MATI{EMAîICA

### -

REVUE D'ANAI{YSE^{NUMÉRIQUE}ÞT

^{DE }TIIÉORIE DE IT'APPROXIMATION

### L'ANALYSE

^{NUMERIOUE }

### ET LA THÉORIE DB

L'APPROXIMATION Tome### B,

No### 2'

1979,### PP.99-109

### oNTHECLASSIFICATIONoFDYNAMICALSYSTEMS

by

GH. TOADER (Cluj-Napoca)

### 1.

Infioduction### The set of

dYnamical^{sYstems}

### problem" is

unsolvable. How-### åittg so-" Partial

^{answels' }

^{So}ontäins

### a

nãcessary and' suffi-### In thjs

^{papef we }

^{pfopose }

^{á.methocl }

^{similar } to

^{l:hat }

^{of }

Vrublevskaya
### but

basecl dn### -a

### simpler

^{definition}

### the

Pompeiu-Hausdorff### metric in

^{th'}

### similar

^{dne }

### in the

case### of the

^{GH'}

clefined

### for the set of

continuous### in

_{[11] }

### and

have used### then in a

_{Pr}

i

i

¡

100 GII, TOADËR

2. Basic notations and delinitions

I-,et

### X be a metric

space### with a fixed metric d,:

^{d,*. }

### Êor

^{a. }

### e X,

### r>0 and A (X we

denote### by:

### (1)

d'(ø,### A) : ial

_{{d,(a, }x)

### ; x ^{e } ^{A}}

### the

distance### from ø to A

and### by:

### (2) V(ø,r) : _{{x e } X;

^{d,(a, }

### x) < r}

### the ball of radius r and center a. For a

function### f

^{: }

### X* Y ^{and } ^{a }

^{set}

### AUX we

denote:### (3) _{Í(A) } : {Í(x);

^{tc }

^{e } ^{A}.}

### Def inition

1.### A

d,ynømicø|, system on### X'i,s a

continuous funct'ion### n: X X R- X ^{that }

søti,sfi,es the fol'l'owing ^{øxioms :}

### (Ð n(x,0) : x, foÍ every x in ^{X;}

### (ä)

æ(rc(x,l),s)### :

æ(x,t### I ^{s), } ^{for } x in X,

ú ### and

s### in

R.### Def inition 2. For eny x ^{e } X

one d'efines:
### a) tke

mot'ion### (through x) n,: R-* X by:

### f4) ^{n,(t) } :

æ(x,t);
### b) the

trajectory### of x ^{by:}

### (5)

_{[email protected]) }

### : _{{æ(x,t); } I ^{e }

R} i
### c)

the þositiue### l,imit

set### of x

^{by }:

### (6) L*(x) : _{{y } _{= } X;

^{Jtn }

^{---+ }

### | ^{æ, }

n(x,t,)----y}.
### Def inition 3. A _{þoint } x ^{e } X

^{(and }

^{its } trajectory) is

^{søi'd' }

### to

be:### a)

cri'ticøl',### iÍ y(*) : _{{x};}

### b)

_{þeriod,ic }

### if

^{tkere }

### is ø þ10,

^{such }

^{that }

^{æ(x, }

^{t } ^{+ } þ) :

rc(r,t) ### for

^{øll}

### teR; c) L

^{støbl,e, }

### iÍ _{[email protected]) }

i's rel'atiael'y ^{comþact;}

### d) þ

^{Poisson }

^{støble, }

### i'f x e L+(x);

### e)

e-støbl,e (in### the

sense of M.### BEr

or{rNo### lll) if ^{øny }

e-ne'ighbourhood'
### oÍ [email protected])

contø'í,ns### at

leøst### ø

trajectory d'i'stinct### oÍ

_{[email protected]).}

### Def inition 4. Let n

^{ønd, }

### o

be d'ynamical' systems on### X

oel,y

### on Y. Tkey

øre NS-'ísomorþkic (respectively GH-isomorphic) ex,ists### ø

homeomorþhdsm### h: X----Y,

^{uhích' }

_{þreserues }trajectories,

### (7)

^{y(h(x)) }

### : [email protected]\, ^{for } all n e X

resþect'i-

### if

^{there}

tkøt is :

