On the comparison problem of the stability for non lineaVề bài toán so sánh tính ổn định của hệ động...

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OF SCIENCE, Nat.Sci., t.XII, n94 - 1996.

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VNU. JOURNAL

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ON THE COMPARISON PROBLEM OF THE STABILE
FOR NON LINEAR DYNAMICAL SYSTEMS
PERTURBED BY SMALL NOISE
Nguyen Huu Du

Faculty of Mathematics , Informatics and Mechanics

Hanoi National University, Viet Nam*
ABSTRACT

This paper deals with the comparison problem of stability of differential equ
perturbed by non - linear small noise. We suppose that the the linear system

dZ, = a(t,w)Z, dt + A(t,w)Z, dW; ;

Zo =2€ Rt

is strictly stabler than the system

dY, = W(t,w, ¥;)dt + B(t,Y;,w)dM%;

Yo=ye R?

then, under the assumption of the regulity of (1), it is proved

that the system

dX, = (a(t,w)Xe + f(t, Xt) dt + A(t,w)ZedWM; = Zo =z Rt
is strictly still stabler than System (2) provided f(t,x) satisfies the condition

[f(t,2)| < & min{|z|%, z|""?};

œ>1>>0

I. INTRODUCTION
As is known, investigating of the fact whether a given dynamical system is sti

unstable is important in both theory and application.

Therefore, many definitions

stability of systems are given (see, for example, [6], [7],[3]) and there are a vast a
of works dealing with criteria by which we know whenever a given differential equz

stable ( see [6], (7], (3],...). Among these criteria, the Lyapunov exponents of solutic
a powerful tool mainly because of its importance for explaining chaotic behaviour

systems (see [1], [2],...). Furthermore, in order to study the stability of linear syste

general, we have only to consider their Lyapunov exponents.
their trivial solution X = 0 must be stable.

If they are negativ

But as to our knowledge, there is no definition which allows us to compa

”degree” of the development of systems even they are defined in a same space an
the same dimension.

In some cases, this comparison is necessary because many

te:

problems require us to choose a system which is the less chaotic the better ama

given systems.

On the other hand, studying the Lyapunov exponent of a function means t
compare this function with exponential functions. However, the class of expo
Hoa,

* The work is done under the support of Seminar of ” Numerical Analysis” monitored by Ph.D Ngu

Faculty of Mathematics,

Mechanics

and Informatics.

30


contains not many informations of growth rates because they are monotonous.

re, if we

replace

this class

by

a larger one,

we

hope

to have

more

informations

behaviour of the considered function.
‘
ng on this idea we give a concept for comparing the growth rate of two systems.

sical definition of stabilities can be obtained by comparing the considered system
trivial system

X = 0.

ides, by Lyapunov Theorem for the. Stability (see [4],pp. 267), if the linear system
jentialy stable then it is still stable under small noise. We want here to generalise

ult in the point of view of preserving the “order” of stabilities. It is proved that
m (1) is stabler than (2), then it is stabler than (2) under small non linear noise.

2 article is organized as follows: Section II introduces a definition for comparing
vility between two systems whose states are described by stochastic equations in

ce of real noise or white noise and we give some remarks on this definition. In
IIIf, we formulate the main result. It is shown that under the small noise f(t,z)
: regulity of the linear system, System (3) is stabler than (1).

Il. COMPARISON

OF GROWTH

RATE OF DYNAMICAL

(Q,Z,. t > 0, P) be a stochastic basis

SYSTEMS

satisfying the standard conditions (see [5])

1.4 > 0) be ad- dimension wiener process defined on (9, Z¿, t > 0, P). We consider
astic system

described

by the following equation

{ dX,
Xo

=a(t,X,,w)dt + A(t, X;,w)di

(2.1)

=reRrt

or all x € R4 , ((a(t,r)) and (A(t, x) are two stochastic processes F,- adapted with
n ?? and in the space of dx d- matrices respectively such that

a(,0Z0 —
pose

A(0)=U

that for any x € Rt, Equation

P=as

(2.2)

(2.1) has a unique strong solution.

