t/u.s paper we gwe a covdition for which the charateristic spectrum o f tÃc sequence of l i n t a r drjfereĩittaỉ equati on systeTUS are s t a b l e

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V N U J O U R N A L OF SCIEN C E. N a t . S c i . . . \ XV . n‘^5 - 1999

O N U N I F O R M S T A B I L I T Y O F T H E C H A R A T E R I S T I C S P E C T R U M F O R S E Q U E N C E S O F L I N E A R

D I F F E R E N T I A L E Q U A T I O N S Y S T E M ’

N g u y e n T h e H o a n

Faci l it y o f Mỉitheiiìatic^, M e ch a ni cs a n d InibriuHtics College o f Natiiial Scicjices - V N Ư

D a o T h i L ie n

TeHchei \s T r yi ng Coilege. Th ai N g u y e n U n i v e r s i t y

A b s t r a c t : / / ? t/u.s paper we gwe a covdition for which the charateristic spectrum o f tÃc sequence of l i n t a r drjfereĩittaỉ equati on systeTUS are s t a b l e . T h is c on d it i on IS i m p o s e d 071 the coeffi.cemts o f s y s t e m s . The ob tai ne d results are appli ed f o r s tu d y i n g un tf on n roìighĩiess.

I. I N T R O D U C T I O N

C o n s i d e r a s e q i i e i i c e o f s y s t f ' i n s f o n s i s t i i i g OÍ l i n e a r c l i f f e r e i i t i a l e q u a t i o n

^ = . 4 „ ( n . T , T e n ' \ n = ( 1 )

fit

where An{f ) a X - m atrix contimious 01Ì [^O'Oo) and satisfies th e condition

sup ^ < ^{C X ) ,}'// — 1, 2, • • • (2)

t>fa

denote by the chaiatoristic sp e ctru m of th e system (1).

Let MS a s s o c i a t e w i t h (1) a s c q u e u r e of n o n - li i u' a r syyteiiis

^ = A „ ( / ) . r + / „ ( / , . ' ■ )

(if

p e rtu rb e d by the function ỷ n ự - x ) satisfying th e relation

| / n ( ^ - r ) l | < <^».l|-i'||.0 < Ố,. < Ỗ < CO. (4) As well known, Ishe above assum ptions imply t h a t the ch arateristic spectrum s of th e sequence (3) are a b ou n d ed set.

Denote by A„ the charateristic sp ectru m of (3).

* T his paper was supported in part by the National Research Program in N atural Scieces, K T 04, 137

28

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D e f i n i t i o n . T h e c h a r i i t e n s t i c s p c c t r u u i o f (1) is'saici t o i>e ii iii fonui y ì ỉ p p c ỉ - s t ỉ i h ỉ e i f for Hu y g i v e n e > i) t h e r e e x i s t s Ò = ố{ f ) Sììch t h a t t h e H s s ỉ u i i p t ỉ u n (4) i m p l i e s

O n u n i f o r m s t a b i l i t y o f the c h a r a t e r i s t i c spectr^um f o r , . . 29

for all // € A „ .

If the assiiniption (4) implies

th en the ch a ia ti'iistic sp(‘ctru n i of (1) is said to be uniformly lower-stahlo.

If bo th the inoqualitios (5)-(6)hold. thoỉỉ tlif* charateristic sp ectru m of (1) is said to be uniformly stable.

Tho notion of uniforin staỉ)ility of a charateristic sp ectru m for th e sequence of differential equation syst(niis is used in tho s tu d y of uniform roughness of this sequence and, in tu rn , the Iinifonii loughiioss of the sequence of dirft'rential equation systems is used in estiin atio n of Iiunxber of stab le periodic solutions of th e differential system s [1.

II. S P E C I A L C A S E / „ ( / , . r ) -

First of all. wo roiisiiler tlii‘ special case when* is linear in x ,th a t is:

fn{t . -r) - 'I'hen th(' system (3) is of tilt' fonn

tỉ-i'

- .4,,(/)./■ + / ? , ( 0 . / - , // - | | ữ „ ( / ) | ị < < (S V / > / o (7)

ih'iiuti \>\ \ \ , aÍ, ^ < Aj the (. li ai nt t 1 i^it ii 11 liiii of (T).

