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12. Auxiliary Statements

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Consider a multifrequency system of the form

dτ

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

dϕ

dτ

= ω(τ )

+ b(x, ϕ, τ, ε), (12.1)

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

domain, and the real vector functions a, _a, A, ω, and b are defined and 2π-

periodic in each variable ϕν, ν = 1,m, on the set G = D×Rm × R × (0, ε0].

Without loss of generality, we can assume that the function _a(x, ϕ, τ ) averaged

with respect to ϕ over the cube of periods is identically equal to zero [otherwise,

it can be included in a(x, τ ) in system (12.1)].

Assume that

[a, _a, b] ∈ C1

τ (G, σ1) ∩ C2

x,ϕ(G, σ1),

∂a

∂x

∈ C1

τ (G, σ1), A∈ C2

x,ϕ(G, σ1), (12.2)

k_=0

_k_2 sup

_ck_ + _k_

sup

___

∂ck

∂τ

___

+ sup

___

∂ck

∂x

___

≤ σ1,

and

∂

∂τ

A(x, ϕ, τ, ε) is continuous in (x, ϕ, τ ) ∈ D×Rm × R for every fixed

ε ∈ (0, ε0]. Here, σ1 is a certain positive constant, ck = ck(x, τ, ε) are the

Fourier coefficients of the harmonics exp{i(k,ϕ)} in the Fourier expansion of

the function c(x, ϕ, τ, ε) = [_a(x, ϕ, τ ); b(x, ϕ, τ, ε)], i is the imaginary unit,

(k,ϕ) = k1ϕ1 + . . . + kmϕm is the scalar product of vectors k = (k1, . . . , km)

133

134 Integral Manifolds Chapter 3

and ϕ = (ϕ1, . . . , ϕm), _k_ = |k1|+. . .+|km|, and Cl

x,ϕ(G, σ1) (Clτ

(G, σ1))

denotes the set of vector functions that have partial derivatives with respect to all

variables x and ϕ (τ ) up to the lth order inclusive that are continuous in x, ϕ,

and τ and bounded in G by the constant σ1. Unless otherwise stated, the norm

of a matrix is understood as the sum of the absolute values of its elements.

We also impose certain restrictions on the coordinates 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 )_ ≤ σ2 = const ∀τ ∈ R, (12.3)

where Wp(τ ) and WT

p (τ ) denote the matrix

(ω(μ−1)

ν (τ ))m,p

ν,μ=1

and its transpose, respectively.

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

averaged with respect to all angular variables ϕ

dτ

= a (x, τ ), (12.4)

and assume that it has a solution x = x(τ ) defined on the entire numerical

straight line and lying in D together with its ρ-neighborhood.

Lemma 12.1. If conditions (12.2) and (12.3) are satisfied and ϕ = ϕt

τ (ψ, ε)

is a solution of the Cauchy problem

dϕt

= ω(t)

+ b(x(t) + Y (ϕt

τ , t, ε), ϕt

τ , t, ε), ϕττ

= ψ ∈ Rm, (12.5)

where Y (ϕ, t, ε) is continuously differentiable with respect to (ϕ, t) ∈ Rm × R

for every fixed ε,

___∂

∂t

+ ∂Y

∂ϕ

ω(t)

___

≤ d1,

___

∂Y

∂ϕ

___

≤ d2εα

∀(ϕ, t, ε) ∈ Rm × R × (0, ε0] ≡ G1,

Section 12 Auxiliary Statements 135

d1, d2 = const, and α =

, then there exist constants c1 and c2 independent

of ε and such that

___

∂

∂ψ

(ϕt

τ (ψ, ε) − ψ)

___

≤ c1εα(1 + d2)ec2(1+d2)εα|τ−t|(1 + |τ − t|)

for sufficiently small ε0 > 0 and all (ψ, t, ε) ∈ G1 and τ ∈ R.

Proof. We rewrite problem (12.5) in the form

ϕt

− ψ =

_ω(l)

+ b(x(l) + Y (ϕl

τ , l, ε), ϕl

τ , l, ε)

dl.

Then, denoting ztτ

= ∂

∂ψ

(ϕt

− ψ), we obtain

ztτ

∂b

∂x

∂Y

∂ϕ

(zlτ

+ Em)dl +

∂b

∂ϕ

(zlτ

+ Em)dl, (12.6)

whence

_ztτ

_ ≤ nσ1d2εα

m|τ − t| +

___

_ztτ

_ dl

___

k_=0

___

Bk(x(l) + Y (ϕl

τ , l, ε), l, ε)(zlτ

+ Em)

× exp{i(k, θlτ

)} exp

_ i

(k, ω(r)) dr

__ _

(

12.7)

Here, Em is the m-dimensional identity matrix, Bk(x, τ, ε) are the Fourier coefficients

of the function

∂

∂ϕ

b(x, ϕ, τ, ε), and θtτ

= ϕt

− 1

ω(r)dr.

