Some remarks on chaos of a coupled lattice system related with the Belusov–Zhabotinskii reaction

Abstract

Guirao and Lampart (J Math Chem 48:159–164, 2010) presented a lattice dynamical system stated by Kaneko (Phys Rev Lett 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction. Motivated by Guirao and Lampart (J Math Chem 48:159–164, 2010) and Li and Zhao (J Math Chem. doi:10.1007/s10910-015-0538-y), in this paper we study the following more general lattice dynamical systems:

$$\begin{aligned} y_{i}^{j+1}=(1-\eta )h_{i}\left( y_{i}^{j}\right) +\frac{1}{2}\eta \left[ h_{i}\left( y_{i-1}^{j}\right) -h_{i}\left( y_{i+1}^{j}\right) \right] , \end{aligned}$$

where j is discrete time index, i is lattice side index with system size H, $\eta \in J=[0, 1]$ is coupling constant and $h_{i}$ is a continuous selfmap on J for any $i\in \{1, 2, \ldots , H\}$. In particular, we obtain that the following hold:

• (1) For zero coupling constant, if $h_{i}$ is topologically exact for any $i\in \{1, 2, \ldots , H\}$, then so does the above system.

• (2) For zero coupling constant, if $h_{i}$ is topologically mixing for any $i\in \{1, 2, \ldots , H\}$, then so is the above system.

• (3) For zero coupling constant, if $h_{1}\times h_{2}\times \cdots \times h_{H}$ is topologically transitive, then the above system is chaotic in the sense of Devaney.

Moreover, we give two examples. These results extend the results of Guirao and Lampart (J Math Chem 48:159–164, 2010).

1 Introduction

By a topological dynamical system (t.d.s. for short) (Q, h) we mean a compact metric space Q together with a continuous map $h:Q\rightarrow Q$. Since Li and Yorke [1] introduced the term of chaos in 1975, topological dynamical systems were highly considered and studied in the literature (see [2, 3]).

Coming from physical/chemical engineering applications, such as digital filtering, imaging and spatial vibrations of the elements which compose a given chemical product, an extension of classical discrete dynamical systems is the so called Lattice Dynamical Systems or 1d Spatiotemporal Discrete Systems which has recently appeared as an important subject for investigation. In [4] we can find the importance of these type of systems.

An open problem is to try to analyze when one of these type of systems has a complicated dynamics or not by the observation of one topological dynamical property (see [5]). In [5], by using the chaotic property, the authors characterized the dynamical complexity of a coupled lattice system stated by Kaneko [6] which is related to the Belusov–Zhabotinskii reaction. They also got that this CML (Coupled Map Lattice) system is chaotic in the sense of both Devaney and Li-Yorke for zero coupling constant. At the same time, some problems on the dynamics of this system were given by them for non-zero coupling constants. In [7] we generalized results obtained by Guirao and Lampart [5].

Inspired by Guirao and Lampart [5] and Li and Zhao [7], we will study the following more general lattice dynamical systems which extend the system presented by Guirao and Lampart [5]:

$$\begin{aligned} y_{i}^{j+1}=(1-\eta )h_{i}\left( y_{i}^{j}\right) +\frac{1}{2}\eta \left[ h_{i}\left( y_{i-1}^{j}\right) -h_{i}\left( y_{i+1}^{j}\right) \right] , \end{aligned}$$

(1)

where j is discrete time index, i is lattice side index with system size H, $\eta \in [0, 1]$ is coupling constant and $h_{i}$ is a continuous map on J for any $i\in \{1, 2, \ldots , H\}$. In particular, it is proven that the following are true:

• (1) For zero coupling constant, if $h_{i}$ is topologically exact for every $i\in \{1, 2, \ldots , D\}$, then so does the above system.

• (2) For zero coupling constant, if $h_{ni}$ is topologically mixing for any $i\in \{1, 2, \ldots , H\}$, then so is the above system.

• (3) For zero coupling constant, if $h_{1}\times h_{2}\times \cdots \times h_{H}$ is topologically transitive, then the above system is chaotic in the sense of Devaney.

Moreover, we give two examples. Our results extend the existing ones.

2 Preliminaries

Throughout this paper, Q is a compact metric space with metric $\pi$, (Q, h) is a topological dynamical system and $J=[0, 1]$.

A pair $(a, b)\in Q$ is called a Li-Yorke pair of a given map $h:Q\rightarrow Q$ or a given system (Q, h) if the following hold:

• (1) $\limsup \limits _{i\rightarrow \infty }\pi (h^{i}(a), h^{i}(a))>0$.

• (2) $\liminf \limits _{i\rightarrow \infty }\pi (h^{i}(a), h^{i}(b))=0$.

A subset $T\subset Q$ with at least two points is called a LY-scrambled set (Li-Yorke set) of a given map $h:Q\rightarrow Q$ or a given system (Q, h) if any pair of distinct points in T is a Li-Yorke pair. A given system (Q, h) or a given map $h:Q\rightarrow Q$ is said to be chaotic in the sense of Li-Yorke if it has an uncountable scrambled set.

The state space of LDS (Lattice Dynamical System) is the set

$$\begin{aligned} \mathcal {A}=\{a: a=\{a_{j}\}, a_{j}\in \mathbb {R}^{c}, j\in \mathbb {Z}^{d}, \Vert a_{j}\Vert <\infty \}. \end{aligned}$$

where $c\ge 1$ is the dimension of the range space of the map of state $a_{j}$, $d\ge 1$ is the dimension of the lattice and the $l^{2}$ norm

$$\begin{aligned} \Vert a\Vert _{2}=\left( \sum \limits _{j\in \mathbb {Z}^{d}}|a_{j}|^{2}\right) ^{\frac{1}{2}} \end{aligned}$$

is usually taken ($|a_{j}|$ is the length of the vector $a_{j}$) (see [5]).