Let us

he classical definition of stability in Lyapunov’s sense. Denote by X(t,z,w) the
1 of (2.1) starting from x at t= 0. From (2.2), it follows that X =0 is a solution
ation (2.1).
jon 2.1.

The trivial solution X = 0 is said to be stable if for any «> 0

lim p( ott,
sup | |X(t,r,w)|
jim
(t.2,w)| >> ¢ ) =0

5] pp.

limcans

206 ).
we

K(t,2,w)| <
|

pret

(2.3) 2.3

It is known that in fact considering whether a system

compare its ‘solutions

means

with

|X(t,2,w)| < ¢(t)

constant

functions

because

for any 1 > 0 where

that this definition gives no information

when

the

is stable
relation

(() =cVt>0.

the solution X(t,z) tends to

ye or to 0. Thus it requires us to consider a larger class of functions to know more
avior of systems. We now realise this idea.

ide of Equation (2.1) we consider the equation

|

{ dY,
Yo

= (t,Yi,w)dl + BL, Y;, wd

44

= yo € Ke”

oe y)) and (B(t,y)) satisfy the same hypothesis as (a(t,y)) and (A(t, y)), ie.
6(t,0)=0

H(,0)8U
31

W>0

P-as

(2.5)

We writé for Y(t, x) thé solution of (2.4) starting from „ at £ = 0
Let C the set of all’ positive continuows furtctions from

[0,00) into Rt and

subset of C.

Defitiitfoti 2.2. The trivial solution X = 0 of System (2.1) is said to be stabler t
solution Y = 0 of System (2.4) in the comparing class M if for any q € M, the rel

follows that

Definition

Tiny P{lY(,v)|<œ

for all. t>0} =1

lim P{\X(t,z)|
forall

2.2 is an extension

following theorem.

44

|

t>0}=1

of the classical one of stability.

Indeed, we h

Theovem 2.3; System (2.1) is stable in sense of (2.3) if it is stabler than the trivial

Ÿ=0,

Yo=ueR“2

on the class C.
Proof: If (2.1) is stabler than (2.8), then it is easy to see that (2.1) is stab
every solution of (2.8) is constant. Inverselly, suppose that (2.1) is stable and q
dat

0
q: = 0 then Equality (2.6) does not hold.

Meanwhile

if

inf

Hods which implies that
18 lim P(

ice System
Theorem

sup

0
[X(t,z)|
b> 0)

(2.1) is stabler than System (2.4). Moreover, it is easy to prove that
2.2:

If MCC

consists of all functions having

every stable system is stabler than any unstable system.

Exaniple:

q=k>0th

0
the exact limit as t —¬

Both two systems

X-X4+2X =0
Ÿ-9Ÿ+2Y
=0
are unstable. But it is easy to see that (A) is stabler than (B) in C.
Ill. LINEAR REGULAR

SYSTEM.

We introduce the so-called regular system as in [4]. Let us consider the lineal

dZ,=AZidt+ Bi2dW,
‘

where

A;,

condition.

B,

are two

ptf

T
0

stochastic

processes

[Ail dt < 00} = Pf f

T
0

with

2a=zeRt
vatues

in

|Bildt
dx

d- matrices

forany

“This condition ensures the existence of strơng sịlutions of (3.1).
32

T>0

satisf

Ì

Z(,z)

be the solution of (3.1) starting from

t. of Z(t,z) defined

z. WWe write for À[z] the Lyapunov

by

Ä{4= lraup : In |Z(t, 2)|

(3.2)

toro

ase where the limit in (3.2) exists, we say that

hs known

that

s of n random

(see

[1], [6]...)

variables, namely,

the Lyapunov

MS A2 Ss
tion 2.3.

(See [4] pp.

165).

