Some criteria of chaos in non-autonomous discrete systems
aa r X i v : . [ m a t h . D S ] M a r Some criteria of chaos in non-autonomous discrete systems
Hua Shao † , Guanrong Chen † , Yuming Shi ‡† Department of Electronic Engineering, City University of Hong Kong,Hong Kong SAR, P. R. China ‡ Department of Mathematics, Shandong UniversityJinan, Shandong 250100, P. R. China
Abstract.
This paper establishes some criteria of chaos in non-autonomous discretesystems. Several criteria of strong Li-Yorke chaos are given. Based on these results, somecriteria of distributional chaos in a sequence are established. Moreover, several criteria ofdistributional chaos induced by coupled-expansion for an irreducible transition matrix areobtained. Some of these results not only extend the existing related results for autonomousdiscrete systems to non-autonomous discrete systems, but also relax the assumptions of thecounterparts. One example is provided for illustration.
Keywords : non-autonomous discrete system; strong Li-Yorke chaos; distributional chaos;coupled-expansion; irreducible transition matrix.2010
Mathematics Subject Classification : 37B55, 37D45, 37B10.
1. Introduction
Chaos of the non-autonomous discrete system (briefly, NDS) x n +1 = f n ( x n ) , n ≥ , (1 . { f n } ∞ n =0 is a sequence of maps from X to X , with ( X, d ) being a metric space. Manycomplex systems of real-world problems in the fields of biology, physics, chemistry andengineering, are indeed non-autonomous, putting the model (1.1) into focus. The positiveorbit { x n } ∞ n =0 of system (1.1) starting from an initial point x ∈ X is given by x n = f n ( x ),where f n := f n − ◦ · · · ◦ f , n ≥ δ -chaos forsome δ > δ ′ -chaos in a sequence for some δ ′ > X, d ) is a compact metric space and f n arecontinuous maps for all n ≥
2. Preliminaries
In this section, some basic concepts and useful lemmas are presented.For convenience, denote f ni := f i + n − ◦ · · · ◦ f i and f − ni := ( f ni ) − , i ≥ n ≥
1. Let
A, B be nonempty subsets of X . The boundary of A is denoted by ∂A ; the diameter of A isdenoted by d ( A ) := sup { d ( x, y ) : x, y ∈ A } ; and the distance between A and B is denotedby d ( A, B ) := inf { d ( a, b ) : a ∈ A, b ∈ B } . The set of all nonnegative integers and positiveintegers are denoted by N and Z + , respectively. Definition 2.1 ([14], Definition 2.7). System (1.1) is said to be Li-Yorke δ -chaotic for some δ > δ -scrambled set S in X ; that is, for any x, y ∈ S ⊂ X ,2im inf n →∞ d ( f n ( x ) , f n ( y )) = 0 and lim sup n →∞ d ( f n ( x ) , f n ( y )) > δ. Further, it is said to be chaotic in the strong sense of Li-Yorke if all the orbits starting fromthe points in S are bounded. Definition 2.2 ([11], Definitions 2.1 and 2.2). System (1.1) is said to be distributionally δ -chaotic in a sequence P = { p n } ∞ n =1 for some δ > δ -scrambled set D ⊂ X in P ; that is, for any x, y ∈ D ⊂ X and any ǫ > n →∞ n n X i =1 χ [0 ,ǫ ) (cid:0) d ( f p i ( x ) , f p i ( y )) (cid:1) = 1 and lim inf n →∞ n n X i =1 χ [0 ,δ ) (cid:0) d ( f p i ( x ) , f p i ( y )) (cid:1) = 0 , where χ [0 ,ǫ ) is the characteristic function defined on the set [0 , ǫ ). Further, if P = N , thensystem (1.1) is said to be distributionally δ -chaotic.The relationship between Li-Yorke chaos and distributional chaos in a sequence forsystem (1.1) is shown below. Lemma 2.1 ([11], Theorem 3.6).
