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2. Justification of Averaging Method for Oscillation Systems with ω = ω (τ )

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Consider the nonlinear system of ordinary differential equations

dτ

= a (x, ϕ, τ, ε),

dϕ

dτ

= ω(τ )

+ b (x, ϕ, τ, ε), (2.1)

where x = (x1, . . . , xn) ∈ D, ϕ = (ϕ1, . . . , ϕm) ∈ Rm, n ≥ 1, m ≥ 2,

τ ∈ [0, L], L is a positive constant, (0, ε0] ε is a small parameter, D is a

bounded domain, and Rm is the m-dimensional real Euclidean space.

Let ω(τ ) ∈ Cl

[0,L], l ≥ m − 1. Furthermore, assume that the function

c(x, ϕ, τ, ε) = [a(x, ϕ, τ, ε); b(x, ϕ, τ, ε)] is continuously differentiable with respect

to (x, ϕ, τ ) ∈ D×Rm ×[0, L] for every fixed ε ∈ (0, ε0], 2π-periodic in

each variable ϕν, ν = 1,m, and such that

sup

_c0_ + sup

___

∂c0

∂x

___

+ sup

___

∂c0

∂τ

___

_k_>0

sup

_ck_ +

_k_

sup

___

∂ck

∂x

___

+ sup

___

∂ck

∂τ

___

_k_q

≤ σ1 = const, q≥ 0. (2.2)

Here, G = D×[0, L]×[0, ε0], ck = c(x, τ, ε) are the Fourier coefficients of the

harmonics exp {i(k,ϕ)} of the Fourier expansion of the function c(x, ϕ, τ, ε),

and k = (k1, . . . , km) is a vector with integer coordinates.

The following conditions are sufficient for the validity of (2.2):

c(x, ϕ, τ, ε) ∈ Cl1

ϕ (G, σ),

∂

∂τ

c (x, ϕ, τ, ε) ∈ Cl2

ϕ (G, σ),

∂

∂x

c(x, ϕ, τ, ε) ∈ Cl3

ϕ (G, σ), min {l1 − 1; l2; l3} ≥ m + q,

where Clϕ

(G, σ) denotes the set of functions f(x, ϕ, τ, ε) that, for every fixed

ε, have partial derivatives up to the order l inclusive continuous with respect to

x, ϕ, and τ, and uniformly bounded by a constant σ on the set (x, ϕ, τ, ε) ∈ D×

22 Averaging Method in Systems with Variable Frequencies Chapter 1

Rm[0, L] × [0, ε0] ≡ G. Indeed, under this assumption, the following estimates

hold for all k _= 0 [BMS]:

sup

_ck_ ≤ σml1

_k_l1

, sup

___

∂ck

∂τ

___

≤ σml2

_k_l2

, sup

___

∂ck

∂x

___

≤ σml3

_k_l3

Consequently,

k_=0

_k_q sup

_ck_ ≤ σml1

k_=0

_k_q−l1 ≤ σml1

∞_

s=1

sq−l1

_ _

_k_=s

≤ σml12m

∞_

s=1

sq−l1+m−1 ≤ σml12m

1 +

∞ _

tq−l1+m−1dt

= σml12m

1 +

l1 − q − m

k_=0

_k_q−1

sup

___

∂ck

∂τ

___

+ sup

___

∂ck

∂x

___

≤ σ2m

ml2

1 +

l2 − q − m + 1

+ ml3

1 +

l3 − q − m + 1

In the proof of the last inequalities, we have used the fact that the number of mdimensional

vectors with integer coordinates whose norm is equal to s does not

exceed 2msm−1 [GrR3].

Along with (2.1), we consider the following system averaged over all angular

variables ϕ :

dτ

= a (x, τ, ε),

dϕ

dτ

= ω(τ )

+ b (x, τ, ε) , (2.3)

where

[a(x, τ, ε); b(x, τ, ε)] = (2π)−m

_2π

. . .

_2π

a(x, ϕ, τ, ε),

b(x, ϕ, τ, ε)

dϕ1 . . . dϕm = [a0(x, τ, ε); b0(x, τ, ε)] = c0(x, τ, ε).

Section 2 Justification of Averaging Method for Systems with ω = ω (τ ) 23

For Eqs. (2.1) and (2.2), we specify the initial conditions

x|τ=0 = y ∈ D1 ⊂ D, ϕ|τ=0 = ψ ∈ Rm, (2.4)

where D1 is a certain domain, and denote by (x(τ, y, ψ, ε); ϕ(τ, y, ψ, ε)) and

(x(τ, y, ε); ϕ(τ, y, ψ, ε)) solutions of problems (2.1), (2.4) and (2.3), (2.4), respectively.

