Every autonomous dynamical system （ X , f ） induces a set-valued dynamical system
on the space of compact subsets of
X
. In this paper we have investigated some chaotic relations between a nonautonomous dynamical system and its set valued extension.
There are two main types of dynamical systems: differential equations and iterated maps(also called difference equation). Differential equation describes the continuous time evaluation of the system, whereas difference equation describes the discrete time evaluation of the system. Iterated maps are the tools for analyzing periodic and chaotic solution of differential equation. Again, there are two types of difference equation: autonomous and nonautonomous, called as autonomous and nonautonomous discrete dynamical system. During the past few decades, there has been increasing interest in the study of discrete dynamical system (or difference equation) of the form,
where 
is a map and 
is a metric space or all
. In particular, if 
and 
for all
, then (1) reduces to,
where 
is a map.
The system (1) is called a nonautonomous discrete dynamical system, which is governed by the sequence of maps
. While the dynamical system (2), governed by the single map f, called an autonomous discrete dynamical system.
Chaos of system (2) or a time-invariant map 
has been discussed thoroughly in [1-5]. However, evolutions of certain physical, biological, and economical complex systems are necessarily described by a nonautonomous systems whose dimensions vary with time in some cases. In [
and 
for all
) by introducing several new concepts. In 2009, Chen and Shi in [
The main task to investigate the dynamical system 
is, how the points of X move under the iterate of
. Nevertheless, in many fields or problems such as biological species, demography, numerical simulation and attractors, etc, it is not enough to know only how the points of 
move, one should know how the subsets of 
move. So it is also necessary to study the set valued dynamical system 
associated to the system
, where 
is a continuous map on a compact metric space 
and 
is a natural extension of 
on 
(collection of all non-empty compact subsets of
). Many papers [9-13] has been devoted to the study of chaotic relation between autonomous system 
and its set valued extension
. Normally, we come across so many natural phenomena which explicitly depend on time where the starting point is just as important as the time elapsed. We would like to know what would be the collective dynamics of such system in relation to the individual dynamics. This paper is an endevour to investigate the relations between the individual dynamics and collective dynamics for time dependent discrete systems.
So, here we have considered the set-valued extension of a nonautonomous system (1), as
where
. It is clear that this system is governed by the sequence of maps 
on setvalued extension of
, i.e.
. So far, there is no investigation has been done on the chaotic relationship between systems (1) and (3).
In present paper, we investigate the relation between 
and 
in the related chaotic dynamical properties such as transitivity, sensitivity, dense set of periodic points, weak mixing, mixing and topological exactness, with 
and 
for all 
Let 
be a continuous self map on a compact metric space
.
Definition 2.1 A map 
is said to be transitive if for every pair of open sets 
and
, there exists an 
such that
.
Definition 2.2 A point 
is said to be periodic if there exists 
such that
. The least such 
is called the period(prime period) of the point
.
Definition 2.3 f is said to have sensitive dependence on initial conditions (sensitive), if there exist 
(sensitivity constant) such that for every point 
and for each 
there is 
and 
such that 
and 
A continuous map f is chaotic in the sense of Devaney (Devaney chaotic) if:
1) f is topological transitive;
2) f has dense set of periodic points;
3) f has sensitive dependence on initial conditions.
It is known that condition (1) together with (2) implies (3) on compact metric spaces, see [
Definition 2.4 Map f is weakly mixing if for any pair of non-empty open sets
, 
in
, there exists a positive integer k, such that 
and
.
Definition 2.5 f is topologically mixing(mixing) if for any pair of non-empty open sets U, V in X, there exists an integer 
such that
, for all 
.
Definition 2.6 A map 
is topological exact or locally eventually onto(leo) if for any non empty open set 
there exist an integer 
such that 
Define the hyperspace 
as the collection of all the non-empty compact subsets of
. If 
we define the 
-neighbourhood of 
as the set
where 
The Hausdorff metric on 
is defined as
It is well known that 
is a compact metric space, if 
is a compact metric space.
For any finite collection 
of non empty subsets of 
define
Collection of these kind of sets form a base for the topology on
, called Vietoris topology (also called hit and miss topology [
is a compact metric space then Hausdorff topology coincide with Vietoris topology.
Let A be a subset of X. Define the extension of A to 
as,
.