Let ( X, d ) be compact and f n be continuous in X , n ≥ .Then, system (1.1) is Li-Yorke δ -chaotic for some δ > if and only if it is distributionally δ ′ -chaotic in a sequence for some δ ′ > . A matrix A = ( a ij ) N × N ( N ≥
2) is said to be a transition matrix if a ij = 0 or 1 forall i, j ; P Nj =1 a ij ≥ i ; and P Ni =1 a ij ≥ j , 1 ≤ i, j ≤ N . A transitionmatrix A = ( a ij ) N × N is said to be irreducible if, for each pair 1 ≤ i, j ≤ N , there exists k ∈ Z + such that a ( k ) ij >
0, where a ( k ) ij denotes the ( i, j ) entry of matrix A k . A finitesequence ω = ( s , s , · · · , s k ) is said to be an allowable word of length k for A if a s i s i +1 = 1,1 ≤ i ≤ k −
1, where 1 ≤ s i ≤ N , 1 ≤ i ≤ k . For convenience, the length of ω is denoted by | ω | and ω n := ( ω, · · · , ω ) | {z } n , n ≥ + N := { α = ( a , a , · · · ) : 1 ≤ a i ≤ N, i ≥ } is a metricspace with the metric ρ ( α, β ) := P ∞ i =0 d ′ ( a i , b i ) / i , where α = ( a , a , · · · ) , β = ( b , b , · · · ) ∈ Σ + N , d ′ ( a i , b i ) = 0 if a i = b i , and d ′ ( a i , b i ) = 1 if a i = b i , i ≥
0. Note that (Σ + N , ρ ) is a compactmetric space. Define the shift map σ : Σ + N → Σ + N by σ (( s , s , s , · · · )) := ( s , s , · · · ). Thismap is continuous and (Σ + N , σ ) is called the one-sided symbolic dynamical system on N symbols. For a given transition matrix A = ( a ij ) N × N , denoteΣ + N ( A ) := { s = ( s , s , · · · ) : 1 ≤ s j ≤ N, a s j s j +1 = 1 , j ≥ } . Σ + N ( A ) is a compact subset of Σ + N and invariant under σ . The map σ A := σ | Σ + N ( A ) : Σ + N ( A ) → Σ + N ( A ) is said to be a subshift of finite type for matrix A . For more details about symbolicdynamical systems and subshifts of finite type, see [9, 24].The following two lemmas will also be useful in the sequel.3 emma 2.2 ([18], Lemma 2.2). Σ +2 has an uncountable subset E such that for any differentpoints α = ( a , a , · · · ) , β = ( b , b , · · · ) ∈ E , a n = b n for infinitely many n and a m = b m forinfinitely many m . Lemma 2.3 ([22], Theorem 2.2).
Let A = ( a ij ) N × N be an irreducible transition matrix with P Nj =1 a i j ≥ for some ≤ i ≤ N . Then (i) for any ≤ i, j ≤ N and any M ∈ Z + , there exists at least one allowable word ω = ( i, · · · , j ) for A such that | ω | > M ; (ii) for any given allowable word ω = ( b , b , · · · , b k ) for A , if k > N ( N − N + 2) , thenthere exists another different allowable word ω ′ = ( c , c , · · · , c k ) for A with c = b and c k = b k . Next, the definition of weak coupled-expansion for a transition matrix is introduced.
Definition 2.3.
Let A = ( a ij ) N × N be a transition matrix. If there exists a sequence { V i,n } ∞ n =0 of nonempty subsets of X with V i,n ∩ V j,n = ∂V i,n ∩ ∂V j,n ( d ( V i,n , V j,n ) >
0) for any1 ≤ i = j ≤ N and n ≥ f n ( V i,n ) ⊃ [ a ij =1 V j,n +1 , ≤ i ≤ N, n ≥ , then system (1.1) is said to be (strictly) weakly A -coupled-expanding in { V i,n } ∞ n =0 , 1 ≤ i ≤ N .In the special case that V i,n = V i for all n ≥ ≤ i ≤ N , it is said to be (strictly) A -coupled-expanding in V i , 1 ≤ i ≤ N . Remark 2.1.
Definition 2.3 is a slight revision of Definition 2.4 in [21].
3. Some criteria of strong Li-Yorke chaos
In this section, two criteria of strong Li-Yorke chaos for system (1.1) are established.Further, other two criteria of strong Li-Yorke chaos are given by applying the relationshipof Li-Yorke chaos between system (1.1) and its induced system.The following result can be easily verified.
