Countable Systems of Differential Equations

,

where

Si = —(

3

% -Ν (2ρ°βη^)

7 V7 — av/(u + 1)

v y

^ '

2 J

= const > 0.

72

Chapter 2

Invariant Tori

By using the inequality \\ο(φτ(φ)) - ο(ψτ(φ))\\

<

(2Κ1ο°)ϊ\\φ-φ\\^βχρ{

au 2(ι/+ 1)

we conclude that oo h

<

<

au

- ί^ΙΙ^ΤΤΤ e x p { - (7

J Ν(2Κ\ο°)^\\φ —oo

2(ι/+ 1)

) Iτ I

}dr

S2\\φ-φ\\^,

where

2Ν αζ/

S2 = 7 -

(2c°(2cV)1^T)

2

= const > 0.

2(i/+1)

The required estimate now follows from inequality (8.7): u<

(S1 + S2)h

-

•

The following assertion establishes conditions for the differentiability of the invariant torus with respect to φ : Theorem 8.3. Assume that the system of equations (8.1) satisfies the conditions of Theorem 8.2 and α(φ), Ρ (φ), and ο(φ) belong to C 1 (J-m).

and i=1

Σ t=l

Let the series

dCi((p)

dips

be uniformly convergent for s — 1, m and let

Σ max ψ t=l

dPsj (φ) 9ψί

< ps = const < oo,

= p* < oo. a=l

If, in addition, 7 > a = max max V |M|=1

da(ip) d(p

(8.8)

then the invariant torus of the system of equations (8.1) is differentiable with respect to φ.

Section 8

Existence of a Smooth Invariant Torus

73

The proof of this assertion is similar to the proof of Theorem 7.3 on the differentiability of the Green function with respect to a parameter. We divide equality (8.6) by φτ — ψτ, assume that ψ differs from φ only on the rth coordinates, and pass to the limit as φ τ —> ψ τ . Let us now show that ψν^ψτ

φΓ — φΓ = α0(ι1,

9 φ )

-?ψΜο, αφτ

Λ τ

,

(8.9)

φ )

in the weak sense. The element standing in the pth column and Ith row of the matrix under the limit sign on the left-hand side of this equality has the form f > *

= f > l * («1,
k=l

k=1

ψτ

1=1

(8.10)

ψτ

where G0(ti,
and

Gtl(r^)

=

[gik(tur,ip)]^k=l.

The following inequalities are true: Μ

< f>exp{-

7

! t , I } f j l ^ f f ' ' I « p { « I «11 >

xLiVexp{—7 11\ - τ | } <

^

Σ Σ j = ι i=ι

dPkA
τη oo oo fc=l

j=l fc=l i=l τη oo < j=lk=l

^ LN2J2p* j=1

=

LN2mp*.

They enable us to pass to the limit as φΓ — ψτ —> 0 in the repeated series (8.10) term by term. In this case, L = const < oo is determined from the inequality [MSK2] dipt(
74

Chapter 2

Invariant Tori

Hence, by passing to the limit, we arrive at relation (8.9). Then oo lim

Go(T,y>)-Go(r,y)

ψν-*ψν

φΓ — φΓ

= /

Οο{8,φ)

9Ρ(φ3(φ))

9ψ τ

Gs(t,

φ) ds

whenever the integral on the right-hand side converges uniformly. This condition is obvious. The expression G0(T,y>) -Gq{t,
φ)

determines the function — ^

0φτ

-. For this function, we obtain

oo 9Gq(t,

φ)

9ψτ

<

J N2 exp{—(7 — α) | r | — 71 s — τ \ }ds

<

*βχρ{-(7-α)|τ|},

(8.11)

where Κ = Ν2 ρ* and τ e R1. Let us show that d —

Γ J

ΰ0(τ,φ)ο(φτ{φ))άτ

—oo 00 =

J ^ ÖGQ (Τ, φ) <ψτ{ψ))

9ψ τ

+ Gq{t,

φ)

9ο{ψτ(φ))

0ψ τ

)dr.

For this purpose, it suffices to prove that the integral

—00

dtp·,

is uniformly convergent and d dip,

(Go(τ,φΠφτ(φ)))

χ

=

dGo(τ,φ) Ψ)

ο(φτ(φ))

+

G0(r,
9ο(φτ(φ)) 9ψτ

Section 9

Cl-Differentiability

of the Invariant Torus

75

We represent the Ith coordinate of the vector Go (τ, φ ) ε ( φ τ ( ψ ) ) in the form of a series Y^9oli{T, ip)Ci(
Σ( i=l

dg0ii(r,

ψ)

dipr

£ί(ψτ(φ))

+ g0u(r,

ψ)

θα{φτ(φ))· δφΓ

is uniformly convergent for all / = 1 , 2 , 3 , . . . . This follows from the boundedΘΡ(ω) ness of the derivative „ for i = 1, m and the fact that the series d*Pi

Σΐ<*(ν)ΐ

and

i= 1

Σ zi1

dci{
,

s = 1, m ,

are uniformly convergent. Finally, by using inequalities (8.5) and (8.11) and the conditions of the theorem, we obtain

II /

0^{αθ(τ,φ)ο(φτ(φ)))άτ

<

—σο

Kx 7 — α'

where Κ ι is a constant independent of τ and ψ.

2.9.

Cl -Differentiability of the Invariant Torus

In this section, we present conditions under which the invariant torus of the system of equations (8.1) has continuous derivatives with respect to ψ up to the Ith order inclusive. Parallel with (8.1), we consider the following system of equations: dx

n

= ^2Ρη(ψι{.φ))Χΐη i=1

+ Cs(Vt{
s = l,n.

(9.1)

(n) Let Ω τ { φ ) be the matrizant of the corresponding homogeneous system doc ^ - ^ L = ^psi{(pt((p))xin,

s = T~n.

(9.2)

76

Invariant Tori

Chapter 2

Assume that system (9.1) possesses the invariant torus T ^ · . x(

n

) = υ,(η)(φ),

φεΤτη,

where

x ( n ) = ( x l n , x2n, • • •, xnn)·

We now prove the following assertion: Theorem 9.1. Assume that the system of equations (8.1) satisfies the following conditions: (i) α(φ) e Cg,p ( f

m

) and Ρ (φ), ο(φ) e

oo

(ii)

Σ max \ρ3^(φ) | < Κε(η), j=n+1 Ψ and ε(η) 0 as η oo;

s = 1 , 2 , . . . , where Κ = const < oo

(n)

(iii)

|| Ω°(<^)|| < iVexp{7r}, τ < 0, where Ν and 7 are positive constants independent of η and φ.

Then the system of equations (8.1) possesses the invariant torus T.x — «(<£>), where =u i

JSä,

i( P)i

3 = 1» 2> 3, · · ·,

uniformly in φ (Ξ Tm. Proof. We now show that, in the domain H, the right-hand side of the system of equations dx — = P((pt{· · ·}• For any s ~ 1 , 2 , 3 , . . . , the following inequalities are true: 00

η

oo

i=l

i=l

i=n+1

00

<

Σ i=n+1

\Ρ3χ(φί(φ))\\χχ-χί\ 00

< Δηχ

Σ max i=n+1

\Ρ5ί{ψί{ψ)) I <

ΚΔηχε(η).

Cl-Differentiability

Section 9

of the Invariant

Torus

77

In this case, the following limit exists coordinatewise: lim { Ω ι τ { φ ) = & τ { φ \ η—>oo

(9.3)

where fΫτ{φ) is the matrizant of the system of equations

To show that ||fij(y>)||

< N e x p i ^ r } ,

τ

< 0,

Ν,

7 = const > 0,

we assume the contrary. Then there oo exist τ < 0 and ψ £ jFm such that *

Hence,

3=1

oo Σ ι α;^· (τ, 0,^)1 > i V e x p { 7 r } j=ι

for some i*. But then there exists a number N° such that N° Σ j=ι

l^*j(T>°>v)l

> N e x p { y r } ,

which is impossible because <*>i'j ( τ , 0 , φ)

and

where

= JKn^

N°

N°

j=1

j=1

and

( τ , 0,

φ)

t, ψ) are elements of the matrizants Ω\{φ)

and

Ω \ (φ), respectively: ω) •' ξ 0 for i > η or j > n. By Go (τ, φ) and G ο (τ, φ) we denote the Green functions of the problem of invariant tori for systems (8.1) and (9.1), respectively. Under the conditions of the theorem, they have the form

{

0,

r > 0,

78

Chapter 2

Invariant Tori

and (η) {Τ, ψ) = <

(η) Ω ο (ν?),

Τ < 0,

0,

τ > 0.

(η)

We denote c (φ) = (ci(y>), c2 (φ), ...,οη(φ)}.

Let

(η)

=

{{ΐ£(φτ{φ)),οη+1(φτ(φ))}.

Theorem 2.1 implies the equality Jim η\{φ)ο\φτ(φ))

= #{
(9.4)

where the limit transition is performed coordinatewise. According to Theorem 8.1 and Remark 7.2, the invariant tori of the system of equations (8.1) and truncated system (9.1) can be represented in the form ο Τ·.η{ψ)=

J

η°τ{φ)ο{φτ(φ))άτ,

ψG

—oo and 0

: {ΐ\φ)

= J

(«) η°Τ{7ί{φΤ{φ))άτ,

φ €

—oo respectively. In view of (9.4), to prove the theorem, it remains to show that

lim ί ι—•OO / n-

η°Τ{7ο{φΤ(φ))άτ

=

f

lim ü f c {φτ{φ))άτ.

/ Tt ' oo

(9.5)

—oo

To simplify our calculations, we denote the vector functions under the limit signs on the right-hand and left-hand sides of equality (9.4) by

and

Cl-Differentiability

Section 9

(«)/

V

of the Invariant

V

rW /

79

Torus

(") ,

respectively. In this case, the expression (»),

.

f (n)

(n)

(η)

(n)

(η)

is regarded as the vector { ζ ι(τ,φ),..., ζ η ( τ , φ), 0 , 0 , 0 , . . . } . Let A be an arbitrarily small negative number. According to the theorem on the limit transition under the sign of Lebesgue integral, we have o o ο l i m [J ^ζ\(τ,φ)άτ η—>οο

=

[J η—too l i m ^ζ\(τ,φ)άτ

Denote

=

[J ζΑτ,φ)άτ.

(9.6)

υ F(n,A)

Ι(ζ\(τ,φ)άτ.

=

Since

J ^ζ\(τ,φ)άτ - j

φ) dr

ο

(η) z i(r,

f φ)άτ

<

J

cN cN exp{77-}

dr =

exp{7^},

/

in view of equality (9.6) we conclude that

lim F(n, A) = lim [ ^ζ\(τ, φ) dr — f ζΛτ, φ) d ο

n—foo

n—*oo J

and

ο / A

J

(τι) Ζί(τ,φ)άτ=

ο

Ρ /

Jo -o

Zi(r,ip)dr

uniformly in n. Further, according to the theorem on the permutation of limit transitions, we obtain

80

Invariant Tori lim lim F(n,A)= Π-+ΟΟΛ—>-oo

lim lim A^-oon-*oo

Chapter 2 Fin.A).

which yields relation (9.5). We now consider a subset (!F m ) of the set Cl (Tm) whose elements satisfy the Lipschitz condition in φ together with their derivatives with respect to φ up to the Ith order inclusive. Theorem 9.2. Assume that the system of equations (8.1) satisfies the conditions of Theorem 9.1, the functions α (φ), Ρ{ψ), and ο(φ) belong to the set C l ^ m ) , and inequality (8.8) is true. Then the invariant torus Τ of the system of equations (8.1) belongs to the space Cl{Tm). Proof. All conditions of Theorem 9.1 are satisfied. Hence, the invariant torus T\ χ = u(ip), φ G Tm, of the system of equations (8.1) can be represented as the (n)(n) 71 coordinatewise limit of the sequence of invariant tori Τ ·, χ u (φ), φ G Ττη·, for systems of equations of the form (9.1): (n) lim u (ψ) = η(ψ). η—»oo

(9.7)

Consider a sequence of vector functions du (φ) du (φ) d[u{
ι G l , τη,

(9.8)

and the sequence of their sth coordinates Λ(1)

, χ Ä(2) , , Λη) s = (9.9) du 3(φ) du 3(φ) ο u 8(φ) δψί ' dipi ' " ' ' Θψί If sequence (9.8) is uniformly bounded and equicontinuous in norm, then the same is true for sequence (9.9) for any s = 1 , 2 , . . . . By virtue of equality (9.7), dus(ip) this yields the existence of —_ for any natural s and, hence, the existence of Οψχ du(
Cl-Differentiability

Section 9

of the Invariant Torus

81

In [MSK2], it is shown that, under the conditions of Theorem 9.2, the partial (n) derivative of the matrizant Ω "(y?) of the truncated system of equations (9.2) with respect to φι, i = 1, m, is continuous and satisfies the inequality (n) teR1.

< ATi e x p { - ( 7 - ct) | r | } ,

d
Moreover, the solution ψί{φ) of the first equation in system (8.1) satisfies the inequality
9ψί

where K\ and K2 are positive constants independent of n, r , and ψ. It is clear that if ||c(y>)|| < c and

l^ll < c

dipi

< R, then, for any natural n,

9 c {φ)

and

dtpi

< R.

Similarly, for any n, we have (n) Ω {{φ)

< Ν exp{-7(t - r) }

for

τ < t.

The uniform boundedness of sequence (9.8) follows from the inequality / Μ Λ ο u {φ) 9ψί ο

(η)η

dn°T(
» ο

dr

ο'(φΤ(φ))+η»(φ)

(η)„ W, ΟΩ°τ(φ)(η) β\φ (φ))+η° (φ)Σ τ τ 9ψί

Π υ

< J Äicexp{(7 —

Kxc + N^Rm 7 —a

»

d c (<ρτ(φ)) 9ψτό{φ) }dr

υ

a)r}dr + j NmRK? exp{(7 — a)r}di

82

Invariant Tori

Chapter 2

Prior to proving the equicontinuity of this sequence, we present the following auxiliary statements: Lemma 9.1. The inequality d(

p-|l=

pt Μ 9ψι

fytAtp) δψί where Φ = const > 0 is independent oft, φ, and φ, and u is an arbitrary positive constant, is true under the conditions of Theorem 9.2. <

Φ | | < / ?

-

ψ \ \

ν

+

ι

e

x

p

{

(

a

+

-

ί

^

·

)

| t i

|

} ,

Proof. It is easy to see that the difference X is a solution of the equation

fda\ -— \dipj

where the matrix

and the vector function / have the form fleifoifc)) 9φίι2(φ)

daij^tM) \ d
dam ((ptl (φ))

dam{
( 9α\{φιΑψ)) fytAv) \0φ) dam ((ptl (φ))

d
_

J{ υ

9α

(ψίΛψ))\9φ^(φ) 0φ^{φ) ) 0ψί •

We denote the matrizant of the system of equations (9.10) by U^1 (φ). By using condition (8.8) in Theorem 9.2, we obtain < Mexp{a\ti

- τ | },

r, t\ € i? 1 ,

Μ = const < oo.

In view of the fact that X(0) = 0, we conclude that 0 r\ Since

/

Οψϊ

0 \r>j

' ^ ' j - A δφ,,(φ)

Οφ,3(φ)

>

βφ,

\

£ C y (Fm), i = 1, m, one can find constants

of φ and such that

δα(φ) 9ψί da( φ) Οψϊ

and η independent

< A0 and da(
<η\\φ-φ\\,

i = l,m.

Cl-Differentiability of the Invariant Torus

Section 9

83

This yields d α(φ) 9ψι Vi

δα(φ) 9φ υψίι

< (2Α?ην)

\\ψ ~ ^ll17"*71) v = const

> 0,

and, hence, ti ||X||

<

J M e x p { a | ίχ — s | }τη(2Α°ηι/)τ+τ

β χ

ρ { ~ τ γ

I

s

I }

ο 1/

χ \\φ — φ\\ "+ι Κ<ι e x p { a | s | }

τηΜΚ2(2Α°ην)—ι{ν For Φ = au

+1)

ds.

. , this implies the statement of Lemma 9.1.

Lemma 9.2. The inequality (n)

UGH =

(n)

dipi

difi

< Φο||V — ^11 ^

ex

p{(7 — ö — ~ ~ γ ) τ } >

+

where Φο is a constant independent of η, τ, and φ, and τ < 0, holds under the conditions of Theorem 9.2. Proof. By using the equality A An) d_ 7Α^Λψ)~ ~dt

(η) ν (n) η*Λν)) =

/(n) (n) Ρ{Μ<ρ))(η*Λ<ρ)-η*ΛΦ) /(n)

(η)

+ (Ρ(φ^φ))~

(n) Ρ(φί(φ)))^τ(ψ) N

and the properties of the matrizant established in Section 1.4, we obtain η°τ(φ)-Ω°τ(φ)

=y τ

(

Ω

}

?

χ

( ν » ) - ^ ( V i t ! ? ? ? A l .

We set \\Ρ(φ) - Ρ(φ)\\ < β\\φ - φ\\,

\\Ρ(ψ)\\<Ρ°,

where β and p° are positive constants independent of φ.

(9.11)

84

Chapter 2

Invariant Tori It is clear that (n)

(n)

II Ρ(φίι(φ))-

Ρ(φΗ(φ))\\

< ( 2 ρ ° β ) "+ 1 \\φ -

e x p { ^ |h | },

^ = const > 0 .