3 CLASSIFICATIoN oF DYNAMICAL SYSTEMS 101

(respectively

### which

makes commutative### the following

diagram:### X x R

^{n }

### -X

## (B)

_{lz }

^{* } r. _{I,}

### v i ^{n } _{-'--Y}

### that is:

### (B') k(n(x,t)): "(h(x),/), for

^{a71 }

### x ^{e } X ^{and }

### ' ^{e }

^{R)'}

### In what follows we shall ^{use } the set

1 ### :

_{[0, }

### 1]

^{and }

### the

function### ).:[0, æ]*.I

defined bY:### I' Îorte

[0,oo)### (e) ^{r(t) } :{t+,

### I I forl:co

### on the set of non-empty

subsets### of x

^{we }

### shall

use### the

premetric### p oi Þoãpeiu-Hausclorff

[-61- definea### for anv M, N C X bv:

(10)

### P(M, N) : I(sup {sup {d(ø,N) ; ^{x } ^{e } ^{M\, }

^{sup }

### {d'(y,M); y _{= N}})'} I,et c(R,X) be the set of all

continuous functions ### from R to X.

### We

^{need }sòverál metrices

### for

C(R,### X)

^{:}

### a) the uniform metric T,

defined bY:### (11) T(f,e): I

^{(sup }

### {d(Í(t),s(t));t = ^{R}), } for

anv ### Í' ^{g } = ^{C(R' } x)

;
### b)themetricK,whichgeneratesthecompact-opentopology,

defined

### by:

### K(Í, Ð :i r-"

^{r(max }{d(f Ø,e(¿))

### ;

l¿l### < n\);

### the metric S, of

Pompeiu### - ^{Hausdorff } ^{type, } ^{which } ^{we }

^{defined}

### I ^{bv:}

### s(l

^{g) }

### :

tr(sup### {so(l s),

^{so(g, }

_{"f)})'}

2

(r2)

### in _{tl}

c
(13)
) 1

where

### (13') S,(l

^{g) }

### :inf _{{r } _{> } 0;Vr ^{e } ^{R, } inf {d(f(t),

^{e(s)) }

### ;

^{l¿ }

### -

^{sl }

### { ^{r\ } <r}'

### with the usual

convention### : ^{inf }

^{Ø }

### :

æ_{'}

### ft ^{is }

^{easy }

^{to }

^{check }

^{that } ^{for } ^{any.f,g } = ^{C(R,X):}

### (14) ^{K(f, } e) < ^{?Lt } e) ^{and } ^{s(/, }

^{s) }

^{< } T(l

_{e)}

702 GH, ToADER

### and, as we ^{proved } in _{[11], } the identity

function :
### i

: (C(R, X),### S)*

^{(C(R, }X), K)

### is

continuous.### If it is

necessary,### we indicate by a

lower### index the

space used### in the definition of a certain metric (for

exemple### P¡).

### L

^{e }

^{m }

### ma ^{2, } If _{f } ^{: X---,Y } is a uniform

continuous ### function uith

^{the}

þroþerty that

### f(M)"e"w Íor ^{any } M e

^{(J }then the ind,uced, functioø

### f :u- -W ^{is }

continuous.
Proof

### . we

proceed### by

contradiction.### Le!

