ON POINT DISSIPATIVE N-DIMENSIONAL SYSTEMS OF DIFFERENTIAL

ON POINT DISSIPATIVE N-DIMENSIONAL SYSTEMS OF DIFFERENTIAL

EQUATIONS

WITH

^QUADRATIC

NONLINEARITY

ANILK.BOSE ALANS.COVER JAMES A.RENEKE

Department

ofMathematical Sciences

Clemson University

Clemson,

SC 29634-1907

(Received October 17, 1991)

ABSTRACT. Known

sufficientconditionsforquadratic dynamical system

x’= Ax + f(x)

tobe point dissipative givenin termsof

A

and f fordimensions 2 and 3 are extendedtoallowfor more general forms for the nonlinear term

f(x).

Furthermore, the conditions extend to n dimensionswhen fisquadraticwithzerosetan(n 1)-dimensional

hyperplane.

0.

INTRODUCTION

We

are concerned with a class ofvectorequations of the form

x’= Ax + f(x)

^{where the} nonlinearterm

f(x)

isquadratic of the form

/Tc

x

r(x)

xTCn

^x

The n^)<n matrices Ci} are assumed symmetricwiththe orthogonalityproperty

xTf(x) O

for all x. If

xTf(x)

0 for all x we

say

that

f

isaconservative function.

Note

thatif

x’ f(x)

withf conservative then Ilxll²isconstant. The

problem

istodetermine conditions on

A

andfsufficient to havethe system

point dissipative,

i.e., which guarantee theexistenceof a bounded region

R

withthe property that

every

trajectory of the system

eventually

entersandremains within

R [2].

Consider the

Lyapunov

function

V(x) (x ^{(x)T(x (x)}

for the system.

For large

Ilxll the

d(V(x(t)))

Therefore our questistofindconditions on quadratic ^terms

xTAx- aTf(x)dominate

_dt

A

andffor which there is an admissibletzwhich makesthesetermsanegativedefinite function.

Ifthere exists an admissibleothen a classic result 4 impliesthe system ispoint dissipative.

A necessary

condition for the existence of an admissible tis that

xTAx <

0foreach nontrivial x in the zerosoff.

We

have shownforn 2 and 3 that this condition is also sufficient. Settling the obvious conjecture for all n

bogs

down in a proliferation of^cases.

However,

we haveshown in this

paper

thatthe

necessary

condition is sufficientfor the simplest n-dimensional case,

namely,

whenthe zerosoff forman

(n

1)-dimensional

hyperplane.

Finally,weextend the notionofanadmissibletttoprovide sufficientconditionsfor systems

x’= Ax +

fix) +

g(x)

tobe point dissipative when fisquadratic butnotconservativeand

g

isnot quadratic. Ifthere is apositivedefinite matrix

S

such thatSfis conservative then avectortis admissible for thedynamical system

x’ Ax +

fix)if

xTSAx tTHf(x)

is anegative definite function where

HTH S.

If

z

is an admissiblevector for the_system

x’= Ax

+

f(x)

then the system

x’= Ax

+ fix) +

g(x)

ispoint dissipativeifthereisanordered triple (e,

C, M)

such that

-czTHg(x)

^<

C

Ilxll^2-eforallxwithIlxll>_

M.

Ourinterest in systems oftheform

x’= Ax

+ fix)was stimulatedby all of thework in the literature based on systems of the sameform originally studied by

Lorenz

5 ].

We

hope to understand the richness of the class of chaotic systems, especially of dimension n > 3, by classifying a sufficiently rich class of point dissipative systems in terms of their compact attractors. This paper represents a step forward in that program by enlarging the class of systems that can be first classifiedas point dissipativeintermsof their coefficients.

1.

PRELIMINARIES

The proof of the prinicipal result Theorem 3 uses some properties of skew symmetric matrices. Theseare matrices such that

A

^T -A. Herearepropertiesthat are needed.

1.1.

We

notethat k^T isinthekernel ofaskew symmetric

B

relativetoleft multiplication if and only if k is in the kernel of

B

relative to right multiplication. This follows since

T

(k

T

B) =-Bkwhen B

isskewsymmetric.

1.2. Therankofaskewsymmetricmatrix is even 3,

page 217].

1.3. Againlet

B

be an n-dimensional(n

> 2)

skewsymmetricmatrixand

C

be the (n 1)- dimensionalprincipalsubmatrixobtainedfrom

B

by removingthe first rowandthe firstcolumn.