Theorem 2.1. Suppose that the following conditions are satisfied:

(i) det (WT

p (τ )Wp(τ )) _= 0 ∀τ ∈ [0, L] for certain minimal p ≥ m, p ≤

l + 1;

(ii) condition (2.2) is satisfied for q = 0;

(iii) for all τ ∈ [0, L], y ∈ D1, and ε ∈ (0, ε0], the curve x = x(τ, y, ε) lies

in D together with its ρ-neighborhood.

Then one can find a constant σ2 independent of ε and such that, for sufficiently

small ε0 > 0 and every τ ∈ [0, L], y ∈ D1, ψ ∈ Rm, and ε ∈ (0, ε0],

the following estimate holds:

_U(τ, y, ψ, ε)_ ≤ σ2 ε1/p, (2.5)

where U = (x(τ, y, ψ, ε) − x(τ, y, ε); ϕ(τ, y, ψ, ε) − ϕ(τ, y, ψ, ε)).

Proof. Since the right-hand side of system (2.1) is smooth, a solution of the

Cauchy problem (2.1), (2.4) exists. Denote by [0, T), T = T(y, ψ, ε), the maximum

half-interval of the segment [0, L] for which the curve x = x(τ, y, ψ, ε)

lies in the ρ-neighborhood of the curve x = x(τ, y, ε). Then Eqs. (2.1) and (2.3)

and the Gronwall–Bellman lemma [FiS] imply that, for any τ ∈ [0, T),

_U(τ, y, ψ, ε)_ ≤ eσ1L sup

τ∈[0,L]

k_=0

____

_τ

ck(x(t, y, ε), t, ε)

× exp{i(k, θ)} exp

(k, ω(z))dz

____

24 Averaging Method in Systems with Variable Frequencies Chapter 1

where

θ = ϕ(t, y, ψ, ε) − 1

ω (z) dz.

Note that, under the conditions imposed on the functions ω(t) and f(t) =

ck(x(t, y, ε), t, ε) exp{i(k, θ)}, all conditions of Theorem 1.3 are satisfied. Consequently,

using this theorem for the estimation of each integral on the right-hand

side of the last inequality, we get

___

_τ

ck(x(t, y, ε), t, ε) exp{i(k, θ)}dt

___

≤ σ2ε

2 + sup

_b(x, τ, ε)_

1 + sup

_a(x, τ, ε)_

sup

_ck_ +

_k_ sup

___

∂ck

∂τ

___

_k_ sup

___

∂ck

∂x

___

_U(τ, y, ψ, ε)_ ≤ eσ1L(2 + σ1)(1 + σ1)σ1σ2ε

p ≡ σ2ε

p ∀τ ∈ [0, T).

Let σ2ε

≤ 1

2ρ. Then inequality (2.5) implies that x(T, y, ψ, ε) ∈ D together

with its

2ρ -neighborhood. Therefore, T = L and estimate (2.5) holds for all

τ ∈ [0, L]. Theorem 2.1 is proved.

We now study in more detail the dependence of the function U(τ, y, ψ, ε)

on the initial data y and ψ. Below, using properties of oscillation integrals, we

establish estimates for the partial derivatives

∂

∂y

U and

∂

∂ψ

U, which are substantially

used in the Chapter 2 for the solution of boundary-value problems. For

this purpose, we impose on c(x, ϕ, τ, ε) a stronger restriction than (2.2). Assume

that the function c(x, ϕ, τ, ε) is twice continuously differentiable with respect to

x, ϕ, and τ for every fixed ε, and its Fourier coefficients satisfy the inequality

sup _c0_ + sup

___

∂c0

∂τ

___

+ sup

___

∂c0

∂x

___

j=1

sup

___

∂2c0

∂x∂xj

___

Section 2 Justification of Averaging Method for Systems with ω = ω (τ ) 25

k_=0

_k_ sup _ck_ + sup

___

∂ck

∂τ

___

+ sup

___

∂ck

∂x

___

_k_

sup

___

∂2ck

∂x∂τ

___

j=1

sup

___

∂2ck

∂x∂xj

___

≤ σ1. (2.6)

Here, the supremum is taken over all (x, τ, ε) ∈ G.

Theorem 2.2. If conditions (i) and (iii) of Theorem 2.1 and inequality (2.6)

are satisfied, then, for all τ ∈ [0, L], y ∈ D1, ψ ∈ Rm, and ε ∈ (0, ε0] (where

ε0 is positive and sufficiently small), the following estimate holds:

___

∂

∂y

U(τ, y, ψ, ε)

___

∂

∂ψ

U(τ, y, ψ, ε)

___

≤ σ3ε

p , (2.7)

where the constant σ3 is independent of ε.