This yields (»L (n) Ω°τ(φ)- njeoil

e

xp{(

7

--^)r},

r < 0,

where Φ ι = Ν2(2ρ°β)τϊϊ^1

= C onst. au We assume that φ and φ differ only by their zth coordinates, divide both sides of equality (9.11) by φι — φi, and pass to the limit as ψι —> ψχ. This gives

(n)

(n)

0ψ{

=

(n)

η°Λφ)-ϊΐ°Λφ) φι - φ ί

lim

φί^φί

- °ί ο'ο

ν

τ

A f t W ) W * > ) ' ο ' , , , , ,

ι,

J -1

After obvious transformations, we arrive at the following representation:

τ

J 1

+ r

J=1 ^

dvtute)

d{

Pi

J

We denote the integrals in this equality by J\ and J2, respectively, and assume that

ΘΡ{φ) d
< Po = const < 00.

Section 9

Cl-Differentiability

of the Invariant Torus

85

For Ji, we obtain ο || Jill

< J τ

$ι\\φ-φ\\^πιΡοΚ2Ν

x exp{(7 -

-

Φ2 = *imP0K2N

at

1 α

+ ιΗ-1

i - 7(^1 ~ r ) } d « i

τ < 0.

Similarly, (η)

Μ

-

< }ΙΙΩ^ΜΙΙΙΙΡΚΜ)

ρ^^))!!!!^

1

^)!!^

τ

<

Ν2(2ρ°β)ττϊ\\φ-φ\\ττϊ t χ J e x p { - 7 ( i - ti) - 7(ίι ~

Τ)

~ ^""ϊ*1}^1

τ \\ψ - (^||^ΐβχρ{-7ί + ( 7 -

< = ΦχΙΙν? -

β χ ρ { - 7 ί + (7 -

for τ < t < 0. Let us now estimate the difference C.

For this purpose, we

represent it in the form

_

dP

{φίχ{ψ))

(η) <9Ρ

K M ^ i i M o i i r ^ Ι Ι Λ Ψ fl^-fcO θφι

)

86

Invariant Tori

Chapter 2

(η)

dP

ι

(η) +

dPi- '(ν*Λ<ρ)) /d
)

(dPjiftM) _

ö

+

V 0φ^{φ)

ö

p

0Ψίιύ{φ)

^

n

)

T

^

H

δΨί

We denote the terms on the right-hand side of this equality in the order of their appearance by J3, J4, and J5. In view of Lemma 9.1, for τ < t\ < 0 we obtain ||J 3 || < Ρ0Κ2*ι\\φ-φ\\«^

expj-αί! + (7 -

II J 4 || < Ρ 0 Ν * \ \ φ -

μ5|| < ^Κ Ν\\φ 2

- 7*1},

e x p { - 7 ( i i - Τ) - ( a +

- φ\\τπ exp{(-a -

- 7(*ι - r)},

where Ψ 3 = (2PoC) ^ = const < 00 and ζ > 0 is the Lipschitz constant for 9Ρ{φ) , the matrices — , ζ = l,m. 9ψί By using these representations, we get ||£|| < WM + ΙΙΛΙΙ + ΙΙΛΙΙThis guarantees the validity of the inequality 0 IIΛ II < / i V m e x p { 7 i 1 } ( | | J 3 | | + ||J4|| + lk5||)rfii τ

where mNP0K2Vi

T

mN2P0Φ

are constants independent of η, φ, and r.

T

mN2$3K2

Cl-Differentiability of the Invariant Torus

Section 9

87

Thus, we get IIGII < m

where Φο =

+

+ WM

+

+

which proves Lemma 9.2.

We now prove that sequence (9.8) is equicontinuous. Indeed,

dipi

δψί 0 , ,

A

ο —oo

0 {

J

^τ{φ) ο{ψΛψ))άτ-^

(n)n ΟΩ°τ(φ)(η) c d(fi

J

^Τ(φ){α\φτ(φ))άτ

{φτ{φ))άτ

d


έ ί

—oo

{

J

9Ψί

(n)

0 —oo

(n) J

— OO

—oo

-

»

Η

J—1

δφι 0

9ψτΛΨ)

J —oo

θψί

θψί

ν

0

f dn°T{ψ)(n),

J

c

f

J

.

dQ°T{φ)(n),

- Q j T

c

{ φ Λ φ ) ) ά τ

Invariant Tori

88

+/

r (η)

η , Μ Σ d c (ψτ(φ)) dipTj{
fr{

0 In)

m

/

— OO

0 (n) /

-οο

0

<-I

9φτί{φ)

d
d{c\
m

m

9{θ\φτ(φ))

δψτό{φ) dr dipi

Ξι

ι ι ί Μ Σ

0 (n) /

(n)

m

—oo

+

Chapter 2

π ! ( *

τη J( ηn)) 9 c (<φτ{ψ)) δφτί(φ) dr 0 Σ ^ dipi 3=1

(η) , V

0Ω°Λτ(φ) dipi

I

{ηο{ψ

τ{φ))-

{ηο{ψ

(η) + J II {c\
τ{φ))\\άτ

(9.12)

Μ

9φϊ

dr

(9.13)

0

+ δ{ο\φτ(φ)) 0φτά(φ) +

" m / Σ Ι

-οο i=i

0φτό(φ) dr θψί (n)r

dipTj(ip)

dtpi

n S o ^ . ^ H ^

(9.14)

(9.15)

We now separately estimate each term on the right-hand side of the last inequality. For this purpose, we denote these terms by Jq, J7, and Jg (in the order of their appearance). The required estimates for integrals (9.12), (9.13), and (9.15) are obvious. Indeed, let ||c(ip) - c(
Cl-Differentiability

Section 9

of the Invariant Torus

υ JQ <

J

<

K l

exp{( 7 - a ) T } < W ) * \ \ < p -

e x p { - ^ r } dr

Ζ^φ-φΙ]—!,

where ζ

ι = 7 — α — OLvjxy // +TT1)=

const

> °>

and υ J-j <

/ ο Φ

0

| | ^ | | ^ β χ

Ρ

{ (

7

- α

—oo

where Z2 = ctf 0

77 TT = const > 0 7 — a — a u / { u + 1)

and ζ/ > 0 is chosen from the inequality

Further, Jg <

0 JmRKM\
where Z3 =

77 -τ- = const > 0. 7 — α — a v j y y + 1)

It is clear that the constants Z\, Z2, and Z3 are independent of r , φ, and order of truncation n.

Invariant Tori

90

Chapter 2

It remains to estimate integral (9.14). We have =

d{rc (φτ(ψ)) 0φτύ(φ) 0ψτί{φ) dipi 0{ο\φτ(φ)) 9ψΤ3{ψ) |

Assume that inequality

dc{ip) 9
d{c\
/0φτ,{φ) ^ dip*

Μηο{ψτ{ψ)) V δφτό{φ)

dtp* _

)

δ{ηο{φτό(φ))\δφτ5(φ) d
dc(
3{ο\φτ(φ)) _ 0(c (y>TJ-(y)) d
we obtain P i l l < ( Λ Φ + K 2 ( 2 R S V ) ^ ) \\φ -

β

χρ{-(α +

This yields ο

J 8 < J mN\\Li || e x p { 7 r } dr < Ζ4\\φ -

,

where Z4 =

rnNm+

πιΝΚ2{2ϋδι/)τ+ϊ

,, , 7 - a — a v j i y + 1)

= const > 0

is independent of n, r , and ψ. In view of the estimates for integrals (9.12)-(9.15), we conclude that

0^η(φ) d(fi

d^u(
<Φ\\φ-φ\\f+1

where Φ = Σ Zs is a positive constant independent of η, φ, and r. 3=1

Cl-Differentiability of the Invariant Torus

Section 9

91

We now establish conditions under which the invariant torus Τ of the system of equations (8.1) is I times differentiable. By ϋ^(Φ(φ)) we denote any sth derivative of the function Φ (φ) with respect to φ = ( φ ι < p m ) . The following assertion is true: Theorem 9.3. Assume that the system of equations (8.1) satisfies the conditions of Theorem 9.2 and the functions α(φ), Ρ (φ), and c(ip) belong to {Fm)· If, in addition, 1 δα(ψ) — > a = max max —-—-η I
= η{φ), ψ (Ξ Tm, of system (8.1) belongs to the

Proof. According to [MSK2], under the conditions of the theorem, the func(n) tions ψί{φ) and satisfy the following inequalities (for r < 0): ^>φψτ{ψ)II <

Kexp{-sar},

Κ \\ϋ8φΩ°τ{φ)\\ < Ü T e x p { ( 7 - a s ) r } ,

Κ = const > 0,

»n . Κ \\^ψ(η0τ(φ) - Ω°ΛΨ))\\ Γτ)7"}'

< Κ\\φ - ψ\\ «+ι exp{ ( 7 -

ζ +

- w

I* -

Λ *

s =

°> 1>

1'

« * { - ( « + ^

s = 0,1 — 1, where φ, ψ G T m , and ζ is a positive constant such that 7 >

(ί

+

ΓΤΐ)α'

To prove the theorem, it suffices to show that, for all s = 1, DjftW

D'J8(
...

the sequence (9.17)

is uniformly bounded and equicontinuous. This can be proved by induction.

Chapter 2

Invariant Tori

92

Indeed, let s — 1. In this case, uniform boundedness is established just as in the proof of Theorem 9.2. The property of equicontinuity is also established as in the proof of Theorem 9.2 with the only difference that ο J7<

J Kcexp{(^ —oo

- α

-

-

άτ <Κ3\\φ-φ\\^.

(9.18)

In view of (9.18), we obtain g ί

Ό /η\φ)

-

< £

J, < KA{\\φ -

+ \\ψ -

φ φ ] ,

i=6

where ν > 0 is chosen from the inequality 7 - .- · ^ v + For s = 1, the required assertion is proved Let us now assume that sequence (9.17) is uniformly bounded and equicontinuous for s < / — 1. (n)

Then, for D ' u (φ), we can write υ ϋ1φ{η\φ)= where £(φ,τ)

and s — 0,1,...

J

C(
is the sum of finitely many terms of the form

,1. (n)

In turn, the quantity Dιψ c (φτ(φ)) finitely many terms of the type

m where 1 < s < l, Σ \i = I, i=1 Then (n)

admits a representation in the form of

e {1,2,...

,1}, ki € l,m,

and i = 1 , 2 , . . . ,m.

D^ c (φτ{φ))\\ < K4 exp{ —λχατ - . . . - Λ m a r } = K4 e x p { - i a r }

Cl -Differentiability

Section 9

93

of the Invariant Torus

for r < 0, whence it follows that ο \\ν1φ{ν(φ)\\

< Ks J

e x p { ( 7 — as)r

— {I — s)ar}dr

=

—oo This proves that sequence (9.17) is uniformly bounded for s — I. We have ο 1

{

{

ϋ φ( Ζ\φ)

- Ζ\φ))

= J (£(φ, τ) - £{φ, τ))

dr,

—oo where the integrand is the sum of finitely many terms of the form ϋ*φ(η°τ(φ)

-

Ή°ΛΦ))ΐ%··(ϊ\φΛφ)))

Using expression (9.19), we conclude that Όιφ ( ^ ο \ φ τ ( φ ) ) — ^ ο \ φ τ ( φ a d mits a representation in the form of finitely many differences of the form s z — Dψτ{φ) Zs - Ο

{ c

( ^ )^ f i h M

- dm
^

m

Γ)3 (tyf.n (,n\\ D ^ c i M v ) ) -

-

dXm

Vrkm{v)

. ..

Q f t

• ••

(,Τ,λλλ^Ι^Μ c(M
d A m

··•

^ V r k M ^

where Z(\)

=

9Χι<Ρτ kM

δψ^ =

,0χΐφτ]ΐλ{φ) \

δφ^

dXm

0Χιψτ^(Ψ)

^rkM

' ''

dXm(PTkm($)

0φ Χ ι

dipfr

'''

0^φτ}ΐι(φ)γχ*φτ}ΐ2{φ) >

0χ™φτ}ΐτη{φ)

δφ^

'' '

Invariant Tori

94

+ χ

δφϊAl

Chapter 2

)

δφλλ2 2

0φγ

θΧ>φτΙ<3(φ)

_ _ g^PrfcJp)

ö^3

dipt νψπι

+

^^ΨτΗτη-Λψ) f9XmiPTkm(
...

0λΐ

+

Vrfc! ( ? ) ,λι

dX™
)·

B y using inequalities (9.16), we obtain l ^ l l < Κβ\\φ -

e x p { - (la + J ^ y ) ^ } ·

Consequently,

Further, we get

\\£(φ,τ) - C(
expj^y - sa -

+ e x p { ( 7 — as)r}

- {I ~ s ) a r }

[exp j — (l — s -I

—
+ e x p { - ( i - s + — L - ) dr)\\φ - φφ]

< Κ8[\\φ - ΦΐΙ'ϊ+ϊ exp{(p/+ \\φ -

exp{(y

}

la - la - J^L)r}],

τ

and, hence,

ν1ψ(η{φ)

-

(η) / _

< Κ9{\\φ -

+ \\φ -

<

Cl-Differentiability of the Invariant Torus

Section 9

95

This means that sequence (9.17) is equicontinuous for s = I. Theorem 9.3 is proved. Remark 9.1. Instead of the Lipschitz condition, for the validity of a similar statement, one can require that the Ith derivatives must be continuous with a modulus of continuity μ(σ) such that u

/ d

dc -r-^ —• - o o , μ(σ) u^+o

L

J

'

d> 0.

The method used in this section for the truncation of the original system of equations can be applied to the investigation of the problem of differentiability of the invariant torus with respect to the parameter ρ appearing in this system. Consider a system of equations of the form ^

=

^

= Ρ(φ,ρ)χ

+ α(ψ,ρ),

(9.20)

where χ £ 991, φ G Rm, ρ G [p\,p2\ C R1, and both the vector functions α(φ) and ο(φ, p) and the infinite matrix Ρ(φ, ρ) are 27r-periodic in φι, i — 1, m, and continuous in φ. By ClUp{p) we denote the set of functions and matrices Φ (φ, ρ) bounded in norm and satisfying the Lipschitz condition together with their partial derivatives with respect to ρ up to the Ith order inclusive. Theorem 9.4. Assume that the system of equations (9.20) satisfies the following conditions: (i) α(φ) <= Clip(Tm) (ii)

Σ j=n+1

and Ρ(φ, ρ), ο(φ, ρ) € C°{Tm) Π <^ ρ (ρ);

max \ ρ3ί(φ,ρ) \ < Κ ε (ή), where ε(η) —» 0 as η —» oo uniformly V.P

in ρ and the constant Κ does not depend on p\ (n)

(Hi)

Ω ® ((/?, p) < K\ exp{7r}, τ < 0, where the constants Κ and η are independent of n and p.

Then the invariant torus T.x — η(φ, ρ), φ Ε Tm, of the system of equations (9.20) belongs to the space <7£ίρ(ρ).

96

Invariant Tori

Chapter 2

Proof. By using Theorem 9.1, we represent the invariant torus T \ χ = η(φ, ρ) of the system of equations (9.20) as the weak limit υ(φ,ρ)

=

Jüm.Ju{
where T ^ - . ^ x ^ u {φ, ρ) is the invariant torus truncated to the nth order for the system of equations corresponding to system (9.20). To prove the theorem, it suffices to show that the sequence δ{ιί(φ,ρ) dp

0{η(φ,ρ) dp

(9.21)

dp

is uniformly bounded and equicontinuous. In the analyzed case, equality (9.11) takes the form (n)

(n)

Ω?(<Μ-

η°τ(φ,ρ)

= J

ΡΜ<Ρ),Ρ)\^(ψ,Ρ)<ϋι,

(9.22)

whence we easily obtain the representation

(9.23)

This yields (n)

0Ω°τ(φ,ρ) dp

< K2(—T) exp{7r},

r < 0.

(9.24)

This inequality and the representation d(n) — η(φ,ρ)

=

} j'ΘΩ°{<ρ,ρ)(η), J { — c (ψτ(φ),ρ) dp

(9.25)

Section 9

Cl-Differentiability

of the Invariant

Torus

imply the uniform boundedness of sequence (9.21): ο —

{

u(ip,p)

<

K3

ο ( - τ ) exp{"/r}dT

J

+

K4

exp{*yr}d

J

^(i + i)·

*

Τ

Relation (9.25) implies that 9{ν!(ψ,ρ)

0{η(φ,ρ)

_

dp

dp

ο

_(»)c 2

dp

(n)

(n)

+ {—q-dp 0

0ϊ—)

€ { φ τ { φ )

dp

+ Ωτ0Μ(

'

ρ )

-

) (τι)

By using the conditions of the theorem, we obtain max veFm-,p,pe[pi,P2}

IIIpW) -

II, || ^ ( φ , ρ ) -

{

Ρ(φ,ρ)

Ψ

{φ, ρ)

*·

(n) dP{
(n) dP(
dp

_

(n) dc(
dp

(n) Οο{φ,ρ)

dp < K

6

dp

\ p - p \ ,

where Kq is a positive constant. Moreover, relation (9.22) implies that (η) \\η°

(η) τ

(φ,ρ)-η°

° τ

(^Ρ)\\

<

K

7

j

\p - p

\

exp{7,r}dtx

τ

< K 7 \ p - p \ (-r)exp{7r}.