Ifthe

ker(B)

is

nondegerate

andis contained in

{xlx

0 then

C

is singular. This follows since k inthe

ker(B)

implies k 0 thatis k

(0,

k₂

kn)T. ^Now

^{if we}project k by

P

definedby

Px

x₂

Xn)

^Tthen

^P

^{is a}projection fromnton dimensionalvector

spaces.

We

notethat k inthekernelof

B

implies that Pkis in

ker(C).

Since k^T

(0,

k₂ k we have

kT

B (kTB1 kTB2 kTBn

(0, pkTCI pkTCn_I ^(0,

⁰

⁰⁾

where

Bj

^and

Cj

^refertothe

^jth

^columnof

^B

^and

^C,

respectively.

1.4. Iftheker(B)is contained in

{xlx

0 thenthe

dim(ker(B))

< dim(ker(C)).

To

see this let n bethedimensionof

B. We

seefrom the

proof

of

3)

thatthedim(ker(C))^>_dim(ker(B)) anddim(ker(C))

>

1.

Suppose

that

dim(ker(C))

dim(ker(B)). Then

rank(C) +

dim(ker(C)) n andrank(B) + dim(ker(B)) n. and so rank(C)

rank(B)

1.

But

thisimpossiblesince the ranks of both the skewsymmetricmatrices

B

and

C

mustbe even.

Withoutlossofgenerality we canassumethat the

hyperplane

of zerosoff is

{xlx 0}.

For

consider the dynamical system

x’ Ax

+

f(x)

where the zeros of f are an (n 1)- dimensional

hyperplane,

callit

H. Let R

be a rotation so that

H

goesontothe

hyperplane Y1

{y

y

^{0 under}

R -. ^Let

^x

^Ry.

^Thentheorginal dynamical system is

represented by y’= R

^-1

ARy

+

R

^-1f(Ry).

Note

that the zeros of

R

^-1

f(Ry)

are precisely the zeros of under the rotation

R -.

^{This is true for}

any

rotation

1. We

note that

Z(R.

^-1

fiR.y)) R.(Z(f(x))).

Thisappliestoboth

R

and

R -.

Let

x

:

0 be in

H.

Thehypothesisof our theoremrequires that

xoTAx ^<

^{0. Then}

Yo Rx

and

yo

^T

(R-1AR) Yo xoTAxo ^<

^{0 Since}

^R

^has an inverse the hypotheses for the dynamical system holds whetherrepresentedintermsof x or y.

Now

assuming that

Z(f) {x

x

0}

we notice that f has a convenientrepresentation.

Each coordinate functionoff,

fk,

^k 1,2 n, musthave

{x

x

0}

contained in its zeroset.

If we represent

fk,

^{k 1,2 n}by

n n

fO,) b ^x.x.

j=l ij tj

then

n n

fk(0’x2 Xn)=E E

^bk

x.x.=0

i=j j=l ij tj

forallx_2,x₃ x_n Therefore, b_k 0if neither nor is1.

Or

n n

fk (x)=E

j=l ^bkl

_Xl xj _=Xl jlb

klj ^x.

and

f (x)

Xl _1 ^bllj

^x

j:l ^b2

Xl-=

^lj

^xj

x b_n

j! lj

xj

=x

blll bl12-.- bll

n

b211 b212 b21

ⁿ

bnll bnl2 bnln

X

)n

ⁿ Since

xTf(x)

^x

xTBx

0for all x Let

B

denotethe man’ix(b

k 1"

n

xTf(x)

"1"= ^{bjlj xl x2j}

n n

+

E E (bkl ⁺ bjlk)

^x

xj

^xk 0 which is the zero polynomial having

k= j=

zerocoefficients.

B

is skew symmetric andwedrop the second subscript which isalwaysone and we have therepresentation

f(x) x

0

b12 b13 bin

-b12

⁰

b23 b2n

-bln -b2n -b3n

⁰

x2

We

use the

Lyapunov

function

V(x)= (x )T

(x

a)

then

d(V(x)(T)))

dt

xTAx Tf(x) +

d(V(x(t))

linearterms. For large llxll thequadratic terms

xTAx otTf(x)

dominate

dt quest is find an

x

whichturnsthe quadraticintoa

negative

^definitefunction.

therefore our

2.