A]>plyiiig P fM o n ’s tian sfo iin a tio n

■ r ^ U „ { t ) y , ( 8 )

where ư„{t ) is an o rth o g o n a l m atrix , tlu' system (1) is reduced to the trian gu lar one

§ = p»(')!/- (9)

wliere Pn{t) = { f ) A„{t ) Un{t ) - It is easy to verify th a t

\\Pn{f)\\ < M l , n = by th e traiisfo niiatio n (8), th e system (7) becomes

dy

(if = P n [ f ) y + Qn{f ) y, (10)

where Qn{ t ) - ^ { f ) Bn{ t ) UnỰ) - D enote by P u \ t ) , p \²\ f ),P^{2 2} Ự) elements of the m atrix Pn{t).

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3() N g u y e n The H oan , D a o T h i L ie n w<‘ rewrite the syst('iii (10) a.s

<i)l

w h o I P

(h

K ( t ) =

— { ^ ) // + Q i i ( n fl ■ (1 1 )

V 0

i o Q „ ( 0 = Q n ( 0 +

Obviously, if B „ ( f ) < ^Ò: ⁿ = 1,2,... \vr liavc iic ? „ { /) l| < ^{I I =} 1.2,...

From tlu' rolatioii |1P„(0|| < Mị - 11= 1.2,... and bv applviiifi, th(> tra nsfo n n atioii

/— — VVO’ can verify tliat

V " V

Q,At)\\<

²

s/Al^.

⁽¹²⁾

Since Pn{f) is a diagonal m atrix, the solution \'(ĩ ) ot tli(' systf'iii (11 ) with the initial condition y{fo) = yo of the form

^ ( / ) = e x p ( I Pn{ r ) dr ) .ự(, + I ('xp ( - I Pn{s)<ls)Qn{r)y(T}<ỈT

■'to ^ - / / u 'U)

O r equivalent Iv.

e x p ( - I P n { r ) ( Ỉ T ) t j ị f ) = : (jo I ( ^ x p ( - I f \ { ' ^ ) d ' ^ ) Q n { r ) y { T ) < l r .

■Itu no -flu

Henc(\ WT obtain

( ' x p ( I P „ ( . s ) i / . s ) / / ( / ) < IIi/oII + I | | f ' x p ( - I f \ ( s ) ( l s ) C ^ „ { T ) o x p ( I F „ { s ) ( l s ) \

' Ita ■'/„ ^ h o 'h,

/■' - M

X

I

('xp( - / P,As)(ls) ' !

' -'U)

Denote by (i,i= L 2 ) elements of the m a trix Qn ự) - T h en bv stiaightforwaK calculations, \V(' liaví'

e x p ( ^ - Ị Ạ , ( r ) ( / r ^ ộ „ ( 0 < ' x p ( y Ạ , ( r ) f / r ) =

_ / '7Ỉ2’ (0<'>;P ( , / , „ [ / 4 2 ^ ) - /'ii V ) ] ^ /t) ^

Fioin the proof of Penoii'^3 thoon'iii \V(' (huhu'o i==l-2 ; U=L2,... wliPK^

{t), i = l,2 art* diagonal ('Irnieiits of the m atrix U~^( t ) A^, {t ) Un( t ) . As an orthogonal t raiisfoniation in the plaiio. P erron's m atrix u „ ( f ) in this case is of t he form

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O n u n i f o r m s t a b i l i t y o f the c h a r a t e r i s t ị c s p e c t r u m / o r . . . :n

o r

n __ { - s i n v j ( ' ') ( 0 ^

C O S ^ ( " ) ( 0 ,

whe re is t he aii^le betwe(‘n a s o l u t i o n o f (1) and t hv axis Tị. a direct c o m p u t a t i o n s h o w s t h a t

l > \ V i ^ ) ~ P 2 2 Ì ^ ) = P n ^ ( ^ ) - P 2 2 ^ ( 0 = < ' o s 2 < ^ o ^ " > ( f ) Ị n ị " ’ ( f ) - a ị 2 ’ ( 0 ì + ^ i n 2 ự > < " > ( f ) l a ị ' Ị ^ + o . ị : ^ > ( f )

/ 4 2 V ) V , ”*(^o = /í Ì2 V )-P Í" V ) = cos2^<”)(0[o.i:]>(0-«iiV)l-sni2v^('')(0[4i^+«i”V )

Th(MefoK\

p ' j ; ' ( n - / : ; > i t ) = ự í i ^ l V í ' ) - + [ « y ; ' + , / , ' ; ' ( 0 P ) ^ < ™ | 2 i ' " ' ( / i + 4 -,.{ ()

in which

A " ' ư ì - " ' ă m and

- p\:ht) = V {!"n V ) -

⁴

;;V)]^ + [

⁴

" ^ + «i;;’(

⁰

in- C'OS [

²

^<")(

⁰

+ ^„{t)