Proof. First, we establish estimates for the first-order partial derivatives with

respect to y and ψ for a solution of the Cauchy problem (2.3), (2.4). The smoothness

conditions for the right-hand side of system (2.3) yield

x (τ, y, ε) = y +

_τ

a (x(t, y, ε), t, ε) dt,

∂x(τ, y, ε)

∂y

= En +

_τ

∂

∂x

a (x(t, y, ε), t, ε) ∂x(t, y, ε)

∂y

dt,

whence

___

∂x(τ, y, ε)

∂y

___

≤ n + σ1

_τ

___

∂x(τ, y, ε)

∂y

___

dt.

Solving this inequality, for all τ ∈ [0, L], y ∈ D1 and ε ∈ (0, ε0] we get the

estimate ___ ∂x(τ, y, ε)

∂y

___

≤ neσ1L, (2.8)

which, together with the first equation in (2.3), yields

___

dτ

∂x(τ, y, ε)

∂y

___

≤ nσ1eσ1L. (2.9)

26 Averaging Method in Systems with Variable Frequencies Chapter 1

Since

ϕ(τ, y, ψ, ε) = ψ +

_τ

b(x(t, y, ε), t, ε)dt +

_τ

ω(t)dt,

we have

___

∂ϕ(τ, y, ψ, ε)

∂y

___

≤ Lσ1neσ1L,

___

∂ϕ(τ, y, ψ, ε)

∂ψ

___

= m,

dτ

∂ϕ(τ, y, ψ, ε)

∂ψ

≡ 0,

___

dτ

∂ϕ(τ, y, ψ, ε)

∂y

___

≤ σ1neσ1L (2.10)

for all τ ∈ [0, L], y ∈ D1, ψ ∈ Rm, and ε ∈ (0, ε0].

We now differentiate Eqs. (2.1) and (2.3) with respect to y. Then we obtain

the following integral equation for

∂

∂y

U(τ, y, ψ, ε) :

∂

∂y

U(τ, y, ψ, ε)=

_τ

A(x(t, y, ψ, ε), ϕ(t, y, ψ, ε), t, ε) ∂

∂y

U(t, y, ψ, ε)dt

_τ

_ ∂

∂x

c0(x(t, y, ψ, ε), t, ε) − ∂

∂x

c0(x(t, y, ε), t, ε)

_ ∂

∂y

x(t, y, ε) dt

_τ

∂

∂x

˜c(x(t, y, ψ, ε), ϕ(t, y, ψ, ε), t, ε) ∂

∂y

x (t, y, ε) dt

_τ

∂

∂ϕ

˜c (x(t, y, ψ, ε), ϕ(t, y, ψ, ε), t, ε) ∂

∂y

ϕ(t, y, ψ, ε) dt, (2.11)

where

˜c (x, ϕ, t, ε) = c (x, ϕ, t, ε) − c0 (x, t, ε),

A(x, ϕ, t, ε) =

_ ∂

∂x

c (x, ϕ, t, ε); ∂

∂ϕ

c (x, ϕ, t, ε)

Taking into account the estimate for the error of the averaging method (2.5) and

condition (2.6), we obtain the inequalities

Section 2 Justification of Averaging Method for Systems with ω = ω (τ ) 27

____

_τ

_ ∂

∂x

c0(x(t, y, ψ, ε), t, ε) − ∂

∂x

c0(x(t, y, ε), t, ε)

_ ∂

∂y

x(t, y, ε)dt

____

≤ σ1σ2nLeσ1Lε

p ,

_A(x, ϕ, t, ε)_ ≤ σ1 (2.12)

for all τ ∈ [0, L], y ∈ D1, ψ ∈ Rm, and ε ∈ (0, ε0].

We estimate the last two integrals on the right-hand side of (2.11) with the use

of Theorem 1.3 for λ = k and conditions (2.6) and (2.8)–(2.10). As a result, we

obtain

___

_τ

∂˜c

∂x

∂y

___

≤

k_=0

___

_τ

∂

∂x

ck(x(t, y, ψ, ε), t, ε) ∂

∂y

x (t, y, ε)

× exp{i(k, θ)} exp

_ i

(k, ω (z)) dz

___ ≤

σ2ε

k_=0

sup

___

∂ck

∂x

___

2neσ1L(1 + σ1)

_k_

sup

___

∂2ck

∂x∂τ

__ _

σ1

j=1

sup

___

∂2ck

∂x∂xj

___

neσ1L

≤ 3(1 + σ1)σ1σ2neσ1Lε

p . (2.13)

Similarly,

___

_τ

∂˜c

∂ϕ

∂y

___

≤ nσ2

1σ2(1 + 2L + Lσ1) eLσ1ε

p . (2.14)

In view of (2.12)–(2.14), Eq. (2.11) yields the integral inequality

___

∂

∂y

U(τ, y, ψ, ε)

___

≤ σ1

_τ

___

∂

∂y

U(t, y, ψ, ε)

___

dt + ε

p σ2,

whose solution satisfies the estimate

___

∂

∂y

U(τ, y, ψ, ε)

___

≤ σ2eσ1Lε

p ≡ σ3ε

p (2.15)

28 Averaging Method in Systems with Variable Frequencies Chapter 1

∀(τ, y, ψ, ε) ∈ [0, L]×D1 × Rm × (0, ε0]. Here,

σ2 = nσ1[σ2 + σ2(3 + 4σ1 + 2σ1L + σ2

1L)]eσ1L.

