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International Journal of Bifurcation and Chaos, Vol. 16, No. 9 (2006) 2615–2636 c World Scientific Publishing Company


CHAOTIFICATION OF DISCRETE DYNAMICAL SYSTEMS IN BANACH SPACES

Int. J. Bifurcation Chaos 2006.16:2615-2636. Downloaded from www.worldscientific.com by UNIVERSITY OF WESTERN ONTARIO WESTERN LIBRARIES on 07/24/12. For personal use only.

YUMING SHI∗ and PEI YU Department of Applied Mathematics, The University of Western Ontario, London, Ontario N6A 5B7, Canada ∗Department of Mathematics, Shandong University, Jinan, Shandong 250100, P. R. China [email protected] GUANRONG CHEN Department of Electronic Engineering, City University of Hong Kong, P. R. China Received May 17, 2005; Revised June 27, 2005 This paper is concerned with chaotification of discrete dynamical systems in Banach spaces via feedback control techniques. A criterion of chaos in Banach spaces is first established. This criterion extends and improves the Marotto theorem. Discussions are carried out in general and some special Banach spaces. All the controlled systems are proved to be chaotic in the sense of both Devaney and Li–Yorke. As a consequence, a controlled system described in a finitedimensional real space studied by Wang and Chen is shown chaotic not only in the sense of Li–Yorke but also in the sense of Devaney. The original system can be driven to be chaotic by using an arbitrarily small-amplitude state feedback control in a certain space. In addition, the Chen–Lai anti-control algorithm via feedback control with mod-operation in a finite-dimensional real space is extended to a certain infinite-dimensional Banach space, and the controlled system is shown chaotic in the sense of Devaney as well as in the sense of both Li–Yorke and Wiggins. Differing from many existing results, it is not here required that the map corresponding to the original system has a fixed point in some cases. An application of the theoretical results to a class of first-order partial difference equations is given with some numerical simulations. Keywords: Chaotification; discrete dynamical system; Banach space; snap-back repeller; firstorder partial difference equation.

1. Introduction Chaos control has been developed so far mainly in two different directions: control of chaos and anticontrol of chaos (or called chaotification). A process of making a chaotic system nonchaotic or stable is called control of chaos. Over the last decade,

∗

research on control of chaos has been rapidly developed (cf. [Chen & Dong, 1998; Fradkov & Pogromsky, 1999; Judd et al., 1997; Kapitaniak, 1998], and many references cited therein). This control is traditional, regarding chaotic motions as harmful. Anti-control of chaos, on the other

Author for correspondence. 2615

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hand, is a process that makes a nonchaotic system chaotic, or enhances a chaotic system to present stronger or different type of chaos. In recent years, it has been found that chaos can actually be very useful under some circumstances, for example, in human brain analysis, heartbeat regulation, encryption, digital communications, and liquid mixing (cf. [Brandt & Chen, 1997; Ditto et al., 2000; Freeman, 1995; Jakimoski & Kocarev, 2001; Kocarev et al., 2001; Schiff et al., 1994]). Therefore, sometimes it is useful and even important to make a system chaotic or create new types of chaos. Due to the great potentials of chaos in many nontraditional applications, there is growing interest in research on chaotification of dynamical systems today. In research on chaotification for discrete dynamical systems, a mathematically rigorous chaotification method was first developed by Chen and Lai from a feedback control approach [Chen & Lai, 1996, 1997, 1998]. They showed that the Lyapunov exponents of a controlled system are positive [Chen & Lai, 1996], and the controlled system via the mod-operation is chaotic in the sense of Devaney [1989] when the original system is linear, and is chaotic in a weaker sense of Wiggins [1990] when the original system is nonlinear [Chen & Lai, 1998]. Later, chaos generated by their method was shown to be also in the sense of Li–Yorke [Wang & Chen, 1999]. This method plays an important role in studying chaotification problems of discrete dynamical systems. Recently, Wang and Chen [2000] and Zhang and Chen [2004] and Zheng and Chen [2004], applied the Marroto theorem [Marotto, 1978] to prove that some slightly modified controlled systems are also chaotic in the sense of Li–Yorke. Zheng et al. [2003] used the anti-integrable limit method, proposed by Aubry and Abramovici [1990], to show that such controlled systems are chaotic in the sense of Devaney. More recently, in [Shi & Chen, 2005] we studied chaotification for some discrete dynamical systems governed by continuous maps, and showed that the controlled system is chaotic in the sense of Devaney. We have noticed that all the chaotification problems for discrete dynamical systems studied so far are formulated in finite-dimensional real spaces. To the best of our knowledge, no results have been reported on chaotification for infinite-dimensional discrete dynamical systems in the present literature. Clearly, dynamical systems in Banach spaces are very important in both theory and applications. For example, partial difference