System

Z(t,z) has an exact exponent.

spectrum

of the solution of (3.1)

San

(3.3)

(3.1) is said to be regular if there exists a

nental system of solutions Z(t) such that the column
ent and takes all values \;, i= 1,2,...,d in (3.3)

vectors of Z(t) has the exact

t the comparing class M consist of elements ¢ € C having the exact limit

1 SES

(3.4)

teeoe t

y that (2.1) is strictly,stabler than (2.4) if the condition (2.7) is replaced by: There
M

such that ạ* < ÿ and

lim PUX(L2) sasy to see that
ently stable.

(2.1) is strictly

Wi>0)=1

stabler than

(2.7)

(2.8) in M

if and

only

if (2.1) is

‘em 3.2. Suppose that (3.1) is regular and strictly stabler than the system

dy; =a(t,¥i)dt+o(t,YdWi

= Yo

ye Rt

(3.5)

the perturbed system

AX, =(AdXi + ft, Xd] t+ BX dW, = Xo =z RE
| strictly stabler than (3.5).
ndition:

There

|

Where f(t,r) is a locally Lipchitz function satysfying

are constants œ >1

> >0;

K>0

such

that

J/ứŒ.,z)| < K. min(|z|*, |zl'~)
roof:

From

the assumption

(3.6)

(3.7)

of the regulity of (3.1), we can find a fundamental

n of solutions of (3.1), namely Z(t), such that: if

&(t) = Z(t).exp{-At},

A = diag{A1, Ao)... Aa}

. 7 In |(¢)| = jim [In
aO-"(0)| = 0
fim,
fore, for any y > 0 there is a random

(3.8)

variable N such that

|Z().Z—!@@)|< N.exp[(u+3t~Ou—+)3) — Pa
33

(3.9)


_

-

=—_

Let ạ€ A4 such that

Jìm P(Y@,9| <&

a

Wt>0}=1

Since (3.1) is strictly stabler than (3.5), then there exists g* € 4,

lim P{|Z(9z| <4

đ° < đ anl

Wr>0}=1

This equality implies Ag < 7 < q. Therefore, we can choose + in the inequaliv (3.
that
(*)

<1

and

(**)

Àa+>+<1

Aa++y<
and

when

Àz>0

(a—1)Ag+(a+1)y<0

when

Az<0

It is easy to see that (3.16) is equivalent to

X.=Z()z+ ƒ Z().Z~'(s)ƒ(s, X,)ds
Therefore

IXi<lZ(9z|+K [ ˆ |Z0).Z^(6)|min(|X,l*,|X;l!>#)4s
We consider two cases:
a). Aa>0
By (3.13) and (3.9) we get

'
1X1)
Ix,1-Pa

= exp{(Àa+ +)t}[out + Kw f , exp{((2- B)y — Bda) .8}-le" OF X,!~8a,
By virtue of of Bihari’ s inequality ( see [4] pp.

LX¡| < exp{(Aa + +)} [er
where

z:

o =: (2 — 8)y — Ba.

|z|<1

we have

Hence,

there

|Xi|< M.exp[(Au+++Ø))

110 ) we get

+BK.N | exp{os} as}

Ỷ

0

exists

a random

Peas

variable

where

P-as
M

such

that

F= max(0.5)

For any ¢ > 0 fixed, it follows from (3.12 («)) that there exists a random 7; >0 su

P{M.exp{(Aa+++ỡ)t} < &

Wt>T;} >1-<«/2

On the other hand, on (0,7;], the solutions X(t,z) depend continuously o1 the
condition z, then we can choose an 6 > 0 such that