ug.supp-ggg.### that there

exists### u cottvetg"nt

^{sé^quence }

### M**M

^{such }

### lhat.f(M") *Í(A), ^{that is }

^{one }

^{can}

### find a r )

^{O }

^{sucñ }

^{that }

for-any no ### thete

exists### n)

tto### for what P"(Í(M")' Í(M))> r. ^{There are } ^{two }

possibilities:
### a) sup _{{d"(f }

^{(*), }

### f(MD; ^{x } ^{e } M"o)> r;

### b) sq {d'(f(M"u), fU))', ! ^{e } ^{M}2r,}

### for

some### frh*æ. In the first

^{case, }

^{for } ^{any } å

one can ### find a

xnhe M'rh such### that:

### (15)

dt([email protected]"*),### Í(x)) ^{> } ,, ^{for }

^{anY }

^{,í } ^{e } ^{M.}

But

### /being

^{uniformly }continuous there exists

### a

s### > ^{0, }

^{such }

^{lhal }

^{d,v(x,y) }

### 1 ( s'impliãs d't(f(x),Í(Y)) <r,

^{and}

### with the ^{property } that for

h ### )

^{hu,}

exists

### lo = ^{M }

^{wít]¡ }d'v(x,*, Yo)

### <

^{s, }

^{he}

### dicts

^{(15). }

### fn the

^{case }

### b) one fall the simple continuity of /.

### Consequence ^{1. } ^{Let } X

^{ønd' }

### Y

be comþact### mlllt-sþaces,-V o!

### W famities

oi.subsets### of X

resþectiael,y### Y. IÍ Í(M) =

^{W }

### for

^{ønynM }

### e

_{U}

a.nd,

### Í-1(N) = ^{IJ } for ^{any } N ^{e } W,

^{then }

^{the }

^{ind,uced' }function

### f ^{: } U"-W

^{is}

### a

homeomorphism.### I,et us äpply

^{these }

### results to

dynamical systems-### For a

dynamical system### tr ott if

^{we }

^{denote }

^{by } læ the family of all

trajectories. ### Ifow

^{one}

### kîows [A],

^{1.æ }

### is a ^{partition } of X

^{and }

^{we }

^{consider }

^{on } it,

as before,
### the premetric P ^{of }

Pompeiu ### -

Hausdorff and### the

corresponding equiva- lencË### relation which

^{we- }

^{name } in this

case 1-equivalence. One obtains
### the ^{quotient }

space ### îr, und the

following:TIrEOREM

### L If

^{tke }d,ynami,cal systems ¡c

^{ønd, }a d,efined,-on tke comþøct

rnetric sþacets X^rbsþectiael,y

### Y

are-NS-isomorþkic, tken the corresþond'ing quoti,ent sþaces### læ

ønd,### lo

øre homeomorþhic.Rernarh

### 2. This condition is not sufficient for

Ns-isomorphism how shows### the

following:ExemþIe

### l. Let X:Y : {(x,i) e Rzi x'*y1.5-3}

_{?s- }

^{subspaces}

of R2. The dynamical systems defiàed.-by autonomous

### differential

equations### (in polar

coordinates)^{:}

### ä:p(p-1)(p-s);

### g:1.

4 5 CL,A.SSIFICATION OF DYNAMICAL SYSTEMS 103

### 3.

Conditions### lor

NS### -

isomorphism### I.et X and Y be two metric

spaces, and, æ### and o two

dynamical systems### on X

respectively### on Y. For theirs

NS-isomorphism### the

two### families of trajectories (of

æ### and of o) must

correspond. each### to

other### by o

homeomorphisrn### k: X-"Y. ^{But } the families

being ### infinite, it

^{is}

### dilficult to check this

correSpondence.### Thus it is natural to look for criterions

which contains simpler conditions,### at least

necessary,### for

equi- valence.### We begin with

some### lrrore

general considerations.On

### any family of

subsets### of X we

consider### the

premetric### P

defined### by

(10).### I.et U be

such### a

family.### Def inition 5.

We say### that tuo

sets### M,N C{ ^{are }

U-equivalent
### (and

denote### by M - N ^{(rel UD } ^{i"f }

^{there }

^{exists }

^{a }

continuous ### function h:

### I* U

such### thøt

h(0)### : M

ønd,### h(l) : _{¡¡.}

Remørh, 1.