Remark 3.1 [
if and only if
.
Result 3.2 [
. Then 
is a non-empty open subset of
.
Result 3.3 [
and 
be two non-empty subset of 
and 
is continuous. Then1) 
2) 
3)
, for every 
It has been proved that the collection 
generate a topology on
, called 
- topology(also called Upper Vietoris topology). So if 
is any non-empty open set in 
(with 
- topology) then by the above result, there exists non-empty open subsets 
of 
such that,
.
Let 
be a metric space and 
be a sequence of maps, 
For a point 
define a sequence as follows:
then the sequence 
is said to be an orbit of the sequence 
of the maps (starting at x0)
or an orbit of 
in the iterative way.
In addition, for any point
, define a sequence as follows:
then, the sequence 
is said to be an orbit of the sequence 
of the maps (starting at x0)
or an orbit of 
in the successive way.
Now on for convenience, for any sequence
of maps defined on a metric space
denote maps 
for any
, by
and
, for any
. It is obvious that any orbit 
of 
in the iterative way is an orbit 
of 
in the successive way.
Definition 4.1 
is said to be transitive in iterative(or successive) way if for every open set 
of
, there exists a positive integer 
such that 
(or
).
Definition 4.2 Let
, 
is said to be periodic in iterative(or successive) for
, if there exists a integer 
such that 
(or 
) and for any interger
, 
(or
).
Definition 4.3 If there exists a constant 
such that for any point 
and any
, the ball 
contains a point 
and there exists a positive integer 
such that 
or
, then the sequence 
of maps is said to be sensitive dependence on initial condition (on
) in the iterative or successive way.
The sequence 
of maps is said to be chaotic (on
) in the iterative(or successive) way, in the sense of Devaney, if 1) F is transitive (on X) in the iterative (or successive) way.
2) The set of periodic points of F is dense in X in iterative (or successive) way.
3) F has sensitive dependence on initial condition in the iterative(or successive) way.
Definition 4.4 If for any pair of non-empty open sets
, 
in
, there exists a positive integer
, such that 
and 
(
and
), then the sequence of maps 
is said to be weakly mixing in iterative (successive) way.
Definition 4.5 If for any non-empty open sets U and 
in
, there exists a positive integer 
such that,
Then the sequence 
of maps is said to be mixing (on X) in the iterative or successive way.
Definition 4.6 If for any non-empty open set 
in
, there exists a positive integer 
such that,
then the sequence 
of maps is said to be topologically exact (on
) in the iterative or successive way.
It easy to see that that chaotic properties defined for a autonomous system (2) which is governed by the single map f on a metric space X, is a particular case for the chaotic properties defined for nonautonomous system (1) in successive way.
Consider a compact metric space 
and its setvalued extension
. Let 
and
be the sequence of continuous maps representing the nonautonomous systems 
and 
respectively, where 
and 
for all
. Here we will take 
-topology on 
for proving all our results and examples.
Theorem 5.1 Sequence of maps 
is transitivity in iterative (or successive) way on X iff
is transitive in iterative (or successive) way on
.
Proof. We will do the proof for iterative way, for successive way it would be similar.
Take a pair of non-empty open sets
,
in
, where Ui, 
are open in X for 
and 
Fix 
and 
Since F is transitive in iterative way, we can find a 
and 
such that
, implies
, where
. Consequently,
.
Conversely, take a pair of non-empty open set U and V in X. Since X is compact, so for U open in X we can find a non-empty open set
, such that 
Clearly, 
and 
will be open and non-empty in
, there exist an positive integer 
such that
, therefore,
Hence
. 
Example 5.2 Consider the sequence of maps
on the unit circle
, defined as
, for
where 
is an irrational.
Then,
. It is not difficult to prove that F is transitive in iterative way but not in successive way. Hence the sequence
on 
is transitive in iterative way but not in successive way (by Theorem 5.1).
Theorem 5.3 The sequence of maps 
is topologically mixing in iterative (or successive) way on X iff 
is topologically mixing in iterative (or successive) way on
.
Proof. The proof is similar to proof done for transitivity, with slight modifications.
Theorem 5.4 Let 
and 
be the sequences of continuous maps on 
and 
respectively. If 
is sensitive in iterative (or successive) way, then 
is sensitive in iterative (or successive) way.
