Пресс-релиз популярных книг

Авторы: 111 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 164 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

На сайте 111 авторов, 92 книг, 72 статей, 5913 глав.

21. Weakening of Conditions in the Theorem on Integral Manifold

Back

In this section, we return to the problem on the integral manifold of the oscillation

system

dτ

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

dϕ

dτ

= ω(τ )

+ b(x, ϕ, τ, ε).

(21.1)

Note that, in Sections 12–14, we have proved the existence and established properties

of the integral manifold x = X(ϕ, τ, ε) in the case where the right-hand

234 Integral Manifolds Chapter 3

sides of Eqs. (21.1) are twice continuously differentiable with respect to x, ϕ,

and τ and the norm of the matrix P = ∂

∂x

_a(x, ϕ, τ ) is sufficiently small [inequality

(13.3)]. In what follows, we omit the condition that _P_ is small and

study analogous problems. Note that the method proposed here requires an increase

in the smoothness order by one and certain additional restrictions on the

Fourier coefficients of the function _a(x, ϕ, τ ).

Assume that

[a, b,A] ∈ C1

τ (G, σ1) ∩ C2

x,τ (G, σ1),

∂a

∂x

∈ C1

τ (G, σ1), _a ∈ C3

x,τ (G, σ1),

k_=0

_k_2 sup _bk_ + _k_

sup

___

∂bk

∂τ

___

+ sup

___

∂bk

∂x

___

≤ σ1,

k_=0

_k_2 sup _ak_ + _k_

sup

___

∂ak

∂τ

___

+ sup

___

∂ak

∂x

___

+ sup

___

∂2ak

∂x∂τ

___

j=1

sup

___

∂2ak

∂x∂xj

___

_k_

sup

___

∂2ak

∂τ2

___

j=1

sup

___∂

3ak

∂x∂xj∂τ

___

j,s=1

sup

___

∂3ak

∂x∂xj∂xs

___

≤ σ1. (21.2)

Here, the supremum is taken over all (x, ϕ, τ, ε) ∈ G. Note that the notation used

in this section is the same as in Sections 12–14.

Assume that the components ων(τ ), ν = 1,m, of the frequency vector ω(τ )

and their derivatives with respect to τ up to an order p −1 (p ≥ m) are uniformly

continuous on the entire axis and

_(WT

p (τ )Wp(τ ))−1WT

p (τ )_ ≤ σ2, _ω(τ )_ +

___

dτ

ω(τ )

___

≤ σ2, (21.3)

where, as above, Wp(τ ) and WT

p (τ ) denote the matrix

_ ds−1

dτs−1 ων(τ )

_m,p

ν,s=1

and its transpose, respectively, and σ2 is a constant.

Section 21 Weakening of Conditions in the Theorem on Integral Manifold 235

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

averaged over all angular variables ϕ, namely

dτ

= a(x, τ ),

and assume that there exists its solution x = x(τ ) defined for all τ ∈ R and

such that x(τ ) ∈ Dρ for certain ρ > 0 (Dρ is the set of points that belong to

the bounded domain D together with their ρ-neighborhoods). Assume that the

variational system

dτ

= H(τ )z, H(τ) = ∂

∂x

a(x(τ ), τ),

is hyperbolic and the Green matrix Q(τ, t) satisfies the inequality

_Q(τ, t)_ ≤ Ke

−γ|τ−t| ∀τ, t ∈ R, (21.4)

where γ > 0 and K ≥ 1 are certain constants.

In system (21.1), we set

x = z + εu(z, ϕ, τ, μ), u=

k_=0

1 − hμ((k, ω(τ )))

i(k, ω(τ ))

ak(z, τ)ei(k,ϕ), (21.5)

where i is the imaginary unit, k = k

_k_,

hμ(t) =

∞ _

−∞

ν2μ(l)ωμ(t − l)dl,

ν2μ(l) ≡ 1 for |l| ≤ 2μ, ν2μ(l) ≡ 0 for |l| > 2μ, and ωμ(l) is the averaging

kernel [Mik], namely

ωμ(l) =

⎧⎪⎨ ⎪⎩

0, |l| ≥ μ,

σ3e

− μ2

μ2−l2 , |l| < μ,

σ3 =

−1

− 1

1−l2 dl.

We fix the averaging radius μ < 1 in what follows. The function hμ(t) thus

constructed is infinitely differentiable for all t ∈ R and finite, 0 ≤ hμ(t) ≤ 1

236 Integral Manifolds Chapter 3

for any t ∈ R, hμ(t) ≡ 1 for |t| ≤ μ, hμ(t) ≡ 0 for |t| > 3μ, and, for all

integer q ≥ 0, the following estimates are satisfied:

___

dtq hμ(t)

___

≤ cqt

−qhμ(t), (21.6)

where cq are constants, hμ(t) ≡ 0 for |t| ≤ μ and |t| ≥ 3μ, and hμ(t) ≡ 1

for μ < |t| < 3μ.