Lemma 3.1.
Let A = ( a ij ) N × N be a transition matrix and V i , ≤ i ≤ N , be disjointnonempty compact subsets of X . Assume that f n is continuous in S Ni =1 V i , n ≥ , and { V i,n } ∞ n =0 is a sequence of nonempty closed subsets of V i , ≤ i ≤ N . Then, V m,nα is compactand satisfies that V m +1 ,nα ⊂ V m,nα ⊂ S Ni =1 V i,n for all m, n ≥ and all α = ( a , a , · · · ) ∈ Σ + N ( A ) , where V m,nα := m \ k =0 f − kn ( V a k ,n + k ) . (3 . Further, ( T ∞ m =0 V m,nα ) ∩ ( T ∞ m =0 V m,nβ ) = ∅ for any α = β ∈ Σ + N ( A ) and m, n ≥ . Theorem 3.1.
Let all the assumptions of Lemma 3.1 hold and suppose that A is irreduciblewith P Nj =1 a i j ≥ for some ≤ i ≤ N . If V m,nα = ∅ for all m, n ≥ and all α = ( a , a , · · · ) ∈ Σ + N ( A ) , where V m,nα is specified in (3.1) ; (ii) there exist β = ( b , b , · · · ) ∈ Σ + N ( A ) , an increasing subsequence { m k } ∞ k =1 ⊂ Z + , and ≤ s ≤ N , such that b m k = s , k ≥ , and d ( V m k ,n k β ) converges to as k → ∞ forall increasing subsequence { n k } ∞ k =1 ⊂ N ;then system (1.1) is chaotic in the strong sense of Li-Yorke. Proof.
Since A is a transition matrix, there exist 1 ≤ t , r ≤ N such that a t b = a b r = 1.By Lemma 2.3, there exist four allowable words for matrix A : ω = ( r , · · · , t ) , ω = ( b , · · · , t ) , ω = ( b , · · · , t ) , ω = ( s , · · · , b ) , where | ω | = l , | ω | = | ω | = l with ω = ω , and | ω | = l (if s = b , then set ω = ( b )with length 1). DenoteΩ := { γ = ( B , B , · · · ) : B i = ω or ω , i ≥ } . (3 . ⊂ Σ + N ( A ) is uncountable. Note that a b mk − s = 1, k ≥
1, since b m k = s . For any γ = ( B , B , · · · ) ∈ Ω, setˆ γ := ( b , · · · , b m − , ω , ω , B , b , · · · , b m − , ω , ω , B , B , · · · ) . (3 . γ ∈ Σ + N ( A ) and ˆ γ = ˆ γ if and only if γ = γ . By assumption (i) and Lemma 3.1,one has that V m, γ is nonempty, compact, and satisfies that V m +1 , γ ⊂ V m, γ , m ≥
0. Thus T ∞ m =0 V m, γ = ∅ . Fix one point x ˆ γ ∈ T ∞ m =0 V m, γ , and denote S := { x ˆ γ : γ ∈ Ω } . It follows from Lemma 3.1 that x ˆ γ = x ˆ γ if and only if ˆ γ = ˆ γ , which holds if and only if γ = γ . So, S is uncountable.Next, it will be shown that S is a Li-Yorke δ -scrambled set for system (1.1), where δ :=min ≤ i = j ≤ N d ( V i , V j ) >
0. For any x ˆ γ = x ˆ γ ∈ S , one has that ˆ γ = ˆ γ . By (3.1)-(3.3), thereexist an infinite sequence { h k } ∞ k =1 and 1 ≤ s = s ≤ N such that f h k ( x ˆ γ ) ∈ V s ,h k ⊂ V s and f h k ( x ˆ γ ) ∈ V s ,h k ⊂ V s , k ≥
1. So, d ( f h k ( x ˆ γ ) , f h k ( x ˆ γ )) ≥ δ, k ≥ . Thus, lim sup n →∞ d ( f n ( x ˆ γ ) , f n ( x ˆ γ )) ≥ δ. (3 . σ n k A (ˆ γ ) = ( b , · · · , b m k , · · · ) for any γ ∈ Ω, where n k = ( k − l + k ( k − l + ( k − l + k X t =1 m t , k ≥ . f n k + i ( x ˆ γ ) , f n k + i ( x ˆ γ ) ∈ V b i ,n k + i , 0 ≤ i ≤ m k . Thus, f n k ( x ˆ γ ) , f n k ( x ˆ γ ) ∈ V m k ,n k β , k ≥
1, implying that d ( f n k ( x ˆ γ ) , f n k ( x ˆ γ )) ≤ d ( V m k ,n k β ) , k ≥ . This, together with assumption (ii), yields thatlim k →∞ d ( f n k ( x ˆ γ ) , f n k ( x ˆ γ )) = 0 . So, lim inf n →∞ d ( f n ( x ˆ γ ) , f n ( x ˆ γ )) = 0 . (3 . S is an uncountable δ -scrambled set of system (1.1) by (3.4) and (3.5). Moreover, forany x ˆ γ ∈ S , its positive orbit { f n ( x ˆ γ ) } ∞ n =0 ⊂ S Ni =1 V i by (3.1) and (3.3). Thus, all the orbitsstarting from the points in S are bounded. Therefore, system (1.1) is chaotic in the strongsense of Li-Yorke. The proof is complete. Remark 3.1.
