Chaos for Discrete Dynamical System
We prove that a dynamical system is chaotic in the sense of Martelli and Wiggins, when it is a transitive distributively chaotic in a sequence. Then, we give a sufficient condition for the dynamical system to be chaotic in the strong sense of Li-Yorke. We also prove that a dynamical system is distributively chaotic in a sequence, when it is chaotic in the strong sense of Li-Yorke.
Since Li and Yorke first gave the definition of chaos by using strict mathematical language in 1975 , the research on chaos has greatly influenced modern science, not just natural sciences but also several social sciences, such as economics, sociology, and philosophy. The theory of chaos convinced scientists that a simple definite system can produce complicated features and a complex system instead possibly follows a simple law. However, scientists in different fields, finding different chaotic connotations, gave different definitions of chaos such as Li-Yorke chaos, distributional chaos, and Devaney chaos. In order to establish a satisfactory definitional and terminological framework for complex dynamical systems that are based on strict mathematical definitions, these concepts with less ambiguous are necessary, and their interdependence has to be clarified. There is no doubt that the mathematical definition of Li-Yorke chaos has a large influence than any other one, whereas distributional chaos possesses some statistical connotations besides the uncertainty of long-term behaviors. So, comparing distributional chaos with Li-Yorke chaos is a meaningful and significant problem.
In order to reveal the inner relations between Li-Yorke chaos and distributional chaos, the author brought up the definition of distributional chaos in a sequence in . In this paper, we mainly prove the relations between some different chaoses in discrete dynamical systems.
The main theorems are stated as follows.
Theorem 1. If a dynamical system exhibits transitive distributional chaos in a sequence, then,(1)it is chaotic in the sense of Martelli;(2)it is chaotic in the sense of Wiggins.
Theorem 2. Let with and be a sequence of positive integers. If for any sequence where or there exists , such that for each then system is chaotic in the strong sense of Li-Yorke.
Theorem 3. If a dynamical system exhibits chaotic in the strong sense of Li-Yorke, then it is distributively chaotic in a sequence.
2. Problem Statement and Preliminaries
Throughout this paper will denote a compact metric space with metric .
2.1. Several Definitions and Lemmas
is said to be a chaotic set of if for any pair , , we have
Definition 4. is said to be chaotic in the sense of Li and Yorke (for short: Li-Yorke chaotic), if it has a chaotic set which is uncountable.
Let be an increasing sequence of positive integers, . Let where is 1 if and 0 otherwise. Obviously, and are both nondecreasing functions. If for we define , then and are probability distributional functions.
Definition 5. Let , If , we have
then is said to be a distributively chaotic set in a sequence. The two points are said to be distributively chaotic point pair in a sequence. is said to be distributively chaotic in a sequence, if has a distributively chaotic set in a sequence which is uncountable.
Definition 6. Let . If there exist two strictly increasing sequences of positive integers and such that for any , , then is said to be a strong scrambled set. is said to be chaotic in the strong sense of Li-Yorke, if has an uncountable strong scrambled set.
Definition 7. Let be an increasing sequence of positive integers, then,
is called proximal relation with respect to .
Thus is called asymptotic relation with respect to .
Thus is called distal relation with respect to .
So is called distributively chaotic respect to .
Definition 8 (see ). is (topologically) transitive if for any two nonempty open sets there exists such that . is (topologically) weakly mixing if for any three nonempty open sets there exists such that and .
Definition 9 (see [4–6]). Let be a continuous map from a compact metric space into itself. The orbit of a point is said to be unstable if there exists such that for every there are and satisfying inequalities and . The map is said to be chaotic in the sense of Martelli if there exists such that has dense orbit which is unstable.
Definition 10 (see ). Let be a continuous map from a compact metric space into itself. We say has sensitive dependence on initial conditions if there exists such that for any and , there is some and a nonnegative integer satisfying and . is said to be chaotic in the sense of Wiggins, if is transitive and has sensitive dependence on initial conditions.
Definition 11. Let be a compact metric space, be a continuous map, and be an uncountable distributively scrambled set in a sequence.
We say that exhibits dense distributional chaos in a sequence if the set may be chosen to be dense. If is not only dense but additionally consists of points with dense orbits, then we say that exhibits transitive distributional chaos in a sequence.
Lemma 12. Let be an infinite sequence set of . Then, there exists an uncountable subset such that for any different points for infinitely many and for infinitely many .
Proof. For a proof, see .
Lemma 13. If and are both infinite increasing subsequences of which is a sequence of positive integers, then there exists an infinite increasing subsequence such that
Proof. For a proof, see .
Lemma 14. is weakly mixing if and only if for any is transitive.
Proof. For a proof, see .
3. Proof of Main Theorem
Proof of Theorem 1. (1) There is no isolated points in as otherwise the set of points with dense orbit is at most countable. But in the case of compact set without isolated points, the existence of dense orbit implies transitivity.
Let be a dense scrambled set in the sequence consisting of transitive points, and let be such that for all distinct . Let us fix any . Because orbit of is dense, for any , there exists and satisfying the inequalities and . This implies that for some . This shows that the orbit of is unstable. So, is chaotic in the sense of Martelli.
(2) Fix any . In -neighborhood of any point , we can find points such that . Then, or .
Proof of Theorem 2. Let be an uncountable subset of , as in Lemma 12. For each , by the hypotheses, we can choose a point such that for any , if then,
Put . Clearly, if then . It follows that being uncountable implies so is .
Let be any different points, where . By the property of , we know that there exist sequences of positive integers such that for all , and for large enough , we have . Thus, this shows Meanwhile, for large enough, and lie in the same ball of diameter less than . Thus, , so This shows
Above all, is chaotic in the strong sense of Li-Yorke.
Proof of Theorem 3. Because is chaotic in the strong sense of Li-Yorke, there exists an infinite increasing sequence and uncountable set , such that for any with , we have
so that .
Again, by the definition of chaos in the strong sense of Li-Yorke, there exists , such that so that .
Hence, , where . Then by Lemma 13, there exists a subsequence such that . This shows that is a distributively chaotic set in the sequence of .
Corollary 15. If system satisfies conditions of Theorem 2. then it is distributively chaotic in the sequence.
Corollary 16. Let be a locally compact metric space containing at least two points. If system is weakly mixing, then it must be chaotic in the strong sense of Li-Yorke.
Proof. Let be weakly mixing, with . Take arbitrarily a nonempty open set such that is compact. Since is weakly mixing, there exists such that and . Thus, we find points such that . Assume that there exist positive integers such that for each finite sequence , where , there is a point satisfying for , the set of all such points will denoted by . By continuity of , each has an open nonempty neighborhood such that , if , it follows from Lemma 14 that there exists such that for each and . Thus by induction, we know that there exists a sequence of positive integers such that for any finite sequence , there is a point satisfying .
Let be an infinite sequence, where For each , we can take a point such that Since is compact, the infinite sequence has a limit point in , say , it is not difficult to show .
Thus by Theorem 2, is a strong chaos in the sense of Li-Yorke.
This work is supported by the NSFC no. 11271061, the NSFC no. 11001038, the NSFC no. 61153001, and the Independent Research Foundation of the Central Universities no. DC 12010111.
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2, Springer, New York, NY, USA, 1990.View at: MathSciNet