II

98

Chapter 2

Invariant Tori

Let us estimate the difference (»)„

θη°τ{φ,ρ) dp

(n) n

ΟΩ°τ(φ,ρ) dp

In view of relation (9.23), we can rewrite this difference in the form (n)

(n) dn°T(
θη°τ(φ,ρ)

dp

dp

τ

+ » Μ γ ' θ ψ ϊ <„, ρ) -

«)}

By using the properties of the matrizant, we obtain (η),

(«),

V (η).

= J τ

Λη) ηϊ(φ,ρ)(Ρ(φ8(φ),ρ)-

(η)

Ρ (φ3(φ),

. (η)

ρ)) η°τ(φ,

ρ) ds

and, hence, ||oV (φ, ρ) -

(φ,ρ)\\<κ8\ρ-ρ\

for τ < t\. Further, we get (η)

(η)

0Ω?(ιΜ dp

οη°τ(φ,ρ) dp

ο rt1-'y(t1-T)}

(t\ - τ ) β χ ρ { - 7 ( ί ι -

τ)}

Section 10

The Case of Infinitely Many Angular

Variables

99

+ exp{7*i — 7(
_ d{u(
0

< τ 1

3

—oo

2

where Κi2 is a constant independent of η and p. Theorem 9.4 is proved.

2.10. The Case of Infinitely Many Angular Variables We consider a system of differential equations of the form (10.1) where χ £ SDΐ, φ = (φ\, φ ι , . . . ) £ 9Jt, and the components of the vector functions α(φ) and ο(φ) and the infinite matrix Ρ (φ) are functions 27r-periodic in φ^ ζ = 1 , 2 , . . . , and continuous in φ. Assume that oo y : s u p \paj(
s = 1,2,... .

(10.2)

*

Then the matrix Ρ(φ) is a linear operator in the space SOT of bounded numerical sequences mapping a vector χ eVJl into a vector Ρ (φ), x G We say that a function f ( x ) satisfies the strengthened Cauchy-Lipschitz conditions with respect to φ with a coefficient e(m) if /(Vl) · · -,ψτη, • • • ) < e(m)sup{|
\m+2|, · · ·}>

100

Invariant Tori

Chapter 2

where ε(τη) —> 0 as m —» oo and (Vl,...,^mVm+l»Vm+2»···)

and

• · · > Ψτη, Vm+b Vm+2, • · •)

are arbitrary points from VJl whose first coordinates coincide. The set of bounded vector functions and matrices whose components are 2πperiodic in φι, i = 1 , 2 , . . . , and continuous in φ = {ψ\,ψ2, · · ·) is denoted by C°(^oo). The set of elements from C°(!F,oo) satisfying the Lipschitz condition with respect to φ is denoted by Here, Too is a countably dimensional torus. In the set ClipO^»)' w e s e ^ e c t a subset of elements satisfying the strengthened Cauchy-Lipschitz conditions with respect to ψ and denote this subset by ^ip(^oo). Let α(φ) e ^Χφί-^οο)· According to Theorem 1.3, the first equation in system (10.1) possesses a unique solution φ =
Therefore, the matrizant Ω* (v?) of equation (10.3) exists. We define the Green function of the problem of invariant torus for the system of equations (10.1) by setting η0τ(φ)Ο(φτ(φ)),

τ < 0, (10.4)

and assuming that the matrix 0{ψ) belongs to C°(JF m ) η Οφ and is such that ||σ0(τ,^)||<ίΤβχρ{-7|τ|},

tgR1.

(10.5)

Then the invariant torus T'.x — u[}p), φ € Too, of the system of equations (10.1) is specified by the function oo (10.6) —oo Theorem 10.1. Assume that the system of equations (10.1) satisfies the following conditions:

Section 10

The Case of Infinitely Many Angular Variables

(i) α(φ), Ρ(φ),ο(φ)

101

6 C f t ^ ) and ||α(φ) - α(0)|| < α\\φ - φ\\·,

(ii) inequality (10.2) is true; (iii) the Green function of the problem of invariant torus GQ(T, φ) exists and satisfies inequality (10.5). Then the system of equations (10.1) possesses the invariant torus Τ admitting representation (10.6) and such that \\η(φ) - η(φ)\\ < \\φ -

φ \ \ ,

where ν > 0 satisfies the condition 7 -

α

>

ν ν+ 1

and Κ ι is a positive constant. The proof of Theorem 10.1 is similar to the proof of Theorem 8.2. Denote (m)

Ψ = (
¥>< (( · • •. Ψτη, 0, 0, . . . ) , φ2{ί, φι, . . . , ψτη, 0, ο , . . .), . . . }, ΨΛφ) = {φι(ι,φ),·

•• ,φτηψ,φ),ο,ο,

·•·},

Tßt = { 0 , 0 , . . . , 0,
}.

Let us now study the possibility of reducing the problem of invariant torus for the system of equations (10.1) to the finite-dimensional case. First, we prove some auxiliary statements. Parallel with (10.1), we consider an equation of the form § = P M ( £ > ) ) * + cM<£>)),

(.0.7)

where is the solution of the first equation in (10.1) such that φο = (m.) φ . By Ω' ( φ J we denote the matrizant of the homogeneous system corresponding to (10.7).

Lemma 10.1. Assume that the system of equations (10.7) satisfies the following conditions:

102

Invariant Tori

Chapter 2

(i) α (φ), Ρ (φ) Ε Cy p(Foo) with coefficients a(m) and e(m), respectively; (ii) inequality (10.2) is satisfied; (Hi) Ω®( a(0). Then lim ^r ( V ) — ^τ(ψ) τη—too τ e r = (-00,0].

=

0 uniformly with respect to φ G Too and

Proof. First, we estimate the following quantity:

For t > 0, we have t _ v

f(a(
(t) =

-a^s^*?)))^!

t < J

α(φ3(φ))

-

ds

ι

< J

-

α(ψ8(ψ)

0 + 1|'0 {ψs (Ψ + Ψ*

) - α (<Pa

ο According to the Gronwall-Bellman lemma, \\
< βχρ{α(0)|ί|}Δ πιψj

where t e R1 and Am

) 11}

Section 10

The Case of Infinitely Many Angular

103

Variables

Hence, for V(t), we obtain t

V{t) <

exp {a(0)t}

+ J a ( 0 ) F ( s ) ds,

ο where, without loss of generality, we can set In this case, inequality (10.8) takes the form

t > 0,

(10.8)

< 2π.

t

V(t) <

exp { a ( 0 ) t j + J a ( 0 ) V ( s ) ds,

t > 0.

ο We solve this inequality to obtain V(t) <

+ α(0)ί] exp {α(0)ί},

α(0)

t > 0.

Similarly, for t < 0, we conclude that y ( t )

"

2 7 Γ

^

[ 1

+

α(0)|ί|] β Χ Ρ { α (

°)ί}'

t e R l

(10 9)

'

'

Then I p ^ C ^ - P ^ ^ ) ) !

+

||p(^)+pt((5)))-p(^((5>))||

< ε(τη)βχρ{-α(0)ί}2π + ε ( 0 ) | | ^ Μ < 2π{ε(τη) +

-

- ε(0)α(τη)ί} exp { - a ( 0 ) t }

at

t < 0.

In this case, for τ < 0 and ψ € Too, we can write

ο < j

N2 exp { j s — ^(s — τ)}

τ

χ 2Tr{e(m) + Ä p )

-

e(0)a(m)i}

exp {—a(0)s}ds

104

Invariant Tori

<

1

2τrJV2

Chapter 2

f ε(0)α(πι)

,S(öj(-W

. A

J + e ( m

V

,.

e x p { ( 7

, -

a ( 0 ) ) T }

υ + e(0)o;(m) βχρ{7τ} J exp {—q(0)s}(—s)d& τ <

2τχ Ν2 Γ —r— ε(0)α(τη) + ε(τη) + ε(0)α(τη) βχρ{7τ} α(0) L χ ((-τ)βχρ{-α(0)τ} +

<

2πΝ2 Γ /Μ α(0)

=

η(τη) —> 0

. ./

^ )

1

1

\

,

.

as 77ΐ —> οο uniformly in φ G ^ , ο and τ £ By u(
ψ ) we denote functions of the form (10.6) specifying the

invariant tori of the systems of equations (10.1) and (10.7), respectively. Theorem 10.2. Assume that the conditions of Lemma 10.1 are satisfied and ) 6 Cgip(^oo). Then = o

lim

m—KX>

uniformly in φ (Ξ TooProof. The estimates Ι ω ο ^ - Ω ? ^ ) ! ^ ^ ) , |ω°(*0 -

< 2iVexp{ 7 T},

|ω?(*0 -

< (2Νη(πι)γ

r e R~,

yield expj^r}·

(10.10)

We denote a constant for which c(y?) e 6*° ίρ (^ Γ 00 ) by β im). Further, in view of inequality (10.9), we obtain

Section 10

The Case of Infinitely Many Angular

Variables

< ß{m) exp { - α ( 0 ) ί } Δ 7 η ν ? + 0 ( 0 ) || φ^φ) -

ψ | |

< 2ttlß{m) + I

- ß(0)a(m)t\

"(Ο)

exp { - a ( 0 ) t } ,

J

105

t < 0.

Finally, by using this inequality and inequality (10.10) and setting ||c(
+ —oo 0 <

J

Nex-p{^ys}2n

—oo x {ß(m)

+

/?(Q

J ^ m ) - 0 ( O ) a ( m ) e } exp { - q ( 0 ) s } d s

0

+

J

! ο°(2Νη{πι)Υ

exV{^}ds

—oo < ~

2π Ν

\ß(m) 7-a(0)rv 2πΝ

+

1

+ ß(0 Wm)( * + \ 'Va(O)

7-a(0)/J

—(2Νη(τη)Υ2=ζ(τη),

where C ( m ) —> 0 as m —• oo uniformly in ψ e Too. Theorem 10.2 is proved. Parallel with equation (10.7), we consider the equation /M

_

/Kh

(10.11)

Invariant Tori

106

Chapter 2

By Ω ' (^ρ^ we denote the matrizant of the equation

(10.11) A function of the form (10.6) specifying the invariant torus of equation (10.11) is denoted by

φ

Theorem 10.3. Assume that the conditions of Theorem 10.2 are satisfied and, moreover, |^((^)|
r < 0,

m = l,2,3,...,

where Ν is a positive constant. Then lim l l w f V ^ — η(φ) v ' τη—»ooll V /

= 0

uniformly in ψ Ε Τ^· Proof. In view of Theorem 10.2, it suffices to show that

By setting ||a(y>)|| < a 0 = const < oo, for φ e Too and t < 0 we obtain IMvOII < 2π - oPt.

(10.12)

Then

f P ^ ) )

- P^C^))!

<

< s(m)(2*

whence it follows that

υ < J Ν2 βχρ{7τ}ε(τη)(2π ί α°τ21 = ε(πι)Ν2 | 2 f f ( - r ) + -γ-

a°s)di

|exp{7r}.

-

Λ),

The Case of Infinitely Many Angular Variables

Section 10

107

By using estimate (10.12), we get ο u

—

—oo

0 <

J Ν exp{7s}β(τη) (2π —oo + J £(m)N2c0^2ir(-s) —oo

-a°s)ds

+

^-Jexp{7s}ds

where C ° ( m ) —> 0 as m —> oo. Theorem 10.3 is proved. Consider the equation — = pi
))x

+ cl(pt

[φ ) J ,

(10.13)

is the solution of the finite-dimensional system of equations (10.14) (m) /(m)\

(m)

such that φο ( φ ) = φ . The matrizant of the equation dx

is denoted by Ω^.

/(τη)

/(rn)\\

Mr

, and a function of the form (10.6) specifying the invariant

torus of equation (10.13) is denoted by u ( ^ ) · Theorem 10.4. Assume that the conditions of Theorem 10.3 are satisfied and < iVexp{7r},

r < 0,

m = 1,2,....

108

Invariant Tori

Chapter 2

Then mlimJ§(

φ ) - η(φ) = 0

uniformly in φ £ Proof. It suffices to show that (m)>

l m j g ^ ) - ^ ) and use Theorem 10.3. For t < 0, we have V m

=

\Ψι(ψ)

-

(τη) /(τη) ΨΑΨ) ~

< j ja(tps(
+ J t

0

-

( ψ ))\\ds

0

< I α(τη)\\φ.(φ)\\ά3 + J a ( 0 ) | | ? » t t

®

By using estimate (10.12), we obtain .0 V°(t) < a(m) (-2ττί + y t 2 ) + J a(0) t

ds,

t < 0.

We now solve this inequality with respect to V°(t). This gives V°(t)

<

V?(t) α(πι) ct(m) r Λ αίΟ) i(Ö)" I —

n

α " αα(ί (0)

2ττ-

α,ο exp {—α(0)ί}|, a(0)J

where V{°(ί) = 0 for t = 0 and Vi0(t) > 0 for t < 0. In this case, ΐ(τη)\ (τη) (m) /(τη) Ι /(πι)\ /(τηι \
( m )

II + Ι ψι{ψ) -

/(m)> ( y )

ί<0,

Section 10

The Case of Infinitely Many Angular

<

a(m)

2π(1 - α(0)ί) exp { - α ( 0 ) ί } + a°t

a(0)

+

109

Variables

(27r+^y)exp{_a{0)i}

,

t<

0.

(10.15)

Then φ ) ) - c(
Jρ

(m)

^ afrwW

< 0(0)

Ψ ) - φι {

ψ)

< a(m) [ P i ( - t ) exp { - a ( 0 ) t } + P2t + P 3 exp { - α ( 0 ) ί } ] , where M

(

o ) ,

»

-

Ά t(0)

'

and

Λ

α(0)ν

•>)

a(o).

are positive constants. Since t < 0, we have IK^C^)) " K ^ C ^ ) ) ! <«(H[-At +

ft]exp{-a(0)t}.

Further, we get

< J

<

( φ ) ds

A r 2 g

^

( m )

ß(ß)

exp{7T} J [-pis

+

p3}exp{-a(0)s}ds

Ν2ε( 0) <

ot(m)

βχρ{7τ}

ß(0) υ

χ J [Pi(-s)

exp { - a ( 0 ) s } + P 3 exp { - a ( 0 ) s } }

110

Invariant Tori

+

e x p { 7 T }

^ )

Chapter 2

} ·

This yields the following estimates:

Η

φ

uä / M φ

)~ \ )

< /

llnsC^llllc^^^-cfÄf?»))!^

+

ds —oo

0 < j iVexp{7s}a(m) —oo χ { P i ( - s ) exp { - a ( 0 ) s } + P 3 exp { - a ( 0 ) s } } d s , ,c°N2e( v 0) + / a(m)' m ?

Pi -

<

W ) )

e

x

p

^

(

7

~

a

(

0

)

)

s

}

+

e

x

p

{

7

s

}

α(τη)κ,

where κ = const > 0 and a ( m ) —> 0 as m —> oo. Theorem 10.4 is proved. Let us now present the principal result of this section. Parallel with the system of equations (10.1), we consider a finite-dimensional system of the form (n)

dTp

dx <£),_v(») . (n) /— ~jj;= Ρ(φ)χ + ο{φ),

(10.16)

Section 10

The Case of Infinitely Many Angular

111

Variables

obtained by truncation with respect to ψ up to the mth order and with respect to χ up to the nth order inclusive. The invariant torus of this system of equations is denoted by T: h —

^φ G Tm, and the matrizant of the equation

d{X

1

Γ

=

(j/(m)/(m)u(n)

,ιηπΛ

Ρ {
(10.17)

is denoted by Ω* . Here, ^p} {^ψ^ is the solution of the first equation in system (10.16) such that (m) φο l/(m)\ φ ) = (m) φ , (n) χ = (.χ ι , χ 2 , . . . , x n ).» (n) c = (η) (ci, c 2 , . . . , c n ) , and Ρ {ψ) = [Pij()]ij=i· Theorem 10.5. Assume that the system of equations (10.1) satisfies the following

conditions:

(i) α(φ),

(ii)

constant

a(m)·,

oo Σ j=n+1

IbsjCvOH < Κδ(η),

δ(ή) (in)

Ρ {φ), ο{ψ) G ^ ί ρ ( - ^ ο ο ) ' for

SUP

φ

function

α(φ),

this is true with a

s — 1 , 2 , . . . , where Κ = const < oo and

—> 0 as η —> oo;

the matrizant of the system of equations (10.17) satisfies the inequality |^((^)η|<ΛΓβχρ{7ί},

τ < 0,

where Ν and 7 are positive constants independent of n, m, and φ\ (iv) 7 > a ( 0 ) . Then the invariant torus T'.x

= η(φ),

ψ G !F0o, of system (10.1) can be

represented in the form of repeated limit as follows: ιι(φ)

=

lim lim τη—>οο τι—>οο

\

/

(10.18)

where the outer limit is understood in the strong sense, the inner limit is understood in the weak sense, and, moreover, η(φ)

G

C^^Too)·

112

Chapter 2

Invariant Tori

Proof. It is easy to see that Theorem 9.1 is true under the conditions of Theorem 10.5. Therefore, the function uy φ 1 can be represented in the form of the following coordinatewise limit:

It remains to use Theorem 10.4 and complete the proof of equality (10.18). For this purpose, it suffices to show that ||ω?((^)| <

linJfoOH < iVexp{ 7 r}, | Ω ° Τ ( ( ^ ) | < Ν exρ { 7 τ } ,

Νβχρ{Ίτ},

< iVexP{7r},

where τ < 0. We denote these inequalities in the order of appearance by O3, and O4. According to Theorem 9.1, the limit

0\,02,

exists in the weak sense (see 9.3) and, hence, the inequality O4 is true. We set po = Κδ(0), i.e., ||P(y>)|| < P°- BY u s i n S inequality (10.15), for r < 0 we can write < τ 0

< J exp{-p0s}e(0)|^t((!?) Τ

(

||iVexp { - 7 ( s - r)}ds

0 <

/ exp{-p°s Τ

7s

+ 7 r } [ P i ( - s ) + P 3 ] exp { - a ( 0 ) s } d s

< a(m)/(r), where / ( τ ) > 0 for τ < 0 is a continuous function and a(m) —• 0 as m —» 00. This estimate implies that

Section 10

T h e

C a s e

o f

I n f i n i t e l y M a n y

K U )

^

A n g u l a r

K

113

V a r i a b l e s

M

for any τ < 0, and, hence, the inequality O3 is satisfied. The inequalities O2 and 0\ are proved by analogy. Equality (10.18) is proved. Further, everywhere in the proofs of Lemma 10.1 and Theorem 10.2, instead (m)

of φ = ( φ ι , . . . , φ Τ η , 0 , 0 , . . . ) , w e s e t Φ

=

(
• • • , φ τ η , φ ' τ η + 1 > φ ' π ι + 2 >

• •

•)•

As a result, we arrive at inequality (10.9) with max{|<^ m+ i —
=

II φ — φ II instead of 2π. This yields INv) -

u

< CMIIv-^ll,

( v ) \ \

where ζ{τη) is a positive constant such that ζ(τη) —> 0 as m —> 00. This means thatu^eC&pC^oo). Finally, we note that the proof of Theorem 9.1 remains valid in the case where (n)

φ G 9H and Theorem 4.10 is true for χ £ Rn, i.e., χ = commutativity of the limit.

