Some Chaotic Properties of a Kind of Coupled Map Lattices

This paper is concerned with some chaotic properties of a kind of coupled map lattices, which is proposed by Kaneko. First, this research discussed the sensitivity, infinite sensitivity, transitivity, accessibility, densely Li-Yorke sen-sitivity and exact of coupled map lattices. Then, some sufficient conditions under which ( ) , , d F ∞∞ ∞∞ ∆ ∆ is Kato chaotic, positive entropy chaotic and Ru-elle-Takens chaos are obtained.


Introduction
In 1983, Kaneko [1] proposed coupled map lattices (Short for CMLs). Then, in biophysics, materials, chaos, image processing, CMLs are intensively discussed (Refer to literature [2]- [8] and others). In 2005, the literature [9] showed that CMLs have some topology and ergodic properties. In 2010, Juan Lu [10] presented a definition of distributional chaos on a sequence (DCS) for CML systems and stated two different sufficient conditions for having DCS. In 2010, Juan Luis [11] proved that this CML system has positive topological entropy for zero coupling constant. In 2016, Risong Li [12] [13] had obtained some relevant conclusions for the zero coupling constant and proved that the system has three kinds of chaos. In this paper, the following CML from [14] is considered.    Then, one can see that the above system (1) is equivalent to the following system For the above system (3), the map F is said to be induced by the system (1).
Obviously, a double-indexed sequence { } Next section, the definitions of sensitive, infinite sensitive, transitive, accessibility, densely Li-Yorke sensitive and exact will be reviewed. And then, in section 3, it is proved that the system ( ) satisfies three definitions of chaos (Kato chaotic, positive entropy chaotic and Ruelle-Takens chaos) under the conditions that f is chaos in these sense.

Preliminaries
After T. Y. Li and J. A. Yorke [15] first put forward the mathematical definition of "chaos", many other definitions of chaos appeared later. For example, sensi-tivity, infinite sensitivity, transitivity, accessibility, densely Li-Yorke sensitivity, Kato chaotic, positive entropy chaotic, Ruelle-Takens chaos, and so on.
, X ρ be a metric space and : f X X  be a continuous function. f is said to be 1) transitive if for any nonempty open subsets 1 2 , [16]).

Remark 1. [19]
There is another equivalent definition of transitivity. f X X → ) is Kato chaotic if it is sensitive and accessible (see [20]).
2) A dynamic system ( ) , X f (or the map : f X X → ) is chaotic in the sense of Ruelle and Takens (short for R-T chaotic) if it is transitive and sensitive (see [21]

Main Results
In this section, let X I = . The metric ρ in I is defined by Theorem 1. If f is transitive, then the system ( ) Proof. Since f is transitive, then there exist a I .
Theorem 2. If f is sensitive, then the system ( ) It is easy to know that, for is chaotic in the sense of Ruelle and Takens. Proof. According to Theorem 1, Theorem 2 and the definition of R-T chaos, the conclusion is obvious.
According to Proposition 2 and Theorem 2, the following Corollary is hold.
Corollary 2. If f is infinitely sensitive, then the system ( ) In fact, there is a stronger conclusion.
Theorem 3. If f is infinitely sensitive, then the system ( ) Proof. Since f is infinitely sensitive, then there exists a Corollary 3. If f is Kato chaotic, then the system ( ) Proof. According to Theorem 2 and Theorem 4, the conclusion is obvious.
Theorem 5. If f is exact, then the system ( ) In [23] we had proved that, f is Li-Yorke sensitive implies that the system ( ) is Li-Yorke sensitive. Inspired by this, the following conclusion can be drawing.