We focus on the above systems (1) which extend the following Coupled Map Lattice system stated by Kaneko in [6] which is related to the Belusov–Zhabotinskii reaction (see [7–10]):

$$\begin{aligned} a_{i}^{j+1}=(1-\eta )h\left( a_{i}^{j}\right) +\frac{1}{2}\eta \left[ h\left( a_{i-1}^{j}\right) -h\left( a_{i+1}^{j}\right) \right] \!, \end{aligned}$$

(2)

where j is discrete time index, i is lattice side index with system size H, $\eta \in (0, 1)$ is coupling constant and h is the unimodal map on J.

In general, one of the following periodic boundary conditions of the system (1) is needed:

• 1) $x^{m}_{n}=x^{m}_{n+L}$,

• 2) $x^{m}_{n}=x^{m+L}_{n}$,

• 3) $x^{m}_{n}=x^{m+L}_{n+L}$,

standardly, the first case of the boundary conditions is used.

3 Main results

The system (2) was investigated by many authors, mostly experimentally or semi-analytically than analytically. The first paper with analytic results is [11], where the authors established that this system is Li–Yorke chaotic. In [5] the authors obtained an alternative and easier proof of this conclusion.

Let $\pi$ be the product metric on the product space $J^{H}$, which is defined as

$$\begin{aligned} \pi ((a_{1}, a_{2}, \ldots , a_{H}), (b_{1}, b_{2}, \ldots , b_{H}))=\left( \sum \limits _{j=1}^{H}( a_{j}-y_{j})^{2}\right) ^{\frac{1}{2}} \end{aligned}$$

for any $(a_{1}, a_{2}, \ldots , a_{H}), (b_{1}, b_{2}, \ldots , b_{H})\in J^{H}$.

For a given system (Q, h) on a metric space $(Q, \pi )$, a map $h:Q\rightarrow Q$ is said to be:

• (1) topologically transitive if $h^{i}(\mathcal {U})\cap \mathcal {V}\ne \emptyset$ for any nonempty open sets $\mathcal {U}, \mathcal {V}\subset Q$ and some integer $i>0$ (see [5]).

• (2) topologically exact if $h^{i}(\mathcal {U})=X$ for any nonempty open set $\mathcal {U}\subset Q$ and some integer $i>0$ (see [5]).

• (3) topologically mixing if $h^{j}(\mathcal {U})\cap \mathcal {V}\ne \emptyset$ for any nonempty open sets $\mathcal {U}, \mathcal {V}\subset X$, some integer $i>0$ and any integer $j\ge i$.

A periodic point of period i of h is a point $a\in Q$ with $h^{i}(a)=a$ and $h^{j}(a)\ne a$ for $0<j<i$.

In [5] the authors declared that for non-zero couplings constants, this lattice dynamical system (2) is more complicated.

Inspired by Guirao and Lampart [5] we establish the following results.

Theorem 3.1

For zero coupling constant, if $h_{i}$ is topologically exact for any $i\in \{1, 2, \ldots , H\}$, then so does the system (1).

Proof

Clearly, for $\eta =0$ the system (1) is equivalent to the system $(J^{H}, h_{1}\times h_{2}\times \cdots \times h_{H})$. By the definition and hypothesis, the system $(J^{H}, h_{1}\times h_{2}\times \cdots \times h_{H})$ is topologically exact. $\square$

Remark 3.1

Theorem 3.1 extend Lemma 2 in [5]. Moreover, the proof of Theorem 3.1 is simpler and easier.

By Theorem 3.1 the following example is true.

Example 3.1

Let $h_{i}=\Lambda ^{l_{i}}$ for any $i\in \{1, 2, \ldots , H\}$ and any $l_{i}\in \{1, 2, \ldots \}$ and $\Lambda$ be the tent map. Then the system (1) is topologically exact.

Theorem 3.2

For zero coupling constant, if $h_{i}$ is topologically mixing for any $i\in \{1, 2, \ldots , H\}$, then so is the system (1).

Proof

Since the system (1) is equivalent to the system $(J^{H}, h_{1}\times h_{2}\times \cdots \times h_{H})$ for $\eta =0$, by the definition and hypothesis the system $(J^{H}, h_{1}\times h_{2}\times \cdots \times h_{H})$ is topologically mixing. $\square$

Theorem 3.3

For zero coupling constant, if $h_{1}\times h_{2}\times \cdots \times h_{H}$ is topologically transitive, then this system is chaotic in the sense of Devaney.

Proof

As $h_{1}\times h_{2}\times \cdots \times h_{H}$ is topologically transitive, by the definition $h_{i}$ is topologically transitive for any $i\in \{1, 2, \ldots , H\}$. This implies that the set of periodic points of $h_{i}$ is dense in J. So, the set of periodic points of $h_{1}\times h_{2}\times \cdots \times h_{H}$ is dense in $J^{H}$. By hypothesis and the definition, $h_{1}\times h_{2}\times \cdots \times h_{H}$ is chaotic in the sense of Devaney. $\square$

Remark 3.2

Theorem 3.3 extend Theorem 4 in [5].

From Theorem 3.3 we know the following example is true.

Example 3.2

Let $h_{i}=h^{l_{i}}$ for any $i\in \{1, 2, \ldots , L\}$ and any $l_{i}\in \{1, 2, \ldots \}$ and h be a topologically mixing continuous map of J. Then the system (1) is chaotic in the sense of Devaney.

For any coupling constant $\eta \in (0, 1]$, the dynamical properties of the system (1) is more complicated. Consequently, we give the following problem.

Problem 3.1

For any coupling constant $\eta \in (0, 1]$, are the above three theorems true?