5. Averaging over All Fast Variables in Multifrequency Systems of Higher Approximation

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Consider the case where system (4.1) can be represented in the form

dx

dτ

=

_r

s=0

As(x, τ )εs + εr+1a(x, ϕ, τ, ε),

dϕ

dτ

=

_r−1

s=−1

Bs(x, τ )εs + εrb (x, ϕ, τ, ε), (5.1)

where r is a nonnegative integer and B−1(x, τ ) ≡ ω(x, τ ) _≡ 0, m ≥ 2. The

principal difference between system (5.1) and (4.1) lies in the fact that the functions

As and Bs−1, s = 0, r, in (5.1) depend only on the slow variables x and

64 Averaging Method in Systems with Variable Frequencies Chapter 1

τ and do not depend on the angular variables ϕ. For r = 0, Grebenikov and

Ryabov [GrR3] justified the method of averaging with respect to the time variable

along a solution of the generating system under the assumption of isolated

resonances. Since, in the case of the existence of resonances, the values obtained

by averaging with respect to time and with respect to all fast variables do not

coincide, the averaging scheme proposed in [GrR3] is, in fact, a scheme of averaging

with respect to a part of fast variables. Below, we justify the averaging

method for (5.1) with respect to all angular variables and establish the quantitative

dependence of estimates on the value of the small parameter.

Assume that

[As(x, τ );Bs−1(x, τ )] ∈ Cl

x,τ (D×[0, L], c1), s= 0, r,

[a(x, ϕ, τ, ε); b(x, ϕ, τ, ε)] ∈ Cl

x,τ (G, c1), l≥ m, (5.2)

_

k_=0

_k_q

_

sup

G

_ck_ +

1

_k_

_

sup

G

___

∂ck

∂τ

___

+ sup

G

___

∂ck

∂x

___

__

≤ c1, q≥ 0,

where c1 is a constant independent of ε, ck = ck(x, τ, ε) are the Fourier coefficients

of the function c(x, ϕ, τ, ε) = [a(x, ϕ, τ, ε); b(x, ϕ, τ, ε)] 2π-periodic in

ϕ, G = D×[0, L] × (0, ε0], and G = G × Rm.

Consider the system averaged with respect to all variables ϕ :

dx

dτ

=

_r

s=0

As(x, τ )εs + εr+1a(x, τ, ε),

dϕ

dτ

=

_r−1

s=−1

Bs(x, τ )εs + εrb(x, τ, ε). (5.3)

We denote by (x(τ, y, ψ, ε); ϕ(τ, y, ψ, ε)) and (x(τ, y, ε); ϕ(τ, y, ψ, ε)) the solutions

of (5.1) and (5.3), respectively, that take a value (y; ψ) ∈ D1 × Rm for

τ = 0; here, D1 is a certain domain in D.

Assume that, 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 (ρ is a constant

independent of ε and y ).

Using the smoothness conditions (5.2) and the Gronwall–Bellman lemma, we

deduce from (5.1) and (5.3) the a priori estimates

_x(τ, y, ψ, ε) − x(τ, y, ε)_ ≤ 2Lc1eLnc1(r+1)εr+1 ≡ cεr+1,

_ϕ(τ, y, ψ, ε) − ϕ(τ, y, ψ, ε)_ ≤ [Lnc1c(1 + r) + 2c1]εr ≡ cεr (5.4)

Section 5 Averaging over All Fast Variables in Multifrequency Systems 65

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

(ρ(2c)−1) 1

r+1 guarantees that the solution (x(τ, y, ψ, ε); ϕ(τ, y, ψ, ε)) of system

(5.1) is defined for all τ ∈ [0, L], and the curve x = x(τ, y, ψ, ε) lies in D

together with its

1

2ρ-neighborhood. The order of the second inequality in (5.4)

with respect to ε is less by one than the order of the first inequality because, in

system (5.1), ω depends on x, and the rate of the variation of angular variables

is proportional to ε−1.

The problem is to improve estimates (5.4) under certain additional restrictions

by replacing r in these estimates by r + α, α = const > 0. Assume that, for

all (x, τ ) ∈ D×[0, L] and certain p, m ≤ p ≤ l, the following inequality is

true:

det(WT

p (x, τ )Wp(x, τ )) _= 0, (5.5)

where

Wp(x, τ) =

_ dj−1

dτj−1 ων (x, τ )

_p,m

j,ν=1

and the total derivatives of the functions ων(x, τ ) with respect to τ are calculated

along the solutions of the equation

dx

dτ

= A0(x, τ ).

Theorem 5.1. If x = x(τ, y, ε) lies in D together with its ρ-neighborhood

∀(τ, y, ε) ∈ [0, L]×D1 × (0, ε0] and conditions (5.2) for q = 0 and (5.5) are

satisfied, then there exists a constant c2 such that

_U(τ, y, ψ, ε)_ ≤ c2εr+1+1

p ∀(τ, y, ψ, ε) ∈ [0, L]×D1 ×Rm ×(0, ε0], (5.6)

where ε0 is sufficiently small and U = (x(τ, y, ψ, ε)−x(τ, y, ε); εϕ(τ, y, ψ, ε)−

εϕ(τ, y, ψ, ε)).

