I nt. J. Math. Math. Sei.
No. 2 (1980) 267-274
Vol.
267
ON A CLASS OF CONTINUOUS FLOW
RAMON MOGOLLON
Deparmento de Matemticas
Escuela de Ciencias
Universidad Centro Occidental
arquisimeto Venezuela
(Received August 20, 1979 and in revised form November 9, 1979)
ABSTRACT.
This paper involves characterizations of a class of continuous flows.
The flows considered are those in which the positive prolongation of each
coincides with the trajectory through the point.
The characterizations are based
on the theory of prolongation.
KEF WORDS AND PHRASES. F., prolongation, trajectory, pidie,
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
1
point
.
Pmay 4C35.
INTRODUCTION.
The purpose of this paper is to study dynamical systems in which the
posi-
rive prolongation of each point coincides with the trajectory through the point.
Ahmad
I] studied similar flows--flows in which the positive prolongation of each
point coincides with the closure of the positive semi-trajectory through
point.
He referred to such flows as flows of characteristic 0
+.
Other
the
authors
R. MOGOLLON
268
I], [2], [3], [5], [6]).
have studied these flows or similar ones (see
6],
Knight
for example, gave characterizations of flows of characteristic 0,
i.e.,
flows in which the prolongation of each point coincides with the closure of the
trajectory through the point.
The author in
7] characterized flows
in
which
the positive prolongation of each point coincides with the positive semi-trajec-
tory through the point.
posi-
In this paper we give several characterizations of flows where the
rive prolongation of each point coincides with the trajectory through the point.
making
We show that several seemingly different flows are all equivalent, thus
it unnecessary to study all such flows.
We also give a simple characterization
of our flows for the case where the phase space is restricted to the two dimensional space R 2.
2.
DEFINITIONS AND NOTATIONS.
+,
We shall let R, R
R-
real
represent the real numbers, the non-negative
numbers, and the non-positive real numbers, respectively.
or a continuous flow is meant a pair
(X,),
By a dynamical system
where X is a topological space, re-
ferred to as the phase space, and : XR- X is a mapping satisfying the following three axioms:
(1)
(x,O)
(2)
((x, t) ,s)
(3)
x,
(x, t+s),
is continuous.
For convenience, we shall let (x,t) be denoted by x.t, or simply xt.
x
X, C(x)
x.R,
(C+(x)
x.R+),
(C-(x)
For each
x.R-) represents the trajectory (pos-
itive semitraectory) (negative semitraectory) through x.
called a critical point if xt--x for all t in R.
a periodic point if there exists a number t, t
> 0,
A point x in X
is
A non-critical point is called
in R satisfying xt
x.
A
269
A CLASS OF CONTINUOUS FLO
variant) if
C(x),
C+(M)
M) (C(M)
M).
C+(x), and K-(x) C-(x).
denoted by L+(x), i. e., L+(x)
R+ with t.
1
-
+ oo}. Similarly,
is denoted by
point x of
and (t i) of
R+
,
For each x in
K(x)
we let
The positive limit set of a point x
:
{y
xt. -+ y for some net
1
(t i) of
of x
the negative limit set and the limit set
L-(x) and L(x), respectively.
are denoted by
of
M(C-(M)
K+(x)
is
in
i-
is said to be positively invariant (.negatively invariant)
subset M of
D+(x),
such that x.
1
i.e.,
-
The positive prolongation
D+(x)
x, x.t.
i 1
-+
:
{y e
y}.
of a
there exist nets (x
Similarly,
D-(x)
i)
D(x)
and
denote the .negative prolongation and the prolongat.ion of x, respectively. The
is denoted by
positive prolongational limit set of a point x in
J+(x)
t
I.
-
:
{y
+0%
there exist nets (x
x.tl i -+ y}
Similarly,
i)
of
J-(x)
and (t i) of
and J(x) denote the negative
is said to be positivel7 Poisson stable if x
lateral versions are defined similarly.
.
such that
se____t and the prolongational limit set_, respectively.
tional limit
dering if x e
J+(x)
R+
J+(x).
A point x in
L+(x).
A point x in
J+(x),
x.1
i.e.,
-x’
polongaA point x in
The negative and biis said to be nonwan-
is said to be positively dispersiv e
if
The negative and bilateral versions are defined similarly.
For more information on the above concepts and related notions pertinent
to this paper one is referred to the bibliography.
Throughout this paper, the phase space is always assumed to be Hausdorff.
3. CHARACTERIZATIONS OF FLOWS SATISFYING
D+(x)
C(x).
THEOREM 3.1. Let (X,) be any continuous flow.
tions are equivalent:
(I)
(2)
D+(x)
J+(x)
C(x)
for all x in
C(x)
for all x in
(3) J-(x)’-- C(x)
..
for all x in X.
Then the following condi-
2?0
R. HOGOLLON
(4)
J(x)
C(x)
for all x in
(5)
D(x)
C(x)
for all x in
.,
and there are no positively dispersive
points.