EXTENSION OF PREVIOUS RESULTS

A

sufficient condition for aquadratic dynamical system tobe point dissipativehas been given when the dimension is two orthree. This condition uses a relation between the quadraticand linear partsofthe systemwhen f is conservative. Thefollowinglemmaallows us toextendthe conditiontothe casewhere thereexists apositivedefinite matrix

S

such that Sf is conservative.

Lemma

1.

Let

x’ Ax + f(x) (2.

be aquadratic dynamicalsystemforwhichthere exists apositive definite matrix

S

suchthat Sf is conservative. Then there exists matrix

H

for whichthe change ofvariablesy

Hx

transforms the dynamical system

(2.1)

into

y’ By

+ g(y) which has a conservative quadratic term.

Furthermore,

xTSAx yTBy.

Proof.

We

canfactor

S by S HTH [31. Let y I-Ix

or x

H-ly.

Thesystem transforms into

y’ (HAH-1)y

+

H f(H-y)

or

y’ By

+

g(y) (2.2)

The system

(2.2)

hasaconservative

quadratic

term since

yTg(y) xTH

^T

H f(x) xTSf(x)

0.

Notethat

xTSAx yT(H-I)THTHA H-ly yTHA H-ly yTBy.

When Sf isconservative then we say that avectorct is admissible for the dynamical system

x’ Ax

+ fix) if-

xTSAx

+

aT H fix)

is apositivedefinite functionwhere

S HTH.

When f is conservativethen the condition for an admissiblectreducesto

xTAx + cgTf(x)

is positive definite. The proofs of the theorems when the quadratic part of the system is conservative entaildemonstrating theexistenceof anadmissible

o

for the system.

Theseresults can berestatedasfollows:

Theorem1.

A

quadratic dynamical system

x’ Ax + f(x)

ispoint dissipative when thereexists a positive definite matrix

S

such that Sf isconservative and thereexists an admissible^0t. Ifthe system has dimension 2or3, Sfis conservative and

zTSAz <

0for

any

z which is anontrival zeroof fthen the system ispoint dissipative.

Proof.

We

cantransformthedynamicalsystemby y

Hx

where

S HTH

andby

Lemma

the resulting dynamical system satisfies the hypothesis of the previous theorem.

Hence,

the resulting dynamical system as well asthe original system are bothpoint dissipative.

Another direction of generalizing the past results is to consider nonlinear dynamical systems which have nonquadratic nonlinearterms as well as quadratic terms. Relative to the nonlineartermsthereagainmustexitsapositivedefinite matrix

S

such that the nonlinearterms premultiplied by

S

areconservative.

Theorem 2. When there exists a positivedefinite matrix

S

such that

Sg

and the quadratic functionSfareconservative and

Condition

(A)

For some admissibleot for x

Ax

+

f(x)

there exists an orderedtripleof numbers (e,

C, M)

such that-0t

THg(x) < C

Ilxll^2-efor all xwith Ilxll

> M.

then

x’ Ax + f(x) + g(x)

is point dissipative.

Note

that condition(A)canbe

replaced

byeither of the stronger conditions

(B)

or

(C).

Condition(B) There is an admissible0tforx

Ax

+

f(x)

and

g

o x

2.

Condition

(C)

There is an admissibleotforx

Ax

+

f(x)

and

g

is bounded.

3.

THE PROOF OF THE THEOREM

Theorem 3.

A

quadratic dynamical system x

Ax + f(x), A

^{a matrix and f a} quadratic function, ispoint dissipative when

and

(1) there exists apositivedefinite matrix

S

suchthat Sf is conservative,

(2)

the zerosof f are an

(n

1)-dimensional hyperplane,

(3) zTSAz <

0for

any

z which is a nontrivial zero of f.

Proof.

From

Lemma the system can be transformed into a system which has a conservative quadratric term.

Moreover,

the zeroproperties are preservered.

So

we assume that f is conservative. Wecan assume that

Z(f)

x lx 0 andf(x)

xlBx

^where

^B

^is

skew symmetric as shown in Section 1.3. Since

Z(f)

^x[ x 0

},

k in thekernel of

B

implies (0, k_ k

n)

^{orthat k 0.}

Let

h beavectorwhich isnotinthe kernel of

B

but

(h

₂

hn)

^{T is in the}

^ker(C),

see Section 1.4.

Here

again

C

is the

principal

submatrixof

B

formed

by

removing thefirstrowancolunm of

B. Moreover,

ifthe rank of

C

0, we can

choose

h sothat h 0.

