statement, in fact, repeats the proof of Theorem 2.4.

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