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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 sensitivity and exact of coupled map lattices. Then, some sufficient conditions under which is Kato chaotic, positive entropy chaotic and Ruelle-Takens chaos are obtained.

In 1983, Kaneko [

x m + 1, n = ( 1 − ε ) f ( x m , n ) + ε f ( x m , n − 1 ) (1)

where x m , n ∈ I , m ∈ ℕ 0 = { 0,1,2, ⋯ } , n ∈ ℤ = { ⋯ , − 1,0,1, ⋯ } , I is a non-degenerate compact interval, f is a map on I, and ε ∈ [ 0,1 ] is a constant.

For t ∈ ℤ , let ℕ t = { t , t + 1, ⋯ } and Ω = { ( 0 , n ) : n ∈ ℤ } = { ⋯ , ( 0 , − 1 ) , ( 0 , 0 ) , ( 0 , 1 ) , ⋯ } . For any sequence ϕ = { ϕ 0, n } ∞ ∞ on Ω , by induction, one can obtain a double-indexed sequence x = { x m , n : m = 0 , 1 , 2 , ⋯ ; n = ⋯ , − 1 , 0 , 1 , ⋯ } , which is said to be a solution of the above system (1) with initial condition ϕ .

Let I be a subset of real number set, write

I ∞ ∞ = { { a n } n = − ∞ ∞ = ( ⋯ , a − 1 , a 0 , a 1 , ⋯ ) : a n ∈ I , n ∈ ℤ }

and

Δ ∞ ∞ = { ( ⋯ , a − 1 , a 0 , a 1 , ⋯ ) : a i = a j ∈ I , i , j ∈ ℤ }

which is called the diagonal set of I ∞ ∞ .

For arbitrary, two sequences x 1 = { x 1 , n } n = − ∞ ∞ , x 2 = { x 2 , n } n = − ∞ ∞ ∈ I ∞ ∞ , it is easy to prove that

d ( x 1 , x 2 ) = sup { | x 1 , n − x 2 , n | : n = ⋯ , − 1 , 0 , 1 , ⋯ } (2)

is a metric on I ∞ ∞ .

Let f : I ↦ I be a continuous map and x = { x m , n : m ∈ ℕ 0 , n ∈ ℤ } be a solution of the above system (1) with initial condition ϕ = { ϕ 0 , n } ∞ ∞ ∈ I ∞ ∞ .

Let

x m = { x m , n } n = − ∞ ∞ = ( ⋯ , x m , − 1 , x m , 0 , x m , 1 , ⋯ ) , ∀ m ∈ ℕ 0 ,

and

x m + 1 = { x m + 1 , n } n = − ∞ ∞ = ( ⋯ , x m + 1 , − 1 , x m + 1 , 0 , x m + 1 , 1 , ⋯ ) = F ( x m ) , ∀ m ∈ ℕ 0 ,

where x 0 = ϕ = { x 0 , n = ϕ 0 , n } n = − ∞ ∞ and

x m + 1, n = ( 1 − ε ) f ( x m , n ) + ε f ( x m , n − 1 ) , ∀ m ∈ ℕ 0 , n ∈ ℤ .

Then, one can see that the above system (1) is equivalent to the following system

x m + 1 = F ( x m ) , x m ∈ I ∞ ∞ , m = 0 , 1 , 2 , ⋯ (3)

For the above system (3), the map F is said to be induced by the system (1). Obviously, a double-indexed sequence { x m , n : m ∈ ℕ 0 , n ∈ ℤ } is a solution of the above system (1) if and only if the sequence { x m = { x m , n } n = − ∞ ∞ : m ∈ ℕ 0 } m = 0 ∞ 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 ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) satisfies three definitions of chaos (Kato chaotic, positive entropy chaotic and Ruelle-Takens chaos) under the conditions that f is chaos in these sense.