First, we consider the case t ≥ τ + 2. We represent the segment [τ, t] as a

union of segments, namely

[τ, t] =

q_−1

s=0

[τ + s, τ + s + 1] ∪ [τ + q, t],

136 Integral Manifolds Chapter 3

where q is the integer part of the number t−τ −1, 1 ≤ t−(τ +q) < 2. Then

we represent the integral over [τ, t] under the summation sign on the right-hand

side of (12.7) as the sum of integrals over the segments indicated. Estimating the

integral over the segment [τ + s, τ + s + 1] of unit length by using condition

(12.3) and the uniform estimate (1.20), we get

Δs,k ≡

___

τ+_s+1

τ+s

Bk(zlτ

+ Em) exp{i(k, θlτ

)} exp

_ i

(k, ω(r))dr

___

≤ c3εα

(1 + σ1)(m + max

[τ+s,τ+s+1]

_zlτ

_) + max

[τ+s,τ+s+1]

___

zlτ

___

× sup

_Bk(x, τ, ε)_ + (m + max

[τ+s,τ+s+1]

_zlτ

_k_ sup

___

∂

∂τ

Bk(x, τ, ε)

___

+ (σ1 + d2εα0

σ1 + d1)(m + max

[τ+s,τ+s+1]

_zlτ

_k_ sup

___

∂

∂x

Bk(x, τ, ε)

___

where c3 is the constant corresponding to the constant σ3 in estimate (1.20).

Since

dzlτ

_ ∂b

∂x

∂Y

∂ϕ

+ ∂b

∂ϕ

(zlτ

+ Em),

the inequality

max

[τ+s,τ+s+1]

___

zlτ

___

≤ (m + nd2εα0

)(m + max

[τ+s,τ+s+1]

_zlτ

_) (12.8)

yields the following estimate for d2εα0

≤ 1:

Δs,k ≤ c4εα(1 + max

[τ+s,τ+s+1]

_zlτ

sup

_Bk_

_k_

sup

___

∂

∂τ

___

+ sup

___

∂

∂x

___

where c4 = mc3[(1 + σ1)(1 + σ1(n + m)) + 2σ1 + d1].

Further, we consider the differentiable norm

_z_1 =

__m

i,j=1

Section 12 Auxiliary Statements 137

of the matrix z = (zij)m

i,j=1. It is obvious that

_z_1 ≤ _z_ ≤ m2_z_1,

___

dτ

_z_1

___

≤

___

dτ

___

, z= z(τ ).

Let l1 and l2 be, respectively, the maximum point and the minimum point of a

continuously differentiable function _zlτ

_1 of a variable l on the segment [τ +s,

τ + s + 1]. Then the following inequalities hold:

max

[τ+s,τ+s+1]

_zlτ

_ ≤ m2 max

[τ+s,τ+s+1]

_zlτ

_1 = m2[_zl1

_1 − _zl2

_1 + _zl2

_1]

= m2

_ _l2

_zlτ

_1dl + _zl2

≤ m2

τ+_s+1

τ+s

____ d

zlτ

___

+ _zlτ

≤ σ1(m + nd2εα0

)m2

τ+_s+1

τ+s

_zlτ

_dl + m3(m + nd2εα0

)σ1,

In view of these inequalities, the estimate for Δsk takes the form

Δs,k ≤ c4εα

1 +

τ+_s+1

τ+s

_zlτ

_dl

sup

_Bk_ +

_k_

sup

___

∂

∂τ

___

+ sup

___

∂

∂x

___

, (12.9)

c4 = c4(1 + m3(m + n)), s= 0, q − 1.

Since the length of the segment [τ + q, t] is not less than 1 and less then 2,

we conclude that the expression

Δq,k =

____

τ+q

Bk(x(l) + Y (ϕl

τ , l, ε), l, ε)(zlτ

+ Em) exp{i(k,ϕl

τ )}dl

____

138 Integral Manifolds Chapter 3

can also be estimated using inequality (1.20). Repeating the scheme of the proof

of estimate (12.9), we get

Δq,k ≤ _c4εα

1 +

τ+_s+1

τ+s

_zlτ

_dl

sup

_Bk_ +

_k_

sup

___

∂

∂τ

___

+ sup

___

∂

∂x

___

. (12.10)

Combining (12.9) and (12.10) and using condition (12.2) for the Fourier coefficients,

we deduce from (12.7) for t ≥ τ + 2 that

_ztτ

_ ≤ mn

1 +

_zlτ

_dl

σ1d2εα0

+ σ1(c4 + _c4)εα

__t

_zlτ

_dl + t − τ

≤ c5εα(d2 + 1)

_ _t

_zlτ

_dl + t − τ

, (12.11)

c5 = mσ1(n + c4 + _c4).