Let G x Ix

h, arealnumber and we willshow

G

iscontained inthesetof

a

is in

G

then

ctTf(x)

x

thTB

x

xl

^{(q, 0, 0}

⁰⁾

^x

^q x

admissible

Note

that if

sincePh isinthe

ker(C).

Also

q

0 since h isnotinthe

ker(B).

Noticethat

q

0 whetheror nottherank(C) 0 furthermore,we canchoose so that

a

^T

f(x) >

0 forall x.

We

can restrict our attentiontothe sphere ^[Ixll which iscompact. Thereis aclosed cone on aclosedcone which containsthe

hyperplane

x _x] 0 suchthat

xTAx <

0 forall x 0in this cone.

Hence,

there is an13

>

0suchthat

xTAx

< 0 on

{x

Ilxll =1 and-13<

x ^{< 13}.}

Let M

max(

xTAx

on x Ilxll

}. By

picking

large enough

inmagnitudewecanassure thattq13²>

M. Hence,

0tT

f(x)

>_tq132

> M

on

{x

Ilxll and

Ixll ^>

¹³

^}.

Thus for allof x Ilxll

}, xTAx

^0tT

f(x) <

0. Andsoforallnonzerox wehave that

xTAx 0tTf(x) <

0which implies thatthe system ispoint dissipative.

This result can be generalized

by

adding to the differential equation

any

conservative function

g(x)

whose

growth

isrestricted. The corollarystatesthis condition.

Corollary Let

g

andthequadraticfunction be conservative. If

Z(f)

isan (n 1)-hyperplaneand x in

Z(f)

implies

xTA

^x

<

0 and

Condition(A) for some admissible0t thereexistsan orderedtriple of numbers (13,

C,

M) such that ot

Tg(x) <

Ilxll^2-efor all x with Ilxll >

M.

then

x’ Ax

+

f(x)

+ g(x) ispoint dissipative.

Note

that condition(A)canbe

replaced by

eitherof the strongerconditions

(B)

or

(C).

Condition(B) There is an admissible0t for

x’ Ax

+ fix)and

II g II

o

II

x

II 2.

Condition

(C)

There is an admissibleot for

x’ Ax + f(x)

and

g

is bounded.

4.

EXAMPLES

Considerthe dynamicalsystem

X

XlX

2+

XlX

3

+ XlX

⁴

2 -1

-x+ ^XlX

^3+x

^Ix

⁴

-1-3 -1

X

+

₂

-1

-I

-3 -1

-Xl XlX2 ⁺ XlX4

-1 -1 -1 -3 2

"x

1- XlX

2

XlX

3

In

thisexample the conservative quadraticfunction canbewritenas

f(x) Xl

0

-1 0 x=x

IBx

1-10 1-10

B

isanonsingularmatrixand

C B

₁₁ issingular. The kernelof

C

is

generated by

(1, -1,

1).

Since

B

isnonsingularthezerosetof

f(x)

is

Z(f)

z z (0,z2,z3,z

4)

^x

x=

⁰

^}.

^The

hypothesisof the theorem hold since 0 (0,z_2,z

3,

z4) ^A z3

^(z_2,^z_3,

z4) AI

^{z3 and}

^A

z4 z

4

-3 -1

-1]

-1 -3 -1 -1 -3

is anegativedefinitematrix.

We

can useotT (0,t,-t,

t)

andtx"r

f(x) x

^0,t,-t,

^t).(x

2+x₃+ X4, ^X +X +X4, ^X +_X2 X_4, X X2

x3)T tXl2. ^So

the quadratic termofthe

Lyapunov

function is

T

txT

x^{T 2} xT

x

Ax- f(x) Ax-tx!xT(A-Q)

^x=

2-t -1 -1 -1

-1 -3 -1 -1

-1 -1 -3 -1

-1 -1 -1 -3

Thisquadraticfunction isnegativedefinitewhen > 2.6.

By

thetheorem this nonlinearquadratic

dynamical

system ispoint dissipative.

Indeed inthe

example q

turnsout tobe 1. Therefore,

txTf(x)

can

only change

the

a

element of

A.

Thisturns outbejust what weneedandwant.

Because

the zerosetof f is

{(0,

z2, z3,

z4)}

wemusthave that

-AI

^{ispositivedefinite. If can}be choosen so thatdet(-A + Q)^ispositive thenthat willbe

enough

toinsurethat

A + Q

ispositivedefinite.