Let us estimate the norm of the matrix

∂

∂ψ

U(τ, y, ψ, ε). Since

∂

∂ψ

x (τ, y, ε) ≡ 0,

∂

∂ψ

ϕ (τ, y, ψ, ε) = Em,

it follows from Eqs. (2.1) and (2.3) that

∂

∂ψ

U(τ, y, ψ, ε) =

_τ

A(x(t, y, ψ, ε), ϕ(t, y, ψ, ε), t, ε) ∂

∂ψ

U(t, y, ψ, ε)dt

_τ

∂

∂ϕ

c (x(t, y, ψ, ε), ϕ (t, y, ψ, ε), t, ε) dt.

This yields

___

∂

∂ψ

U(τ, y, ψ, ε)

___

≤ 2σ1(1 + σ1)σ2eLσ1ε

p ≡ σ3ε

p . (2.16)

Combining (2.15) and (2.16), we get estimate (2.7) with the constant σ3 = σ3 +

σ3. The smallness of ε0 > 0 is determined by the possibility of the application

of Theorems 1.3 and 2.1.

Remark 2. If p = m, then the condition det (WT

p (τ )Wp(τ )) _= 0 ∀τ ∈

[0, L] can be reduced to the condition Δ(τ ) = det Wm(τ ) _= 0. If the function

Δ(τ ) has zeros of multiplicity not higher than r on the segment [0, L], then,

using Theorem 1.4, instead of estimates (2.5) and (2.7) we obtain estimates of the

form

_U(τ, y, ψ, ε)_ +

___

∂

∂y

U(τ, y, ψ, ε)

___

∂

∂ψ

U(τ, y, ψ, ε)

___

≤ σ4ε

m+r .

The investigation of oscillation systems becomes more complicated if Δ(τ )

is identically equal to zero on some segment [α, β] ⊂ [0, L]. In this case, the

solution of system (2.1) may deviate from the solution of the averaged system

(2.3) at time Δτ = L by a distance proportional to unit, i.e., the scheme of

Section 2 Justification of Averaging Method for Systems with ω = ω (τ ) 29

averaging over all angular variables is, generally speaking, inapplicable. As an

example, we consider the Cauchy problem

dτ

= 1− cos(ϕ1 + ϕ2 − ϕ3),

dϕ1

dτ

= ω1(τ )

dϕ2

dτ

= ω2(τ )

dϕ3

dτ

= ω1(τ_____________) + ω2(τ )

x(0) = ϕ1(0) = ϕ2(0) = ϕ3(0) = 0

(where ω1(τ ) and ω2(τ ) are twice continuously differentiable functions on the

segment [0, 1] ) and the corresponding problem averaged over all angular variables

ϕ1, ϕ2, and ϕ3, namely,

dτ

= 1,

dϕ1

dτ

= ω1(τ )

dϕ2

dτ

= ω2(τ )

dϕ3

dτ

= ω1(τ) + ω2(τ )

x(0) = ϕ1(0) = ϕ2(0) = ϕ3(0) = 0.

It is obvious that x(τ) = τ, x(τ ) ≡ 0, Δ(τ ) ≡ 0 ∀τ ∈ [0, 1], and |x(1) −

x(1)| = 1. Hence, the condition Δ(τ ) ≡ 0 leads to the violation of the efficient

estimate of the error of the method of averaging over all angular variables on the

segment [0, 1].

In this case, it is convenient to perform averaging over a part of angular variables.

Below, we describe this method in brief and give its justification.

Assume that

|Δ(τ )| ≥ σ5(α − τ )r1 ∀τ ∈ [0, α),

|Δ(τ )| ≥ σ5(τ − β)r2 ∀τ ∈ (β,L],

Δ(τ ) ≡ 0 ∀τ ∈ [α, β], (2.17)

where r1, r2, and σ5 are certain positive constants. The linear dependence of

the functions ω1(τ ), . . . , ωm(τ ) on the segment [α, β] is a sufficient condition

for the validity of the identity Δ(τ ) ≡ 0 ∀τ ∈ [α, β]. Further, we assume

that there exist h < m linearly independent vectors k(j) = (k(j)

1 , . . . , k(j)

m ),

j = 1, h, with integer coordinates for which

(k(j), ω(τ )) ≡ 0 ∀τ ∈ [α, β] . (2.18)

