Chaotic Properties on Time Varying Map and Its Set Valued Extension ()
1. Introduction
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,
(1)
where
is a map and
is a metric space or all
. In particular, if
and
for all
, then (1) reduces to,
(2)
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 [6], Chen and Tian study the chaos of system (1) (with
and
for all
) by introducing several new concepts. In 2009, Chen and Shi in [7] introduced some basic concepts, including chaos in the sense of Devaney, Wiggins and in a strong sense of Li-Yorke and studied their behavior under topological conjugacy. In [8], the author introduced a new type of subsystem of a nonautonomous discrete dynamical system, which is a partial compositions of a given sequence of maps(from which nonautonomous dynamical system is generated), and the concept of chaos in the strong sense of Wiggins is introduced. Also, some Li-Yorke and Wiggins chaotic connections (in the strong sense) between a given dynamical system and its subsystems have been studied.
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
(3)
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 ![](https://www.scirp.org/html/8-5300418\9fa39031-328a-43f9-ba26-c50bd01d69e9.jpg)
2. Basic Definition and Notation
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 ![](https://www.scirp.org/html/8-5300418\9b2a1063-df45-4a9f-a360-6aba2f8c849d.jpg)
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 [3]. Further, for interval maps it is known that transitivity alone implies chaos [2].
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 ![](https://www.scirp.org/html/8-5300418\7b44085b-c6c1-4ed8-b958-ab740923685b.jpg)
3. Set-Valued Extension
Define the hyperspace
as the collection of all the non-empty compact subsets of
. If
we define the
-neighbourhood of
as the set
where ![](https://www.scirp.org/html/8-5300418\1b0ec4e0-d2e5-49a2-a828-a669fc6921ad.jpg)
The Hausdorff metric on
is defined as
![](https://www.scirp.org/html/8-5300418\5d977a44-7dc2-4653-950c-4a62ccfbc607.jpg)
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
(4)
Collection of these kind of sets form a base for the topology on
, called Vietoris topology (also called hit and miss topology [9] given by Leopold Vietoris). It is worth noting that if
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 [12] It is clear that
if and only if
.
Result 3.2 [12] Let A be a non-empty open subset of
. Then
is a non-empty open subset of
.
Result 3.3 [12] Let
and
be two non-empty subset of
and
is continuous. Then1) ![](https://www.scirp.org/html/8-5300418\a6ca45c2-4e39-45e3-934c-a879c8bb3b69.jpg)
2) ![](https://www.scirp.org/html/8-5300418\4baa052d-807f-40b0-82dc-0f5d2a6b507e.jpg)
3)
, for every ![](https://www.scirp.org/html/8-5300418\2a9439bf-9103-47c7-a734-9f642b30af81.jpg)
It has been proved that the collection ![](https://www.scirp.org/html/8-5300418\7a8acaf8-9e03-4f8e-913d-e4a0fcc12427.jpg)
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,
.
4. Dynamical Properties on Time-Varying Map
Let
be a metric space and
be a sequence of maps,
For a point
define a sequence as follows:
![](https://www.scirp.org/html/8-5300418\630e6872-af8a-44c9-a835-ae6382f4936b.jpg)
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:
![](https://www.scirp.org/html/8-5300418\deae54d6-4290-492a-bcf9-66751a3e7ca4.jpg)
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
![](https://www.scirp.org/html/8-5300418\b8c07bbf-28af-4d3c-b259-cee185a4b31f.jpg)
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 ![](https://www.scirp.org/html/8-5300418\090a3a8e-6704-4af0-8bab-af8826151fd7.jpg)
) 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 ![](https://www.scirp.org/html/8-5300418\5256d126-245b-450f-b8b1-c538251403a3.jpg)
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
(![](https://www.scirp.org/html/8-5300418\315f160a-e329-48d8-8d01-e23e29fc6f75.jpg)
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,
![](https://www.scirp.org/html/8-5300418\a9aef831-eeb2-47ad-abea-0e7b21dcda25.jpg)
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,
![](https://www.scirp.org/html/8-5300418\189e5ebc-465c-4612-a766-7fdfec7682f7.jpg)
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.
5. Main Results
Consider a compact metric space
and its setvalued extension
. Let
and
be the sequence of continuous maps representing the nonautonomous systems ![](https://www.scirp.org/html/8-5300418\e1b2fd39-2abf-43c2-ac9a-3e9f2e20ee48.jpg)
and
respectively, where ![](https://www.scirp.org/html/8-5300418\fc176c2a-5da1-4b3a-96c4-6846acf838ff.jpg)
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,
![](https://www.scirp.org/html/8-5300418\fdc3d260-93a0-4d85-9bc1-2b2a93f78dab.jpg)
Hence
. ![](https://www.scirp.org/html/8-5300418\afcdb2a6-f92c-4461-96d8-98f3332c37ab.jpg)
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. ![](https://www.scirp.org/html/8-5300418\d3533e27-e029-4714-938f-96fa00fab741.jpg)
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. ![](https://www.scirp.org/html/8-5300418\1517e44e-45f3-4d0b-b055-9ea90939ead1.jpg)
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 ![](https://www.scirp.org/html/8-5300418\a9c01517-23c3-49fd-aa48-eb2fe33163d6.jpg)
,
be any two elements of
. Define distance between them as ![](https://www.scirp.org/html/8-5300418\94e84070-c7b4-4041-b55b-74af9803ac33.jpg)
. 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 ![](https://www.scirp.org/html/8-5300418\8394f59c-8941-419e-baa3-304859b060ff.jpg)
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,
![](https://www.scirp.org/html/8-5300418\43eb2550-27ca-4279-a9e3-7cdcb57af315.jpg)
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 ![](https://www.scirp.org/html/8-5300418\adcf203b-9501-43c3-acf5-7f35c27be7f7.jpg)
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 ![](https://www.scirp.org/html/8-5300418\d8857dfe-2a8b-49cc-ba0c-112a3a678dfd.jpg)
and
Now
![](https://www.scirp.org/html/8-5300418\09288296-def1-4180-99de-1a5faf1073bb.jpg)
and
![](https://www.scirp.org/html/8-5300418\32ece80d-f752-4126-b981-6ec6263d9c1a.jpg)
Hence
and ![](https://www.scirp.org/html/8-5300418\fc0a4504-ec54-4578-8365-4556c44ab2e1.jpg)
The proof in iterative way can be done likewise. ![](https://www.scirp.org/html/8-5300418\3dbc7873-7629-443e-8c0d-29afc2b08f7c.jpg)
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
![](https://www.scirp.org/html/8-5300418\e11bd941-b626-48e2-95db-527aebe743af.jpg)
Define a sequence of maps
on X, as
, where
![](https://www.scirp.org/html/8-5300418\73909afb-80cc-4b4b-9ad7-c0672e298a08.jpg)
It is clear that for every non-empty cylinder set
,
![](https://www.scirp.org/html/8-5300418\59f477b6-f740-431f-9fbb-960a4d4624ef.jpg)
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.
6. Conclusion
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 ![](https://www.scirp.org/html/8-5300418\3b57d431-2ac7-41fd-a3ee-76cfd6683e40.jpg)
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).
NOTES