### It ^{is }

easy ### to verify that

U-equivalence### is an

equivalence### relation,

hence### it

^{induces }

### a partition of U. We

denote### the

equivalence class### of M by û _{^na } ^{the } ^{quotient }

space ### bv

Û.### I,emma

1.### If U

ønd,### W

are### fømilies of

subsets### of X

resþectiael,y### Y, tken øny

continuous funct'i,on### F: UnW

^{d'efines }

### a

continwous function### ^^^^¡A

### F:UnW by F(M):F(M).

Proof.

### It ^{is }

^{obvious }

### that M - N ^{(rel U) }

^{implies }

^{F(M) } -

^{F(.N)(rel}

### W) so

^{Llnat }

### F is well

defined. Using### the

continuous canonical projections### i:U-U and j:W*W we obtain the following

diagram:### (JF-W

### l¿ ^{l¡}

### I ^ l'

### Ù ' -û,

F'rom

### his commutativity

results### the continuity of Ê b"""or"

_{7 }

^{o }

### F is

con-### tinuous

_{(see t5l).}

### (")

r04 and

(")

### p:

GH, TOADER

### *: ^{x(x } - ^{l)}

### !:0

6

CLASSIFICATION OF DYNAMICAL ^{SYSTEMS} 105

I

### Then the

l,agrange### with the

^{trajectorY}

### that is the

theorem### LemmaS'Iftkecløssof.y-equiuølenceoføcriticøl'þoi'ntxisnot a

singleton, them### x is

^{e-støbl'e'}

### The condition is not sufficient as

^{shows }

### the

following:Exemþle

### 3. In ^{the } ^{dynamical }

^{system }

^{generated }

^{by } ^{the }

differential
### system (iå _{Polar }

coordinates)
p

### sinl _{, }

^{f.or p }

### *0i

P

### 0 , for P:Q' 0:1;

### the origin is an

e-stable### critical point

whose class### of

y-equivalence rs^{a}singleton.

### 4.

Conilitions### for

GH### -

isomorphism### 1

because### the GII-

### irs

NS-isomorPhism### [13]). But

^{we look}For a dynamical system

^{7r }on

### X,leL us

consider### the

space### of

motions^{:}

### fI:{zcn; xeX)

### and the

maP### æ*:X* [I

defined bY:### (16)

^{æ*(x) }

### :

nn### The

use### of the metric T ^{f.or } fI

Pro
### two ^{motions } ^{on the }

same trajecto
### the following:

Exemþle

### 4. Let

^{æ }

### be a

dynamical system### on R definedby n(x't):

### :

rc### ' exp (t).

1L### x* _{Y } ^{we }

^{have:}

T(æ',

### nv):

tr(sup### {lr - ^{yl } ^{exp } ^{(t); } t ^{e } ^{R}) } :

1'
stable

### traiectory {(x,l); ^{0 } < x < ^{1} is }

'¡-equivaleut
### {(x.O\'. d < ^{x } < i} ^{wtrich } ^{is } ^{not }

^{l,agrange }

^{staole'}

là-

### ;å¿ true if

one renollnces### at

completness' t### p(l-p) if0<p(1;

### 0 if ^{1 } < ^{p } ( 2;

### (p-2)(p-3) íf 2<p<3;

### 0:1

afe not Ns-isomorphic although the quotient spaces l1rc

### and Ilo

are homeo- morphic.### As

concerns### the

equivalence classes, we have### the

following:### the systenr,

^{ed, }

^{on } ^{a }

comþIet,loc.øl'Iy
### X,

^{ais }

### of

^{ce }

^{of }

^{ø }

^{Løgrønge }

^{støbl'e}

### nly

^{stable}

Proof.