D-(x)
(6)
PROOF.
implies that
C(x).
C(x)
.
for all x in X.
J+(x)
Assume that (I) holds.
Let x
X.
D+(x)
C+(x).
It follows from (I) that
C+(x)
Therefore, x.(-l)
U
J+(x)
+,
R
x.t for some t
Then
For,
and hence x.(t+l)
J+(x)
C+(x)
x.
This
L+(x)
that J+(x)
shows that x i either a critical or a periodic point; in either case
.
C+(x)
we
J+(x)
We note that J+(x) C D+(x)
must also have C(x) C J+(x).
#
implies that
Assume that (2) holds.
J+(y)
in general, x
J-(x)
which contradicts the assumption
C(x).
.
J+(x).
C C(x).
Then, x
J-(y)
Hence, we have J(x)
Let x
.
U
J+(x).
J-(x)
C C(x).
C(x).
C C(x).
Since
J+(y)
Then x
J’(x)
Therefore,
J-(x)
Then C(x)
J-(x)
set,
J’(x)since,
Hence, x
J-(x).
Now, let y
C(y), and hence y
J+(x)
is an invariant
for any two points x, y in X.
This shows that
Assume that (3) holds.
J+(x)
C(x), and (2) holds.
Then x
C(x) C J-(x).
C(x), and (3) holds.
y
J+(x)
J-(x)
implies y
C(x).
But, since
Hence,
Let x
is invariant, we have
C(y), and hence y
,
C J(x).
Now,
This shows that
let
J+(x)
Consequently, J(x)
C(x), and (4) holds.
Assume that (4) holds.
C(x) U J(x).
J+(x),
(4),
J+(x)
contradicting
Assume that (5) holds.
J+(x)
X.
Obviously, D(x) -C(x), since
To see that x is not positively dispersive, i. e.,
we note that by
that x
Let x
.
implies that x
J+(x)
Let x
X.
is a nonempty invariant subset of
J-(x). But
x
J+(x)
J-(x)
D(x)
,
implies
Thus, (5) holds.
Then,
J+(x)
C D(x)
C(x), we must have C(x) C
C(x).
Since
J+(x). Hence
A CLASS OF CONTINUOUS FLOW
x
J+(x),
J-(x).
and, consequently, x
fore, C(x) C V-(x) since J-(x) C V-(x).
C(x).
D-(x)
Thus, we have
is obvious from (5)
It
.
Let x
We note that
C-(x) U J-(x). But,
C(x) since D-(x)
.
This shows that C(x) C
J-(x).
J-(x)
-
C(x) would imply that x is a critical or a periodic point, thus
dicting
J-(x)
is invariant, we must have C(x) C
x
J+(x).
Therefore, C(x) C
x
D-(y)
C(y).
But, x
Therefore,
D+(x)
C C(x).
J-(x) C D-(x)
E
J-(x).
J+(x)
C(x).
Now,
C(y) implies that y
y
let
C(x).
-C(x), and hence (I) holds.
would
contra-
Again, since
J-(x),
This implies that x
D+(x).
C
C
as was shown earlier,
C-(x)
Thus, we have
There-
V-(x)
that
C(x), and (6) holds.
Finally, assume that (6) holds.
imply that C-(x)
271
D+(x).
J-(x)
and hence,
Then,
This shows that
D+(x)
This completes the proof
of the theorem.
We note that if D()
C(x), then D(x)
K(x) since C(x) C K(x) C D(x).
Thus, the equivalence of statements (I) and (5)
in the preceding theorem shows
that the class of flows that we are studying here form a subclass of flows
characteristic 0
D(x)
(recall [6] that a flow is said to have
K(x) for all x in the phase space).
ceding theorem that
D+(x)
C(x) C
variant, we must have
COROLLARY 3.2.
If
D+(x)
J+(x).
.
characteristic 0 if
In addition, we showed in the
C(x) implies that
J+(x)
of
Since
J+(x)
is
prein-
Therefore, we have
C(x) for each x in
,
then the flow
is
a non-
wandering flow of characteristic 0.
EXAMPLE 3.3.
Consider the flow
(R2,)
defined by the system of
differ-
ential equations
It is easy to verify that this flow satisfies the
condition D(x)
C(x)
for
R. MOGOLLON
272
all x in R2 and is of characteristic 0.
D+(x)
But any point x has the property
This shows that our flows form a proper subclass of flows of character-
# C(x).
istic 0.
A trivial example of a flow of characteristic 0 that
R%RK 3.4.
isfies the condition
THEOREM 3.5.
D+(x)
C(x) is a global Poincar8 center.
Let (,) be a dynamical system where the phase space
D+(x)
C(x) for each x in
.
is
Further, as-
either a locally compact metric space or a complete metric space.
sume that
sat-
Then the set of periodic and critical
points is dense in X.