It can be easily verified that the weak A -coupled-expansion in { V i,n } ∞ n =0 ,1 ≤ i ≤ N , implies the condition in assumption (i) of Theorem 3.1. However, the converseis not true in general, even in the special case that f n = f and V i,n = V i , n ≥
0, 1 ≤ i ≤ N (see Example 3.1.1 in [4]). Hence, assumption (i) of Theorem 3.1 here is strictly weakerthan assumption (ii ) of Theorems 3.2 and 3.3 in [21]. Moreover, both assumption (iii ) ofTheorem 3.2 and assumption (iii ) of Theorem 3.3 in [21] imply assumption (ii) of Theorem3.1 above (see, for example, the proof of Corollary 4.1 below). Hence, Theorems 3.2 and 3.3in [21] are corollaries of Theorem 3.1 here.The following result is a direct consequence of Theorem 3.1. Corollary 3.1.
Let all the assumptions of Theorem 3.1 hold except that assumption (i) isreplaced by that system (1.1) is weakly A -coupled-expanding in { V i,n } ∞ n =0 , ≤ i ≤ N . Then,system (1.1) is chaotic in the strong sense of Li-Yorke. Let { k n } ∞ n =1 ⊂ Z + be an increasing subsequence. The following system:ˆ x n +1 = ˆ f n (ˆ x n ) , n ≥ , (3 . { k n } ∞ n =1 [16], whereˆ f := f k − ◦ f k − ◦ · · · f , ˆ f n := f k n +1 − ◦ f k n +1 − ◦ · · · f k n , n ≥ . Let { x n } ∞ n =0 be the orbit of system (1.1) starting from x and { ˆ x n } ∞ n =0 be the orbit of theinduced system (3.6) starting from ˆ x := x . Then, ˆ x n = f k n ( x ), n ≥
0, and thus the orbit { ˆ x n } ∞ n =0 of system (3.6) is a part of the orbit of system (1.1) starting from the same initialpoint x .Next, recall the relationship of (strong) Li-Yorke chaos between systems (1.1) and (3.6).6 emma 3.2 ([16], Theorem 3.1). If system (3.6) is Li-Yorke δ -chaotic for some δ > through { k n } ∞ n =1 , so is system (1.1) . Further, if system (3.6) is chaotic in the strong senseof Li-Yorke through { k n } ∞ n =1 , so is system (1.1) in the case that X is bounded. The following result follows directly from Theorem 3.1 and Lemma 3.2.
Theorem 3.2.
Assume that there exists an increasing subsequence { k n } ∞ n =1 ⊂ Z + such thatall the assumptions of Theorem 3.1 hold for system (3.6) . Then system (1.1) is Li-Yorke δ -chaotic for some δ > . Further, system (1.1) is chaotic in the strong sense of Li-Yorke inthe case that X is bounded. Applying Lemma 3.2 to Corollary 3.1, one obtains the following result.
Corollary 3.2.