After T. Y. Li and J. A. Yorke [

Definition 1. Let ( 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 U 1 , U 2 ⊂ Y , f n ( U 1 ) ∩ U 2 ≠ ∅ for some integer n ∈ ℕ (see [

2) sensitive if there exist η > 0 such that for any x ∈ X and ε > 0 , there exists y ∈ B ( x , ε ) and n ∈ ℕ such that ρ ( f n ( x ) , f n ( y ) ) (see [

3) infinitely sensitive if there exist η > 0 such that for any x ∈ X and ε > 0 , there exists y ∈ B ( x , ε ) and n ∈ ℕ such that lim sup n → ∞ ρ ( f n ( x ) , f n ( y ) ) ≥ η (see [

4) accessible if for any ε > 0 and any two nonempty open subsets U 1 , U 2 ⊂ X , there are two points x ∈ U 1 and y ∈ U 2 such that ρ ( f n ( x ) , f n ( y ) ) < ε for some integer n > 0 (see [

5) exact if for any open subset U ⊂ X , there is m ∈ ℕ such that f m ( U ) = X (see [

Remark 1. [

Definition 2. 1) A dynamic system ( X , f ) (or the map f : X → X ) is Li-Yorke sensitive, if for any x ∈ X has x ∈ Q δ ( f ) for some δ > 0 .

2) A dynamic system ( X , f ) (or the map f : X → X ) is densely Li-Yorke sensitive if Q δ ( f ) is dense in X for some δ > 0 . Among them,

Q δ ( f ) = { x ∈ X : ∀ ε > 0 , ∃ y ∈ B ( x , ε ) such that ( x , y ) ∈ L Y ρ ( f , δ ) }

L Y ρ ( f , δ ) = { ( x , y ) ∈ X × X : lim sup n → ∞ ρ ( f n ( x ) , f n ( y ) ) > δ and lim inf n → ∞ ρ ( f n ( x ) , f n ( y ) ) = 0 }

Definition 3. 1) A dynamic system ( X , f ) (or the map f : X → X ) is Kato chaotic if it is sensitive and accessible (see [

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 [

Proposition 1. A dynamic system ( X , f ) (or the map f : X → X ) is Li-Yorke sensitive if and only if P δ ( f ) ¯ = X for some δ > 0 . Among them,

P δ ( f ) = { x ∈ X : ∀ ε > 0, ∃ y ∈ B ( x , ε ) , ∃ n ∈ ℕ such that ρ ( f n ( x ) , f n ( y ) ) > δ }

Proposition 2. [

Proposition 3. [

In this section, let X = I . The metric ρ in I is defined by ρ ( a , b ) = | a − b | ( ∀ a , b ∈ I ) . The metric d in I ∞ ∞ is defined by (2).

Theorem 1. If f is transitive, then the system ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) is transitive.

Proof. Since f is transitive, then there exist a ∈ I satisfying O r b f ( a ) ¯ = I . Then for any b ∈ I and any ε > 0 , B ( b , ε ) ∩ O r b f ( a ) ≠ ∅ . That is, there exists a k 0 > 0 such that ρ ( f k 0 ( a ) , b ) = | f k 0 ( a ) − b | < ε . Take x 0 = ( ⋯ , a , a , a , ⋯ ) ∈ Δ ∞ ∞ . It is easy to see, for any k ∈ ℕ , F k ( x 0 ) = { f k ( a ) } n = − ∞ ∞ . Then, O r b F ( x 0 ) = { f k ( a ) } n = − ∞ ∞ | k ∈ ℕ . For any y = ( ⋯ , b , b , b , ⋯ ) ∈ Δ ∞ ∞ and above k 0 > 0 ,

d ( F k 0 ( x 0 ) , y ) = sup { | f k 0 ( a ) − b | : n ∈ ℕ } = | f k 0 ( a ) − b | < ε .

So, B ( y , ε ) ∩ O r b F ( x 0 ) ≠ ∅ .

Thus, the system ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) is transitive.

Theorem 2. If f is sensitive, then the system ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) is sensitive.