PROOF.
L+(x)
Obviously, if x is either periodic or critical, then x e C(x)
L-(x).
On the other hand, suppose that there is a point x,
Poisson stable but neither periodic nor critical.
L+(x)
C(x).
+(x)
is invariant, we have C(x) C L
C D
+(x)
If x e
L+(x),
C(x).
lqerefore,
But this contradicts a known result (see, e. g., [4]
which is
then
since
L+(x)
that if X is el-
ther locally compact and metric or complete metric, then every point that is
neither periodic nor critical satisfies
gument shows that if x e
L-(x),
L+(x)
C(x)
L+(x).
A similar
then x is either periodic or critical.
ar-
Thus,
we have shown that the set of points that are either periodic or critical coincides with the set of Poisson stable points.
lary 3.2, all points of X are non-wandering.
follows from a known result (see, e. g.,
4])
We further note that by CorolThe proof of our theorem
that if the phase space
either locally compact-and metric or complete metric, and if all points
nonwandering, then the set of Poisson stable points is dense in X.
now
X is
are
273
A CLASS OF CONTINUOUS FLO
Let (X,) be any dynamical system, where, X C R2.
THEOREM 3.6.
D+(x)
Then
C(x) for all x in X if and only if the flow has characteristic 0 and
there are no positively dispersive points.
PROOF.
The "only if" part follows from Corollary 3.2.
J+(x)
the flow has characteristic 0 and
C D(x)
J+(x)
K(x).
C C(x).
# @
.
for each x in X.
First let us assume that L(x)
J+(x) C C(x) implies that
that D+(x)
C(x). Now, let
@ #
But
C(x), which implies
Now, suppose that
We have
C(x), and hence
Then, K(x)
C(x) C
J+(x).
Hence,
us assume that L(x)
y s L(x). Then y s J(x) and, hence, x s J(y) C D(y)
J+(x)
.
J+(x)
But, since L(x)
K(y).
is a closed invariant set, we must have K(y) C L(x). Therefore, x e L(x).
x is either a critical or a periodic point.
C(x).
Thus,
D+(x)
C C(x)
L+(x)
C
D+(x).
This implies that D(x)
This shows that
Let
D+(x)
Hence
K(x)
C(x),
and
the proof is complete.
NOTATION.
Let
(R2,)
be any continuous flow and S be the set
of critical
For any s e S we shall hence forth let
points.
N
s
{x e
R2:
REMARK 3.7.
x
s or x is periodic and S
Knight showed in
flows having characteristic 0.
int C(x)
{s}}
5] that there are six basic types of planar
These are
(I) parallelizable flows,
(2) flows having a global Poincar center,
(3) flows where S consists of one local Poincar center s, N s is unbounded and N is a single trajectory. The restriction of the flow
s
to R 2
N
is parallelizable,
(4) flows similar to Example 3 of [5],
(5) flows similar to Example 3 of [5], except that Ns
Ns where
2
R. MOGOLLON
274
S-
{Sl, s2}
and
(6) flows having only critical points
From this and from theorem 3.6, it follows that there are three basic
types of flows
the flows in
(R2,w)
satisfying
D+(x)
C(x) for each x
R 2.
These are
(2), (5) and (6).
The author is grateful to Professor Shair Ahmad for suggesting this prob-
lem.
His guidance and suggestions have been invaluable.
The author also
thanks the referee for his appropiate suggestions concerning the revision of
this paper.
References
I.
Ahmad, Shair, "Dynamical Systems of Characteristic 0+’’
3! (1970), 561
574
2.
Ahmad, Shair, "Strong Attraction and Classification of Certain Continuous
163.
Flows", Math Systems Theory, 5_. (1971), 157
3.
Bhatia, Nam, "Criteria for Dispersive Flows", Math. Nachr.
89
93.
4.
Bhatia, Nam and Hajek, Otmar, Theory o__f Dynamical Sy,stems, Parts I and II,
Technical Notes BN-599 and BN-606, University of Maryland, 1969
5
Knight, Ronald, "Dynamical Systems of Characteristic 0"
4_.1
6.
(1972), 447
Jour
Mogoll6n, Ramn,
of Math
32 (1960),
Pacific J
Math
457.
Knight, Ronald, "Structure and Characterizations Of Certain
Flows", Funkcial.
7.
Pac
Ekvac.
Continuous
17 (1974), 223- 230.
Characterizing Certain Continuous Flows", to appear
in
J. Math. and Phys. Sc.
8.
9.
Seibert, Peter and Tulley, Patricia, "On Dynamical Systems in
Arc___h. Ma.th. I.8 (1967), 290- 292.
the Plane",
Ura, Taro, "Sur le Courant ExtSrieur a une Rgion Invariante; Prolongements d’une Caracteristique et l’order de StabilitY", Funkcial.
Ekvac.
2 (1959), 143
200.