Assume that there exists an increasing subsequence { k n } ∞ n =1 ⊂ Z + such thatall the assumptions of Corollary 3.1 hold for system (3.6) . Then, system (1.1) is Li-Yorke δ -chaotic for some δ > . Further, system (1.1) is chaotic in the strong sense of Li-Yorke inthe case that X is bounded.
4. Some criteria of distributional chaos in a sequence and distributional chaos
In this section, several criteria of distributional chaos in a sequence and of distributionalchaos are established for system (1.1), respectively.Applying Lemma 2.1 to Theorems 3.1 and 3.2 and Corollaries 3.1 and 3.2, respectively,the following result can be obtained.
Theorem 4.1.
Let all the assumptions of Theorem 3.1 or Corollary 3.1 or Theorem 3.2 orCorollary 3.2 hold. If X is compact and f n is continuous in X for all n ≥ , then system (1.1) is distributionally δ -chaotic in a sequence for some δ > . The next result gives a criterion of distributional chaos for system (1.1), which is inducedby weak coupled-expansion for an irreducible transition matrix.
Theorem 4.2.
Let all the assumptions of Lemma 3.1 hold, { f n } ∞ n =0 be equi-continuous in S Ni =1 V i , and A be irreducible with P Nj =1 a i j ≥ for some ≤ i ≤ N . If (i) system (1.1) is weakly A -coupled-expanding in { V i,n } ∞ n =0 , ≤ i ≤ N ; (ii) there exists a periodic point γ = ( r , r , · · · ) ∈ Σ + N ( A ) such that d ( V m,nγ ) uniformlyconverges to with respect to n ≥ as m → ∞ ;then system (1.1) is distributionally δ -chaotic for some δ > . Proof.
Since A is a transition matrix, there exist 1 ≤ l , m ≤ N such that a l r = a r m = 1.By Lemma 2.3, one has that there exist three allowable words for matrix A : ω := ( m , · · · , l ) , ω := ( r , · · · , l ) , ω := ( r , · · · , l ) , | ω | = | ω | = l and ω = ω . Let E ⊂ Σ +2 be the set satisfying the property in Lemma2.2. For any β = ( b , b , b , · · · ) ∈ E , setˆ β := ( r , ω , ω p b , r , · · · , r m − , R , ω , ω p b , r , · · · , r m − , R , ω , ω p b , · · · ) , (4 . p := 2 | ( r , ω ) | , p n = 2 n | ( r , ω , · · · , R n , ω ) | for n ≥ m n := 2 n | ( r , ω , · · · , ω p n − b n − ) | for n ≥
2; and for any n ≥ R n := ( r ) if r m n − = r , and otherwise, R n := ( r m n − , · · · , r ),while the fact that there exists an allowable word ( r m n − , · · · , r ) for the matrix A since A is irreducible has been used in the case that r m n − = r . Clearly, ˆ β ∈ Σ + N ( A ) and ˆ β = ˆ β if and only if β = β . By assumption (i) and Lemma 3.1, V m, β is nonempty, compact, andsatisfies that V m +1 , β ⊂ V m, β , m ≥
0. Thus, T ∞ m =0 V m, β = ∅ . Fix one point x ˆ β ∈ T ∞ m =0 V m, β ,and denote D := { x ˆ β : β ∈ E } . Using Lemma 3.1 again, one has that x ˆ β = x ˆ β if and only if ˆ β = ˆ β , which holds if andonly if β = β . Hence, D is uncountable since E is uncountable.In the following, it will be shown that D is a distributionally δ -scrambled subset forsystem (1.1), where δ := min ≤ i = j ≤ N d ( V i , V j ). Fix any x ˆ β , x ˆ β ∈ D with x ˆ β = x ˆ β . Then, β = β ∈ E . Suppose that β = ( b (1)0 , b (1)1 , b (1)2 , · · · ) and β = ( b (2)0 , b (2)1 , b (2)2 , · · · ). Thus, thereexists an increasing subsequence { t i } ∞ i =1 ⊂ Z + such that b (1) t i = b (2) t i , i ≥