If a positive ε0 is sufficiently small, then, for all ε ∈ (0, ε0], the change of

variables (21.5) reduces system (21.1) to the form

dτ

= a(z, τ) + δ(z, ϕ, τ, μ) + εv(z, ϕ, τ, μ) + εA1(z, ϕ, τ, ε, μ),

dϕ

dτ

= ω(τ )

+_b(z, ϕ, τ, ε, μ), (21.7)

where

= b(z + εu, ϕ, τ, ε), v= −∂u

∂τ

, u= u(z, ϕ, τ, μ),

δ =

k_=0

ak(z, τ)hμ((k, ω(τ )))ei(k,ϕ),

A1 = B − ∂u

∂z

En + ε

∂u

∂z

_−1

(a(z, τ) + δ + εv + εB),

B = A(z + εu, ϕ, τ, ε) − ∂u

∂ϕ

[a(z + εu, τ )

− a(z, τ) + _a(z + εu, ϕ, τ ) − _a(z, ϕ, τ )].

Taking conditions (21.2), (21.3), and (21.6) into account, one can easily establish

the existence of a constant σ4 such that

_u_ +

___

∂u

∂z

___

∂u

∂ϕ

___

≤ σ4

, _v_ ≤ σ4

μ2 , _A1_ ≤ σ4

(1 + ε_v_) (21.8)

for all (z, ϕ, τ, ε) ∈ D1

2 ρ

× Rm × R × (0, ε0]. Note that the restriction z ∈ D1

2 ρ

and the inequality σ4εμ−1 <

2ρ guarantee that the point z + εu belongs to the

domain D.

Section 21 Weakening of Conditions in the Theorem on Integral Manifold 237

Lemma 21.1. Suppose that f(t) = (f1(t), . . . , fm(t)) ∈ D and θ(t) =

(θ1(t), . . . , θm(t)) ∈ Rm are arbitrary continuous (for t ∈ R ) functions and

conditions (21.2)–(21.4) are satisfied. Then there exist constants σ5 and σ6 independent

of μ and such that the following estimates hold for all τ ∈ R:

___

∞ _

−∞

Q(τ, t)δ(f(t), θ(t), t, μ)dt

___

≤ σ5μ

p−1 , (21.9)

___

∞ _

−∞

Q(τ, t)v(f(t), θ(t), t, μ) dt

___

≤ σ6

. (21.10)

Proof. According to condition (21.3) and the results of Section 1, for arbitrary

τ ∈ R there exist Δ > 0 independent of τ and k and an integer r = r(τ, k) ∈

[0, p − 1] such that the following inequality holds for all t ∈ [τ − Δ, τ +Δ] :

___

dtr (k, ω(t))

___

≥ 1

2pσ2

. (21.11)

If r ≥ 1, then the last inequality implies that the functions (k, ω(t)) + c and

(k, ω(t)) (c = const) can take the zero value on the segment [τ−Δ, τ+Δ] at

at most 2p−1 points. Moreover, [τ−Δ, τ+Δ] can be divided into two sets M(τ )

and N(τ ) of segments such that M(τ ) consists of l1 ≤ 2p−1 − 1 segments

whose lengths do not exceed 2μ = const, and N(τ ) consists of l2 ≤ 2p−1

segments on each of which the following inequality is satisfied:

|(k, ω(t))| ≥ 1

2pσ2

μp−1. (21.12)

First, we prove estimate (21.9). We have

___

∞ _

−∞

Qδdt

___

≤

∞_

s=−∞

k_=0

___

τ+2_(s+1)Δ

τ+2sΔ

Qakhμ((k, ω(t))) dt

___

≤ K

∞_

s=−∞

−2|s|γΔ

k_=0

sup

_ak_

& _

M(τ+(2s+1)Δ)

hμ((k, ω(t)))dt

N(τ+(2s+1)Δ)

hμ((k, ω(t)))dt

. (21.13)

238 Integral Manifolds Chapter 3

We set μp−1 = 7pσ2μ. It follows from inequality (21.12) and the definition of the

function hμ(t) that hμ((k, ω(t))) ≡ 0 on the set N(τ +(2s+1)Δ); therefore,

___

∞ _

−∞

Qδdt

___

≤ 2K

∞_

s=0

−2sγΔ2p

k_=0

sup

_ak_ ≤ 2p+1σ1K

1 − e−2γΔμ.

This yields estimate (21.9) with the constant

σ5 = 2p+1Kσ1

1 − e−2γΔ(7pσ2) 1

p−1 .

