Internat. J. Math. & Math. Sci.
Vol. 10, No. 3 (1987) 613-614
613
POISSON STABILITY IN PRODUCT OF DYNAMICAL SYSTEMS
A. KUMAR and R.P. BHAGAT
Satgharwa, P.O. Buniad Ganj
GAYA(INDIA) Pin 823003
(Received July I0, 1986 and in revised form October 14, 1986)
ABSTRACT.
This paper deals with Poisson stable and distal dynamical system.
It is
shown that the product of Poisson stable motions is a Poisson stable motion in
dynamical systems.
KEY WORDS AND PHRASES.
Distal dynamical syst, compact, Poison stable, reck%rent
motions.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
34C35.
INTRODUCTION.
Product of dynamical systems is a dynamical system and product of semidynamical
systems [I] is a semidynamical system.
s e I
In [I] Bajaj has shown that if fIR
R
and
be
the
factor
positive
systems
be the product of positive limit sets of the
x
cannot be replaced
IIR
In general,
limit set of the product system then R
X
X
E I
is nonempty. He
may be empty even if each of
by equality. Indeed
x0
x
has found the conditions under which
I
x2 2
Xl
x
and
x
could be nonempty
and compact for a finite product.
We studied Poisson stable and distal dynamical systems in [2].
shall mention the conditions under which
x
IIRx
e I
In this paper we
and the product of
Poisson stable motions is a Poisson stable motion in dynamical systems.
2.
DEFINITIONS AND NOTATIONS.
We shall use the definition and notations of [2] and [3].
3.
MAIN RESULTS.
PROPOSITION 3.1.
trajectory
y(x)
THEOREM 3.2.
(x,t)
If a motion
is a closed set.
is Polsson stable and distal then its
[2, theo.2.1].
If a compact motion is Poisson stable and distal then it is a
compact recurrent motion.
PROOF.
Let the motion
y(x) is closed hence
y(x)
(x,t)
cl
be Poisson stable and distal then its trajectory
y(x)
x
As the motion is compact each of the above sets is compact and minimal, and thus by
Birkhoff recurrence theorem w(x,t) is compact recurrent.
THEOREM 3.3.
then
X
Let
(X,)
is distal iff for every net
{zeX
x3
z for some
be Lagrange stable,
be a semldynamlcal system, let
xeX
R+,
the phase space
t
in
i
and some subnet
tj of
tl}.
[4, theorem 2.6].
A. KUMAR AND R.P. BHAGAT
614
One can show that positive distal dynamical system are distal whenever all positive
Thus the above result for semidynamical systems
trajectory closures are compact.
is applicable in present dynamical systems as well.
cly(x)
y(x)
is compact and minimal.
x
If a compact motion
PROOF.
is Poisson stable and distal iff
(x,t)
A compact motion
THEOREM 3.4.
y(x)
cly(x)
is Poisson stable and distal then
(x,t)
is compact and minimal.
x
be compact and minimal then it is compact
x
recurrent and therefore compact Poisson stable.
y(x)
Conversely, let
cly(x)
is compact and minimal, for every
x
X
in
the net
xt
in
i
(x)
has a subset
in y(x).
z
xt.
3
Here y(x)
is distal we shall use theorem 3.3.
(X,)
To show the dynamical system
{z e X
X
Thus
z for some net t
xt.
i
in
and t. is some subnet of
T
in T}.
t
i
Which completes the proof.
THEOREM 3.5.
of a compact Poisson stable distal motion is
y(x)
The trajectory
complete and totally bounded.
PROOF.
Let
Now we shall consider product of such dynamical systems.
Let
be a family of dynamical systems.
{x}.
xX, x
The dynamical system
of the family
(Xa,a),
(X,)
THEOREM 3.7.
{x
then
(X,)
(X,)
THEOREM 3.8.
by
{xat},
(x,t)
ae I
and
be a family of dynamical systems and
I
{xa}.
be a family of
Then
x
(X,)
II x
{Lagrange stable} {distal}
is the product dynamical system.
(X,)
a E I
Let
x
E
X
and
be Lagrange stable distal dynamical systems.
their product dynamical system, then the product
is Poisson stable motion in the product space
-
(X,)
iff each
(xa,t),
is Poisson stable motion.
PROOF.
(x)
X
{Lagrange stable} {distal}.
(Xa,),
Let
{x
x X, x
(x,t)
is
ae I
xeX, x
Let
(Xa,),
Let
dynamical systems and
I
Let
obtained above is called direct product or product
(Xa,a)
the product dynamical system.
a
into
ae I
(Xa,),
be the product space.
a I.
Let
PROPOSITION 3.6.
Let
X x T
from
Define a map
IIXa
X
(X,) is a dynamical system.
then
x
X.
is compact in a Hausdorff uniform space
y(x)
Since
We have
llya(xa)
Therefore
y(x)
y(x)
II
cly(x
llya(xa)
x’
II cl
y=(xa)
cl
llya(xa)
cly(x).
Also
each being compact minimal and closed.
REFERENCES
"Product of Semidynamical Systems", Recent Advances in Topological
i.
BAJAJ, P.N.
2.
Dynamics, Springer Verlag lecture notes, No. 318, p. 23-29.
PRASAD, S.S. and KUMAR, A. "Stable P and Distal Dynamical Systems", Internat. J.
Math. & Math. Scl. 7(1984), 181-185.
KUMAR, A. "Almost Periodic Motions in a Complete Space", Internat. J. Math. &
3.
Math. Sci. 9(1986), 621-622.
4.
BHATIA, N.P. and NISHIHAMA, M. "Distal Semidynamical Systems", Dynamical Systems,
An International Symposium, Vol. II, Edited by Cessari, Hale & Lassale,
A.P., 1976, p. 187-190.