30 Averaging Method in Systems with Variable Frequencies Chapter 1

Without loss of generality, we can assume that the hth-order minor γ in the left

upper corner of the matrix K = colon (k(1), . . . , k(h)) is nonzero. Denote

= colon (k(1), . . . , k(h), eh+1, . . . , em),

where eν is a basis vector of the space Rm. Then, in the variables

_ Kϕ = (ψ; θ), ψ= (ψ1, . . . , ψh),

θ = (θ1, . . . , θm−h) = (ϕh+1, . . . , ϕm),

ϕ =

(S1ψ + S2θ),

where S1 and S2 are matrices whose elements are integer numbers, system (2.1)

takes the form

dτ

= A(x, ψ, θ, τ, ε),

dψ

dτ

Kω(τ) + B(x, ψ, θ, τ, ε),

dθ

dτ

_ω(τ) + C (x, ψ, θ, τ, ε). (2.19)

Here,

_ω(τ) = (ωh+1(τ ), . . . , ωm(τ )),

A(x, ψ, θ, τ, ε) = a

S1ψ +

S2θ, τ, ε

B(x, ψ, θ, τ, ε) = Kb

S1ψ +

S2θ, τ, ε

C(x, ψ,θ, τ, ε)

bh+1

S1ψ +

S2θ, τ, ε

, . . . , bm

S1ψ +

S2θ, τ, ε

The corresponding system averaged over all variables θ has the form

dτ

= A(x, ψ, θ, τ, ε),

dψ

dτ

Kω(τ) + B(x, ψ, τ, ε),

dθ

dτ

_ω(τ) + C (x, ψ, τ, ε), (2.20)

Section 2 Justification of Averaging Method for Systems with ω = ω (τ ) 31

where

[A;B;C] = (2πγ)h−m

_2πγ

. . .

_2πγ

A(x, ψ, θ, τ, ε),

B(x, ψ, θ, τ, ε);C(x, ψ, θ, τ, ε)

dθ1 . . . dθm−h.

Theorem 2.3. Suppose that the following conditions are satisfied:

(i) there exists a solution (x(τ, ε); ψ(τ, ε); θ(τ, ε)) of the averaged system

(2.20) that lies in D×Rm together with its ρ-neighborhood ∀τ ∈ [0, L],

ε ∈ (0, ε0];

(ii) ω (τ ) ∈ Cm−1

[0,L] ;

(iii) conditions (2.2) for q = 1, (2.17), and (2.18) are satisfied and, furthermore,

det (ω(j−1)

h+ν (τ ))m−h

ν,j=1

_= 0 ∀τ ∈ [α, β]. (2.21)

Then there exist constants ε∗ > 0 and σ6 > 0 such that ∀(τ, ε) ∈ [0, L] ×

(0, ε0] (ε0 ≤ ε∗), the following inequality holds:

v(τ, τ1,ε)

≡ _x(τ, τ1, ε) − x(τ, ε)_ + _ψ(τ, τ1, ε) − ψ(τ, ε)_

+ _θ(τ, τ1, ε) − θ(τ, ε)_

≤ σ6g(τ, τ1, ε), (2.22)

where (x(τ, τ1, ε); ψ(τ, τ1, ε); θ(τ, τ1, ε)) is a solution of system (2.19) that coincides

with the solution (x(τ, ε); ψ(τ, ε); θ(τ, ε)) of the averaged system (2.20)

for τ = τ1 ∈ [0, L] and, furthermore, g(τ, τ1, ε) = ε

m+r, r = max{r1; r2},

for all τ ∈ [0, L] and τ1 _∈ [α, β] and

g (τ, τ1, ε) =

⎧⎨

⎩

m−h, τ ∈ [α, β],

m+r, τ ∈ [0, L]\[α, β],

for τ1 ∈ [α, β].

32 Averaging Method in Systems with Variable Frequencies Chapter 1

Proof. Denote

F (x, ψ, θ, τ, ε) = [A;B;C]

Fk(x, τ, ε) exp

_ i

(ST

1 k,ψ)

exp

_ i

(ST

2 k, θ)

Fk(x, τ, ε) = [ak(x, τ, ε); _ Kbk(x, τ, ε)],

F(x, ψ, τ, ε) =

2 k=0

Fk(x, τ, ε) exp

_ i

(ST

1 k,ψ)

≡ [A;B;C].