### Let

To### be

aÍ"agtange stable,trajectory, 11### :

^{Y0. }

^{and }

^{h: } I*læ

continuoús

### and,'ðuch thaí nP\ :

_{"¡o, }

### h(l): _{Yr: }

^{D-eno-te }

### lv I : ^{{t-=*! }

^{|}

### nftf i,

Lagrange stable).### of

còúrse 6".=### i,'thai ^{is } ¡+Ø' ^{Let }

^{to }

^{e } J' T¡.

space

### X ^{being } ^{locally }

^{compact }

^{and } ffi

^{c:gmpact, }

^{k,(to) }

^{has }

^{a }

^{compact}

### "ãi*rr¡å"tr'àãål ur"l

^{iå }

_{7 } ir

ãpen ### in /. ^{Èdî } /

^{is-also }

^{closed }

^{in } ^{1. }

^{rndeed,}

### ii;,;";;pp;."'tir" "onúrty,'thcle ^{exists } ^{a }

^{sequence }

^{(1,) }

### in / ^{which }

^{has}

### a limit point

to### ê J.

^{That}

sequence

### (ø,) which has

^{no}

### hai'e also k(t,)-

^{71¡0]r }

^{(!n th}

### the set of

subsets### of X). If

bers,

### then for

everY### þ,

^{thete }

^{is } ^{a }

^{n}

implies P(k(t^),h(to))

### <!'t'.tm1)Ntanð'yl, =îñsuchthat

^{d(y"' }

^{x,) } ^{<}

### < 3 for ^{afly } n.

^{Because }

### TAÃ is

compact, there### is a

subsequence^{(yl,o)}

### of ^{(yt-) } ^{which } ^{is }

convergent. ### We may

assume### that a0'"0,

y|r)### < 3

for any### h

^{anð. }

### l.

^{Denoting }

^{x,o tty }

^{x'0, }

### t'. ^{obtain } ^{the }

subsequence ### (øl) oÍ

(x*) such### that

d,(xte,### x!) (

e,### for

any### h

and### l'.

Step### by

step,### for þ :2, 3,

_{' }

_{' }

_{'}

### we obtain the

sequence @!,)### such that d(x!, xt) (

e¡### for

^{any }

### h

^{anð' l,}

### (x!"\ being

subsequence### of lxl-'). ^{So } ^{çxi) is } a

Cauchy. sequence. and,
### Ï"f"itrg--Ëo-piãi=it ^{is } "

convärgãnt subiequ_"ttc". ### of (1,) in

contradiction### *itrr--iË" urän*þtiou- Att"t uil, _{J } : I, ^{^that } is

^¡f ### is

l,agrange stable.ExemþIe

### 2. Let ^{X } :

R2 ### - ^{{(0,0)} } ^{with } ^{the } usual Euclidean

^{metric}and. rc

### thå

dynamical system gèierated### by the differential

^{system:}

### p:

t

106 ^{GH. }^{TOADER} ^{B}

### I

CLASSIFICATION OF DYNAMICAL SYSTEMS^{107}

### *xo

such

### that is no motion

is. equivalent### with

another### (taking the metric T

^{for}

### the

space### of

motions),^{-}

### ln what follows we shall

^{use}

### avoid

these### difficulties. With this

cD

### ef initio n

G. We say that### Ient (and nII,

suclr denote### thøt

h(0)### by : ¡- - ^{n,,\'if }

^{the}

### ".

;;Nd"h(l)### As

above### we

denote### the

class### of

Il-equivalence### of

nn### by i

^{and }

^{the}

### quotient

_{space }

### II/_by û.

### L

e m### tna 4. If

y(x)### : y(y), then

nn### -

^{ær.}

### _ ?o-tf.By ^{the }

hypothesis ### !:n(x, ^{s) } ^{with }

^{some }

### s e R. Definig å:

### /*II by

h(t)