equations and functional difference equations are often investigated in Banach spaces, which can be either finite-dimensional or infinite-dimensional. In this paper, we investigate chaotification for discrete dynamical systems in Banach spaces (particularly, infinite-dimensional ones), and prove that the controlled system is chaotic in the sense of both Devaney and Li–Yorke. Since all finite-dimensional real spaces are Banach spaces, our results simultaneously show that some controlled systems in finitedimensional real spaces studied in [Wang & Chen, 2000] and [Chen & Lai, 1998] are chaotic in the sense of Devaney, as well as in the sense of both Li–Yorke and Wiggins. Two criteria for the existence of chaos for discrete systems in complete metric spaces and Banach spaces, established recently in [Shi & Chen, 2004a, 2004b], are employed in this paper. Derivatives of maps in Banach spaces are assumed to be Frech´et derivatives in this paper (cf. [Pugachev & Sinitsyn, 1999, Sec. 9.1] or [Rudin, 1973, Definition 10.34]). The rest of the paper is organized as follows. In Sec. 2, the chaotification problem under investigation is described, and some concepts and lemmas are introduced. Especially, a criterion of chaos for maps in Banach spaces is established in this section, which can be regarded as a generalization and improvement of the Marotto theorem. In Sec. 3, the chaotification problem in general Banach spaces is studied and two different simple controllers are designed, where the map corresponding to the original system is only required to be continuous in a closed neighborhood and continuously Frech´et differentiable in an open neighborhood of a fixed point of the map. In Sec. 4, the chaotification problem in some special Banach spaces is considered with two feedback controllers. The map corresponding to the original system is only required to have a fixed point and satisfy the Lipschitz condition in a closed neighborhood of the fixed point of the map with one feedback controller. Especially, this control can be arbitrarily small in norm in this case. With the other feedback controller, the map corresponding to the original system is only required to satisfy the Lipschitz condition in a closed ball of the domain of interest, but not required to have a fixed point. This is different from and weaker than many existing results. The map corresponding to one designed controller is an extension of the sawtooth function used in [Wang & Chen, 2000], which is further shown to be chaotic in the sense of both Devaney and Li–Yorke (Corollary 4.1). In addition,

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Chaotification of Discrete Dynamical Systems in Banach Spaces

the Chen–Lai anti-control algorithm via feedback control with mod-operation in finite-dimensional real spaces [Chen & Lai, 1998] is extended to a certain infinite-dimensional Banach space. It is shown that the controlled system is chaotic in the sense of Devaney, as well as in the sense of both Li–Yorke and Wiggins (Theorem 4.3 and Corollary 4.3). In Sec. 5, a chaotification scheme established in Sec. 4 is applied to a class of first-order partial difference equations. In view of time evolution, first-order partial difference equations are first reformulated into a specific class of finite-dimensional and infinitedimensional normal discrete dynamical systems in the two cases: the system size is finite and infinite, respectively. Subsequently, stability and expansion of fixed points of some specific first-order partial difference equations are discussed and a chaotification problem is studied. Numerical simulations are given for the case of finite system size (it is impossible to give a numerical simulation for the case of infinite system size). Finally, Sec. 6 concludes the paper.

for the controlled system (2) with the controllers (3) and (4), respectively. Denote the Frech´et derivative of f at x by Df (x) and the closure of a subset A of X by A.

2.2. Some concepts and lemmas Since Li and Yorke [1975] first introduced a mathematical definition of chaos, several different definitions of chaos have been proposed; some are stronger and some are weaker, depending on the requirements in different problems [Devaney, 1989; Martelli et al., 1998; Robinson, 1999; Wiggins, 1990]. For convenience, we list three definitions of chaos in the sense of Li–Yorke, Devaney, and Wiggins below. Consider the following system: xn+1 = F (xn ),

n ≥ 0,

(5)

where F : X → X is a map and (X, d) is a metric space. Definition 2.1. Let (X, d) be a metric space and

2. Preliminaries In this section, we describe the chaotification problem and introduce some concepts and lemmas, which will be useful in the following sections.

2.1. Description of the chaotification problem Consider the following discrete dynamical system: xn+1 = f (xn ),

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n ≥ 0,

(1)

where f : D ⊂ X → X is a map and (X,  · ) is a Banach space. The objective is to design a (simple) control input sequence, {un }, such that the output (state) of the controlled system xn+1 = f (xn ) + un ,

n ≥ 0,

(2)

exhibits chaos in the sense of Devaney. The controller to be designed in this paper is in the form of un = g(µxn )

(3)

or

F : X → X be a map. A subset S of X is called a scrambled set of F if, for any two different points x, y ∈ S,

(i) lim inf n→∞ d(F n (x), F n (y)) = 0; (ii) lim supn→∞ d(F n (x), F n (y)) > 0. F is said to be chaotic in the sense of Li–Yorke if there exists an uncountable scrambled set S of F . Note that there are three conditions in the original characterization of chaos in Li–Yorke’s theorem [1975]. Besides the above conditions (i) and (ii), the third is for all x ∈ S and for all periodic points p of F , lim sup d(F n (x), F n (p)) > 0. n→∞

But conditions (i) and (ii) together imply that the scrambled set S contains at most one point x that does not satisfy the above condition. So, the third condition is not essential and can be removed. [Devaney, 1989]. A map F : V ⊂ X → V is said to be chaotic on V in the sense of Devaney if

Definition 2.2

un = µg(xn ),

(4)

where µ is a positive parameter, and the map g : D  → X is expected to be very simple with D  being a suitable subset of X. For simplicity, introduce the same notation Fµ (x) := f (x) + g(µx) and Fµ (x) := f (x) + µg(x)

(i) F is topologically transitive; (ii) the periodic points of F are dense in V ; (iii) F has sensitive dependence on initial conditions.