Let us prove estimate (21.10). Taking into account the definition of the function

v and relation (21.6) for q = 1, we get

___

∞ _

−∞

Qvdt

___

≤ K

∞_

s=−∞

−2|s|γΔ

k_=0

_k_

sup

___

∂ak

∂τ

___

+ sup

_ak_

_ τ+2_(s+1)Δ

τ+2sΔ

gμ(k, t)dt,

where

gμ(k, t) = [1 − hμ((k, ω(t)))]

_ 1

|(k, ω(t))|

___

(k, ω(t))

___

+ c1hμ((k, ω(t)))

___

(k, ω(t))

___

If inequality (21.11) holds for r = 0, then, obviously,

τ+2_(s+1)Δ

τ+2sΔ

gμ(k, t)dt ≤ 4pσ2Δ(1 + 2pσ2(1 + c1)). (21.14)

Assume that inequality (21.11) holds for r ≥ 1. According to the arguments

presented above, the segment [τ +2sΔ, τ +2(s+1)Δ] can be decomposed into

finitely many segments [αj, βj ] on each of which the functions μ − |(k, ω(t))|

and

(k, ω(t)) do not change their signs. If μ − |(k, ω(t))| ≥ 0 ∀t ∈ [αj, βj ],

then

Section 21 Weakening of Conditions in the Theorem on Integral Manifold 239

_βj

αj

gμ(k, t)dt = 0, (21.15)

and if μ − |(k, ω(t))| ≤ 0 for t ∈ [αj, βj ], then

_βj

αj

gμ(k, t)dt ≤ βj − αj

+ (1 + c1)

_βj

αj

___

(k, ω(t))

___

= βj − αj

+ (1 + c1)

___

_βj

αj

(k, ω(t))

___

≤

βj − αj + 2(1 + c1)

_ 1

. (21.16)

Combining inequalities (21.13), (21.14), and (21.16) and equality (21.15), we

obtain estimate (21.10). The lemma is proved.

We now transform system (21.7) using the change of variables z = x(τ)+y,

_y_ ≤ 1

2ρ as follows:

dτ

= H(τ )y + F(y, τ) + δ(x(τ) + y, ϕ, τ, μ)

+ εv(x(τ) + y, ϕ, τ, μ) + εA1(x(τ) + y, ϕ, τ, ε, μ), (21.17)

dϕ

dτ

= ω(τ )

+_b(x(τ) + y, ϕ, τ, ε, μ),

where F(y, τ) = a(x(τ)+y, τ)−a(x(τ ), τ)−H(τ )y and _F_ ≤ 1

2n2σ1_y_2.

We define the integral manifold of Eqs. (21.17) as the limit of the following

iterations as j →∞:

Yj(ψ, τ, ε, μ) =

∞ _

−∞

Q(τ, t)[F(Yj−1, t) + δ(x(t) + Yj−1, ϕt

τ,j, t, μ)

+ εv(x(t) + Yj−1, ϕt

τ,j, t, μ)

+ εA1(x(t) + Yj−1, ϕt

τ,j, t, ε, μ)]dt, j ≥ 0, (21.18)

240 Integral Manifolds Chapter 3

where Y0 ≡ 0, Yj−1 = Yj−1(ϕt

τ,j, t, ε, μ), and ϕt

τ,j = ϕt

τ,j(ψ, ε, μ) is a solution

of the Cauchy problem

ϕt

τ,j = ω(t)

+_b(x(t) + Yj−1, ϕt

τ,j, t, ε, μ), ϕτ

τ,j = ψ ∈ Rm.

Theorem 21.1. If conditions (21.2)–(21.4) are satisfied, then one can find

constants ds, s = 1, 6, independent of ε and μ = μ(ε) and such that, for

sufficiently small ε0, the functions Yj = Yj(ψ, τ, ε, μ(ε)) are 2π-periodic in

each component ψν, ν = 1,m, of the vector ψ, twice continuously differentiable

with respect to ψ and τ, and such that the following inequalities hold for

all (ψ, τ, ε) ∈ Rm × R × (0, ε0] = G1 :

_Yj_ ≤ d1ε

p ,

___

∂Yj

∂ψ

___

≤ d2ε

p ,

ν=1

___

∂2Yj

∂ψ∂ψν

___

≤ d3ε

p , (21.19)

___

∂Yj

∂τ

___

≤ d3ε

−1,

___

∂2Yj

∂ψ∂τ

___

≤ d4ε

−1,

___

∂2Yj

∂τ2

___

≤ d6ε

−2. (21.20)

Proof. Consider iterations (21.18). The fact that the functions Yj are smooth

with respect to ψ and τ and periodic in ψν, ν = 1,m, can be established

by analogy with Section 13. Let us prove the first inequality in (21.19). Using

Lemma 21.1 and estimates (21.8), we get

sup

ψ,τ

_Yj(ψ, τ, ε, μ)_ ≤ σ1

Kn2 sup

ψ,τ

_Yj−1(ψ, τ, ε, μ)_2

+ σ5μ

p−1 +

σ6 +

Kσ4

_ ε

+ σ4σ6

ε2

μ2 .