### :

æn1*,¿"¡### it is

continuous because### s(h(tr),

h(tr))### < lr, _ t,l ^{. }

_{lsl}

and _{Remark }_{k(0) }

### :-rr, _{2. }

_{Thus }

_{þ(l) } : _{s } iry, that is n, - ^{v*}

### is

more usefirl### thdn the metric T. Also,

generaily### it

induces### not trivial

quotient._{spaces.. }

### rot

### """-pìã,-"t tt"'dînamicat

system

..appearing - in

. exemple

### 4, óne obtain three'II

jequivalence### "1"r;;;

### (upon the sign

of. x).### ular

classes### of

motions.### First of

all### definsd Iw j(") : _{[email protected]), is }

_{conti-}

zur)..

### Thus every class of

fl-equi- -equivalelce._{So, }

### for

ll-equivaleäce### contained in the

theorem### 2

and### L

e m m### a 5. The set of

motions þosi,tiaety### poisson

stabr,e ,ís crosed,### in II.

Proof.

### r'et

the-### motions

æn,### positivery

^{poisson }

### stable and

æn such### t!^l S(",,, æ,)*0 fot n-co.

TËus,### for any natural þ

^{there }

^{is a } ^{øl }

^{such}

### that:

### (17) [email protected],,,,n.¡ q

### i.e. for soÍre s¿,

_{lsel }

### < ^{ll3þ } ^{we }

^{lnav}

### fi

### being positively

^{poisson }

### stable

so### 1

^{such }

### that:

d,(x,### n(x,

### "¡)) < ^{llþ}

### By the

theorem### of poincaré-B"náixon

### [10]

^{we have }

### the

following:### Conseçluence ^{2. } ^{For a }

d'ynømi,cø|, system, on tke þlane, ^{the set of}þeriod.ic rnot'ions

### is

^{closed' }

### in

^{IL}

### Def initíon7.

The function### f :R*X

^{has }e-þeriod"c

### if for

^{øny,}

### t ^{e } ^{R, } ^{d(Í(t } + r), f(t)) <

^{e.}

### L

^{e }

^{m m }

### a 6. If tke

cløss### of lI- ^{e } of a.

^{motion }

^{æ*0,}

ot

### þass

^{xo, }

^{then }

^{for}

### ry

^{d,o }

### n ^{n }

^{xo, }

^{uhiclt,}

### {e. is uniformly

continuous### on _{[-1,}

,

### 1!, 4l'

_{such }

_{that } for any t,s = ^{[-1,} r + ll, ^{with }

^{l, }

### -

^{sl }

^{< } ^{48, we }

^{have:}

### (18)

^{d'(æ(xo, }

^{t), }

^{n(xo, }

^{t)) } _{< å }

^{.}

### Ry

hypothesis, there### is a x ^{e } X

such ### that

S(æ,., Íç,)### I I

^{and' }

^{æ,(l)}

### lir anï teR. ^{Thus } for any t eR, ^{there } is øs¡, lsr-ll ^{<28}

### that

d,(æ(x,t), æ(xo,^{s }

### )) < ^{2S' } ^{So } ^{we } ^{have }

succesively:
ls,+"

### -

^{(r, }

### *")l(

^{ls,+, }

### -

^{(t }

^{+t)l } *

^{ls, }

### - ^{rl } ^{< }

^{4ò}

### and,

denoting s¿¡"### -

^{na }

^{+ } ^{r0, } ^{with } ^{lo } e _{[0, } r),

^{n, }

^{e } Z:

ls,+"

### - ^{nr } -

^{(s, }

### f

^{c }

### - ^{nr)l } <

^{48,}

### hence

^{s¿+t }

### - ^{tt'r, }

^{st }

### * ^{c(l } - ^{n) } = l-1, t f l]

### antl by

^{(18) }

^{:}

d,(n(xo, s¿¡,), nl()to, s,))