χ . This yields the

R e m a r k 1 0 . 1 . In the case where α { ψ ) = w = (w\,W2,...) € 971, Theorem 10.5 gives a representation of the invariant torus for a system of equations with almost periodic coefficients (the case of infinite frequency basis).

At the end of the section, we consider a system of equations of the form (10.1) dependent on a parameter p, namely, d ( p

d x

~ ^ = α { φ , ρ ) ,

—

=

Ρ { φ , ρ )

+

(10.19)

€ ( φ , ρ ) ,

where ρ G [pi,p2] C Rl is a positive parameter and φ, χ G ÜJI. By w(z) we denote a continuous scalar function nondecreasing on the segment [0,p2 — pi] and such that w(0) = 0.

Theorem 10.6. A l o w i n g

c o n d i t i o n s f o r

00 ( i )

Σ j = l

sup
\ p

s j

s s u m e

t h a t

e v e r y

{ i p , p ) \

<

G

ρ

p

°

t h e [ p i ,

=

s y s t e m

o f

e q u a t i o n s

( 1 0 . 1 9 )

P2]·'

const < 0 0 ,

s

=

1 , 2 , . . .

;

s a t i s f i e s

t h e

f o l -

114

Invariant Tori

Chapter 2

(ii) the inequality χη^{\\α(φ,ρ)-α(φ',ρ')\\,

\\Ρ{φ,ρ)-Ρ{φ',ρ')\\,

\\ο(φ,ρ)-ο(φ',ρ'

< αι\\φ - φ'\\ + α2ω(\ρ - ρ'\), where αϊ and [pi,p2];

are positive constants, holds for all φ, φ' € VJl and ρ' e

(iii) the matrizant of the equation

satisfies the inequality ||Ω;(^,ρ)|| < i V e x p { 7 , r } ,

r < 0,

where Ν and 7 are positive constants independent of φ, ρ, and τ. Then the invariant torus T\ x = u(ip,p), φ e Too, of system (10.19) is continuous with respect to the collection of variables φ and p. Proof. In the analyzed case, we have the following estimate similar to (7.18): f t (Φ) ~ Vt (ψ') <

\\φ-φ'\\

+

—ω{\ν-ρ'\) exp{—<*ii},

By using this inequality, we obtain

< (2ρ°)^ι

[αι\\ψ

2α 2 ω(\ρ - p'\)] ^

exp

< (2c°)^T [ a i \\φ - φ'\\ + 2α2ω(\ρ - p'\)] ^

exp

where u is an arbitrary positive real number. Then

t < 0.

Section 11

Theorem on Convergence of the Sequence of Invariant Tori

Ω»τ(φ,ρ) -

η°τ(φ',ρ') ΛΓ2

<

[αχ\\φ - ψ'\\ + 2α2ω(\ρ - ρ'\) Χβ

Χ Ρ

{ (

7

" ^ )

Τ

~

1 «Η-1

} ·

αχ -j-y > 0, then we get

If we now choose ν > 0 from the inequality 7 Μφ,ρ)

115

η(φ',ρ')\\

ο < J

<

ΙΩ°Λφ,ρ)ο(φΡ(φ),ρ)-Ω°Λφ\ρ')ο(φϊ(φ'),Ρ') Ν

•{οη\\φ-φ'\\ 7 - <*ι/(ι/ + 1) Ι_1 + Χ [(2c°)i.,_•+1

+

dr

2α0ω(\ρ-ρ'\)Υ

1 +1

/Ο«"* „±1 αχ/{ν+1)

which proves Theorem 10.6.

2.11. Theorem on Convergence of the Sequence of Invariant Tori Parallel with the system of equations (8.1), we consider a countable sequence of systems ^=α(φ),

%=Ρ«Ήφ)χ

+ οΜ{φ),

k = 1,2,...,

(11.1)

whose right-hand sides satisfy conditions (i)-(iii) of Theorem 8.1 for any natural k. We now present sufficient conditions under which the sequence of invariant tori of systems (11.1) is convergent and determines the invariant torus of the original system (8.1). We assume that each system in (11.1) has the Green function

[0^(φτ(φ))

(*),

- Ε],

τ > 0,

where Ω τ (φ) is the matrizant of the homogeneous system corresponding to (11.1).

116

Invariant Tori

Chapter 2

We set (k) and denote the invariant torus of system (11.1) by T^-.x = u(k\ip), ψ £ The answer to the question posed is given by the following theorem: Theorem 11.1. Assume that the following conditions are satisfied: (i) ||Go^(r, 0 and 7 > 0 are independent ofk; (ii) in the weak sense, (fc)

Ω°τ(φ) am/

η°τ(φ),

£ C0{Tm) Π

C^{
c^(tp) —> c(tp) as

k —• 00,

where Ω* ( are jointly bounded and the series Σ i=i

Σ Κ ν , * ) j=i

,

< = 1,2,...,

converge uniformly in k. Then = k—*oo ,lim uniformly in φ, and T\ x =

JΜ

'

ψ £ -^m. "

j = 1,2,..., invariant torus of system (8.1).

Proof. If the inequalities ||i2?fo>) C(y?T(v?))|| < ΛΓβχρ{ 7 τ},

τ < 0,

||Ω?( ν ) [0(φ τ (φ)) - Ε]II < Νexp{—7τ},

τ > 0,

are satisfied, then it suffices to use the matrix (Ω0τ(φ)Ο(φτ(φ)), Go {τ, φ) = < ft W(
τ < 0, T> 0,

(11.2)

Section 11

117

Theorem on Convergence of the Sequence of Invariant Tori

as the Green function of the problem of invariant torus for the system of equations (8.1). Consider the case τ < 0. Let

||C(fc)(^)||
||c^(v?)|| < c°,

We now write the equality

oo = lim k—too —: s=1

lim 9^(τ,φ) k—>oo J

e gJ V ( < ? ) ) ,

The series Σ ( r i ψ) ^{ψΛψ)) J s=i are majorized by the series

h3 = 1 , 2 , . . . .

converge uniformly in k because they

oo E K 3=1

f c

W ) | C b ,

i = 1,2,...,

which converge uniformly in k. Then oo

^ φλ) = Σ s=l

fc!™

(r> ^ CS}

= öij (τ,

,

= 1,2,

This means that, for τ < 0, the function Go (τ, φ) is the coordinatewise limit of the sequence GQk\r, 0 is analyzed by analogy. It is easy to see that lim G(0k)(r,
=

Ο0{τ,φ)ο{φτ(φ))

in the weak sense. In view of the integral representation of the invariant torus, it remains to prove the equality oo lim f k:—KX J

G(0k)(T,))d7

— OO

LXJ I

fclim{G

{k\r,
(11.3)

118

Invariant

Chapter 2

Tori

Denote 0

{

* \ τ , φ ) ο Μ (

( φ ) ) =

Ψ τ

ζ Μ ( τ , φ ) =

G o ( r , < p ) c(ipr(
ζ ( τ , φ ) =

{ z [

{ ζ

λ

k )

( r , φ ) ,

( τ , φ ) , ζ

2

ζ ^ ( τ , φ ) , . . . } ,

( τ , φ ) , . . . } ,

where A is an arbitrarily large positive number. By analogy with the proof of Theorem 9.1, we arrive at the following relation valid for all i = 1 , 2 , . . . : A lim κ—too

/ J - A

A ζ ^ \ τ , φ ) ά τ

Denote

—

A

/ J - A

z\k]>(r,φ)άτ fc—»oo

lim

=

/ J - A

ζ

{

( τ , φ ) ( 1 τ .

Λ J

z \

k

\ r , φ ) ά τ

=

F ( k ,

A).

- A

Then

A lim F ( k , A ) fc—>oo

=

/ J - A

Zi(r, φ ) ά τ ,

00 lim

4—too

F ( k , A )

=

ί

ζ ^ ( τ , φ ) ά τ ,

J

where the last limit transition is uniform with respect to k because oo

J

A A

z?\τ,φ)(1τ

-oo

-

J

ζ I ^ \ \τ ,/ φι )/ ά ι

—A oo

-A

<

JI

ζ 1^ ( τ\ , 'φ ' ) /ά τ \ +

oo

\ Ι

ζ ^ ( τ , φ ) ώ

A

—A oo ; J c°Ν e x p { 7 r } d r + J c°Ν exp{-7r}(Zr

—oo <

2

A c°N e x p { ~ 7 A } <

ε

7 1 £7 for all k = 1 , 2 , . . . whenever A > — In ^ n __. 7 2c°iV

Section 12

Invariant Tori of Nonlinear Systems

119

In this case, according to the theorem on changing the order of limit transitions, we obtain lim lim F(k, A) = lim lim F(k,A), k—KX> A—*oo A—>oo k—too which yields the required relation (11.3). Remark 11.1. In the proof of Theorem 11.1, the order of system (11.1) is not used. Therefore, each system in (11.1) can be regarded as finite-dimensional and obtained by using one of the procedures of truncation of system (8.1). It is clear that the validity of the conditions of Theorem 11.1 strongly depends on the choice of this procedure. This can be seen, in particular, if we consider, as an example, a block-diagonal system with Ρ(φ) =

άι^{Ρι(φ),Ρ2(φ),...},

for which it is natural to perform truncation without breaking the blocks.

2.12. Invariant Tori of Nonlinear Systems We consider a nonlinear system of differential equations ^=α(φ,Η,μ),

^

= Ρ(φ, h, μ)Η + β(φ, μ),

(12.1)

where ψ = (φι,..., φ„ι) € Rm, h = (hi,h2,...) € Μ, α(φ,Η,μ) and ο(φ,μ) are bounded vector functions 27r-periodic in φι, i = 1, m, Ρ(φ, h, μ) is an infinite matrix 27r-periodic in φι, μ G (0, μο] is a positive parameter, and ο(φ, 0) ξ 0. We assume that, in a domain D, IM < d, α(φ,Η,μ), Ρ(φ,}χ,μ), inequalities are true:

ο{φ,μ)

μ £ (Ο,μο], € C^Tm)

for h G Cl(Tm),

and the following

I\α(φ,Η,μ) - α{φ~Η,μ)\\ < σ{\\φ - φ\\ + ||/ι - Λ||),

(12.2)

\\Ρ{ψ,Η,μ) - Ρ{ψ,Η,μ)\\<σ\\Κ-ϊι\\, σ = const > 0, οο 5ZSUP\ΡίΑφ,Κμ) I < = const, » = 1 , 2 , . . . , 3=1

(12.3) (12.4)

Invariant Tori

120

Chapter 2

where the elements ρψ(φ, Ιι,μ) ,i,j = 1 , 2 , . . . , of the matrix Ρ are continuous with respect to ψ. The set T\h — η(φ,μ) = {η\(φ, μ), u2((p, μ),...}, φ G Tm, is called the invariant torus of the system of equations (12.1) if there exists μ G (ο, μο] such that, for any μ G (Ο,μ] and η(φ,μ) G C 0 ^ ™ ) , the functions ^ ( ^ , μ ) , i = 1 , 2 , . . . , are continuously different!able with respect to t and = Ρ ( φ ί , υ ( φ ί ι μ ) , μ ) η ( φ ί , μ ) + ο(φ^μ)

(12.5)

for all t G R1 and ψ G Tm, where ipt = ψί{ψ, μ) is the solution of the equation ^

(12.6)

= α(φ,η(φ,μ),μ)

satisfying the condition φο(φ, μ) = φ G TmWe study the problem of the existence of the invariant torus for system (12.1). To do this, we use an iterative process linearizing the problem of finding the invariant torus in each step. The procedure of linearization reduces to the construction of a sequence of invariant tori ... ... (12.7) in which h = 0, ψ G Tm, and the (i + l)th term is determined as the +1 invariant torus !Ρ (μ): h = μ), φ G !Fm, of the system of equations ^

= α(ν?,ω ( ί ) (ν?,μ),μ),

^=Ρ(φ,η®(φ,μ),μ)Η

+ ο(φ,μ).

(12.8)

If the function α(φ,η^\φ, μ), μ) satisfies the Lipschitz condition with respect to φ, then there exists a unique solution φ\ — φ\(ψ, μ) of the first equation in (12.8) satisfying the condition ψτ0 = φ G Tm. According to the results of Chapter 1, the homogeneous system corresponding to the system of equations ^

= Ρ(φί η®{φΙμ),μ)Η

+ cfä, μ)

(12.9)

(0 possesses a matrizant. It is denoted by Ω , ^ φ , μ ) . Assume that system (12.8) has the Green function of the problem of invariant torus Gq \τ,φ,μ)

for any

Section 12

Invariant Tori of Nonlinear

μ G (0, μο]· Then the torus

121

Systems

exists and is given by the function oo

i< r (i+1)=

(12.10)

/ Ο%\τ,φ,μ)ο(φ\,μ)άτ.

The theorem on the existence of the Green function for each system in (12.9) with c = 0 guarantees the possibility of infinite continuation of the iterative process. We now establish conditions for the convergence of this process. First, we prove two auxiliary statements. oo

Lemma 12.1 Assume that the series Σ u\ (0 converges uniformly in t and i=1 k k to a function f^ \t). I f , in addition, lim u\k\t) = Ui(t), i = 1 , 2 , 3 , . . . , fc—• oo

uniformly in t, then the series Σ Ui(t) and the sequence {f^k\t)}

also converge

t=l

uniformly in t and, moreover, 5 > ( t ) = lim /<*>(*) —: fc—»oo i=l Proof. According to the condition of the lemma, for any ε > 0 there exists a n+jTO /L\ number Ν such that Σ u\ '{t) < ε for all natural k and m and all t, provided that η > N . Passing to the limit as k - * oo, we conclude that

n+m

Σ ui(t) < ε for i—n

oo

all t, i.e., the series Σ ui(t) converges uniformly in t. i=l

The following inequality is true: oo

»=1 <

|Σ«5 t=l

w - E ^ w i=l

+1 Σ

^ r w

M t ) . (12.11)

+| Σ

i=n+l

i=n+l

Let ε > 0 be an arbitrarily small number. Then there exists a number N° such that, for η > N° and all t and k, we have I

Σ

i=n+l

< ε~ 3

and

I Σ ί=η+1

(as remainders of series convergent uniformly in t and k).

< ε -

-

3

122

£

Invariant Tori At the same time, u\k\t)

Chapter 2

—»· Ui(t) as k —> oo uniformly in t. Hence, for

— > 0 , there exists a number Ni such that, for all k > Ni and all t, we have 3η ulk) (t) -Ui(t) < Then, for k > max Ni = Nk and all t, oTl t=l,n

t=l

1=1

In relation (12.11), we now fix the number η > N°. In this case, the inequality < ε i=l is true for all k > Nk and all t, which proves Lemma 12.1. Lemma 12.2. Assume that the system of equations (12.8) is such that the sequence (12.7) of invariant tori (μ) is specified by a sequence of functions ι υ,( \φ, μ) € ^(Fm) bounded in the norm || · \\ by a constant d. I f ,forall μ £(0, ß] C (0,μ 0 ]. μ) — η^(φ,μ)\\

—> 0

as

i —> oo

uniformly in φ G Tm, then the function η(φ, μ), φ 6 Tm, specifies the invariant torus of the system of equations (12.1) for μ Ε (0, μ]. Proof. Since (μ) is the invariant torus of the system of equations (12.8), the following identities are true:

dU(l+1

}ft,fl) = Ρ(ψΙη^(φίμ),μ)η^\φί,μ) +

(12.12)

Let t be an arbitrary finite point of time, let To = [0,T°] be an arbitrary segment containing this point, and let | T° | be its length. We integrate equalities (12.12) from 0 to t and obtain t f i =
(12.13)

Section 12

Invariant Tori of Nonlinear

123

Systems

η{ί+1)(ψί,μ)

t + 1 {Ρ(φΙη^(φί,μ),μ)η^1\φί,μ)

+ ο(φΙμ)}ά8.