For ε < μ, this yields

sup

ψ,τ

_Yj_ ≤ σ1

Kn2 sup

ψ,τ

_Yj−1_2+σ5μ

p−1 +

σ6+

Kσ4+σ4σ6

_ ε

. (21.21)

We set μ

p−1 = ε

, i.e., μ = μ(ε) = ε

p−1

p . Taking into account that Y0 ≡ 0 and

using (21.21), for

ε0 ≤ min

__2

Kn2σ1d1

_−p

;

_ ρ

2d1

_p_

d1 = 2

Kσ4 + σ5 + σ4σ6 + σ6

Section 21 Weakening of Conditions in the Theorem on Integral Manifold 241

we get

_Yj(ψ, τ, ε, μ(ε))_ ≤ d1ε

p ∀(ψ, τ, ε) ∈ G1, j ≥ 0.

Note that, for the value of μ = μ(ε) chosen above, the last estimate is the best

order estimate with respect to ε.

By analogy, using Lemmas 12.1 and 12.2, we can establish the last two estimates

in (21.19). Note that, in this case, we essentially use the restrictions imposed

on the Fourier coefficients of the functions _a(x, ϕ, τ ) and b(x, ϕ, τ, ε).

Estimates (21.20) follow from conditions (21.2), (21.3), (21.8), and (21.19) and

the identity

∂Yj

∂τ

+ ∂Yj

∂ψ

_ω(τ )

+_b(x(τ) + Yj−1, ψ, τ, ε, ε

p−1

p )

= H(τ )Yj + F(Yj−1, τ) + δ(x(τ) + Yj−1, ψ, τ, ε

p−1

p )

+ εv(x(τ) + Yj−1, ψ, τ, ε

p−1

p ) + εA1(x(τ) + Yj−1, ψ, τ, ε, ε

p−1

p ),

where Yl = Yl(ψ, τ, ε, ε

p−1

p ) for l = j − 1, j. Theorem 21.1 is proved.

The theorem presented below solves the problem of the existence and smoothness

of the integral manifold of system (21.1). Note that, in its proof, we use Theorem

21.1 and the scheme of the proof of Theorem 14.1. The only difference lies

in the fact that the proof of the convergence of the sequences {Yj} and

_ ∂

∂ψ

is based not on the smallness of the norm of the matrix P, but on properties of

the functions hμ((k, ω(t))). According to Lemma 21.1, the measure of the set of

points of a time interval of length 2Δ for which hμ((k, ω(t))) _= 0 tends to zero

as μ = ε

p−1

p → 0.

Theorem 21.2. Suppose that conditions (21.2)–(21.4) are satisfied. Then, for

sufficiently small ε0 > 0, the following assertions are true:

(i) in the σ1ε

p -neighborhood of the curve x = x(τ ), there exists the integral

manifold x = X(ψ, τ, ε) of system (21.1), where (ψ, τ, ε) ∈ G1,

X(ψ, τ, ε) = x(τ) + Y (ψ, τ, ε) + εu(x(τ) + Y (ψ, τ, ε), ψ, τ, ε

p−1

p ),

Y (ψ, τ, ε) = lim

j→∞

Yj(ψ, τ, ε, ε

p−1

p );

242 Integral Manifolds Chapter 3

(ii) the function X(ψ, τ, ε) is 2π-periodic in ψν, ν = 1,m, continuously

differentiable with respect to ψ and τ, and such that

___

∂X

∂ψ

___

+ ε

___

∂X

∂τ

___

≤ σ2ε

p ∀(ψ, τ, ε) ∈ G1,

and

∂X

∂ψ

and

∂X

∂τ

satisfy the Lipschitz condition:

___

∂X(ψ, τ, ε)

∂ψ

− ∂X(ψ, τ, ε)

∂ψ

___

≤ σ3ε

p _ψ − ψ_ + σ3ε

−1_τ − τ_,

___

∂X(ψ, τ, ε)

∂τ

− ∂X(ψ, τ, ε)

∂τ

___

≤ σ3ε

−1_ψ − ψ_ + σ3ε

−2_τ − τ_;

(iii) on the integral manifold, system (21.1) takes the form

dϕ

dτ

= ω(τ )

+ b(X(ϕ, τ, ε), ϕ, τ, ε).

Here, σ1, σ2, and σ3 are constants independent of ε.

4. INVESTIGATION OF A DYNAMICAL

SYSTEM IN A NEIGHBORHOOD OF A

QUASIPERIODIC TRAJECTORY

Пресс-релиз популярных книг

21. Weakening of Conditions in the Theorem on Integral Manifold

Популярные книги

Популярные статьи

Навигация