1, by Lemma 2.2.Set n i := | ( r , ω , · · · , ω p ti +1 b ti ) | , i ≥ . Then, n i = p t i +1 l + p t i +1 − ( t i +1) , i ≥ . (4 . := { ω : ω is an allowable word for A with | ω | = l } . For any C = ( c , · · · , c l ) ∈ Ω and any n ≥
0, set V n C := l − \ k =0 f − kn ( V c k +1 ,n + k ) . It is evident that d n := inf { d ( V n C , V n D ) : C 6 = D ∈ Ω } ≥ δ , n ≥ . (4 . n i − X j =0 χ [0 ,δ ) ( d ( f j ( x ˆ β ) , f j ( x ˆ β ))) ≤ n i − ( p t i +1 l − l + 1) , i ≥ . This, together with (4.2), implies thatlim i →∞ n i n i − X j =0 χ [0 ,δ ) (cid:0) d ( f j ( x ˆ β ) , f j ( x ˆ β )) (cid:1) ≤ − lim i →∞ p t i +1 l − l + 1 p t i +1 l + p t i +1 − ( t i +1) = 0 , which yields that 8im inf n →∞ n n − X j =0 χ [0 ,δ ) (cid:0) d ( f j ( x ˆ β ) , f j ( x ˆ β )) (cid:1) = 0 . (4 . k i := | ( r , ω , · · · , r , · · · , r m i − ) | , i ≥ . Thus, k i = m i + m i − i , i ≥ . (4 . ǫ >
0. It is claimed that there exists
M > m ≥ M , d ( V m,n + jσ jA ( γ ) ) < ǫ, ≤ j ≤ P − , n ≥ , (4 . P is the period of γ . For simplicity, only consider the case of P = 2. The generalcases can be proved in a similar way. Since { f n } ∞ n =0 is equi-continuous in S Ni =1 V i , there exists0 < δ < ǫ such that, for any x, y ∈ X with d ( x, y ) < δ , d ( f n ( x ) , f n ( y )) < ǫ, n ≥ . (4 . M > m ≥ M , d ( V m,nγ ) < δ < ǫ, n ≥ . (4 . n ≥ m ≥ M . By assumption (i), one has that f n ( V m +1 ,nγ ) = V m,n +1 σ A ( γ ) . (4 . z , z ∈ V m,n +1 σ A ( γ ) , there exist x, y ∈ V m +1 ,nγ such that z = f n ( x ) and z = f n ( y ). By (4.8), one has that d ( x, y ) ≤ d ( V m +1 ,nγ ) < δ . This, together with (4.7), implies that d ( z , z ) = d ( f n ( x ) , f n ( y )) < ǫ. Thus, d ( V m,n +1 σ A ( γ ) ) < ǫ. (4 . P = 2. By (4.1) and (4.6), one hasthat k i − X j =0 χ [0 ,ǫ ) (cid:0) d ( f j ( x ˆ β ) , f j ( x ˆ β )) (cid:1) ≥ m i − M, i ≥ . This, together with (4.5), implies thatlim i →∞ k i k i − X j =0 χ [0 ,ǫ ) (cid:0) d ( f j ( x ˆ β ) , f j ( x ˆ β )) (cid:1) ≥ lim i →∞ m i − Mk i = lim i →∞ m i − Mm i + m i − i = 1 . So, 9im sup n →∞ n n − X j =0 χ [0 ,ǫ ) (cid:0) d ( f j ( x ˆ β ) , f j ( x ˆ β )) (cid:1) = 1 , ∀ ǫ > . (4 . D is an uncountable distributionally δ -scrambled subset for system (1.1) by (4.4)and (4.11). Therefore, system (1.1) is distributionally δ -chaotic. The proof is complete. Remark 4.1.
Theorem 4.2 extends Theorem 3.2 in [6] from autonomous to non-autonomoussystems.
Corollary 4.1.
Let all the assumptions of Theorem 4.2 hold, except that assumption (ii) isreplaced by (ii a ) there exists a constant λ > such that d ( f n ( x ) , f n ( y )) ≥ λd ( x, y ) , ∀ x, y ∈ V j,n , ≤ j ≤ N, n ≥ , (4 . then system (1.1) is distributionally δ -chaotic for some δ > . Proof.