Now let t ∈ [τ, τ + 2). Then equality (12.6) yields

_ztτ

_ ≤ (nσ1d2εα0

+ mσ1)

_zlτ

_dl + 2mnσ1d2εα

k_=0

____

Bk(x(l) + Y (ϕl

τ , l, ε), l, ε)

× exp{i(k, θlτ

)} exp

(k, ω(r))dr

____

. (12.12)

According to (1.20) and (12.2), the last term on the right-hand side of (12.12) is

bounded from above by c6εα. Therefore, inequality (12.12) for d2εα0

≤ 1 yields

_ztτ

_ ≤ _c6εα ∀t ∈ [τ, τ + 2), _c6 = e2(n+m)σ1(c6 + 2mnσ1). (12.13)

Section 12 Auxiliary Statements 139

We return to estimate (12.11) for t ≥ τ + 2. If we represent the segment [τ, t]

as the union of the segments [τ, τ +2] and [τ +2, t] and use inequality (12.13),

then estimate (12.11) takes the form

_ztτ

_ ≤ c5(1 + d2)εα

τ+2

_zlτ

_dl + (1 + d2)c5(1 + 4_c6)εα(t − τ )

for t ≥ τ + 2, d2εα0

≤ 1, and ε0 < 1. Since t − τ increases monotonically as

a function of t, we have

_ztτ

t − τ

≤ c5(1 + d2)εα

τ+2

_zlτ

l − τ

dl + (1 + d2)c5(1 + 4_c6)εα,

Hence, according to the Gronwall–Bellman lemma, we get

_ztτ

_ =

___

∂

∂ψ

(ϕt

− ψ)

___

≤ c5(1 + 4_c6)(1 + d2)εα(t − τ )ec5(1+d2)εα(t−τ)

∀t ≥ τ + 2. Combining the last estimate for t ≥ τ + 2 and estimate (12.13) for

τ ≤ t < τ + 2, we complete the proof of the lemma for t ≥ τ. For t < τ, the

proof is analogous.

Lemma 12.2. Suppose that the conditions of Lemma 12.1 are satisfied. Then

one can indicate constants c7 and c8 independent of ε and such that

___

∂

∂τ

ϕt

τ (ψ, ε)

___

≤ c7

1 +

_ω(τ )_

ec8εα|τ−t|

for all (ψ, τ, ε) ∈ G1 and t ∈ R.

Proof. The Cauchy problem (12.5) yields

∂ϕt

∂τ

= −

_ω(τ )

+ b(x(τ) + Y (ψ, τ, ε), ψ, τ, ε)

_ ∂b

∂x

∂Y

∂ϕ

+ ∂b

∂ϕ

_∂ϕl

∂τ

dl,

∂ϕt

∂τ

_ ∂

∂x

b(x(t) + Y (ϕt

τ , t, ε), ϕt

τ , t, ε) ∂

∂ϕ

Y (ϕt

τ , t, ε)

+ ∂

∂ϕ

b(x(t) + Y (ϕt

τ , t, ε), ϕt

τ , t, ε)

_∂ϕt

∂τ

. (12.14)

140 Integral Manifolds Chapter 3

The first of these equalities yields

___

∂ϕt

∂τ

___

≤

__ω(τ )_

+ σ1

+ nσ1d2εα

___

∂ϕl

∂τ

__ _

___

∂b

∂ϕ

∂ϕl

∂τ

__ _

(

12.15)

If t ∈ [τ, τ + 2), then the last term on the right-hand side of (12.15) can be

estimated from above by the value

mσ1

___

∂ϕl

∂τ

___

dl.

Hence, using the Gronwall–Bellman inequality, we get

___

∂ϕt

∂τ

___

≤

__ω(τ )_

+ σ1

e2σ1(m+n) ∀t ∈ [τ, τ + 2) (12.16)

for d2εα0

≤ 1. If t ≥ τ +2, then we represent the segment [τ, t] as the union of

segments of unit length and the last segment whose length is not less than 1 and

less than 2. Then we decompose the integral under the norm sign on the righthand

side of (12.15) into the sum of integrals over the segments indicated. Taking

into account inequalities (1.20) and (12.2) and the second equality in (12.14), by

analogy with the proof of Lemma 12.1 we get

___

∂b

∂ϕ

∂ϕl

∂τ

___

≤ c9εα

___

∂ϕl

∂τ

___

dl, c9 = const.