Proof. Take Δ ∞ ∞ = { ( ⋯ , x − 1 , x 0 , x 0 , ⋯ ) : x n = a ∈ I , n ∈ ℤ } ⊂ I ∞ ∞ , ∀ x = { ⋯ , a , a , a , ⋯ } , y = { ⋯ , b , b , b , ⋯ } ∈ Δ ∞ ∞ , x ≠ y . It is easy to know that, for ∀ k ∈ ℕ ,

F k ( x ) = { f k ( a ) } n = − ∞ ∞ , F k ( y ) = { f k ( b ) } n = − ∞ ∞ .

So, for ∀ k ∈ ℕ ,

d ( F k ( x ) , F k ( y ) ) = d ( { f k ( a ) } n = − ∞ ∞ , { f k ( b ) } n = − ∞ ∞ ) = sup { | f k ( a ) − f k ( b ) | , k = ⋯ , − 1 , 0 , 1 , ⋯ } = | f k ( a ) − f k ( b ) | .

Since f is Sensitive, so there exists a ε 0 > 0 such that for any p ∈ I and any δ > 0 , there exists a q p , δ ∈ B ( p , δ ) and n p , δ ∈ ℕ such that | f n p , δ ( p ) , f n p , δ ( q p , δ ) | > ε 0 . So for any fixed x = ( ⋯ , p , p , p , ⋯ ) ∈ Δ ∞ ∞ and any δ > 0 , taking y = ( ⋯ , q p , δ , q p , δ , q p , δ , ⋯ ) ∈ Δ ∞ ∞ , one has that,

d ( x , y ) = sup { ⋯ , | p − q p , δ | , | p − q p , δ | , | p − q p , δ | , ⋯ } = | p − q p , δ | < δ ,

that is y ∈ B ( x , δ ) . And because

d ( F n p , δ ( x ) , F n p , δ ( y ) ) = | f n p , δ ( p ) − f n p , δ ( q p , δ ) | > ε 0 ,

so F | Δ ∞ ∞ is sensitive.

Corollary 1. If f is chaotic in the sense of Ruelle and Takens, then the system ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) 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 ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) is sensitive.

In fact, there is a stronger conclusion.

Theorem 3. If f is infinitely sensitive, then the system ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) is infinitely sensitive.

Proof. Since f is infinitely sensitive, then there exists a δ > 0 such that for any a ∈ I and any ε , there exists b a , ε ∈ B ( a , ε ) and n a , ε ∈ ℕ such that lim sup n a , ε → ∞ ρ ( f n a , ε ( a ) , f n a , ε ( b a , ε ) ) ≥ δ . So for any fixed x = ( ⋯ , a , a , a , ⋯ ) ∈ Δ ∞ ∞ , and any ε > 0 , taking x = ( ⋯ , b a , ε , b a , ε , b a , ε , ⋯ ) ∈ Δ ∞ ∞ , one has that

d ( x , y ) = sup { ⋯ , | a − b a , ε | , | a − b a , ε | , | a − b a , ε | , ⋯ } = | a − b a , ε | < ε ,

that is y ∈ B ( x , ε ) . And because

lim sup n a , ε d ( F n a , ε ( x ) , F n a , ε ( y ) ) = lim sup n a , ε → ∞ d ( f n a , ε ( a ) , f n a , ε ( b a , ε ) ) ≥ δ .

So F | Δ ∞ ∞ is infinitely sensitive.

Theorem 4. If f is accessible, then the system ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) is accessible.

Proof. For any open subset

( Δ 1 ) ∞ ∞ = { ( ⋯ , x − 1 , x 0 , x 1 , ⋯ ) , x n = a ∈ U 1 ⊂ I , n ∈ ℤ } ⊂ I ∞ ∞

and

( Δ 2 ) ∞ ∞ = { ( ⋯ , y − 1 , y 0 , y 1 , ⋯ ) , y n = b ∈ U 2 ⊂ I , n ∈ ℤ } ⊂ I ∞ ∞ ,

since f is accessible, then, for the above U 1 , U 2 ⊂ I , there exist a ∈ U 1 , b ∈ U 2 such that

ρ ( f k ( a ) , f k ( b ) ) = | f k ( a ) − f k ( b ) | < ε

for some k > 0 . Take

x = ( ⋯ , a , a , a , ⋯ ) ∈ ( Δ 1 ) ∞ ∞ , y = ( ⋯ , b , b , b , ⋯ ) ∈ ( Δ 2 ) ∞ ∞ ,

then

d ( F k ( x ) , F k ( y ) ) = | f k ( a ) , f k ( b ) | < ε .