It can be shown that d ( V m,nγ ) uniformly converges to 0 with respect to n ≥ m → ∞ for all γ = ( r , r , · · · ) ∈ Σ + N ( A ). Indeed, it follows from (4.11) that, for any x, y ∈ V m,nγ , d ( f mn ( x ) , f mn ( y )) ≥ λ m d ( x, y ) , m ≥ , n ≥ , implying that d ( x, y ) ≤ λ − m d ( f mn ( x ) , f mn ( y )) ≤ λ − m d ( V j ,n + m ) ≤ λ − m d ( V j ) , m ≥ , n ≥ . Thus, d ( V m,nγ ) ≤ λ − m d ( V j ) , m ≥ , n ≥ . This, together with the assumption that λ >
1, yields that d ( V m,nγ ) uniformly convergesto 0 with respect to n ≥ m → ∞ . Hence, all the assumptions of Theorem 4.2 hold.Therefore, system (1.1) is distributionally δ -chaotic for some δ >
0. This completes theproof.The following result is somewhat better since it only requires f n be expanding in distancein one subset for all n ≥ Corollary 4.2.
Let all the assumptions of Theorem 4.2 hold, except that assumption (ii) isreplaced by (ii b ) there exist an integer ≤ j ≤ N and a constant λ > such that a j j = 1 and d ( f n ( x ) , f n ( y )) ≥ λd ( x, y ) , ∀ x, y ∈ V j ,n , n ≥ , then system (1.1) is distributionally δ -chaotic for some δ > . roof. With a similar proof to that of Corollary 4.1, one can show that for γ = ( j , j , j , · · · ), d ( V m,nγ ) uniformly converges to 0 with respect to n ≥ m → ∞ . Hence, all the assump-tions of Theorem 4.2 hold. Therefore, system (1.1) is distributionally δ -chaotic for some δ >
0. The proof is complete.
5. An example
In this section, an example is provided to illustrate the theoretical results.
Example 5.1.
Consider the following non-autonomous logistic system: x n +1 = r n x n (1 − x n ) , n ≥ , (5 . f n ( x ) = r n x (1 − x ), x ∈ [0 , / ≤ r n ≤ µ , n ≥
0, and µ ≥ / | f ′ n ( x ) | = r n | − x | ≤ r n ≤ µ, ∀ x ∈ [0 , . Thus, | f n ( x ) − f n ( y ) | ≤ µ | x − y | , ∀ x ∈ [0 , , which yields that { f n } ∞ n =0 is equi-continuous in [0 , V = [0 , / , V = [3 / , . It is evident that V and V are disjoint nonempty compact subsets of [0 , f n ( V ) = [0 , r n / , f n ( V ) = [0 , r n / . Since r n ≥ / n ≥
0, one can see that V ∪ V ⊂ f n ( V ) ∩ f n ( V ) , n ≥ , (5 . A -coupled-expanding in V and V , where A =( a ij ) × with a ij = 1, 1 ≤ i, j ≤
2, and thus A is irreducible with P j =1 a i j = 2, i = 1 , | f ′ n ( x ) | = r n | − x | ≥ r n ≥ , ∀ x ∈ V , which yields that | f n ( x ) − f n ( y ) | ≥ | x − y | , ∀ x, y ∈ V . (5 . β = (1 , , · · · ) ∈ Σ +2 ( A ), satisfying by (5.3) that d ( V m,nβ ) uniformly converges to 0 with respect to n ≥ m → ∞ . By Theorem 3.1, system (1.1) is chaotic in the strong sense of Li-Yorke.Moreover, all the assumptions in Corollary 4.2 hold for system (1.1) with j = 1, satis-fying assumption (ii b ) by (5.2) and (5.3). By Corollary 4.2, system (1.1) is distributionally δ -chaotic for some δ >
0. 11 emark 5.1.
System (5.1) is an important model in biology, which describes the populationgrowth under certain conditions. Comparing to Example 5.1 in [14], here it not only provesthat system (5.1) is chaotic in the strong sense of Li-Yorke, but also proves that system (5.1)is distributionally δ -chaotic for some δ > f n : X → X is replaced by that f n : X n → X n +1 with compact subsets X n ⊂ X for any n ≥ Acknowledgments
This research was supported by the Hong Kong Research Grants Council (GRF GrantCityU11200317) and the NNSF of China (Grant 11571202).