Nonlinear Analysis: Theory, Methods & Applications
Difference between Devaney chaos associated with two systems
Introduction
Let be a compact metric space with metric and be continuous. Then is called a compact system. For every positive integer , we define inductively by , where is the identity map on .
A compact system is called Devaney chaotic [1] if it satisfies the following three conditions:
(1) is transitive, i.e., for every pair and of non-empty open subsets of there is a non-negative integer such that ;
(2) is periodically dense, i.e., the set of periodic points of is dense in ;
(3) is sensitive, i.e., there is a such that, for any and any neighborhood of , there is a non-negative integer such that .
It is worth noting that the conditions (1) and (2) imply that is sensitive if is infinite [2], [3]. This means that the condition (3) is redundant in the above definition.
In the paper [4] Román-Flores investigated a certain hyperspace system associated to the base system , where is the natural extension of and is the family of all non-empty compact sets of endowed with the Hausdorff metric induced by . He presented a fundamental question: Does the chaoticity of (individual chaos) imply that of (collective chaos)? and conversely?
As a partial response to this question, Román-Flores [4] discussed the transitivity of the two systems, and showed that the transitivity of implies that of , but the converse is not true. Fedeli [5] showed that the periodical density of implies that of . Banks gave an example which has a dense set of periodic points in the hyperspace system but has none in the base system [3]. Gongfu Liao showed that there is an example on the interval which is Devaney chaos in the base system while is not in its induced hyperspace system [6].
In this paper, we show that being Devaney chaos need not imply being Devaney chaos. Further, need not be Devaney chaos even if is mixing and periodically dense. This answers the question posed by Román-Flores and Banks [3], [12].
Section snippets
Preliminaries
is a compact system. If and is -invariant, i.e. , then is called the subsystem of or , where is the restriction of on .
A subset is minimal if the closure of the orbit of any point of is , i.e. , for all . An equivalent notion is that has no proper -invariant closed subset.
A point is said to be an almost periodic point if for any exists such that for any integer exists an integer with property that and
Periodically dense system
Theorem 5 If is the hyperspace system induced by the compact system , then has a dense set of periodic points if and only if for any non-empty open subset of there exists a compact subset and an integer such that .
Proof Since has a dense set of periodic points, every non-empty open subset of has at least one periodic point. Further, every basic element has at least one periodic point. Let be an arbitrary open subset of . Then by the definition of the
Devaney chaos
If is an almost periodic point of and , then is usually a mixing non-trivial minimal subsystem of , which depends on the selection of the point . There are many such examples [11]. From these examples we take an arbitrary one as our research object and this do not affect the proof and our results.
We construct a subsystem of associated to the selected as follows.
For , denote then is
Conclusion
In summary, we showed some difference between the chaoticity of a compact system and the chaoticity of its induced hyperspace system. Devaney chaos in the hyperspace system need not implies Devaney chaos in its base system. Even if the hyperspace system is mixing, the former need not implies the latter. This kind of investigation should be useful in many real problems, such as in ecological modeling, demographic sciences associated to migration phenomena and numerical simulation, etc.
Acknowledgements
The first author is supported by the Youth Foundation of Institute of Mathematics, Jilin University. The second author was supported by a grant from Postdoctoral Science Research Program of Jiangsu Province (No. 0701049C) and Specialized Research Fund for Outstanding Young Teachers of East China University of Science and Technology. The first and third authors were supported by NSFC (No. 10771084).
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