Chaotic Properties on Time Varying Map and Its Set Valued Extension

Every autonomous dynamical system   , f  induces a set-valued dynamical system  X   , X f 


Introduction
has been discussed thoroughly in [1][2][3][4][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 n 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, X X  d d and n  for all n N  ) 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. where is a metric space or all .In particular, and for all , then (1) reduces to, where f X X  is a map.The system (1) is called a nonautonomous discrete dynamical system, which is governed by the sequence of maps n   f .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
The main task to investigate the dynamical system

 
, f is, how the points of X move under the iterate of X f .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 X move, one should know how the subsets of X move.So it is also necessary to study the set valued (collection of all non-empty compact subsets of X ).Many papers [9][10][11][12][13] has been devoted to the study of chaotic relation between autonomous system   , X f and its set valued extension     , X f  .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 . 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 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] as the collection of all the non-empty compact subsets of Collection of these kind of sets form a base for the topology on   X  , called Vietoris topology (also called hit and miss topology [9] given by Leopold Vietoris).It is worth noting that if X 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 then the sequence is said to be an orbit of the sequence of the maps (starting at x 0 )  or an orbit of F in the iterative way.
In addition, for any point 0 , define a sequence as follows: then, the sequence is said to be an orbit of the sequence of the maps (starting at x 0 )  or an orbit of F in the successive way.Now on for convenience, for any sequence . It is obvious that any orbit of in the iterative way is an orbit x is said to be periodic in iterative(or successive) for , if there exists a ) and for any interger of maps is said to be sensitive dependence on initial condition (on X ) in the iterative or successive way.
The sequence of maps is said to be chaotic (on  X ) 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
), then the sequence of maps  then the sequence of maps is said to be topologically exact (on X ) 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.

Main Results
Consider a compact metric space   , X d and its set- . Here we will take -topology on   for proving all our results and examples.

Consider the sequence of maps on the unit circle
, defined as 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  .Proof.The proof is similar to proof done for transitivity, with slight modifications.
Since A is compact and F is continuous, we can find a Similarty, we can prove it for successive way.x U and a positive integer correspondingly, such 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, be any two elements of Define a binary composition of addition on elements of  as and carry 1 to next position.With this composition  is a compact topological group.
Consider a sequence of map on , else 0. where      , , , : forms the basis for the topology on , there exist Hence, has dense set of periodic points in successive way.
Theorem 5. 7 The sequence of maps V , therefore there exist an positive integer such and there is weakly mixing in successive way.Take any pair of non-empty open sets in X , then and will be open in

Hence and n n
The proof in iterative way can be done likewise.
V    Theorem 5.8 The sequence of maps is topologically exact in iterative (or successive) way on X  .Proof.The proof is easy, hence omitted.Example 5.9 Consider 2 Z , the cycle group with two elements and discrete topology.Binary operation of addition ("+") and subtraction ("-") is defined under modulo 2. Let   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 F in not topological exact in successive way on X. Hence, is mixing, weakly mixing, transitive in iterative way on   X and so is F X 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 F is sensitive with sensitivity constant

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 spectively, 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, F is transitive (weak mixing, mixing and leo, respectively) if and only if F is so in iterative (successive) way.Also an example is given to prove that denseness of periodic points for F doesn't imply the same for F , in successive way.The question which is still open is, does sensitivity of F implies sensitivity for F , 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).

Definition 4 . 3
If there exists a constant   such that for any point x F is sensitive in iterative (or successive) way, then F is sensitive in iterative (or successive) way. in iterative way, with sensitive constant  Let x Xand  then as 0,  of periodic points in iterative (or successive) way on X, then of periodic points in iterative (or successive) way on X  .Proof.Let F has dense set of periodic points in successive way.Take any open set

X
in iterative way.

2.3 f is said to have sensitive dependence on initial conditions (sensitive), if there exist 0
.

Dynamical Properties on Time-Varying Map
for every It has been proved that the collection   X X generate a topology on  e w , called topology(also called Upper Vietoris topology).So if  is any non-empty open set in   X  e w (with topology) then by the above result, there exists non-empty     4.
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 0