Let

uschoose 3 andreturn tothe derivative of the

Lyapunov

function.

We

cansetit equaltozeroandhavethe equationof aellipsoidwhich contains theattractorof thesystem. The equationis

-1 -1 -1 -1 2 -1 -1 -1

xT

^{-1 -3 -1 -1} x

(0,3,-3,3)

^{-1 -3 -1} x=0

-1-1-3-1 -1-1-3-1

-1 -1 -1 -3 -1 -1- -1 -3

If we use the rotation

x

=Ry

0.0000 0.0300 0.9370 0.3493

]

0.8165 0.13030 -0.2017

0.54101

^X

-0.4082 0.7071 -0.2017 0.5410 -0.4082 0.7071 -0.2017 0.5410

J

theabove equation becomes

-2 0 0 0

0 -2 0 0

0 0

-3+

⁰

0 0 0 -3

-/’

y + (0.3330,

7.9233, 10.61307,

-2.1149) y 0

or

2 2 2 2

2(x

0.16651) + 2(x

2

3.9617) + 0.3542(x

3

5.3065)

+

5.6458(x

4

-1.0574)

47.7333 The distance to the centeris 6.7082 and the half-diameter of theellipsiod is 11.6070and our boundisthe sum of thesetwo

numbers

is

18.3162

Considerthe dynamical system

[-4 4J ^{lx +} [6xY2-4y2 ⁺ [

²

^ye’x ]-x

^y-y

x’

-4 -6x +

4xy J

^-2xe

wherethelinear part

Ax

andthe quadratic function

[21

^6xy

^4y

fix) .6x²+ 4xy

arerelated by

zTAz <

0 when z is a nontrival zerooff. Theadmissible

a’s

are those for which

xTAx Tf(x)

^{is a}negativedefinite function. This isequivalenttothe matrix

2

3

^+20t2

]

3Oil

⁺

22 41

being poistivedefinite.

For

thediagonialterms tobe positivedefiniteot <

l/,,

^t2

< 2/3

^{and for}

the determiniantofthe matrixtobepositive

2 2

-90t + 120

t2-4tx ^2-

^16t:t

^+60;2-4

^>0

-2

:::::::::::::::::::::

Figure

2

The shadedregionisthesetofadmissible0tTrelativeto

A

and

L Now

iffor oneof these

t’s

condition

(A)

hold thethedynamicalsystemis point dissipative. Thisis thecase when wechoose o^T

(-1/2, 1/2)

21e^xy

1

_di,,(x) =_(,1/21 ^_2ye_X_, j _(x+y)e-x-y <E

andthe value of the determinate is0.75 andthediagonal elementsare and 1.

Here

isthe graph

[6

ofsomeofthe trajectories.

dx_ _4x

₊

4y

⁺ 6x

z _4xy

_{+ 2xe}

^-x-y

dt- dy

d--E ^4x

⁺

^{y 6xy}

⁺

^{4y z 2ye}

x-l Figure 2

Theattractorabovethe originismagnifiedin thefollowing Figure 3.

It

indicatesthat there isanattractor at

(-0.5,

1) surroundedbya limitcyclewhich is anrepeller. There is alsoan attractor at (0,

0)

andasaddle pointnear(1,

2).

Figure 3

BIBLIOGRAPHY

1.

BOSE, A. K., A. S. Cover

and

J. A.

Reneke,

On

Point-Dissipative

Systems

of Differential Equationswith Quadratic Nonlinearity,

Int. J.

Math. andMath. Sci.,

14(1990), 99-110.

2.

HALE, J. K., L. T.

Magalhaes and

W. M.

Oliva,

An

Intrgglucfion

tO lnfinit Dimensional

Dynamical

Systems.

Geometric Theory_.

New

York, Berlin,Heidelberg, Tokyo, AppliedMathematicalSciences,

Spdnger-Verlag.

3.

HORN,

R.

A.

and

C.

R. Johnson,

Mstr.x

Analysis, Cambridge, Cambridge University

Press,

1985.

4. LASALLE, JosephandSolomonLefsehetz,Stability

_

^Liaounov’s

^Direct

Mgthod, with Applicatio.n.s,

New

York,Academic

Press, 19all.

5.

LORENZ, E. N.,

Deterministic non-periodic flow,

J. Atmos.

Sci.,

20(1963),

130-141.

6. Phase Portraits, Version 2.0,

Department

of Mathematics and

Computer

Science, Drexel University, 1988.