W'UoiV ^ t , ự ) - ^I^Tí(0 +

l)(‘llOĨ(‘

- V [ " n V ) - 4 2 * ( 0 ] '’ + ỉ4';^ + Thí' ahovf' rcasoiiiii^ ^iv(‘ us

! | e x p ( - i P n { T ) d T ) Q n { t ) e . \ p { i P n { r ) d r ) < M ^ V S x

'fo -fto

{(>xp( f n „ ( r ) cos [2<ỉ>*'''^(r) - 'i'n ( r ) ] i / r ) + pxp ( / n „ ( r ) COS [-7T + 2i^^” *(r)

■ho . ' t o

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Assimie

VLj,{t)(Ìt < c < oo, n = 1,2,....

then

p x p ( - / Pn { r ) d T ) Qn { f ) e x p

Ị

Pn{r)dT < A/;jV^.

■ ' t o ' ' t o

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(16)

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32 N g u y e n The H o a n , D a o T h i L i e n T h e inqualities (13)-(16) imply th a t

e x p ( - / P u H T ) d T ) y i ( t ) < e x p { A h \ / ố ) { t - t o ) ,

■ho

e x p ( - / P u \ r ) d T ) i j2{ t ) < exp (A/3 v 4 ) (f - ^o)-

■ho

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( 18)

From (17)-(18) and properties of trian g u lar system s we deduce x ị yi ự) ] < A/3V^ + A ị"\

\[yi(0] < A/3V^+

R em ark t h a t the transfoinations used in the above reasoning do not change the charateristic sp ectru m of clifFeretial equation systems. Therefore, if

then

Hence,

x ị y i ự) ] < .^2"^ -h f. (19)

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We f i n i s h t h e s p e c a l c a s o by g i v i n g a l o w e r V)Ouiul f o r For

this purpose we assum e th a t (1) is regular. Let

7 i £ > 2 ’ ^1 ^ 7 - 2

denote th e s p e ctra of th(' adjoint syisteiiis conosponding to (1) and (7). Then hv P m o i r s theorem and Ly ap uno v' s iiioqiiality ^{W ( '}liaví'

A‘" ) + 7 Í ” ^ = 0 , Ã Í ' ' ’ + f , " > > ( ).

Applying (20) to the bigger charateristic exponent it yields

7',” ’ < ^

or

Thus,

ã1'” > À<” > - (21)

Sumiĩig up, we heve the following:

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O n u n i f o r m s t a b i l i t y o f the c h a r a t e r i s t i c s p e c t r u m f o r . . . ₃₃ L e m m a ,

For f sinali ciioĩigb and

i i t , { r ) ( l T < ( ' < oo, 7Ì — 1 , 2 , . . . ,

■ft where

We have

!!» ( ') = + l<4” ' +

Moreover, if the ^{Al l}s y s t e m s o f (1) are regiilm-, we ^{h a v e}also

f ,7i = 1 ,2 ...

Now, t he gpiieial ca.se can be reduced t o the s pecial one by mpans o f t he linear i n c l u s i o n p r i n c i p l e (.S('‘e [3]).

T h e o r e m . Undcj the as sumptions o f the ỉeniiim, the diHiateristic sp ectniin o f the se

quence o f s ys te m s (¹) is uniformly uppei-stĩible .Moreover, i f all s ys te m s (¹) are regiilar the charaxeristic s pcc tn iin ot the seqiience is HÌSO unifoiiniy luwer-stnlile and hence it is unifornily stỉìhlc

Proof. Let .r{t) be a noutri\-ial solution of (3). Accoi'ding to thf> linear iiiclusloa princriple in [3],

x{t)

is a nontrivial s ol ut io n o f linoar s y s t e m

d:r l i t If (4) hold, then

<Ị)„(7) < -^ ,» = 1,2,...

SiiK'c tliP systPiii (22) is linear \V(‘ can a pp ly the a bov e Ipmiiia and then (23) gives us

;tlx(.)| < Aị"> + Í If (1) is regular, then we have

R e m a r k . For aỊ"* = this nnphes the result in [2j.

(2 2 )

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Now wo shall s tu d y the uniform roughness of the following sequence of differetial oquation system:

d x

(If (24)

w h e r e , An{ t ) is a n m X m -ư iat.rix w h ic h is c o n t in u o u s a n d b o u n d e d o n

[^0

^{o o ).}

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34 N g u y e n The H oan , D a o T h i L i e n D e f i n i t i o n . Systenj (24) is said to he uniformly rough if there is H positive ỉiiỉinl)Cỉ Ố, siicii that for every iiiHtrix Bn( t ) sHtisfyiiig the relntioii:

the systems

d r

I t (25

have only nonzero chaiHteristic exponents.