The last estimate, together with estimates (12.15) and (12.16), yields

___

∂ϕt

∂τ

___

≤ [1 + (nσ1 + c9)2e2σ1(m+n)]

__ω(τ )_

+ σ1

+ (c9 + nσ1d2)εα

τ+2

___

∂ϕl

∂τ

___

dl.

Section 12 Auxiliary Statements 141

Solving this inequality, we get

___

∂ϕt

∂τ

___

≤ [1 + (nσ1 + c9)2e2σ1(m+n)]

__ω(τ )_

+ σ1

e(nσ1d2+c9)εα(t−τ) (12.17)

∀t ≥ τ + 2. Estimates (12.16) and (12.17) complete the proof of the lemma for

t ≥ τ. For t < τ, the lemma is proved by analogy.

Lemma 12.3. If the functions

∂2

∂ψ∂ψν

Y (ψ, τ, ε), ν = 1,m, are continuous

in (ψ, τ) ∈ Rm × R for every ε ∈ (0, ε0] and the conditions of Lemma 12.1 are

satisfied, then, for any (ψ, τ, ε) ∈ G1, t ∈ R, the following inequality is true:

ν=1

___

∂2ϕt

τ (ψ, ε)

∂ψ∂ψν

___

≤ c10

εα +

ν=1

sup

ψ,τ

___

∂2

∂ψ∂ψν

Y (ψ, τ, ε)

___

× (1 + |t − τ |2)ec11εα|t−τ|

, (12.18)

where the constants c10 and c11 are independent of ε.

Proof. We prove estimate (12.18) for t ≥ τ (the proof for t < τ is analogous).

Using (12.5), we get

∂2ϕt

τ (ψ, ε)

∂ψ∂ψν

__n

r=1

∂2b

∂x∂xr

∂Y (r)

∂ϕ

_∂(ϕl

− ψ)

∂ψν

+ eν

_∂Y

∂ϕ

μ=1

∂2b

∂x∂ϕμ

δνμ +

∂(ϕl

τ,μ

− ψμ)

∂ψν

_∂Y

∂ϕ

r=1

∂2b

∂ϕ∂xr

∂Y (r)

∂ϕ

_∂(ϕl

− ψ)

∂ψν

+ eν

μ=1

∂2b

∂ϕ∂ϕμ

∂(ϕl

τ,μ

− ψμ)

∂ψν

142 Integral Manifolds Chapter 3

+ ∂b

∂x

μ=1

∂2Y

∂ϕ∂ϕμ

_∂(ϕl

τ,μ

− ψμ)

∂ψν

+ δνμ

Em + ∂(ϕl

− ψ)

∂ψ

∂2b

∂ϕ∂ϕν

∂(ϕl

− ψ)

∂ψ

dl +

∂2b

∂ϕ∂ϕν

_ ∂b

∂x

∂Y

∂ϕ

+ ∂b

∂ϕ

_ ∂2ϕl

∂ψ∂ψν

dl,

where Y = (Y (1), . . . , Y (n)), ϕt

τ = (ϕt

τ,1, . . . , ϕt

τ,m), δν,μ is the Kronecker

symbol, and eν is the unit vector in the space Rm. Using conditions (12.2) and

Lemma 12.1, we get

___

∂2ϕt

∂ψ∂ψν

___

≤ c12

εα +

μ=1

sup

ψ,τ

___

∂2

∂ψ∂ψμ

Y (ψ, τ, ε)

___

× (t − τ + (t − τ )2)ec13εα(t−τ) + nσ1d2εα

___

∂2ϕl

∂ψ∂ψν

___

∂2b

∂ϕ∂ϕν

___

∂b

∂ϕ

∂2ϕl

∂ψ∂ψν

__ _

(

12.19)

where c12 and c13 are constants independent of ε. The last two terms on the

right-hand side of (12.19) can be estimated by analogy with the corresponding

integrals in the proof of Lemmas 12.1 and 12.2. Conditions (12.2) for the Fourier

coefficients of the function b(x, ϕ, τ, ε) and the uniform estimates (1.20) of the

oscillation integrals yield the following inequalities for any t ≥ τ + 2:

____

∂2b

∂ϕ∂ϕν

____

≤ c14εα(t − τ ),

Section 12 Auxiliary Statements 143

____

∂b

∂ϕ

∂2ϕl

∂ψ∂ψν

____

≤ c14

εα +

μ=1

sup

ψ,τ

___

∂2

∂ψ∂ψμ

Y (ψ, τ, ε)