So, the system ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) is accessible.

Corollary 3. If f is Kato chaotic, then the system ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) is Kato chaotic.

Proof. According to Theorem 2 and Theorem 4, the conclusion is obvious.

Theorem 5. If f is exact, then the system ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) is exact.

Proof. Since f is exact, for any open subset D ⊂ I , there exist m ∈ ℕ such that f m ( D ) = I . That is, for any a ∈ D , there exists an m > 0 such that B ( f m ( a ) , ε ) ∩ I ≠ ∅ for any ε > 0 . So there is a b ∈ X such that ρ ( f m ( a ) , ε ) = | f m ( a ) − b | < ε .

Take ( Δ * ) ∞ ∞ is arbitrary open subset of Δ ∞ ∞ , and x 0 = ( ⋯ , a , a , a , ⋯ ) ∈ ( Δ * ) ∞ ∞ . Clearly, for any k ∈ ℕ , F k ( x 0 ) = { f k ( a ) } n = − ∞ ∞ . For any y 0 = ( ⋯ , b , b , b , ⋯ ) ∈ Δ ∞ ∞ , d ( F m ( x 0 ) , y 0 ) = | f m ( a ) − b | < ε . That is to say, there exist an m ∈ ℕ , F m ( ( Δ * ) ∞ ∞ ) = Δ ∞ ∞ . So, the system ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) is exact.

In [

Theorem 6. If f is densely Li-Yorke sensitive, then the system ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) is densely Li-Yorke sensitive.

Proof. Since f is densely Li-Yorke sensitive, then for any a ∈ Q δ ( f ) and any ε > 0 . Then there exists a b ∈ B ( a , ε ) such that ( a , b ) ∈ L Y ρ ( f , δ ) . Take x * = { x n = a } n = − ∞ ∞ , y * = { y n = b } n = − ∞ ∞ . One has that

lim sup n → ∞ d ( F n ( x * ) , F n ( y * ) ) = lim sup n → ∞ d ( f n ( a ) , f n ( b ) ) > δ

and

lim inf n → ∞ d ( F n ( x * ) , F n ( y * ) ) = lim inf n → ∞ d ( f n ( a ) , f n ( b ) ) = 0

Thus there is an x * ∈ Q δ ( F ) .

Any fixed x ∈ Δ ∞ ∞ , write x = ( ⋯ , x m , − 1 , x m ,0 , x m ,1 , ⋯ ) , where x m , p = x m , p + 1 , p ∈ ℤ . Because f : I ↦ I is densely Li-Yorke sensitive, then for any ε > 0 and the above x m ,0 , B ( x m ,0 , ε ) ∩ Q δ ( f ) ≠ ∅ . Take a ∈ B ( x m ,0 , ε ) ∩ Q δ ( f ) , then

d ( x , x * ) = sup { | x m , p − a | } = | x m , p − a | < ε .

So x * ∈ B ( x , ε ) . This suggests that Q δ ( F ) ¯ = Δ ∞ ∞ .

So, the system ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) 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 ( Δ ∞ ∞ , d , F | Δ ∞ ∞ ) it is Topological mixing (or its topological entropy is positive).

Inspired by the literature [

The authors declare no conflicts of interest regarding the publication of this paper.

Yang, X.F., Lu, T.X. and Liu, G. (2020) Some Chaotic Properties of a Kind of Coupled Map Lattices. Journal of Applied Mathematics and Physics, 8, 968-975. https://doi.org/10.4236/jamp.2020.86075