Proof. Denote by D1

2 ρ the closure of the set of points that lie in D together

with their

1

2ρ -neighborhoods. Under the conditions of the theorem, we have

D1

2 ρ

_= ∅. By virtue of the continuity of the functions

dj−1

dτj−1 ων(x, τ ), ν= 1,m, j = 1, p,

on the set D1

2 ρ

× [0, L] and inequality (5.5), there exists a constant c3 > 0 such

that

det(WT

p (x, τ )Wp(x, τ )) ≥ c3 ∀(x, τ ) ∈ D1

2 ρ

× [0, L]. (5.7)

66 Averaging Method in Systems with Variable Frequencies Chapter 1

Further, we consider the matrix

Wp(x, τ, ε) =

_ dj−1

dτj−1 ων (x, τ )

_p,m

j,ν=1

,

where the total derivatives of ων(x, τ ) with respect to τ are calculated along the

solutions of the averaged equations (5.3). It is clear that

det(WT

p (x, τ, ε)Wp(x, τ, ε)) = det(WT

p (x, τ )Wp(x, τ )) + εΔ(x, τ, ε), (5.8)

where Δ(x, τ, ε) is expressed in terms of the functions ων(x, τ ), ν = 1,m,

As(x, τ ), s = 0, r, and a(x, τ, ε) and their derivatives with respect to τ and

x up to the (p − 1)th order. Therefore, according to conditions (5.2), we have

|Δ(x, τ, ε)| ≤ c3 = const ∀(x, τ, ε) ∈ G. It follows from (5.7) and (5.8) for

ε0 ≤ c3(2c3)−1 that

det(WT

p (x, τ, ε)Wp(x, τ, ε)) ≥ 1

2c3

∀(x, τ, ε) ∈ D1

2 ρ

× [0, L] × (0, ε0].

This inequality, together with (5.2), yields

_(WT

p (x, τ, ε)Wp(x, τ, ε))−1WT

p (x, τ, ε)_ ≤ c4 (5.9)

for all x ∈ D1

2 ρ, τ ∈ [0, L], and ε ∈ (0, ε0]; here, c4 is a constant independent

of ε.

Subtracting Eqs. (5.3) from Eqs. (5.1) and multiplying the equality for the

angular variables by ε, we get

_U(τ, y, ψ, ε)_ ≤ 2nc1(1 + r)

_τ

0

_U(t, y, ψ, ε)_ dt

+ εr+1

___

_τ

0

_

k_=0

ck(x, t, ε) exp{i(k, ϕ)} dt

__ _

,

whence

_U(τ, y, ψ, ε)_ ≤ e2nc1(1+r)L

_

k_=0

sup

τ∈[0,L]

____

_τ

0

ck(x, t, ε)

× exp{i(k, θ)} exp

_

i

ε

_t

0

(k, ω(x, t)) dt

           

dt

____

εr+1, (5.10)

Section 5 Averaging over All Fast Variables in Multifrequency Systems 67

where

θ = ϕ − 1

ε

_t

0

ω(x, t)dt, x = x(t, y, ε), ϕ = ϕ(t, y, ψ, ε).

Since the curve x = x(τ, y, ε) lies in D1

2 ρ, condition (5.9) is satisfied for

every solution x = x(τ, y, ε) of the first equation of system (5.3) for τ ∈ [0, L],

y ∈ D1, and ε ∈ (0, ε0]. Moreover, according to conditions (5.2), the total

derivatives of the functions ων(x(τ, y, ε), τ), ν = 1,m, with respect to τ up to

the order l ≥ p inclusive are uniformly bounded from above by a constant independent

of ε and y; therefore, the functions ων(x(τ, y, ε), τ) and their derivatives

with respect to τ up to the order (p − 1) are uniformly continuous in τ

for all (y, ε) ∈ D1 × (0, ε0]. These arguments enable one to apply Theorem 1.2

for ω = ω(x(τ, y, ε), τ) and f = ck(x(τ, y, ε), τ, ε) exp{i(k, θ)} to the estimation

of the oscillation integral on the right-hand side of inequality (5.10). Thus,

according to conditions (5.2) and (5.9), we get

_U(τ, y, ψ, ε)_ ≤ e2nc1(1+r)Lc1σ2[2 + (1 + r)c1]εr+1+1

p

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

which yields estimate (5.6). Here, σ2 is the constant determined by inequality

(1.12). Theorem 5.1 is proved.

Theorem 5.2. Suppose that the conditions of Theorem 5.1 and conditions

(5.2) for q = 1 are satisfied. Then one can find constants c5 > 0 and ε0 > 0

such that

___

∂

∂ψ

U(τ, y, ψ, ε)

___

+ ε

___

∂

∂y

U(τ, y, ψ, ε)

___

≤ c5εr+1+1

p (5.11)

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

The proof of estimate (5.11), in fact, coincides with the proof of Theorem 2.2.