(12.14)

0 Since ||ΐί(ι)(
μ)

HVtiU W (vi.AO.AOII < A = const < oo. In this case, the sequence ψ\ is uniformly bounded and equicontinuous on To. Indeed, +A\t\,

\\φί\\<\\φ\\

and

where

φ (Ξ J-m,

t 11$ - <411 <\J

Ads = A \ t - t \ ,

t , t e To.

t

For I ί — ί I < 4-, we conclude that \[ψ\ — φ\\[ < ε uniformly in t,t Ά

Then the sequence φ\ contains a subsequence φι = φ(ί, μ) uniformly in t G To. We now show that

G To-

that converges to a function

lim α ^ , ^ Ο ^ , ^ , μ ) = a(
(12.15)

uniformly in t e To. The following relations are true: \\α{ψί)ui}Ptiμ)ιμ) < \\a((pt,u(ipt, + I

a(vlk,u(lk)(
~

I

μ), μ) - a((pltk,u((ft, -

μ), μ)\\

a((p]k ,u^k\ip\k,

μ), μ) ||

= Si + S2. Let ε be an arbitrarily small positive number. Since a(tp, h, μ) is a continuous function of φ, there exists δ > 0 such that

124

Invariant Tori

Chapter 2

and (p\k satisfying the inequality

for any

\\φΐ~φϊ\\<δ.

(12.16)

As shown above, there exists a number N\ such that, for all k > N\, this inequality holds uniformly in t £ To. Similarly, the fact that α(φ, h, μ) is a continuous function of h implies that there exists δι > 0 such that

μ) — u(lk\ip\k,

for all ü = \ Η

Ψ

ι

μ)|| < δ\. Further, we denote

, μ ) - ι =

and

\\η^(φί,μ)

-

,μ)\\ = S4.

According to the condition of the lemma, there exists a number N2 such that S3 <

for k > N2 and all t e T 0 .

It is also easy to see that the function 11:

norm || · ||. Indeed, if we denote υί \φ,μ) into account the fact that du{-k)(

μ) is continuous in φ in the — {η^\φ,μ),

r = l,m,

^^(φ,μ),..

.), take

j = 1,2,...,

and use the well-known Lagrange lemma, then we get \η{;*\φ,μ)

- η^\φ,μ)\

< ξ = const,

j = 1,2,...,

whenever lly —
sup { ^ ( φ , μ ) - u f k \ < p ^ ) \ } < f . j—1,2,... g

Then there exists 62 > 0 such that S4 < — whenever Ζ \\φί-φ?\\<δ2.

(12.17)

At the same time, inequality (12.17) holds for all k > N3 and t Ε To. This implies that ü < S3 + S4 < <5i for fc > max{N2, N3} and, hence, S\ + S2 < ε whenever k > max{Ni,

N2, N3}, i.e., equality (12.15) is proved.

Section 12

Invariant Tori of Nonlinear Systems

125

If we now pass to the limit as k —• oo in the integrand in equality (12.13) with i replaced by i\. and take into account the fact that (fg = φ, then we get t
(12.18)

ο The fact that

\φ, μ) are continuous functions of φ and the formula lim η ί ^ ( ^ Κ—HX

+ 1

, μ ) = η(φι,μ)

immediately yield the representation lim κ—*oo

= η(φί,μ),

(12.19)

whence it follows that as

η(φ,μ)

oo.

k

(12.20)

It is also clear that —• ο(<ρί,μ)

as

(12.21)

k o o .

Let us now show that the following limit transition takes place in the weak sense uniformly in t € To: lim Ρ(φlk, k—y oo

, μ), μ)η^(φ\",

μ)

= P((fit, u((pt, μ),μ)η{φι,

μ).

(12.22)

W e prove this fact for the first coordinates of the vector functions appearing on the left- and right-hand sides of equality (12.22). Clearly, the series oo

Σ

plr (φ*" Mik)

γ=1

(<ΡΪ, μ),

μ ) ^ ^

(φ?, μ)

converges uniformly in k and t G To because it can be majorized by the following convergent numerical series: oo

Σ r=l

sup V> h

\pir{(fi,h^)\d.

126

Invariant Tori

Chapter 2

Moreover, P\r {φ1ι . u (lfc) (v l t » μ ) >

}

(<^

f c

, μ) -» Pir(ψί,η(φ,

μ), μ)ιιΓ(φί,

μ)

as k —• οο uniformly in t G To, which follows from the continuity of ριΓ(ψ, h, μ) with respect to φ and h and the fact that —» ^^(φ^,μ) —> η( η(φί,μ) uniformly as k —• oo. Then, according to Lemma 12.1, we have oo £ Pi r

,«

( i f c )

, μ), μ ) * ? * + ι ) ( r f - , μ)

r=l

οο ΣριΛνΗΜψί,μ),μ)ντ{φ^μ) r=1 as A; —»· οο uniformly in ί G To. In this case, the series on the right-hand side also converges uniformly in t e To- This proves equality (12.22). By using equalities (12.19)—(12.22), we pass to the limit as k —• oo in equality (12.14) with i replaced by %k· This gives t ν{ψι,μ)

= η(φ,μ)

+ J{P(ips,u(ip3, ο

μ), μ)η(φ3, μ) +

ο{φ„μ)}ά8.

Together with equality (12.18), this proves the assertion of Lemma 12.2. We now introduce the notion of coarseness of the Green function of a linear system of equations. For this purpose, we set I / ( ν ) lo =

su

p 11/(^)11 ψ

l l / M l l z = s u p 2>j/fo>) o<s
The Green function GO(T, φ) of the system % = α(φ),

^=P(
(12.23)

is called coarse if there exists a constant <5 > 0 such that, for all α1 (φ) G Cl{Tm) satisfying the inequality ΙΙα 1 ^)!^ < δ, the Green function GQ(t, φ) of the system of equations %=α(φ)

+ α\φ),

^

= Ρ{φ)Η

(12.24)

Invariant Tori of Nonlinear Systems

Section 12

exists and is such that Go (τ, ψ)ί{ψ)

G Cl(!Fm)

127

and

ll^o(τ,φ)/(φτ(φ))\\ι < ΚβχΡ{-Ί\τ|

}||/||„

(12.25)

where ψί{ψ) is the solution of the first equation in (12.24), /(φ) is an arbitrary function from Cl(!Fm), and Κ and 7 are positive constants independent of δ, φ, and /.

Lemma 123. Assume that the system of equations (12.23) possesses a coarse Green function Go (τ, φ) and Ρ {φ) G Οφ. Then one can find ρ = ρ{δ) > 0, ρ -» 0 as δ -» 0, such that, for any a1 [φ] G Cl {?m) and Ρ1 (ψ) G Cl (Tm) Π Οψ satisfying the inequality maxilla1^)!!,, I I ^ M I I J < Ρ

and an arbitrary function /(φ) G Cl(Tm), ^

= α(φ) + α1 (φ),

^

the system of equations

= (Ρ(φ) + Ρ\φ))Η

+ f(
(12.26)

possesses the invariant torus T\ h = ιι(φ), φ € Tm, satisfying the condition (12.27)

where K\ is an arbitrary constant independent of ρ and f . Proof. We choose ρ(δ) < δ. In this case, system (12.24) possesses the Green function GQ{T, Φ) satisfying condition (12.25). By using the operator G defined on functions / G by the formula 00

Gf(
J

Go(T,
—00 we introduce the equation

u = GPlu

(12.28)

+ Gf.

For sufficiently small p, the norm of the operator GPl satisfies the inequality

1Κ IK \\GP\<—\\P1\\l<—p
in the space

Cl(!Fm)

128

Invariant Tori

Chapter 2

Therefore, equation (12.28) possesses a unique solution in C ^ F m ) given by the formula oo u(,0,0),

^=P(
(12.29)

possesses a coarse Green function, and the inequality max{||a((£>, h,μ)—α(φ,0,0)||j,

\\Ρ(φ,Η,μ)-Ρ(φ,

0,0)11,, ||c(, μ)|| J

<£ι{ά,μο),

(12.30)

where Ci(d, μο) 0 as d 0 and μο —> 0, holds for h € C^Fjn). Then there exists μ Ε (Ο,μο] such that, for all μ € [0, μ], system (12.1) possesses the invariant torus Τ {μ)'· h = ιι(φ, μ) G Cl~l{Tm), ψ € Tm, I > 1, and, moreover, \\η{φ, μ)|| —• 0 as μ —» 0. Proof. We rewrite the system of equations (12.1) in the form ^ ^

= α0(φ) +

α1(φ>}ι,μ),

= [Ρ0(φ) + Ρ 1 {φ, h, μ)]Λ, + ο{ψ, μ)

by setting αο(φ) = α(φ, 0,0),

Ρ0(φ) = Ρ[φ, 0,0],

h, μ) = α(φ, h, μ) - α(φ, 0 , 0 ) , Ρ\φ,Κ,μ)

= Ρ{φ,Κμ)-Ρ{φ,

0,0).

Invariant Tori of Nonlinear Systems

Section 12

129

To find the invariant torus of this system, we use the iterative process described above. A function u Μ (φ, μ) ξ 0, φ £ Τmi is regarded as the zeroth-order iteration of the process. The first approximation is determined from the system of equations ^

α1(φ,0,μ),

= α0(φ) +

= [Ρ0(φ) + Ρ1 (φ, 0, μ)]/ι + α(φ, μ)

^

(12.31)

as its invariant torus h= μ), φ e !Fm. We choose μ £ (0, μο] to guarantee the validity of the inequalities # ι £ ί ( 0 , μ)
and

Ci{KiCi{0,

fi),fi)<

ρ,

where K\ and ρ are the same constants as in Lemma 12.3. Condition (12.30) guarantees the possibility of this choice. Then, according to Lemma 12.3, the invariant torus of system (12.31) exists. In this case, ^ { ψ , μ ^ Κ Κ Μ 0,fi)
μ£(0,μ}.

We assume that, for all i = 1,2,... ,k, the invariant tori : h = uW (φ, μ), φ G Tm, μ € (0, μ), from sequence (12.7) exist and are such that μ)II, < # ! £ , ( 0 , μ ) . But then, for all μ G (0, μ], we have m a x { | | a W

f c

W ) , Ä

I I P W * W M I I J

<£ι(Κι£ι(0,μ),μ)<ρ, and, hence, Lemma 12.3 can be applied to the system of equations ^

= α0{φ) + α^φ,η^^ψ,

^=[Ρ0(φ)

+ Ρ1(φ,η^(φ,μ),μ)]Η

This means that the invariant torus all μ (Ξ (0, μ] and, in addition, l|u(fc+1W)lh

< Κ\\Ηψ>μ)\\ι

μ), μ),

+ ο(φ,μ).

(12.32)

= u(fc+1)(<£>, μ), φ Ε T m , exists for

< ^ιΑ(ο,μ)

< d.

130

Chapter 2

Invariant Tori

By induction, we can prove that the elements of sequence (12.7) exist for all i — 0 , 1 , 2 , . . . and are jointly bounded in the norm || · \\t by a constant d > 0 for all μ G (0, μ]. It remains to show that sequence (12.7) is convergent and use Lemma 12.2. Since η^+ι\φ,μ) G Cl{!Fm) C C^ {J~TTJ) for any μ £ (Ο,μ] and t G R 1 , we obtain

νψί

V=V>t

= [ P o ( r f ) + Ρ\ψΙη^{ψΙμ),μ)]η^){φΙμ)

+ c(rf^),

(12.33)

where φ \ = φ%(φ,μ) is the solution of the first equation in (12.32) such that ψ0=ψ€·

?τη·

Denote

Σ

Κ

i f f ) + α}(φ,η^(φ,μ),μ)}

=

[a 0 + a 1 ].

It follows from equality (12.33) that ^

}

[°0(ψ) + α1 (φ, u (fc) (φ, μ), μ)]

= [Ρ0(φ) + Ρ1 (φ,

μ), μ)]η^+1\ψ,

μ) + ο(φ, μ)

for any φ £ Tm and μ G (0, μ].

Similarly, we arrive at the formula - [a 0 {ψ) + α 1 (φ, η^^φ, =

[ P o (φ)

+

Ρ \ φ Μ

1 ΐ

-

1 )

μ), μ)} { ψ , μ ) , μ ) ] η

{ 1 ΐ

\ φ ^ μ ) +

whence, by using relation (12.34), for the function fc+1

W(

we obtain

) (φ, μ)-

«<*> (φ,μ)=

ω ^

(φ, μ),

ο(φ,μ),

(12.34)

Section 12

Invariant Tori of Nonlinear Systems

——

[aQ(ip) +

131

a\
+ [Ρ1 {φ, uW ,μ)-Ρ1

{φ, u ^ , μ)]ιχ<*>.

This means that the function h = u)(k+1\
α1(φ,η^{φ,μ),μ),

= αο (φ) +

= [Ρ0(φ) + P1(vMk),ß)]h

+ ck(
(12.35)

where ο^φ,μ)

= + [Ρ 1 (φ, u<*> ,μ) - Ρ 1 (φ, u ^ , μ ) } η ^ .

The system of equations (12.35) has the form of system (12.32) and, therefore,

Since ||Pfc(^,/i)|| < üfio-(m + l j r i i o . Ä i J i i t i W c ^ M ) - u t * - 1 ) ^ , / * ) ! ! ,

we arrive at the estimate <

ρ0\\ω^{φ,μ)\\,

where po = Κγσ{πι + 1)£ι(0, μ). Similarly, we obtain the chain of inequalities \\ω^(φ,μ)\\<ρ0\\ω^(φ,μ)\\

< ρ2\\ω^(φ, <

μ)||

...
132

Invariant Tori

Chapter 2

Since £o(0, μ) —• 0 as μ —> 0, we conclude that there exists μ Ε (0, μ] such that po < 1. This means that the sequence μ), i = 0 , 1 , 2 , . . . , converges as i —> oo uniformly in ψ € Tm. Denote lim υ , ( ι \ φ , μ ) = η ( φ , μ ) . 1—+00 Let us show that μ) G Cl~1(fm) for Ζ > 1. Consider the sequence of sth coordinates of the functions υίι\φ,μ) sequence of their partial derivatives with respect to φ^ :

I

9ψ]

/'

ί =

and the

1 2

< 12 · 36 >

' '···'

where j is a fixed natural number such that 1 < j < m. It is clear that sequence (12.36) is uniformly bounded. Since η ^ \ φ , μ ) e 2 C ( ^ m ) . by virtue of the Lagrange theorem we arrive at the estimate e u f W ) dpj

OuPtod d(fj

s =

li2

,

i.e., sequence (12.36) is equicontinuous in φ. By using the Arzelä theorem, we conclude that there exists a subsequence

t

d
J'

dus((p, μ) of sequence (12.36) that converges to the derivative — ^ uniformly in φ. σφ 3 Thus, for I = 2, the required assertion is proved. Continuing the reasoning presented above and taking into account that the sequence {DPu(k\
Section 13

Exponential Attraction of Motions

133

Thus, for the system of equations

— = a(
= P(
(12.37)

with Ρ (φ) e ^ ( F m ) Π Οψ and α(φ) e Cft (.Fm), the following statement is true:

Theorem 12.2. Assume that the function c(ip, h) satisfies the Lipschitz condition with respect to h for φ G Tm and ||/i|| < d and, for μ = 0, the system of equations (12.37) possesses the Green function of the problem of invariant tori GQ(T, φ) satisfying inequality (8.5). Then there exists μο > 0 such that, for all μ £ [0, μο], system (12.37) has the invariant torus Τ{μ) \ h = η(ψ), φ € Tm, such that lim \\η(φ, μ) II = 0. μ—>0

2.13. Exponential Attraction of Motions in a Neighborhood of the Invariant Torus of a System of Equations to Its Motions on the Torus The Green function studied in the previous section becomes especially simple in the case where the matrizant Ωιτ{φ) of the corresponding linear system satisfies the condition of exponential decay ||Ωο(<ρ)|| < i f e x p { - 7 i } ,

t € R+ = [0, + o o ) .

(13.1)

This condition leads to the exponential stability of the trivial torus of the linear system and guarantees the attraction of its solutions to this torus. In the present section, we formulate conditions under which the behavior of solutions of nonlinear countable systems is similar. First, we consider the quasilinear system of equations (12.37). For this system, we have the following theorem on exponential stability of the invariant torus:

Theorem 13.1. Assume that the system of equations (12.37) satisfies the conditions of Theorem 12.2 and the matrizant of the corresponding homogeneous system satisfies inequality (13.1). Then, for sufficiently small μ, the trajectory of a solution ht = h(ho,t, ψ, μ), ho = h(ho, 0, φ, μ), of this system of equations is attracted as t —> +oo to a trajectory lying on its invariant torus T.h — u(ip, μ), ψ G Tm, according to the law < #ι||Λο-«(ρ,μ)ΙΙ®φ{-7ΐ*},

where K\ and 71 are positive constants.

t> 0,

(13.2)

134

Invariant Tori

Chapter 2

Proof, In system (12.37), we perform the change of variables ht = u(
(ί,χ0,φ),

where xo = x(0, xo, φ)· As a result, we arrive at the following system of equations for xt = x(t,x ο,φ): dx — = Ρ(ψί(φ))χ

+ {c{ipt&),u{
+ μχ) - c(ipt(
μ))}.