### :

d'(rc(xo, Sr+"### - ^{nt), }

^{n(xo, }

^{s, }

### * ^{(l } - ^{n)")) } < ^{tl7'}

### Finally:

d'(n(x,t

### I r),n(x,t)) 4

d(æ(x,t### I

^{r),n(xo, }

^{s,+')) }

### *

d'(æ(xo, s¿1"), æ(ø0, s,))-¡### I

^{d,(rc(xo, }s,), æ(x,r))

### <

^{2S }

### + ; ^{+ }

^{28 }

### (

e,### thus

æ, satisfies### all the

expected' conditions,### Lem ma 7. If ^{the møþ } h:X_*Y is uniformly

coøtinuous, then ^{so}

### is

al,so### tke maþ

h* : C(R,### X)*

C(R,### Y)

^{d'efined }

^{bv:}

### (re) (h.(Í))(t) :

h(f^{(t))}

(using

### the metric S for the

spaces### of

continuous functions)'Proof.

### For any

^{e }

### ) ^{0 } ^{there } ^{is } ^{a..ô, } ^{0 } <-l I

^{e, }

^{with }

^{tEe. }

^{property}

### that d*("x,!) ( ô ^{implies }

^{d, }

^{(k(x), }

^{nOD }

### <e.

^{Tf. }

_{f,g } = ^{C(R,X) }

^{are }

^{such}

108 CH, TOADER

### XxR n

### -X t*

### hx l* -

### YxR

^{o}

### that S*(f,g) <

^{8, }th.en f91. any

### t ^{e } R there is a

s, ### e _{R } _{such } that

_{ls, }

^{_-}

### - ^{tl } < I and d*U(fl,

_{S(s,)) }

### < ^{à. }

Hence:
### min

_{{d.r(k(f }

### (t)), h(g(s)));

_{ls }

### - ^{rl } { ^{e}ç } ^{min }

_{{d,r(k(f }

^{(t)),}

### [email protected]));

_{ls }

### _ _{rl } <

^{s) }

### <

dy(h(Í(t)),_{h(s(s,))) }

### <

^{e.}

Changing

### the role of f

^{and }

^{g }

^{we }

^{get }

^{SyØ*(f), }h*(e))

### <

".

THEoREM S.

### Let îc

^{qnd, }

### o be

d,ttnamical, systems on comþøct s7laces-xorcsþect'iuetv

### y. _{rr }

_{thev }

_{are } cti.-d'"*lïph;;,";tu"; ííí

quoil,rnt
### fI

ønd,### 2 of

cløsscs### of

equiuølent motions are homeomorþhic.### Proof. We have the following

_{diagram:}

10

metric

sþaces

11 ÕLÄSSIFICATION OF DYNAMICAL SYSTEMS

Rernarh

### 5. The condition from

^{theorem }

### 3 is actually

stronger### that of

theorem### 1, and a

class### of

lI-equivalence### may be

^{properly}

### tained." in a

class### of

y-equivalence, as shows### the

following:Exemþtre

### 6. I,et o be

defined### as

above### by:

### (o) ff:o p(l

### l'o: ^{op'}

109

than ,rcon-

One obtaine

### a

single class### of

^¡-equivalence### but two

motions### on

different### trajectories are not fl-equivalent. Thus o is

NS-isomorphic### with _{!h"}

### dynamical system æ from

exemple### 5 but is not GH-isomorphic with it.

h*

### T

I J

### '-fi

.l

h II

### Y

### I lî

### t

### REFERENCDS

[1]

### Bertolino,

^{M,, }

^{Solutions }

^{of }ilifferential equalions ì,n arbàtrøryt neighbourhoods of giuen functions (Setbo-Croatin). Mat. Vestnik, 13 (21Ì),

### l, 2l-33

^{(1976).}

### [2]Bhatia, ^{N,P., } Franklin, M.L.,

Dynørniaalstlsternsuithoutseþaratrices. tr'unkc.
Ekv., 16, 1-12 ^{(1972).}

### i3l Bhatia,

N. P., SzeCö, G.P., Stabi,Iilytheory of dynørnical s)lstems. Spriuger-Verlag, New York, Heiclelberg, Berlin, 1970.t4l ^{H }^{ó }^{j }^{e }

### k,

O., Cølegorial conceþis in dynømicat syslem lheory. fn ,,Topological Dynamics", J.^{Auslander }antl W.