The only difference is that the order of the estimate for

___

∂

∂y

U

___

with respect to ε

is less by one than the order of the estimate for

___

∂

∂ψ

U

___

because the frequencies

ων depend on x and, therefore,

___

∂

∂y

ϕ(τ, y, ψ, ε)

___

∼ ε

−1.

68 Averaging Method in Systems with Variable Frequencies Chapter 1

Remark 6. The main assumption in Theorems 5.1 and 5.2 is inequality (5.5)

[or the equivalent inequality (5.9)], which is a restriction imposed on the averaged

system. Condition (5.5) guarantees the fast passage of the averaged system

[and system (5.1) with regard for the a priori estimates (5.4)] through a small

neighborhood of the resonance surface (k, ω(x, τ)) = 0, k _= 0. Note that, generally

speaking, this is not the case for multifrequency systems of the general form

[Arn4, GrR1, GrR3, Sam5].

Let us formulate a theorem on the justification of the averaging method on a

semiaxis for the system of n + m equations

dx

dτ

= a(x, τε) + εA(x, ϕ, τ, ε),

dϕ

dτ

= ω(x, τ, ε)

ε

+ B(x, ϕ, τ, ε), (5.12)

where a, A, ω, and B are p ≥ m times continuously differentiable with respect

to x, ϕ, and τ for every fixed ε, and all partial derivatives of these functions

are uniformly bounded in G by a constant c6 independent of ε. Assume

that A and B belong to the class of functions almost periodic in ϕj, j = 1,m,

[A(x, ϕ, τ, ε);B(x, ϕ, τ, ε)] =

∞_

ν=0

[Aν(x, τ, ε);Bν(x, τ, ε)] exp{i(λν, ϕ)},

λ0 = 0, λν _= 0 for ν ≥ 1, and

∞_

ν=1

__

1+

1

_λν_

_

sup

G

_Cν_ +

1

_λν_

_

sup

G

___

∂Cν

∂τ

___

+ sup

G

___

∂Cν

∂x

___

__

≤ c6. (5.13)

Here, Cν = [Aν(x, τ, ε);Bν(x, τ, ε)] and (λν, ϕ) is the scalar product of the

vectors (λ(1)

ν , . . . , λ(m)

ν ) and (ϕ1, . . . , ϕm).

The system averaged with respect to ϕ has the form

dx

dτ

= a(x, τ, ε) + εA0(x, τ, ε),

dϕ

dτ

= ω(x, τ, ε)

ε

+ B0(x, τ, ε), (5.14)

where

[A0;B0] = lim

t→∞

t

−m

_t

0

. . .

_t

0

[A(x, ϕ, τ, ε);B(x, ϕ, τ, ε)]dϕ1 . . . dϕm.

Section 5 Averaging over All Fast Variables in Multifrequency Systems 69

As for frequencies, we assume that

_(WT

p (x, τ, ε)Wp(x, τ, ε))−1WT

p (x, τ, ε)_ ≤ c7 ∀(x, τ, ε) ∈ G, (5.15)

where

WT

p (x, τ, ε) =

_ dj−1

dτj−1 ων(x, τ, ε)

_m,p

j,ν=1

and the total derivatives of the functions ων(x, τ, ε) with respect to τ are calculated

with regard for the equation

dx

dτ

= a(x, τ, ε).

We also assume that there exists a solution x = x(τ, ε) of the averaged

equations (5.14) for slow variables that is defined and lies in D together with

its ρ-neighborhood ∀(τ, ε) ∈ R+ × (0, ε0], and the normal fundamental matrix

Q(τ, t, ε) of solutions of the variational system

dz

dτ

= H(τ, ε)z, H(τ, ε) = ∂

∂x

[a(x(τ, ε), τ, ε) + εA(x(τ, ε), τ, ε)],

satisfies the estimate

_Q(τ, t, ε)_ ≤ Kε

−l1e

−γεl2 (τ−t) ∀τ ≥ t ≥ 0, ε∈ (0, ε0], (5.16)

where K > 0, γ >0, r1 ≥ 0, and r2 ≥ 0 are certain constants independent

of ε.

Theorem 5.3 [PeP]. If conditions (5.13), (5.15), and (5.16) for l = l1+l2 <

1

2

+

1

2p

are satisfied, then the solution (x(τ, x(0, ε), ψ, ε); ϕ(τ, x(0, ε), ψ, ε)) of

system (5.12) is defined for all τ ∈ R+, ψ ∈ Rm, and ε ∈ (0, ε0] (ε0 is

sufficiently small) and satisfies the following inequalities:

_x(τ, x(0, ε), ψ, ε) − x(τ, ε)_ ≤ c8ε1+1

p

−l, c8 = const,

_ϕ(τ, x(0, ε), ψ, ε) − ϕ(τ, x(0, ε), ψ, ε)_ ≤ c8(1 + τ )ε

1

p

−l.

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