ABSTRACT

Every autonomous dynamical system 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.

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

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

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

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

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

(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)

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,.

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:

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 spacedenote 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.

5. Main Results

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 U_i, 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

, forwhere 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 constantin 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

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).

REFERENCES

NOTES

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