Let Aj, = < A.2"^ < ... < n = 1 , 2 , . . . . be th(' charactiMistic sptH-trulu of (24). Tho following condition is necessary for th e u n ifo n a roughiiPss of sy stem (24):

P r o p o s i t i o n 1. Sup pose tha t sys tem (24) is uniformly roĩỉgh. Then there is an intcivfii (a,/?), contaiiiing zero, Sĩiclì that:

(o, i?) n =

for every n = 1, 2,...

Proof: We prove this by contradiction. Suppóse th ere is a sequence of characteristic

e x p o n e n t s € Aa:} ( I < 7/, < m ) s u c h t h a t

Consider tho sequence of systems:

(Ix

1ĨĨ

lini — 0, k--oc

A, { f ) - (26)

where I i.s tlu* unit inatiix. For suitably largf' /.■ ||A/ / < hut a / - a / — 0 is in tlií' characteristic spo ctn u ii of (26). This coiitiacts with uniform roui^li of (24). Ộ

Definition of unifoiin stability of spectruni for seqiK^ncí' (24) is .siiiiilai to tho U]|(’ ill section 1: inequality (5) is changed by ỊÁ < Am ^ f .

P r o p o s i t i o n 2. . Assui ii Ji ie that there is ÍÌ¹¹ iiitei vai ( a , /3) co/itaiiiiiig zeio, such that either ( - 0 0, /?) n An — 0 n - 1,2,...

or (a, -f-0 0) n = 0 1) = 1, 2,...

Mo re over , s ỉ i p p o s c t h a t t h e c h a r a c t e r i s t i c s p e c t n u n o f (24) is ỉi/]jfoiiijiy stnhle.

Then the a/jove systeij] is lUiiforiJiIy lOiigh.

Proof.

T he proof follows directly from its hypothe'ses and definition, ộ

Consider now th e case in = 2. T h e proposition 2 and the proved th eo rem give us:

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C o r o l l a r y . Suppose tb ^t there is iiii iỉitcivíìl { n j 3 ) conti^ìiiiiĩig zero Ỉìiiiỉ satisfying condi- tioii o f proposition 2 for the CHse Iii^2. Moicovct , suppose tỉiHt:

On unifoT'rn s t a b i l i t y o f the c h a r a t e r i s t i c s p e c t r u m f o r . . . 35

I ^V + h'ViV) + a[

²

Hr)Y-<lr < c < oo{n ^

¹

,

²

,...)

l l i c n t h e sc(ỊĩieỉiCC UÍ' s v s t ci i i s (Ỉ) is uiiiforinlv roiigh.

REFERENCES

1] V.I. Pliss. IvieẠỊìuì niaiitfolds of per'iodic systems. Moscow. 1977 (Russian).

2] L. A. Aiulii anov a. IJuifonii s t a bi l i t y o f c h a ra t ei i st ic e x po n en t ia l n umbe rs of se- queiK’es of I(‘gular systoiiis. Dỉff. ưraveĩiernịa, Nơ 10, 1974 (Russian).

3] B.F. Bylov, R.E. V'inogiad. D.M. Gi'ohinan, and v . v . Nemyckii. Theory of Lyct- puTiov Exponents. Nauka Press, Moscow, 1966 (Russian).

4] B.P- Di'uiiduvich. Lecf urts ⁰Ĩ) Mafhernaf ical Theory of Sfability. Nauka, 1967 (Rus

sian ).

T A P CHÍ K H O A H O C Đ H Q G H N , K H T N , t . x v , n^5 - 1999

Sir ON ĐỊNH DEIJ CƯA PIIO DẶC TRƯNG CỦA DAY ÍỈẺ P H r ơ N C TRÌNH VI I^HAX TUYEN TÍNH

T l i p H o iin

/ v i j O r i Tuííii - Cơ- Tin học - Dại học K H Tì/ Iihiẻĩi - DH Q G H à Nội Đ à o T h i L i ê u

Khoíì Toán, Đại ^{Ỉ Ì Ọ C}Sìĩ p h ạ m Thái Ngĩìvèn

Trong bài báo này, chung tôi đ ư a ra m ột điều kiện ổn đ ịn h của phổ đặc trư n g của một (lảy hè phvrưiig tiìn h vi phản tu y ến tính. Điều kiện này đ ư ợ c đ ặ t lên các hệ số của he p h ư a n g trìn h tư a n g ứĩig. Kết quả nhận đ ư ự c ílirợc áp (lụng cho việc nghiên cứu sự thò đều.