1. 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.
(1)
where
,
,
, I is a non-degenerate compact interval, f is a map on I, and
is a constant.
For
, let
and
. For any sequence
on
, by induction, one can obtain a double-indexed sequence
, which is said to be a solution of the above system (1) with initial condition
.
Let I be a subset of real number set, write
and
which is called the diagonal set of
.
For arbitrary, two sequences
,
, it is easy to prove that
(2)
is a metric on
.
Let
be a continuous map and
be a solution of the above system (1) with initial condition
.
Let
and
where
and
Then, one can see that the above system (1) is equivalent to the following system
(3)
For the above system (3), the map F is said to be induced by the system (1). Obviously, a double-indexed sequence
is a solution of the above system (1) if and only if the sequence
is a solution of the above system (3).
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.
2. 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, sensitivity, infinite sensitivity, transitivity, accessibility, densely Li-Yorke sensitivity, Kato chaotic, positive entropy chaotic, Ruelle-Takens chaos, and so on.
Definition 1. Let
be a metric space and
be a continuous function. f is said to be
1) transitive if for any nonempty open subsets
,
for some integer
(see [16]).
2) sensitive if there exist
such that for any
and
, there exists
and
such that
(see [17]).
3) infinitely sensitive if there exist
such that for any
and
, there exists
and
such that
(see [17]).
4) accessible if for any
and any two nonempty open subsets
, there are two points
and
such that
for some integer
(see [16]).
5) exact if for any open subset
, there is
such that
(see [18]).
Remark 1. [19] There is another equivalent definition of transitivity.
is said to be transitivity, if there is an
such that
. Where,
is called the orbit of the point
.
Definition 2. 1) A dynamic system
(or the map
) is Li-Yorke sensitive, if for any
has
for some
.
2) A dynamic system
(or the map
) is densely Li-Yorke sensitive if
is dense in X for some
. Among them,
Definition 3. 1) A dynamic system
(or the map
) is Kato chaotic if it is sensitive and accessible (see [20]).
2) A dynamic system
(or the map
) is chaotic in the sense of Ruelle and Takens (short for R-T chaotic) if it is transitive and sensitive (see [21]).
Proposition 1. A dynamic system
(or the map
) is Li-Yorke sensitive if and only if
for some
. Among them,
Proposition 2. [17] A dynamical system
is infinitely sensitive if and only is it is sensitive.
Proposition 3. [22] A dynamical system
is dense Li-Yorke sensitivity, then it is Topological mixing (or its topological entropy is positive).
3. Main Results
In this section, let
. The metric
in I is defined by
. The metric d in
is defined by (2).
Theorem 1. If f is transitive, then the system
is transitive.
Proof. Since f is transitive, then there exist
satisfying
. Then for any
and any
,
. That is, there exists a
such that
. Take
. It is easy to see, for any
,
. Then,
. For any
and above
,
So,
.
Thus, the system
is transitive.
Theorem 2. If f is sensitive, then the system
is sensitive.
Proof. Take
,
,
,
. It is easy to know that, for
,
So, for
,
Since f is Sensitive, so there exists a
such that for any
and any
, there exists a
and
such that
. So for any fixed
and any
, taking
, one has that,
that is
. And because
so
is sensitive.
Corollary 1. If f is chaotic in the sense of Ruelle and Takens, then the system
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
is sensitive.
In fact, there is a stronger conclusion.
Theorem 3. If f is infinitely sensitive, then the system
is infinitely sensitive.
Proof. Since f is infinitely sensitive, then there exists a
such that for any
and any
, there exists
and
such that
. So for any fixed
, and any
, taking
, one has that
that is
. And because
So
is infinitely sensitive.
Theorem 4. If f is accessible, then the system
is accessible.
Proof. For any open subset
and
since f is accessible, then, for the above
, there exist
such that
for some
. Take
then
So, the system
is accessible.
Corollary 3. If f is Kato chaotic, then the system
is Kato chaotic.
Proof. According to Theorem 2 and Theorem 4, the conclusion is obvious.
Theorem 5. If f is exact, then the system
is exact.
Proof. Since f is exact, for any open subset
, there exist
such that
. That is, for any
, there exists an
such that
for any
. So there is a
such that
.
Take
is arbitrary open subset of
, and
. Clearly, for any
,
. For any
,
. That is to say, there exist an
,
. So, the system
is exact.
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.
Theorem 6. If f is densely Li-Yorke sensitive, then the system
is densely Li-Yorke sensitive.
Proof. Since f is densely Li-Yorke sensitive, then for any
and any
. Then there exists a
such that
. Take
,
. One has that
and
Thus there is an
.
Any fixed
, write
, where
. Because
is densely Li-Yorke sensitive, then for any
and the above
,
. Take
, then
So
. This suggests that
.
So, the system
is densely Li-Yorke sensitive.
According to Proposition 3 and Theorem 6 the following is right.
Corollary 4. If f is dense Li-Yorke sensitivity, then the system
it is Topological mixing (or its topological entropy is positive).
4. Conclusion
Inspired by the literature [23], this paper further studies the chaoticity of coupled map lattices. Some sufficient conditions of sensitivity, accessibility and transitivity are obtained. However, the study of coupled map lattices is still a hot topic. Based on the conclusions of this paper and others, one can consider some questions, such as the form of CMLs, the measurement of CMLs, and discuss the chaos of CMLs in other systems, which are worthy of studying.