Proof. Let 
be sensitive in iterative waywith sensitive constant 
Let 
and 
then as 
there exists 
and
such that
.
Since A is compact and 
is continuous, we can find a 
such that
. Clearly
implies
, which implies
. Hence 
is sensitive in iterative way on
.
Similarty, we can prove it for successive way. 
Theorem 5.5 If 
has dense set of periodic points in iterative (or successive) way on X, then
has dense set of periodic points in iterative (or successive) way on
.
Proof. Let F has dense set of periodic points in successive way. Take any open set 
in
, then
, where 
open in
. There exists
and a positive integer 
correspondingly, such that 
for
,
. Take
and
, then clearly 
and
, for all
. Therefore, 
has dense set of periodic points on 
in successive way.
Proof in iterative way can be done likewise. 
Here we give an example where the nonautonomous dynamical system don’t have any periodic points in iterative (and successive) way but its set-valued extension has dense set of periodic points in successive way.
Example 5.6 Consider the sequence space,
on two symbols. Let 
, 
be any two elements of
. Define distance between them as 
. It has been proved that 
is a metric space.
Define a binary composition of addition on elements of 
as
where 
if
, else 
and carry 1 to next position. With this composition 
is a compact topological group.
Consider a sequence of map 
on 
defined as
where
, 
if
, else 0.
for all 
It can be seen that P has no periodic points in iterative and successive way. Consider an open set 
where 
is open in 
Since the cylinder set,
forms the basis for the topology on
, there exist
which is compact in 
hence
. We can find a 
such that
Hence, 
has dense set of periodic points in successive way.
Theorem 5.7 The sequence of maps 
is weakly mixing in iterative (or successive) way iff
is weakly mixing in iterative (or successive)
way.
Proof. Let F is weakly mixing in successive way on X. Consider a pair of non-empty open sets
,
in
, where
, 
are open in
, for 
and 
Fix 
and
, therefore there exist an positive integer 
such that 
and 
there exists 
and 
such that 
and
So, 
and 
consequently implies 
and 
Conversely, suppose that 
is weakly mixing in successive way. Take any pair of non-empty open sets 
in
, then 
and 
will be open in
. We can find 
such that 
and 
Now
and
Hence 
and 
The proof in iterative way can be done likewise. 
Theorem 5.8 The sequence of maps 
is topologically exact in iterative (or successive) way on X iff 
is topologically exact in iterative (or successive) way on
.
Proof. The proof is easy, hence omitted.
Example 5.9 Consider
, the cycle group with two elements and discrete topology. Binary operation of addition (“+”) and subtraction (“–”) is defined under modulo 2. Let
. It is well Known that X is compact, perfect and has countable base containing clopen sets which can be chosen to consist of cylinder sets of the form
Define a sequence of maps 
on X, as
, where
It is clear that for every non-empty cylinder set
,
Therefore, F is topological exact in iterative way, clearly it can be seen that 
in not topological exact in successive way on X.
Hence, 
is mixing, weakly mixing, transitive in iterative way on 
and so is 
on
. Also, in every cylinder set we can find a sequence of repetitive block of symbols, which are periodic in successive and iterative way under F. It is not difficult to see that 
is sensitive with sensitivity constant
in iterative ways.
It is interesting to see that for any open set
, there exists cylinder sets 
and
in
, where
. We can always find a positive integer 
such that
, hence 
is sensitive on
in iterative way.
In this article we have studied some chaotic properties on time-varying map (i.e. a sequence of time-invariant maps). We have investigated the relation between
and 
defined on X and 
respectively, in the related chaotic dynamical properties such as transitivity, sensitivity, dense set of periodic points, weak mixing, mixing and topological exactness. In this endeavour, we proved that, 
is transitive (weak mixing, mixing and leo, respectively) if and only if 
is so in iterative (successive) way. Also an example is given to prove that denseness of periodic points for 
doesn’t imply the same for
, in successive way. The question which is still open is, does sensitivity of 
implies sensitivity for
, which we think may not be possible in general, as for autonomous map sensitivity on original dynamical system doesn’t imply sensitivity on hyperspace dynamical system. These kinds of investigations would be useful in understanding the relationship between the dynamics of individual movement and the dynamics of collective movements for the time-varying map (i.e. a sequence of time-invariant maps).