### II.

Gottschalk (Editors), New York### -

Amstertlam: Benja- min, 1968, 243-258.[5] ^{K }^{e }^{l l }^{e }

### y,

J. L.,General,^{Toþology. }D. Van Nostrantl Company, fnc. Princeton, New Jersey, 1957.

[6]

### Kuratowski, K.,

Toþolog3t,### I-ff.

Academic Ptess, New York and, I,ondon, 1968.### iZl

^{tvt a }

^{¡ }

^{k u }

^{s, }

### \,,

^{Globøt }

^{structuri }

^{oJ }ordinarlt differential equations in

^{the }þlane. Trans. AMS, 76, 127-148 (1954').

[8] ^{M }^{a }^{r }^{k u }^{s, }^{L., }^{Lectuyes }in DliÍfeyentiable Dynamics. Regional Conference Series in ^{IVIathe-}

matics, suppotted by National Science Foundation, Number 3r 1971.

t9l ^{M }^{a }^{r }^{k u }s, I,., Porøltet' d'ynømioal' systems. Topology, B, 47 -57 ^{(1969).}

itÓl

### ¡qemytskii,

V.V.,### Stepanov,

V.V.,Quali'tatiueTheory oJ Diflerentiatr Equations'### [1]

^{T }

^{o a }

### d

^{onsi}

10t1,

tttl orie

L^-l

[13] ^{U }r

### a,

their isomorþhisms' Japan### -

U.S. Sem. OrtI' Diff.cture Notes l!'l:at'ln., 243, Springer Verlag, ^{1971,}
76-90.

[I4]

### Vrublevskaya, I.

N., Oz^{geometric }equiuølenae of the trajeoloriesof ilynamiaal systems (Russian). Mat. Sbornik, 42. 361-424, (1957).

Received s'vrr'

### 1927'

cenrrul leritoriøl ^{dø}
aalaul eleatrcnic
Sh. Reþublicii 107
3 400 Cluj-Nøþocø
o*

where

### the first

rectangle### is from the

GH-isomorphism### of

æ### and o,

æ*### and o*

are defined### rv _{[te¡, k* } b; (rE,

¿
### ""Jitãi"'"Jnäni""l

projections### irr quotient

spaces anð.### h is

defined### by:

### î1î"¡ :

Ç,o,### From the continuity. or h*, one

deduces (sge### rs]) the continuity of î.

### Fr.tt"ttn ^{the proof } it is

eriough ### iJì"p"ut'ttre äuäve-"onsia"i"iió"r-ráï

Remarh,

### 3. v/e

cannot use### the commutativity of the last two

rectan_gles

### and avoid

_{so }

### the proof of ilt"-"ã"tinuilv _{ái } ãi,'îãäo."

rc* ### and

o*### may be

discontinuous.Remarh,

### 4. As

in-

### the

case### of

NS_isomorphism, theorem### S is not sufficient

_{for }

^{- }GÈ_Iso-orpfil.rr¡--'

### wing:

Exemþle

### 5' Let

æ### an. o

be-.dynamicar systems defined### by the

dif-### fererrtial

dystems### (in polar Ã""øi"áiÃ¡r- - -'

### (") _{lþ:o}

### tb:t ^{(P<l)}

### the

condition from how shows### the

follo-### (") p:0

### b:P ^{(p< } ^{l)}

I I

### The

systems### are Ns-is _{morphic } but not

GH-isomorphic, although _{the}

### quotient

_{spaces }