P{lX(tz)|<œ

Wee [0,Ti]}>1-«/2
34

when - |r|<ð


(3.14) and (3.15) it yields
P{|X(t,2)1
Wes 0}

>1-«

when

Iz] <6

ns that (3.6) is stabler than (3.5).
Using (3.13) we have

‘

IXI< |Z(9z|+-+KÍ IZ():Z”!09)11X,If4s
0

< eet IN Jel + Kw [ exp{(A+-1)8 — (da — x)s)|X;|* đa]
0

, X,|* đa]

f

=:(a — LÀ¿ + (ø + 1)y

[4], ‘pp 110), it yields

virtue of Bihari’s inequality which a > 1 (see

N feolexp{(Aa + 04}

Xứ.z)Ì<——————
— ——

=

[1 = (a= D]zol9=!
x fi ee ds] =

p is small.
ng the same argument as above, we conclution that Ve > 0, there is an 6 > 0 such

|z| <6 then

PAX(z)]
E> 0) >1-€

the result follows.

wy 3.3 (See [4] pp.267 ). If the top Lyapunov exponent of (3.1) is negative, then
turbed system (3.6) is stable.
le 3.4

The

assumption

of regulity

of (3.1) is satysfied"when

(A;) and

(B;) are

tionary processes. The matter of fact is that is that in this case (3.2) generates a

2 (Z(t)) and by Floqué’s representation( see [1] and [2])

0

“a

Z(t) = S(t)exp{At + o(t)},

as too

S(t) is a random process with values in the sphere {z € Ré: |z|= 1}

} note that Theorem 3.2 may not be true if conditions 3.7 is violtaed as the following
PP

hs 3.5 Let us conside the logistic equation

dX, = (aX, + YX )dt + 0.X,0dW,

(3.15)

denotes the Stranovich equation. It is easy to see that (3.15) has the solutions
|X| = z.exp{œt + ø.W,} x {IzalŸ +
|

+

ff cxp{~5(as + o.W,)}ds}*

ọ

other hand System (3.15) is a perturbation of

đZ, = d.Ø, + ø.Z, e W,
35

(8.16)


which is stabler than

a
dY, Y= = —.Y,
oY dt

when a <0. Therefore, (3.16) is stabler than (3.17), but (3.15) is not stabbr tha

1e; the assertion of Theorem

are unstable.

3.2 is not true. We remark that both Systen 3.15

REFERENCES

lôi
bị

L. Arnold. Random Dynamical Systems. 1995. Preliminary Version2
L. Arnold and H. Crauel : Random Dynamical Systems Lyapunov Exonet
ceedings, Oberwolfach 1990; Lecture Note in Mathematics 1486 New York

(4)

1991. Springer.
Bylov, R.E. Vinograd, D.M. Grobman and V.V. Neminskii: Theor; of I
Exponents ; Nauka, Moscow 1966 (Rusian).
B.P. Demidovish : Lectures on the Mathematic Theory of Stability; Nauka,

(5)

and A.V. Skorohod Stochastic Differential Equations ; Syringe:

8]

1967.
I.I. Ghihman
1973.

R.S Khaminskii :Stability of Systems of Differential Equations with Rindon

bations of Their Parameters ;Nauka , Moscow 1969 (Rusian ).
H. Kushner : Stochastic Stability and Control; Academic

N.H.Du.

Spectrum
TẠP CHÍ KHOA

Press , Đ.J. -Lone

On the Relation between Lyapunov Exponents of Linear Systems
of Operators.

HỌC,

KHTN,

Accepted in Acta Vietnam

ĐHQGHN.

t.XII

Matematica

1997

n° 4, 1996

VỀ BÀI TOÁN SO SANH TiNH ON DINH CUs

HE DONG HOC CHIU NHIEU NHO
Nguyễn

Hữu

Dư

Đại học Tự nhiên - Dại học Quốc gia Hà Nội
Bài

báo đưa

ra quan

niệm

mới

về sự sỏ sánh

tính

ổn

định

của hai hệ đ

Định nghĩa cổ điển về ổn định có thể nhận được bằng cách so sánh hệ ta ch
tầm thường. Bài báo cùng đề cập tới việc mở rộng định lý Lyapunov vš tính
theo quan điểm bảo tồn thứ tự ổn định của hệ động học chịu nhiều phi tuyết

36