Here, ST

1 and ST

2 are the transposed matrices. Using inequality (2.2) for q = 1,

one can easily obtain the estimate

_F(x, ψ, θ, τ, ε) − F(x, ψ, θ, τ, ε)_ ≤ σ7(_x − x_ + _ψ − ψ_ + _θ − θ_),

σ7 = σ1__K _ 1

|γ| (|γ| + _S1_ + _S2_),

which, together with Eqs. (2.19) and (2.20), yields

v(τ, τ1, ε) ≤ σ7

___

_τ

τ1

v(t, τ1, ε) dt

___

2 k_=0

___

_τ

τ1

Fk(x(t, ε), t, ε)

× exp

_ i

(ST

1 k, ψ(t, ε))

exp

_ i

(ST

2 k, θ(t, ε))

__ _

(

2.23)

Let τ1 ∈ [α, β]. Then inequality (2.21) and the identity Kω(τ ) ≡ 0 are satisfied

for τ ∈ [α, β], i.e., ψ(τ, ε) are slow variables. Therefore, each integral under

the sum sign on the right-hand side of (2.23) can be interpreted as an oscillation

integral. Setting

_θ(t, ε) = θ(t, ε) − 1

τ1

_ω (z) dz,

fk(t, ε) = Fk(x(t, ε), t, ε) exp

_ i

(ST

1 k, ψ(t, ε))

exp

_ i

(ST

2 k, ˜θ(t, ε))

Section 2 Justification of Averaging Method for Systems with ω = ω (τ ) 33

and using estimate (1.20) for p = m − h, we get

v(τ, τ1, ε) ≤ σ7

___

_τ

τ1

v(t, τ1, ε) dt

___

(1 + σ1)_k_ sup

_Fk_

+ sup

___

∂

∂t

___

+ σ1 sup

___

∂

∂x

___

σ8ε

m−h

≤ σ7

___

_τ

τ1

v(t, τ1, ε) dt

___

+ (1 + σ1)σ1__K _σ8ε

m−h ,

where σ8 is a constant corresponding to the constant σ3 in inequality (1.20). The

last inequality proves the following estimate for all τ ∈ [α, β] and τ1 ∈ [α, β] :

v(τ, τ1, ε) ≤ (1 + σ1)__K _σ8eσ7Lε

m−h . (2.24)

Now let τ ∈ [0, α) and τ1 ∈ [α, β]. Then, taking into account condition

(2.17) for τ ∈ [0, α) and inequality (1.21) for p = r1 and using (2.23), we get

v(τ, τ1, ε) ≤ σ7

_τ1

v(t, τ1, ε)dt

2 k_=0

____

_α

Fk(x(t, ε), t, ε) exp{i(k, ˜ϕ(t, ε))}

× exp

_ i

τ1

(k, ω(z))dz

___

_τ1

fk(t, ε) exp

_ i

γε

2 k,

τ1

ω(z)dz

___

≤ σ7

_τ1

v(t, τ1, ε)dt + ε

m−h (1 + σ1)σ1__K _σ8

+ ε

m+r1 (1 + σ1)σ1__K _σ9,

_ϕ(t, ε) = _K

−1(ψ(t, ε); θ(t, ε)) − 1

τ1

ω (z) dz,

34 Averaging Method in Systems with Variable Frequencies Chapter 1

v(τ, τ1, ε) ≤ (1 + σ1)σ1__K _(σ8 + σ9)eσ7L(ε

m−h + ε

m+r1 ) (2.25)

for all τ ∈ [0, α). Here, σ9 is a constant corresponding to the constant σ4 in

inequality (1.21) for p = r1. If τ ∈ (β,L], then v(t, τ1, ε) also satisfies an

estimate of the form (2.25) with r1 replaced by r2 and, possibly, the constant

σ9 replaced by another constant _σ9. Taking this fact and inequalities (2.24) and

(2.25) into account, for all τ1 ∈ [α, β] and τ ∈ [0, L] we obtain estimate (2.22),

where

σ6 = 3σ1(1 + σ1)(σ8 + σ9 + _σ9)__K_eσ7L.

The smallness of ε0 > 0 is determined by conditions for the validity of inequalities

(1.20) and (1.21) and by the estimate σ6ε

m+r

≤ 1

2ρ, which guarantees

that the solution of system (2.19) under investigation does not leave the domain

D × Rm ∀(τ, ε) ∈ [0, L] × (0, ε0]. For τ1 ∈ [0, L]\[α, β], the proof of the

theorem is analogous.

Remark 3. The first two inequalities in (2.17) mean that, at certain times,

resonance occurs in the multifrequency system (2.1), but the system quickly leaves

the resonance state. If identities (2.18) are satisfied, then the system remains in

the resonance state for a sufficiently long time period Δτ = β − α. In this

connection, there arises the necessity of using the method of averaging over a

part of angular variables. The averaging scheme proposed above is not unique.

Efficient estimates for the norm of the difference of solutions of perturbed and

averaged equations can also be obtained by averaging over all angular variables

on the intervals [0, α) and (β,L] and over a part of these variables on [α, β]

and then “glueing” the integral curves in a proper way. Note that the order of the

estimates obtained with respect to ε is the same as in inequality (2.22).