The solution χ = xt of this system of equations satisfies the identity t t

xt = fl 0Xo + J Ω*3{ο(φ3,ιι(φ3,μ) o

+ μχ3) -

c(vs,u(tp3^))}ds

in the entire segment t e [0, Τ] of the existence of the solution. Setting \\c(ip,h) - ο(φ,Κ)\\ < £ | | Λ - Λ | | ,

L = const > 0

and using inequality (13.1), we obtain t

I N I < # { | Ν Ι exp{-7*} +

P JII1«II exp{-7(t ~ L

s)}ds},

t e [0,T].

ο In view of the Gronwall-Bellman inequality, we conclude that

INI


e x p { - ( 7 - t €

[0,T],

For sufficiently small μ, the last inequality implies that Τ = +oo and, hence,

I N I < K\\x0\\ exp{-(7 - ΚΏμ)ί},

t > 0.

Returning to the original variables, we find that IIhi - η{φί{φ),μ)\\

< μΚ\\Η0 - η{φ, μ)\\ e x p { - ( 7 - ΛΧμ)ί},

t > 0,

which yields inequality (13.2). Note that, for sufficiently small μ, the torus T\h = u((p, μ), φ e belongs to the domain D together with its ^-neighborhood (d > p). Thus, if the point ho belongs to the indicated neighborhood, then the semi trajectory of the solution ht for t e R+ also belongs to this neighborhood.

Section 13

Exponential Attraction of Motions

135

We now consider a system of equations of the general form %

= a(
^

=

+

(13.3)

where ψ G Rm, h G VJl, and P{(p,h), c(ip) G Cl{Tm) for h G C define the norm || · U^ of an infinite matrix Ρ(φ, h) by the formula

1

^ ) .

We

oo

WP\\D =

D : φ E Pm, \\h\\ < d.

P ^ suP i j—i
S U

The matrix Α(φ, h) is called Lipschitzian with respect to φ, h G D if, for any φ, h G D, ι\A(
Κ = const > 0.

By CD we denote the set of matrices bounded in the norm || · H^ and satisfying the Lipschitz condition with respect to φ and h in the domain D. Let T- h = u(
= α(φ,η(φ)),

φοΖΤη-

(13.4)

Then u = η(φί(φο)) is a trajectory on the torus T. We represent the solution of the system of equations (13.3) in the form φ = y?t(<po> ho), h = ht(cpo, ho), where φο(φο, h0) = ψο and ho(<po, ho) = ho. We say that the motions of system (13.3) in a neighborhood of the invariant torus Τ are attracted (as t —> + o o ) to its motions on the torus according to the exponential law if there exist positive constants po> Pi £ Rl such that, for any solution {φ, h} of this system of equations satisfying the condition min ||v?o - ψ\\ < Po and \\h0 - η(φ0)\\ < pi, (13.5) one can find a trajectory on the torus η(φ^φο)), where φο depends on ho and ψο, such that the following inequality is satisfied: + where Κ and 7 are positive constants and t > 0.

(13.6)

Invariant Tori

136

Chapter 2

To solve the posed problem, we represent system (13.3) in a neighborhood of the torus Τ in the required form. To do this, we perform the change of variables φ = φ{ί) + μθ,

h = η(φ + μθ) + μ2Ηλ,

where μ is a positive parameter. This change of variables transforms system (13.3) in the neighborhood of the torus Τ into ^

= Α1 (φ, θ, μ)θ + μΑ2(φ, θ, hu μ)Ηι, dkl

dt where

=Ρ1(<Μ,Λι,μ)Λι,

(13.7)

ι A

= j

9{φ

Α*«,, Μι, = / . l.W

J

dT

+ Τμθ)

+

0{ιι{φ + μθ) +

'

U{μ ΗΐΤ) μ2^τ)

Ι

dr,

and Ρ1 (Μ

hu μ) ι

/

+ μθ) + τ μ2 hi) (u{j> + /ig) + τ μ 2 / ΐ ι ) ] άτ θ(η(φ + μθ) + τμ2^)

0[Ρ(φ + μ0,

ο

1 f θα(φ + μθ, η(φ + μθ) + τμ 2 Λχ) J ~~ VT Q /..7.".'7 VT m' Γ'.' ό2 , Τ 3{φ + μθ) J θ(ν(ψ + μθ) + τμ 1ΐι)

0ιι(φ + μθ)

(13·8>

where θ e Rm, hi e 9Jt, A1 is an m χ m square matrix, the matrix A2 has m rows •i duU + μθ) and infinitely many columns, Ρ 1 is an infinite matrix, and the matrix -^7ττδ(φ + μθ) has m columns and infinitely many rows. It follows from expressions (13.8) that there exists 0 < μο < 1 such that, for \\hi\\ < 1

and

0 < μ < μ 0 < m i n { y / p , 1},

(13.9)

the matrices A1, A2, and Ρ 1 are continuous with respect to the collection of variables Θ, h\, and φ in the matrix norm || · || and satisfy the inequalities \\Α\φ,θ,μ)\\

< Μι,

ζ = 1,2,

IIΡ\φ,ΘΜ,μ)\\<Μ2,

Section 13

137

Exponential Attraction of Motions WA1 (φ, θ, μ) - Α1 (φ, θ, μ) || < μΚχ \\θ - θ\\,

\\Α2{φ,ΘΜ,μ)

~ Α2(ψ,θ, hi, μ)II < μΚι(\\θ - 0|| 4- ||fci -

\\Ρ\ψ,ΘΜ,μ)

~ ΡΗΦΜι,μ)\\

Μ),

< μΚ2{\\0 - 9\\ + ||λι -

Μ),

where Μι, Μ2, Κι, and Κ2 are positive constants. By Ω* ( P ( t ) ) we denote the matrizant of a homogeneous linear system of equations of normal form whose matrix of coefficients is equal to P(t). We set

l|Jm = P m . where C R1.

D=T=[a,b]

Lemma 13Λ. Assume that the infinite matrices P(t) and Pi (t) are continuous in t and bounded in the norm || · || for any finite segment T\ C [τ, -|-oo) (T\ c (—00, τ}). If the inequality t \\ntr(P)x0\\ 0 and ß{t) is a function integrable for t > τ (t < τ), holds for all t > τ (t < τ) and XQ € 9ft, then t \fö(P

+ PL)xo\\
t + L\J ||Pi(S)||ds|}||x0|| τ

for any t > τ (t < τ). Under the conditions of this lemma, the matrices Ρ and P + P i satisfy requirement ( P ) from Section 1.3 and, hence, the matrizants Ω*Τ(Ρ) and Ω* ( P + Pi) indeed exist. By virtue of Lemma 12.1, the proof of Lemma 13.1 is performed by analogy with the proof of Lemma 1 in [SamlO, Chap. 4, Sec. 4]. Let Ω* (Po) be the matrizant of the system of equations

and let Ω* (Ao) be the matrizant of the system άθ dt

=

V

(da(ipt,u(iPt))\e dipt )

Chapter 2

Invariant Tori

138

The behavior of solutions of system (13.7) is described by the following theorem: Theorem 13.2. Assume that the system of equations (13.7) is such that the following inequalities hold in domain (13.9): 3 Ι|Ω?(Ρ 0 )Μ < e x p { - J/?(ν>τ)<*τ}||Λο||, t

h 0 € OH,

(13.10)

t ΙΙΩ^Λο^οΙΙ < e x p { J α ( φ τ ) ά τ } \ \ θ 0 \ \ ,

θ0 <Ξ Rm,

(13.11)

s

where 0 < t < s, min β(φ) > β0 > 0, α(φ) < 0, β (φ) + α(φ) > η = const > 0, and α(φί) and β(φί) are integrable functions. Then there exists a positive constant μ* < μο such that, for all μ: 0 < μ < μ*, the system of equations (13.7) possesses a solution exponentially vanishing as t —• +00. Proof. We have Jon ^

= Αί(φ,θη-\μ)θη

+

n

ßA^9

~\hn^)hn,

dhn ^ Γ = Ρ\φ,θη-\Ηη-\μ)Ηη,

(13.12)

provided that 0° = 0 6 Rm and h° — 0 e DJl. For n = 1, we arrive at the system of equations Jßl ^ = ΑοΜΘ1 + μΑ2(ψ, Ο,/»1,μ)/.1,

As a solution of the last system for t > 0, we take a vector function (Θ1, hl) of the form oo Θ1 = ~μ1

^3{Αο)Α2(φ,0,Η\μ)Η1ά8, t h l = Ωο(Ρο)^ο>

(13.13)

Section 13

Exponential Attraction of Motions

139

where \\ΙιΙ\\ < 1. It is clear that, for t > 0, inequalities (13.10) and (13.11) imply the estimates t ll^ll < e x p { - f ο

ß{i>)dr)\\hl\\<\,

\\ηί3{Αο)Α2(φ,0,Η1,μ)Η1\\ t < exp{ Jα{φτ)άτ)\\Α2(φ,

0, h1, μ)Ηι ||

s t

s

< M\ e x p j J α(φτ)άτ

—

Jβ(φΤ)άτ}\\Η10\\

a t

= Mi exp I J α(φτ)άτ ο

- f (β(φτ)

+

α{φτ))άτ}\\Κ&\

s < Mi e x p j — J ( ß ( i P T ) + ο

α(φτ))άτ}||/ij|

< M1e-^\\hl\\,

(13.14)

where 0 < t < s. This yields oo II^OII < μΜι [βχρ{-7β}||Λ5||ώ J

= ^

7

for t > 0, i.e., ^ ( O l l < 1 for μ < min | μ 0 ,

exp{-7i}Kll

(13-15)

and t > 0.

We have the following representations: Pn_ χ =

Ρ1(φ)θη~1,Ηη~1,μ)

= Ρ0(φ) + (Ρ\φ,θη-\Ηη-\μ) An_i = Α1(φ,θη~1,μ)

= Α0(φ) + (Α1(φ,θη~1

-

Ρ0(φ)),

,μ) - Α0(φ)).

(13.16)

Invariant

140

Tori

Chapter 2

The solution (θ η , hn) of the system of equations (13.12) for η > 1 is given by the formulas oo Θη = -μ1

^ί3{Αη-ι)Α2(φ,

0 n ~ \ hn,

h n = Ωο(Ρ η -ι)^ο

(«>0).

ß)hnds,

t (13.17)

Ύ

Let ε > 0 be a constant such that ε < m i n { - ; ßo} and let . r ο ε

ε

7-2ε·«

Assume that, for η = 2 , 3 , . . . , k — 1, solution (13.17) satisfies the inequalities ||0 n || < e x p { - 7 t + 2et}||/iJ||,

7

> 2ε, t > 0,

t ||Λ"|| < e x p { - JßU>T)dr ο

+ et}\\hll

t > 0.

(13.18)

For η = 1, we prove these estimates by using relations (13.14) and (13.15). Let us show that inequalities (13.18) are true for η = k. We have |

|

Po(VOII < μΚ2{\\θ*-1\\ 4- U ^ H ) < 2 μ Κ 2 < ε,

-

WA1 {ψ, θ*-1,μ)

- Α0(φ)II < μΚχΡ^ΙΙ

<μΚλ<

ε,

t> 0, μ < μ*.

By virtue of Lemma 13.1, this yields t \\nts(Ak_1)C\\<exp{Ja^T)dT

+

£(s-t)}\\C\\,

ζζΒ™,

s a

||Ω'(Ρ*_ι)τ/|| < e x p { -

Jβ(<ψτ)άτ

+ e(a -

i)}|M|,

||τ?| < 1, η e Wl, for all 0 < t < s. Further, for t > 0, we can write τ \\hk\\<exp{-J

β{ψΤ)άτ

+ εί}\\Ηΐ\\ < 1,

β0

>ε,

Section 13

141

Exponential Attraction of Motions

t < Mi\\hk\\

e x p j Jα(φτ)άτ

+ e(s

-1)}

8 a

< Mi expj— J{β{φΤ)

+ α(φτ))άτ

+ 2es}||fcJ||

ο < M 1 | | / i ä l | e x p { - 7 s + 2£s}, whence it follows that oo

ll^ll < μΜι J exp{-(7 — 2e)s}ds||/io|| t

^

^«φ{-(7-2ε)ί}||Λ4||<1.

Thus, we have actually proved by induction that the function (θ η , hn) is defined for all η = 1 , 2 , 3 , . . . , t > 0, μ < μ», and \\ho\\ < 1 and, moreover, inequalities (13.18) are satisfied. In what follows, the initial value Hq is denoted by ho. We now prove that the sequences {θη} and {/i n } converge as η —> oo uniformly in t, φο, ho, and μ. We set rn+1

= hn+1

-hn

= [ω5(ρ„)

-

nSiP«-!)]/*.

Then = Pnhn+1

- Ρn—\hn = Pnrn+1

Since r n +i(0) = 0, we have t rn+1

=

0 Hence, for t > 0 and μ» < —^r-, we get

+ (Pn -

Pn_i)/in.

Chapter 2

Invariant Tori

142 ι

||rn+i|| <

τ

J

f

β{<ψΤ)άτ +

ε{ί-8)}μΚ2[\\θη-θη-ι\ 3

+

IIhn - /ι"- 1 !!] e x p { - J β(ψΤ)άτ

+ es}\\ho\\ds

τ

<

jβ{ψΤ)άτ

+ εΐ)[\\θη -

θη~%

(13.19)

+ ||Λη-Λη-1||0]||Λο||ίμ^2, where || · ||0 = sup || • ||. t> ο In view of inequality (13.19), we obtain lkn+l|| < exp^ — Jβ{"Φτ)άτ

+ ei j (||0n — ön_1||0 + ||rn||0) t,

t > 0. (13.20)

Let us now estimate the norm of the difference £ n +i = θη+1 - θη. For t > 0, we find oo ll^+ill

<

μΙ\\[ηί3(Αη)-ηί3(Αη^)]Α2(φ,θη,Ηη+1,μ)Ηη+1\\ά8 oo

+ /i| Ι&β{Α^1)Α\φ,Ρ,Ηη+\μ)(Ηη+1

-

hn) ds

t oo +

-

μΙ\\θίΜη-ι)[Α2(φ,θη,Ηη+\μ)

Α2(φ,θη~1,Ηη,μ) hn

ds.

Further, we denote Δ η = Ω* ( A n _ i ) 0 o and consider the difference Δ η + 1 - Δ η = [Ω'ΛΑΟ - Ω* β (Λ η _ι)1 θ0.

(13.21)

Section 13

Exponential Attraction of Motions

143

This gives d{An+1

Δη)

d~

= A n ( A n + 1 - Δ η ) + (A n - Α η _ ι ) Δ η ,

and, in view of the fact that ( Δ η + ι - Δ η ) = 0 for t = s, we get t Δη+ι - Δη

J ςΐτ{Αη){Αη-

An-xWliAn-^eodr.

This implies the inequality ||Δη+ι — Δη||

t

s

< Jexp{ Ja{il>t)dt + e ( r -

t

Τ

1))μΚχ ||0

n

- θη~ι \

τ Χ e x p j ja(ipt)dt

+ e(s - T)}||0 o ||d7 I

< μΚλ ||0O|| ||Ön - 0"-1110 exp{ Ja{1>t)dt + e(s - t ) } ( s - t ) s

t n

< ||0 - 0

n-1

| | o e x p j J a{ipt)dt + e(s-t)}(s-t)

(13.22)

a

for μ* < - ^ - a n d | | 0 o | | < 1. Λΐ By using inequalities (13.21) and (13.22), we obtain

oo η

t η 1

ll^n+ill < μΙ\\θ -θ - \\0βχρ{Ια(φί)(ϋ

t

+

ε(3-ί)}(3-ί)Μ1

3s Xexpj

- j β(ψτ)<1τ + es}\\ho\\di 0

oo

t

+ μ ^ e x p j J a(^T)rfT + e(s-t)}Mi||/in+1

t

S

-

hn\\ds

Chapter 2

Invariant Tori

144 oo

τ

+ μΙexp{

+ e{s - * ) } # ι ( | | 0 η - θη~ι \

Jα(φΤ)άτ

a

+ ||/ιη+1 <

Λ η | | ) e x p { - Jβ(ψΤ)άτ

+ εβ}||Λ0||ώ

μΜι||ξη||0||/ΐο||^ι + μ Μ ι ^ (13.23)

+ /^ι(||£η|Ιο + ΙΚ + ι|| 0 )||Λομ 3 ) where the integrals J\ and J3 are such that a

00

Ji

<

Jexp{-

J(ß{i)T)

t

0

+ a(^T))dr

+

2£s}{s-t)ds

00

<

/exp{-(7 - 2φ}(

5

- t)ds =

oo

J3 < Jexp{-(7

- 2e)s}ds =

βΧρ{"(7

2e)<} " (7 - 2ε) 2

exp{-(7-2g)*} (7 - 2ε)

for t > 0. In view of inequality (13.20), for the integral J2 we can write 11411

<

(ΙΜ0 +

Μ 0 ) 00

X jexρ

τ

τ

< Ja(ipT)dr

+ e(s — t) — Jβ(ψΤ)άτ

ι

+ es > sds

3 ου

<

(lllnllo + li^llo) / t

exp{ —(7

- 2B)s}sds

<

(IIUo + l l ^ l l o ) ^ ^ + T T 2 i )

ex

P { - ( 7 - 2e)t}

for t > 0. By using representation (13.23), we arrive at the following inequality for£n+i:

Section 13

Exponential Attraction of Motions

145

+ Mit exp{—(7 - 2ε)ί) + Kx (1 + t) e x p { - ( 7 μ

-

7-2e

2e)t}}

< ( U I! J. II. II x M ( 2 e + l ) + K i ( e 7 + l ) < (||ξη|Ιο +' F n l l o ) · e(7 — 2ε)

= CiMIIUo + IMo), where M ( 2 e

±

l )

±

^ =

2 ±

l )

= c o n s t > 0

2

e(7 - 2ε) In view of relation (13.20), we get Ifcn+lllo + Ikn+lllo < μ(
g)

) (||U0 + IMo)·

We now choose sufficiently small μ s u c h that

4 Then the inequality | | ^ + 1 | | 0 + | | r n + 1 | | 0 < i ( | | ^ | | 0 + ||r n || 0 ) means that the sequences {0 m } and {hn} are convergent uniformly in t, ψο, ho, and μ in the domain D°: I I M < 1. Ψο e dm, t > 0, μ < μ* = m i n j - ^ - ;

μ*; μ'„ J .