___

× (t − τ + (t − τ )2)ec15εα(t−τ)

+ εα

___

∂2ϕl

∂ψ∂ψν

___

, (12.20)

where c14, c14, and c15 are certain constants independent of ε. Combining

inequalities (12.19) and (12.20), we obtain

ν=1

___

∂2ϕt

∂ψ∂ψν

___

≤ (c12 + c14 + c14)m

εα +

ν=1

sup

ψ,τ

___

∂2

∂ψ∂ψν

Y (ψ, τ, ε)

___

× (t − τ + (1 − τ )2)ec15εα(t−τ)

+ (nσ1d2 + c14)εα

ν=1

___

∂2ϕl

∂ψ∂ψν

___

dl (12.21)

∀t ≥ τ + 2, c15 = max{c13; c15}.

For t ∈ [τ, τ + 2), relation (12.19) yields

ν=1

___

∂2ϕt

∂ψ∂ψν

___

≤ c16

εα +

ν=1

sup

ψ,τ

___

∂2

∂ψ∂ψν

Y (ψ, τ, ε)

___

(12.22)

where the constant c16 is independent of ε. Therefore, decomposing the integral

over [τ, t] on the right-hand side of (12.21) into the sum of integrals over the

segments [τ, τ + 2] and [τ + 2, t] and using estimate (12.22), we deduce from

(12.21) that

ν=1

___

∂2ϕt

∂ψ∂ψν

___

≤ c17

εα +

ν=1

sup

ψ,τ

___

∂2

∂ψ∂ψν

Y (ψ, τ, ε)

___

× (t − τ )2e(c15+nσ1d2+c14)εα(t−τ) (12.23)

∀t ≥ τ + 2, c17 = const.

Inequalities (12.22) and (12.23) yield estimate (12.18) for all t ≥ τ. Lemma 12.4

is proved.

144 Integral Manifolds Chapter 3

The methods proposed above can be used for the proof of the following statements:

Lemma 12.4. If the conditions of Lemma 12.3 are satisfied, then the following

estimate holds for any (ψ, τ, ε) ∈ G1 and t ∈ R:

___

∂

∂τ

∂

∂ψ

ϕt

τ (ψ, ε)

___

≤ c10εα−1(_ω(τ )_ + 1)ec11εα|t−τ|

where the constants c10 and c11 are independent of ε.

Lemma 12.5. Suppose that the following conditions are satisfied:

(i) conditions (12.2) and (12.3) are satisfied;

(ii) the functions Y1(ϕ, τ, ε) and Y2(ϕ, τ, ε) are twice continuously differentiable

with respect to (ϕ, τ ) ∈ Rm ×R for every ε ∈ (0, ε0], 2π-periodic

in ϕν, ν = 1,m, and such that

_Ys_ ≤ d1εα0

___

∂

∂ϕ

___

≤ d2εα0

ν=1

___

∂2

∂ϕ∂ϕν

___

≤ d3εα0

___

∂

∂τ

Ys + ∂

∂ϕ

ω(τ )

___

≤ d1 ∀(ϕ, τ, ε) ∈ G1, s= 1, 2.

Then there exist constants c18 and c19 such that, for all (ψ, τ, ε) ∈ G1 and

t ∈ R, the following estimates are true:

_ϕt

τ,1(ψ, ε) − ϕt

τ,2(ψ, ε)_

≤ c18(1 + |t − τ |)ec19εα0

|t−τ| max

l∈N(τ,t)

sup

ψ∈Rm

_Y1(ψ, l, ε) − Y2(ψ, l, ε)_,

___

∂

∂ψ

(ϕt

τ,1(ψ, ε) − ϕt

τ,2(ψ, ε))

___

≤ c18(1 + |t − τ |2)ec19εα0

|t−τ|

sup

_Y1(ψ, τ, ε) − Y2(ψ, τ, ε)_

+ sup

___

∂

∂ψ

(Y1(ψ, τ, ε) − Y2(ψ, τ, ε))

___

, (12.24)

Section 13 Construction of Successive Approximations 145

where ϕt

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

ϕt

τ,s(ψ, ε) = ω(t)

+ b(x(t) + Ys(ϕt

τ,s(ψ, ε), t, ε), ϕt

τ,s(ψ, ε), t, ε),

ϕτ

τ,s(ψ, ε) = ψ,

N(τ, t) = [τ, t] for τ < t, and N(τ, t) = [t, τ ] for τ ≥ t.

In what follows, we use the results obtained above in the proof of the existence

of the integral manifold of the multifrequency system (12.1) and in the

investigation of its properties.

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