We give an example of frequencies satisfying conditions (2.17), (2.18), and

(2.21). Let

ω1(τ) = τ + 1 ∀τ ∈ [0, 3],

ω2(τ) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

τ 2 + 3, τ∈ [0, 1),

2(τ + 1), τ ∈ [1, 2],

2τ 2 + 4, τ ∈ (2, 3].

Section 2 Justification of Averaging Method for Systems with ω = ω (τ ) 35

It is clear that ω1(τ ) and ω2(τ ) are continuously differentiable on [0, 3] and

Δ(τ) =

_____

ω1(τ ) ω2(τ )

ω(1)

1 (τ ) ω(1)

2 (τ )

_____

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

−(τ + 3)(1 − τ ), τ ∈ [0, 1),

0, τ∈ [1, 2],

(τ − 2)(τ + 4), τ ∈ (2, 3],

(k(1), ω(τ)) = 2ω1(τ ) − 1 ・ ω2(τ ) ≡ 0 ∀τ ∈ [1, 2].

The functions ω1(τ ) and ω2(τ ) thus chosen satisfy conditions (2.17) for r1 =

r2 = 1, α = 1, β = 2, L = 3, and σ5 = 3 and identity (2.18) for h = 1

and k(1) = (2,−1). In this case, inequality (2.21) takes the form ω1(τ ) ≥ 1 or

ω2(τ ) ≥ 3 ∀τ ∈ [0, 3].

At the end of this section, we justify the averaging method on the semiaxis

[0,∞) = R+. Note that, in Chapter 2, we establish an efficient estimate for the

error of the averaging method on the entire axis.

We assume that

_a(x, τ, ε) − a(x, τ, 0)_ ≤ σ10εδ ∀(x, τ, ε) ∈ D×R+ × [0, ε0], (2.26)

a(x, τ, 0) ∈ C2

x(D×R+, σ10),

and consider the averaged equations of the first approximation for slow variables

dτ

= a (x, τ, 0) . (2.27)

Theorem 2.4. Suppose that the following conditions are satisfied:

(a) _(WT

p (τ )Wp(τ ))−1WT

p (τ )_ is uniformly bounded for certain p ≥ m and

all τ ∈ R+, and the functions ω(j)

ν (τ ), ν = 1,m, j = 0, p − 1, are

uniformly continuous on R+;

(b) there exists a solution x = x(τ ) of Eq. (2.27) that lies in D together with

its ρ-neighborhood for all τ ∈ R+;

variational equation

dτ

= ∂a(x(τ ), τ, 0)

∂x

z satisfies the estimate

_Q(τ, t)_ ≤ Ke

−γ(τ−t) ∀τ ≥ t ≥ 0,

K = const ≥ 1, γ= const ≥ 0; (2.28)

36 Averaging Method in Systems with Variable Frequencies Chapter 1

(d) conditions (2.2) are satisfied for q = 0 and τ ∈ R+, and relation (2.26)

holds for δ ≥ 1

Then there exist positive constants σ11, ε2, and ρ1 < ρ such that the following

assertions are true:

(i) for all τ ∈ R+, ψ ∈ Rm, and ε ∈ (0, ε0], ε0 ≤ ε2, the following

estimate is true:

_x(τ, x(0), ψ, ε) − x(τ )_ ≤ σ11ε

p ; (2.29)

(ii) the slow variables x(τ, y, ψ, ε) of any solution (x(τ, y, ψ, ε); ϕ(τ, y, ψ, ε))

of system (2.1) such that

ψ ∈ Rm, ε∈ (0, ε0],

y ∈ Dρ1(x(0)) ≡ {y : y ∈ Rn, _y − x(0)_ < ρ1}

are uniformly bounded for any τ ∈ R+.

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

Eqs. (2.1) that, for

y ∈ Dρ1(x(0)), ρ1 ≤ 1

ρ, ψ ∈ Rm, ε∈ (0, ε0],

the curve x = x(τ, y, ψ, ε) lies in the domain D2Kρ1(x(τ )) for all τ from a

certain maximum half-interval [0, T). For such τ, the function ξ(τ, y, ψ, ε) =

x(τ, y, ψ, ε) − x(τ ) satisfies the equation

ξ(τ, y, ψ, ε) = Q(τ, 0)ξ(0, y, ψ, ε) +

_τ

Q(τ, t)[F(ξ(t, y, ψ, ε), t, ε)

+ _a(x(t, y, ψ, ε), ϕ(t, y, ψ, ε), t, ε)]dt, (2.30)

where

F(ξ, t, ε) = a(ξ + x(t), t, ε) − a(x(t), t, 0) − ∂

∂x

a(x(t), t, 0)ξ,

_a(x, ϕ, t, ε) = a(x, ϕ, t, ε) − a(x, t, ε), _F_ ≤ σ10(εδ + n2_ξ_2).