We set 0(t) = lim θη(ί) η—>oo

and

hM) = lim hn(t). η—>oo

(13.24)

Clearly, the limiting functions 0(t) and h\(t) satisfy the inequalities ||0(ί)||<βχρ{-(7-2ε)ί}||/ΐο||, t ||Λι(ί)|| < e x p { - Jβ(φτ)dr

+ εί}||Μ·

(13.25)

Invariant Tori

146

Chapter 2

It remains to show that the vector function {#(£), /ii(£)} is a solution of the system of equations (13.17). It is clear that the following identities are true in the domain D°:

=Ρ1(^,βη-1(ί).Λη-1(ί),μ)Λη(ί).

(13.26)

The fact that the (m χ τη) matrix A1 is finite-dimensional immediately implies that lim Α1(φ ,θη-\ή,μ)θη(ή ^ Α1(φ^θ(ί),μ)θ(ί) η—too ί uniformly in t for any finite segment T\ c [0, -f oo). The limit transition in the second vector equality in (13.26) is performed coordinatewise. Thus, for any ith coordinate (i — 1 , 2 , 3 , . . . ) , we obtain JLn/'yA

OO 3=1

Since the matrix Ρ1 defined in the domain D° is bounded in the norm || · H^ on the segment T\ and ||/ι (ί)|| < 1, the series on the right-hand side of the last equality converges absolutely and uniformly in t e T\ and n. Moreover, ton plj^t, en~l(t), hn~\t)^)h<}(t) = p}0t, 9(t), Λι(ί), η

uniformly in t G T\, which enables us to use Lemma 12.1 and write the following formula (for t > 0): lim

n-too

at

=

*( i i m hn{t)\ at >oo /

=

p\ipt,0(t),

Η ^ , μ ^ ) ,

where the limit is taken coordinatewise. Similarly, one can easily show that (coordinatewise) n—>oo uniformly in t G T\. This means that the vector function {#(£), h\(t)} is differentiable with respect to t at any point of the semiaxis [0, +00) and solves the system of equations (13.7) in the domain D°. This completes the proof of Theorem 13.2.

Exponential Attraction of Motions

Section 13

147

By using the theorem just proved, we can easily clarify the behavior of solutions in the neighborhood of the invariant torus of the system of equations (13.3) as compared with the behavior of solutions of this system on the torus. The following theorem is true:

Theorem 13.3. Assume that α(φ, h), P(tp,h), and ο(φ) belong to the set C1 (J-m) for h Ε C1 (JFrn) and the Frechet derivatives 9α(φ, h) WWW ~0ΛΤ' — d h —

e

C d

exist. If the torus J-\h = η{φ), φ Ε J-m, belongs to the domain D together with its p-neighborhood and is an invariant set of system (13.3) and, in addition, u( +oo to its motions on the torus according to the exponential law. Proof. Under the conditions of the formulated theorem, Theorem 13.2 is true. This means that, for any fixed ψο Ε Tm and sufficiently small μ < μ*, there exists a solution

ψι = Φΐ + μθ(ί), h t = u(ik) + [u{$ t + μ θ ( ή ) - U(IIH)] + μ 2 Μ 0

(13.27)

of the system of equations (13.3) such that

\\φι-Μ<μ*Μ-(Ί-2ε)ί}, \\ht - u{ipt)\\ < μ02 e x p { - ( 7 - 2e)t},

(13.28)

where C2 is a positive constant independent of t. The second inequality in (13.28) can be obtained as follows: Since it(i/>) Ε C u M l there exists Ρ = const > 0 such that

<

Ρμ\\θ(ί)\\+μ2\Μί)\\ t

< μ(Ρβχρ{-(7-2ε)ί}

+ μ β χ ρ { - f β(φτ)άτ

ο

< μ(Ρ + μ) exp{-72i} - μ

+

et}}

148

Chapter 2

Invariant Tori

where 72 = min{7 — 2ε, bo - ε } and C2 = Ρ + μ. At the same time, according to the condition of the theorem, we have 7 < ßo and, hence, 7 — 2ε < bo — ε, i.e., 72 = 7 - 2ε. Inequalities (13.28) yield (13.6). In this case, the solution (
and

IIh0 - u(
μ2,

and, hence, for po = μ and p\ = μ2, condition (13.5) is satisfied. Moreover, po and p\ are independent of ψο € Tm· Denote 0(0) = θ(μ, φο, h\(0)). As shown above, the solution (
μθ(μ,φο,^(0)),

ho = h(0) = w(v?o) + μ2hi (0),

φο G T,τη

>

satisfies inequality (13.6) for t > 0. We fix arbitrary values of φο and ho satisfying conditions (13.5). Let us show that, for these φο and ho the equation φο=Φο

+ μθ(μ,φο,

°

k

~

(

1

3

·

2

9

)

possesses a solution φο(φο, ho). This will complete the proof of Theorem 13.3. We introduce an unknown quantity ν = φο — φο and rewrite equation (13.29) in the form ν - μ ί ^ μ , φ ο - ν , ^ ψ ^ ) . μ1 Since μθ(μ,φ0-υ,

ho - η(φο)

(13.30)

< μ, the right-hand side of equation (13.30) μ2 specifies an operator that maps the ball ||υ|| < μ into itself. It remains to show that this operator is continuous with respect to ν and use the Brouwer fixed-point theorem. To do this, by virtue of (13.24), it suffices to prove that the functions 9n(t, φο) given by relations (13.17) with n = l , 2 , 3 , . . . a r e continuous with respect to φο· The function φο) is given by relations (13.13). By virtue of the theorem on continuous dependence of solutions on the initial conditions, φ = Φι{Φο) is a continuous function of φο. The matrices Ao(V't) and Ρο(φί) are continuous in φί and, therefore, in φο· For Λ,1(ί, φο), the following inequality is satisfied:

Exponential Attraction of Motions

Section 13

t <M2

f \ m

149

t 0

) - h i m d s

+

0

j\\Po(s^o)-Po(s^o)\\ds.

(13.31)

0

Further, by using the fact that the matrix Pq(s, φο) is continuous with respect to φο, the Gronwall-Bellman lemma, and inequality (13.31), we conclude that the function hltyo) is continuous with respect to φο uniformly in t for any finite interval [0, To] C R1. Then the matrix Α2(φ, 0, Η\,μ) is also continuous with respect to φο. Hence, the integrand in (13.13) is also continuous with respect to φο in any finite segment of variation of t and, therefore, the function θ1(ί,φο) is also continuous with respect to φο because the indicated integral is uniformly convergent. By induction, using the same reasoning as above, we prove that 0 n (£, φο) are continuous functions of φο for any natural n, which completes the proof of Theorem 13.3. Remark 13.1. Theorem 13.3 remains valid if conditions (13.10) and (13.11) are replaced by the inequalities s ||Ω?(ΡΟ)Μ

< £ i e x p { - Jβ(φτ)άτ}\\}ΐο\\, t

h0 € SW,

t ||Ωί(Λο)0ο|| < L2exp{Jα(φΤ)άτ}\\θ0\\,

θ0 €

Rm,

s

where L\ and are constants greater than one. More precise laws of attraction to the invariant torus of system (13.3) than the law discussed in this section can be obtained in the general case by using the results presented in [Saml2, Saml3] for the case of finite-dimensional systems.

3. REDUCTIBILITY OF LINEAR SYSTEMS

In the present chapter, we study the problem of reducibility of countable linear systems of differential equations, i.e., the possibility of mapping the trajectories of their solutions onto the trajectories of systems with simpler structure. For finitedimensional systems with periodic coefficients, one can especially mention the well-known results of Floquet and Lyapunov establishing the admissibility of this mapping [CoL]. Linear systems with quasiperiodic coefficients are of great interest for applications. Numerous works (see, e.g., [And, Arn3, Art, Bar2, Bel, Gel, MiS2]) are devoted to the problem of reducibility of these systems and construction of their solutions. At present, the problem of reducibility of almost periodic linear systems is of serious practical significance. In this direction, we can mention the works [Blil], [Bli2], [Fil], and [ Par] devoted to the solution of the problem of reducibility for fairly narrow classes of almost periodic systems. We also refer the reader to the works dealing with the reducibility of the original system of equations to a block- diagonal form [Byll, Mil, MSK2]. In the present chapter, we formulate sufficient conditions for the reducibility (in the Lyapunov sense) of a periodic countable system, study the possibility of reducing the problem of reducibility of an almost periodic system of equations to the case of a system with quasiperiodic coefficients, construct a process with accelerated convergence of iterations reducing a quasiperiodic system with unbounded right-hand side, and analyze the problem of decomposition of a countable system of equations.

3.14. Erugin and Floquet-Lyapunov Theorems The theory of reducibility of finite-dimensional linear systems is well developed (see, e.g., [CoL, Pon]). In the present section, we generalize some results of this theory to the case of countable systems. 151

152

Reducibility of Linear Systems

Chapter 3

In the space Wl, we consider a differential equation

UiX

— = A{t)x,

(14.1)

where χ G Wl and A(t) = [α^·(ί)]°^·=1 is an infinite matrix. We assume that the functions a{j(t) (i, j = 1 , 2 , . . . ) are continuous in t in the interval Τ = R1 and, moreover, ||A(i)|| T l < Φτι = const < oo

(14.2)

in any finite segment T\ = [a, b] C T. Then, as shown in Chapter 1, equation (14.1) possesses a unique solution with given initial conditions xq e VJl and to G Τ defined for all t e T. Equation (14.1) is called reducible on Τ if there exist an infinite matrix Φ(ύ) and a differential equation ^

= Py,

yeTl,

||P||
(14.3)

with constant matrix Ρ such that any solution χ = x(t) of equation (14.1) can be expressed via the solution y = y(t) of equation (14.3) by the formula x(t) = Φ ( % ( ί ) valid for t € Γ. A matrix solution X(t) of equation (14.1) is called a fundamental matrix if any vector solution of this equation can be represented in the form X(t)c, where c € 271 is a constant vector. Thus, the matrizant of equation (14.1) is its fundamental matrix. A matrix L(t) is called a Lyapunov matrix if it is continuously differentiable and invertible on Τ and, in addition, TiCT

max|||L(i)||Ti,

on every finite interval. We say that equation (14.1) is reduced to the form (14.3) in the Lyapunov sense if the matrix Φ(ί) realizing reducibility is a Lyapunov matrix. The set of all infinite constant matrices bounded in the norm || • || is denoted by B. It is easy to see that the set Β forms a Banach space over the field of real numbers and the following statements are true:

p2 p

1. If Ρ € Β, then e to ß .

= E + P+

—+

At*

pk ... + — + ... exists and belongs »V»

Erugin and Floquet-Lyapunov Theorems

Section 14

153

2. The matrices Ρ and ep are commutative in the sense of multiplication. 3. The matrix e~p is inverse to ep.

4. The derivative (ePt)' = ePt · P. 5. IfPfc eB(k

t e T , then

= l,2,...),

fc=l

k=2

The following theorem is an analog of the Erugin theorem and establishes the possibility of reducing equation (14.1) to equation (14.3) with the help of the Lyapunov reducing matrix:

Theorem 14.1. Equation (14.1) is reducible to equation (14.3) in the Lyapunov sense if and only if a fundamental matrix X(t) of the original equation admits a representation X(t) = L(t) exp{Pi}, where L(t) is a Lyapunov matrix. Proof. The following assertion is true: If the matrices Z(t) = [^ij(i)]ij=i and G(t) = [ffij(i)]ij=i ^ differentiable on the segment T\ and, in addition, both the matrices and their derivatives are bounded on T\ in the norm || • ||τχ, then

jt{Z{t)G{t))

= jt(Z(t))G(t)

+ Z(t)±(G(t)),

To prove this assertion, it suffices to show that, for all i,j 00

oo

( E ^ w o ) s=l

=

teTi. = 1,2,...,

oo

+

s=l

s=l

t e τι.

This equality is indeed true because the series on its right-hand side can be majorized by the convergent numerical series oo

oo

E^I^WIIIGWIIt! S=1

and

^maxM^IIIG'WIk, 8=1

T l

T l

respectively, and, thus, they converge absolutely and uniformly in t e T\. We assume that there exists a fundamental matrix X(t) of equation (14.1) such that

X(t) =

L(t)ePi,

154

Chapter 3

Reducibility of Linear Systems

where L(t) is a Lyapunov matrix. In equation (14.1), we perform the following change of variables: χ = L(t)y, where L ( i ) = X(t)e~Pt

and y(t) is a vector function such that, on every finite

interval T\ C T , the quantities IIy(t) II and stants Pi and P2 dependent on T\. Then dx d , r . ..

^ ^ dt r

are bounded by positive con-

, .dy

This yields jt(X(t)e~pt)y

+ X(t)e~Pi|

=

A(t)X(t)e~p*y.

The fundamental matrix X(t) is bounded in the norm || · W^. Therefore, dX(t) — a l s o possesses the same property. In this case, we arrive at the equation at X{t)e~PtTt

+

l£e~Pty

- x^e~Ptpy

=

A(t)X(t)e-pty,

which, in view of the fact that the matrix X(t) is fundamental, yields (14.3). It is clear that the solutions of equation (14.3) are bounded in the norm || · U^ together with their derivatives. We now prove necessity. Assume that equation (14.1) can be reduced to equation (14.3) by using the Lyapunov matrix L(t). It is clear that any solution of equation (14.3) with initial values ίο £ Τ and yo e Wl can be represented in the form y(t)=exp{P(t-t0)}y0. We denote e~pt = C e B. Since the matrices Pt and Ρίο are commuting, every solution y(t) of equation (14.3) can be represented in the form y(t) = e x p { P i } c , where c e 931. We denote the product L(t)ePt by X(t). It is clear that X(t) is a fundamental matrix of equation (14.1) because all solutions of this equation are contained in the set X(t)c, c G Wl, since X(t)c = (L(t)ePt)c

= L(t)(ePt

• c)

and e x p { P i } c describes all solutions of equation (14.3). Theorem 14.1 is proved.

Section 15

Periodic Systems

155

Corollary 14.1. For any constant matrix Ρ e B, there exists a Lyapunov matrix that reduces equation (14.1) to equation (14.3). Proof. By virtue of Theorem 14.1, it suffices to prove that the matrizant Ως, of equation (14.1) can be represented in the form Ωβ = L(t)ept, where L(t) is a Lyapunov matrix. We set L(t) = Ω ^ " Ρ ί . (14.4) It is clear that L(t) is a Lyapunov matrix. Now assume that the matrix A(t) is periodic in t with period ω: A(t + ω) = A(t), t e T. In this case, Corollary 14.1 is also true for equation (14.1). The situation becomes more complicated if we require that the reducing matrix must be bounded on the entire real axis. For this purpose, it suffices to demand the periodicity of the Lyapunov matrix L(t) in relation (14.4) as a function of t. In the finite-dimensional case, the existence of this matrix is guaranteed by the FloquetLyapunov theorem [CoL]. At the same time, it is known [DaK, MaS] that, in the infinite-dimensional case, a direct analog of this theorem is absent. In the theorem below, we formulate conditions for the existence of a periodic Lyapunov matrix of the form (14.4). Theorem 14.2. For the ω-periodicity of the Lyapunov matrix L(t) given by equality (14.4), where QQ is the matrizant of equation (14.1) with ω-periodic matrix A(t), it is necessary and sufficient that the real logarithm of the monodromy matrix Ωβ exist and be such that P = - 1ηΩ£. ω This theorem is proved in exactly the same way as in [CoL]. The analysis of conditions for the existence of the logarithm of the monodromy operator in the infinite-dimensional case is in no case a trivial problem [DaK, MaS]. In the next section, we solve the problem of reducibility of equation (14.1) by using a real periodic reducing matrix without finding the logarithm of the monodromy matrix.

3.15. Periodic Systems In the present section, we relate the solution of the problem of reducibility of equation (14.1) to equation (14.3) to the problem of reducibility of finitedimensional systems obtained from infinite systems as a result of truncation.

Reducibility of Linear Systems

156

Chapter 3

Parallel with (14.1), we consider a sequence of systems

—

= A ( t y x n = 1,2,...,

(15.1)

where ^x = ... G i? n and Α (ί) = [α^·(ί)]£ J = 1 . Assume that, for η > N , each equation in (15.1) can be reduced to an equation . (n) d υ ~w

/ \ (") (n) =

p

y

(15,2)

(n) c with the help of a reducing matrix Φ (t) = [φ\*· (£)]" J = 1 such that

V ( t ) = Φ(ί) V ( t ) ·

(15.3)

Let us now formulate conditions that guarantee the reducibility of the original equation (14.1) under the assumptions made above. (n)

Definition 15.1. A sequence of matrices C (t) = [cij(t)]f^J=1 lar if, for every t G Τ, it satisfies the following conditions:

is called regu-

(i) it is jointly bounded, i.e., (n)

|| C ( i ) | | < P = const < oo; (ii) it weakly converges to a matrix C(t) — (iii) the series Σ^ι

[cij(t)]™j-i,

(τι)

|cij(i)|(i = 1 , 2 , . . . ) converge uniformly in n.

If conditions (i)-(iii) are satisfied uniformly in t G T, then the sequence of (n) matrices C is called uniformly regular in T. (n) If condition (iii) is satisfied uniformly in i, then the sequence C is called strongly regular. It is easy to see that the following statements are true: (n)

1. If a sequence C (t) is uniformly regular, then ||C(£)|| < Ρ for all t G T.

Section 15

Periodic

157

Systems

(n)

2. Let C be a sequence of constant infinite matrices. If there exists a ma(n)

trix S = bounded in norm and such that |cjj(t)| < S{j ( i , j = 1 , 2 , . . . ) , then the indicated sequence contains a regular subsequence. (n)

We prove the second statement. Denote ||5Ί| = Ζ. Then || C (ί)|| < Ζ for any (n)

natural n. Therefore, the sequence of matrices C is jointly bounded. It is clear that any subsequence of this sequence possesses the same property. (n)

The validity of condition (iii) in Definition 15.1 for any subsequence of { C } is also obvious. (n)

It remains to show that { C } contains a weakly convergent subsequence. (n)

We write the moduli of the elements of the matrix C in the form of a column as follows: c o l o n | | c n | , |§1|, |c2|, | S i | , iSäl, | c i 3 | , · · · } · (n)

Since every sequence \cij\ of nonnegative numbers is bounded above by a number S{j ( i , j = 1 , 2 , . . . ) , we can always select a convergent subsequence of this sequence. Thus, we select a convergent subsequence |ci*i | (i = 1 , 2 , . . . ) of the sequence |cii|, then a convergent subsequence |c2i| of the sequence |c2*i and so on. By using the procedure of diagonalization, we form the set of sequences ,(«l), |cii|,

,032). |cn|

.(73), |cn|,

|(Ql)i ,(73), I <=21 U |C21 I |C2l|, |(«l)| IC12 ,

|(Ä)| IC121

,(73), IC121,

... ...

... (15.4)

...

each of which converges as a subsequence of a convergent sequence supplemented with finitely many additional elements. We now transform the columns in table (15.4) into matrices and arrive at a weakly convergent sequence of matrices (αϊ) ( Ä ) c , c ,

(73) c , . . .

.

Chapter 3

Reducibility of Linear Systems

158

Theorem 15.1. Assume that equation (14.1) is such that, in addition to the conditions formulated above, the following inequality is true: oo Y^ max |<%·(£)| < Κτχ · ε(η) • I1 j=n+1

(s = 1 , 2 , 3 , . . . ) , Κτχ = const < oo,

where ε (π) —> 0 as η —> oo, T\ is an arbitrary finite segment of Τ, the sequences (n) (rO (n) 1 { Φ (ί)} and {Φ (ί)} are uniformly regular, and the sequence { Ρ } is regular in T\. Then (»)

(n)

lim Φ (t) = Φ(ί) η—»oo

and

lim Ρ = Ρ η—* oo

in the weak sense and equation (14.1) can be reduced to equation (14.3) with the help of the limiting matrix Φ(ί). (η) Proof. First, we note that the sequence of matrices A (t) in equations (15.1) is uniformly regular in T\. Let χ = ( x i , . . . , x n , x n + i , x n + 2 , . . . ) and χ = ( χ ι , . . . , x n , x n + i , x n + 2 , . . . ) be two arbitrary points of the space 371 whose coordinates coincide up to the number η inclusive. Denote Δ η χ = sup{|x n +i - x n + i | , |^n+2 — xn+2\, · · ·}· For any s = 1 , 2 , 3 , . . . , the following inequalities are true: oo η |^aSi(i)xi i=l i=l

-

oo a Σ si(t)xi i=n+1

oo < V] \asi(t)\\xi i=n+1

oo - Xi\ < Δηχ V] max|asi(i)| Jl i=n+l

< ΛΓτι · Δ η χ ε ( π ) , where ε(η) —> 0 as η —> oo. This means that equation (14.1) satisfies the strong Cauchy-Lipschitz conditions with respect to χ and, hence, we can use the following reasoning (see Section 1.2): (π) (η) (η) Let χ = x(t, to,xo) and χ = χ (ί, toi ^o) be solutions of equations (14.1) and (15.1), respectively, and, in addition, x (to, to, xo) =

and

(n) (n) (n) χ (to, to, xo) = ^o,

Section 15

Periodic Systems

159

where x0 = (xi,x2 ,

and

^2, · • · > x n ) ·

xo =

Then lim x (t, n—+oo s

to, XQ) — xs(t, to, xo),

s — 1,2,3,...,

(n) (n) uniformly in t e T\. In other words, the sequence χ (t, ίο, £o) weakly converges to the vector function x(t, to, xo) uniformly in t. According to the condition of the theorem, for each η beginning nwith a certain number, there exists a matrix (n) (n) (n) () (n) Φ (t) such that χ (t, to, xo) = Φ (t) y (t) identically for all t e T, where (n)

y (t) is a solution of equation (15.2). (n)

Let us show that the sequence { y (i)} weakly converges as η —• oo (uniformly in t e Ti). (n) (n) (n) For this purpose, we denote Φ (ί) = Φ (£) = [V>ij(0]£j=i· Then the sth coordinate of the vector (n) y (t) has the form

yj(i) =

ίο, ®o). t=l

If we now replace the products φ31{ί) \'x}(t, to, xo) with i > η by zeros, then we get V» w

=

x

i {t, to, ®o).

i=1 By using the Gronwall-Bellman inequality, we obtain

| | ( * ( i , t o , * o ) | | < ||xo||exp{| J II A ( O I | d i | } to <

||χ0||βχρ{ΦΤι|ί-ίο|}

< Μτχ = const < oo, Then

00

i=l

(n)

( 1

/ -v to, so) <MTl

00

i= 1

t, to, € T\.

(n) ΣΙΓΜΟΙ.

(15.5)

160

Reducibility of Linear Systems

Chapter 3

where the series on the right-hand side converges uniformly in t e T\ and n. (n) According to the condition of the theorem, the sequence Φ (t) is uniformly regular (n)

(n)

and, hence, there exists a matrix Φ(ί) = [zy(t)]ij=i such that lim ipij{t) = i>ij(t)i h j = 1) 2 , 3 , . . . , uniformly in t € T\. By virtue of Lemma 12.1, sequence (15.5) converges, as η —• oo, to a function y3(t), s = 1 , 2 , . . . , uniformly in t e T\. Thus, it is shown that the sequence of vector functions { V ( 0 ) converges weakly to a vector function y(t) uniformly in t and, in addition, y(t) = Ψ(t)x(tt to, xq). Similarly, one can easily show that x(t, to, xq) = Φ(£)ι/(ί), where the matrix Φ(ί) is the weak limit of the sequence (ί). It is also easy to see that, under the conditions of the theorem, the matrix Φ(ί) is invertible. Furthermore, the inverse matrix Φ _ 1 ( ί ) coincides with the matrix Ψ(ί) for all t Ε Τ. It remains to show that y(t) is a solution of equation (14.3), where Ρ = (η)

lim Ρ in the weak sense. n~*oo By using relation (15.2), for the sth coordinate we get dyht) =

^(n)(n)/x

(15-6)

i=l (n) (n)

identically for all t € T. We note that, for i > n, the products p3i yi (t) in equality (15.6) should be set equal to zero. It is obvious that ll ( ?(t)|| < 11^(011 I I ^ V ίο, Si})II (n)

< M T l IIΦ (ί)|| < Κι = const < oo. Then E t ö f c ' w s E l t ö l l l ^ W l s ^ E l f ö l · i=l i=l t=l This means that the series on the right-hand side of equality (15.6) converges uniformly in η and t GT\. In addition, I PsiVi(t) - ρ si yh0 I ^

I Psi I 12h{t) - y}(t) \ + | {${t) \ \ pai - ρ3ί \

< ||P|| ι Ui(t) - ykt) I

161

Periodic Systems

Section 15

(τι) S One can always find a number Ν such that |pSi — pai \ < ~rr and \yi{t) Λ1 Jfo (*) I < "ii^n" f° r all η > iV and t e T\. Here, ε is an arbitrarily small positive (n) (n) number. This means that the sequence {pai Vi (έ)} converges uniformly for all i = 1 , 2 , . . . . In this case, we can apply Lemma 12.1 to the series on the righthand side of relation (15.6). Therefore, 00

„Ü™

=

(n) (n)

00

rί&,ΣΛ Vi (t) = Y^PsiViit) 1=1 i=l and, moreover, the sequences in this equality converge uniformly in t G T\. Hence, = f^PsiViit), a = 1 , 2 , 3 , . . . , i=l which completes the proof of Theorem 15.1. We now apply Theorem 15.1 to the case where A(t) is a matrix periodic in t with period 2π. Theorem 15.2. Suppose that the elements of the matrix A(t) are continuous, periodic in t with period 2π, and such that Σ max|a s i (i)| < Κε(η), j=n+1

s = l,2,..., (n)

where ε(η) —» 0 as η —• oo and the sequence of matrices

Φ(£) reducing (n) equations (15.1) to the form (15.2) satisfies the following conditions: { Φ (£)} (n Li and { Φ x (t)} are uniformly regular and there exists t* € Τ such that the se(n) d (n) quences of matrices { Φ (£*)} and {— Φ (£*)} are strongly regular. Then Φ(ί) = at (n) (n) lim Φ (t) and Ρ = lim Ρ in the weak sense and equation (14.1) can be ren—*oo n—»co duced to equation (14.3) with the help of the matrix Φ(ί) periodic in t with period 4 π. (η) Proof. Clearly, it suffices to show that the sequence of matrices { Ρ } is regular (») and use Theorem 15.1 because the existence of reducing matrices Φ follows from the Roquet theorem for the finite-dimensional case.

162

Reducibility of Linear Systems

Chapter 3

(n) (n) First, we prove that if the sequences of constant matrices { C } and {D} are (n) (n) regular and strongly regular, respectively, then the sequence { C D } is regular. (») (n) The sequence { C D} is jointly bounded because (n) (n) (n) (n) II C D II < II C II II D || < BBi = const < oo. (») (n) In addition, lim C D = CD in the weak sense. Indeed, the element of the n—t-oo ( n ) (n) „ (n) (n) matrix C D located in its ith row and pth column has the form djp · This series converges uniformly in η because („)(«) ^ >),,,(") (η) . M <2> M . ^ djP < Σ I (kj HI D || = || D || | Cij ^ I) j=ι j=ι j=ι and the series on the right-hand side of this formula converges uniformly in τι (n)

(η)

Μ

because the sequence { C } is regular. Clearly, lin^c^· djP = CijdjP and, hence, 2—i Cii jp ~ 2cijajp· 3=1 j=i It remains to prove the validity of condition (iii) in Definition 15.1. Since the oo (n) series Σ I 0, 3=1 (η)

(η)

one £ can find a number I independent of i and η such that | dik | + . . . + | difc+m | < — for all k > I and m — 1 , 2 , . . . . Then the following inequalities are true for all Β natural η and s: k + T n 00 k+m oo ( n ) νW/ (n) (n) ^ ] ^ ^ Csi dip < ^ ] ^ ] I Csi I I dip p=k i=1 p=k j=1

—

Σ I i=1

^ Σ

Csi

fc+m (n) I Σ \dip\ p=k 5 <]gllC|l<e·

Section 16

Systems with Almost Periodic Coefficients

163

OO oo ( n ) (n) Hence, the series Σ I Σ c si diP\ converges uniformly in η for s — 1 , 2 , . . . , p=1i=l (n) (n) i.e., the sequence { C D } is regular. Note that, for any n, we can write d{V s x

= I

d (n) in) (n) d^x(t) (n) φ (t) · (x\t) + Φ { t f - ψ - = A(t)

(n) in) Φ (t) {$(t),

t e T.

This yields d τ (t) (n), r(n) (n) d (n) τω ^ = Φ - Η φ (*) Φ (*) - ^ Φ (*)] (η) (η) (η) (η) (η) ^ (η) Therefore, the constant matrix Ρ = Φ _ 1 ( ί ) A (t) Φ (t) - Φ _ 1 ( ί ) — Φ (t) identidt cally for all t e Τ including t = t*. (») Since A (t) is a strongly regular sequence for any t € Τ and the sum of regular matrix sequences is a regular sequence, by virtue of the statement proved above we conclude that the sequence of matrices (η) (η) (η) (η) (η) Ρ = Φ" 1 ( Ο A(tm) Φ(ί*) -

^ (n) Φ (Ο dt

is regular. Theorem 15.2 is proved. Remark 15.1. The period is doubled only in the case where the period of the (n) reducing matrices Φ (t) is equal to 4π (n = 1 , 2 , . . . ) .

3.16.

Systems with Almost Periodic Coefficients

We consider a system of equations 4f =«(„),

f

=

(16.D

where χ (Ξ Rn, Ρ{ψ) = [Py (<£)]£,·=χ is an η χ η matrix, = (
164

Reducibility

of Linear

Chapter 3

Systems

to the case α(φ) = ω, where ω = (α;ι, . . . ) is the frequency basis of an almost periodic function. The problem of reducibility of systems with almost periodic coefficients was studied by numerous authors, and the most interesting results in this field were obtained for systems with almost constant coefficients. A system of this sort has the form dip

dx

„,

.

= Ax + εΡ(<ρ)χ,

(16.2)

where A is a constant matrix and ε is a small positive parameter. We now study the problem of reducibility of system (16.1) to a system with constant coefficients in normal variables of the form % - φ ) ,

ft=By.

«6.3)

We relate the solution of this problem to the problem of reducibility of a sequence of truncated systems (
— = A(t)x,

(16.4)

where χ = (χι, X2, · · •, χη) £ Rn and t e [a, 6] = if exists a sequence of equations of the form ,( m )

ax

I \ (m) ( (m))

^ = A ( t )

x

C R1. Assume that there

(m = 1 , 2 , . . . )

(16.5)

(m)

reducible (with the help of reducing matrices Φ (t)) in the interval T° to equations of the form ,(m) / \ dy (™)(m) , = B Y (m = 1 , 2 , . . . ) , (16-6) (TO)

where Β are constant matrices, and, in addition, that, for any solution χ = x(t) of equation (16.4), there exists a sequence ^x — ^ x \ t ) of solutions of equations (16.5) that converges to x(t) uniformly in t G (TO)

(TO)

If the sequences of matrices Φ , Φ (m)_

(TO)

(i), and Β converge as m —• oo

and, in addition, the matrix { Φ 1 ( ί ) } converges uniformly in t e I f , then, by

Section 16

165

Systems with Almost Periodic Coefficients

the change of variables χ = Φ(t)y, equation (16.4) is reduced to the following equation: (16.7)

ft=By, where Φ(ί) -

lim

(m)

m—• oo

Φ (ί) and Β =

to—>

(TO)

lim Β .

οο

Indeed, for every natural number m, the change ^x = Φ (t) ^y* reduces (m) H , (to) equation (16.5) to equation (16.6). Since y = Φ (t) χ , we have (TO) (M) , (TO) 1 lim V = J m Φ - 1 ^ ) V = Φ ~ H t ) x ( t ) = y(t), TO—» OO TO—>00

_i

(mL (t) stands for lim Φ (t). In equality (16.6), we pass to the limit as TO—•OO ( M ) (TO) τη —» oo and take into account the fact that, in this case, Β y —> By(t) uniformly in t. This enables us to conclude that y(t) is a solution of equation (16.7). (TO) Instead of the convergence of the sequence { Β }, it suffices to assume that

where Φ

(TO)

^ (TO)

the sequences { A (£)} and { — Φ ( t ) } converge at least at one point t* e This follows from the representation (TO)

(TO)

(TO)

(TO)

Β = Φ " 1 ^ ) A (t) Φ (t) -

(TO)

^ (TO)

Φ - 1 t ) - r Φ (t). at

Let φ = φί(ψ) be a solution of the equation for angular variables in system (16.1). Then we can write dx

-

(16.8)

= Ρ(η{φ))χ.

We denote the solution of this equation taking a value XQ € Rn for t = 0 by χ = xt(xo)· Assume that the matrix P(
eT\.

166

Reducibility of Linear Systems

Chapter 3

To solve the problem of reducibility of the system of equations (16.1) to system (16.3), we write, parallel with (16.1), the truncated system άψί , — = ai{(pi,...,
.

dx — = Ρ{φι,...,φτη,0,0,...)χ,

(16.9)

where m is an arbitrary natural number and i = 1 ,m. Let (m) Am) Am) Am)
(τη) (τη) ψ)}

be a solution of the equation for angular variables in system (16.9), where ^p = {
We set ψ= (0, . . . , 0, V?m+1, ψτη+2, . . .),

Ψι{φ) = {

and (m) Am) φν{φ)=

(m) ψ .

This enables us to write the equation dx n/H,(m) — = P{ipt{