Section 2 Justification of Averaging Method for Systems with ω = ω (τ ) 37

Using the inequality _ξ(τ, y, ψ, ε)_ ≤ 2Kρ1 and relations (2.28) and (2.30), we

get

sup

τ∈[0,T )

_ξ(τ, y, ψ, ε)_

≤ K_ξ(0, y, ψ, ε)_

+ εδσ10

K + n2σ10

K2ρ1 sup

τ∈[0,T )

_ξ(τ, y, ψ, ε)_

+ sup

τ∈[0,T )

___

_τ

Q(τ, t)_a(x(t, y, ψ, ε), ϕ(t, y, ψ, ε), t, ε) dt

__ _

(

2.31)

which, for ρ1 = min

_ ρ

; γ

6n2σ10K2

, yields

sup

τ∈[0,T )

_ξ(τ, y, ψ, ε)_

≤ 3

2Kρ1 +

2γ

Kσ10εδ

sup

τ∈[0,T )

___

_τ

Q(τ, t) _a (x(t, y, ψ, ε), ϕ (t, y, ψ, ε), t, ε) dt

__ _

(

2.32)

We represent the last term on the right-hand side of (2.32) in the form

sup

τ∈[0,T )

___

_τ

Q(τ, t)_a (x, ϕ, t, ε)dt

___ ≤

k_=0

sup

τ∈[0,T )

_s−1

r=0

___

_r+1

Q(τ, t)ak(x, t, ε) exp{i(k, _ϕ)}

× exp

_ i

(k, ω(z)) dz

___

_τ

Q(τ, t)ak(x, t, ε) exp{i(k, _ϕ)} exp

_ i

(k, ω(z))dz

__ _

38 Averaging Method in Systems with Variable Frequencies Chapter 1

where s is the integer part of τ, x = x(t, y, ψ, ε), ϕ = ϕ(t, y, ψ, ε), and

_ϕ = ϕ − 1

ω(z)dz, and estimate each of the integrals over the segments

[r, r + 1] of unit length using inequalities (1.20) and (2.28) as follows:

___

_r+1

Q(τ, t)ak(x, t, ε) exp{i(k, _ϕ)} exp

_ i

(k, ω(z)) dz

___

≤ σ12ε

p (2 + σ1 + σ10n2)Ke

−γ(τ−r−1)

sup

_ak_ +

sup

___

∂ak

∂τ

___

+ sup

___

∂ak

∂x

___

_ 1

_k_

Here, σ12 is a constant corresponding to the constant σ3 in estimate (1.20). Since

τ −s < 1, the integral over the segment [s, τ ] satisfies the same inequality with

the factor e−γ(τ−r−1) replaced by 1. Then, taking into account condition (2.2)

for q = 0 and the inequality

_s−1

r=0

−γ(τ−r−1) <

eγ

eγ − 1,

we get

sup

τ∈[0,T )

___

_τ

Q(τ, t)_a(x, ϕ, t, ε) dt

___

≤ K

1 + eγ

eγ − 1

σ1(2 + σ1 + n2σ10) σ12ε

p ≡ σ13ε

p .

Using the last inequality, we can rewrite estimate (2.32) in the form

sup

τ∈[0,T )

_ξ(τ, y, ψ, ε)_ ≤ 3

2Kρ1 +

2γ

Kσ10εδ +

2σ13ε

p ≤ 3

2Kρ1 + σ11ε

p ,

σ11 =

σ13 +

Kσ10

. (2.33)

Further, setting σ11ε1/p ≤ 1

4Kρ1, we obtain

sup

τ∈[0,T )

_ξ(τ, y, ψ, ε)_ ≤ 7

4 Kρ1 < 2Kρ1, (2.34)

Section 3 Investigation of Two-Frequency Systems 39

i.e., the curve x = x(τ, y, ψ, ε) does not leave the

4Kρ1-neighborhood of the

curve x = x(τ ). Therefore, the solution (x(τ, y, ψ, ε); ϕ(τ, y, ψ, ε)) of system

(2.1) can be extended to all τ ∈ R+. Inequality (2.34) does not change for T =

∞. Thus, relation (2.33) yields the uniform estimate

_x(τ, y, ψ, ε)_ < 2Kρ1 + sup

τ∈R+

_x(τ )_ ≡ σ14

for all τ ∈ R+, y ∈ Dρ1(x(0)), ψ ∈ Rm, and ε ∈ (0, ε0]. Inequality (2.29)

can be obtained from (2.31) and (2.33) for ξ(0, y, ψ, ε) = 0. Theorem 2.4 is

proved.

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