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By the result of [Banks et al., 1992], properties (i) and (ii) together imply property (iii) if F is continuous in V . So, property (iii) is redundant in the above definition. Under some conditions, chaos in the sense of Devaney is stronger than that in the sense of Li–Yorke [Huang & Ye, 2002]. The following result is from [Huang & Ye, 2002, Theorem 4.1] but is somewhat differently presented here.

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Lemma 2.1. Let V be a compact set of a metric

space (X, d), containing infinitely many points. If a map F : V → V is continuous, surjective, and chaotic in the sense of Devaney on V, then it is chaotic in the sense of Li–Yorke. Definition 2.3 [Wiggins, 1990, Definition 4.11.2].

Let S be a compact subset of X. A map F : S → S is said to be chaotic on S in the sense of Wiggins if

(i) F is topologically transitive; (ii) F has sensitive dependence on initial conditions. Obviously, chaos in the sense of Devaney is stronger than that in the sense of Wiggins. Definition 2.4. Let (X, d) be a metric space, F :

X → X be a map, and V be a subset of X.

(i) F is said to be expanding in distance on V if there exists a constant λ > 1 such that d(F (x), F (y)) ≥ λd(x, y),

∀ x, y ∈ V.

The constant λ is called an expanding coefficient of F in V . (ii) F is said to be expanding in set on V if F (V ) ⊃ V . Remark 2.1. These two expansions are different in general (cf. [Shi & Chen, 2004a, Lemmas 2.1 and 2.2, Remark 2.3] for relative discussions).

For convenience, some definitions of relevant concepts given in [Shi & Chen, 2004a] are listed below. Denote by Br (z) = {x ∈ X : d(x, z) ≤ r} and Br (z) = {x ∈ X : d(x, z) < r} the closed ball and the open ball of radius r > 0 centered at z, respectively. Definition 2.5 [Shi & Chen, 2004a, Definitions

2.1–2.6]. Let (X, d) be a metric space and F : X → X be a map.

(i) A point z ∈ X is called an expanding fixed point (or a repeller) of F in Br (z) for some constant r > 0, if F (z) = z and F is expanding in distance on Br (z). Furthermore, z is called a regular expanding fixed point of F in Br (z) if z is an interior point of F (Br (z)). (ii) Assume that z is an expanding fixed point of F in Br (z) for some r > 0. Then z is said to be a snap-back repeller of F if there exists a point x0 ∈ Br (z) with x0 = z and F m (x0 ) = z for some positive integer m. Furthermore, z is said to be a nondegenerate snap-back repeller of F if there exist positive constants µ and r0 such that Br0 (x0 ) ⊂ Br (z) and d(F m (x), F m (y)) ≥ µd(x, y), ∀ x, y ∈ Br0 (x0 ); z is called a regular snap-back repeller of F if F (Br (z)) is open and there exists a positive constant δ0 such that Bδ0 (x0 ) ⊂ Br (z) and z is an interior point of F m (Bδ (x0 )) for any positive constant δ ≤ δ0 . (iii) Assume that z ∈ X is a regular expanding fixed point of F . Let U be the maximal open neighborhood of z in the sense that for any x ∈ U with x = z, there exists k ≥ 1 with / U , F −n (x) is uniquely defined in U F k (x) ∈ for all n ≥ 1, and F −n (x) → z as n → ∞. U is called the local unstable set of F at z and is u (z). denoted by W loc (iv) Assume that z ∈ X is a regular expanding fixed point of F . A point x ∈ X is called homou (z), x = z, and there clinic to z if x ∈ W loc exists an integer n ≥ 1 such that F n (x) = z. A homoclinic orbit to z, consisting of a homoclinic point x with F n (x) = z, its backward orbit {F −j (x)}∞ j=1 , and its finite forward orbit , is called nondegenerate if for each {F j (x)}n−1 j=1 point x0 on the homoclinic orbit there exist positive constants r1 and µ1 such that d(F (x), F (y)) ≥ µ1 d(x, y),

∀ x, y ∈ Br1 (x0 ).

A homoclinic orbit is called regular if for each point x0 on the orbit, there exists a positive constant r2 such that for any positive constant r ≤ r2 , F (x0 ) is an interior point of F (Br (x0 )). Next, we study some properties of continuously differentiable maps on Banach spaces. Denote the norm of a bounded linear operator L on X by L, that is, L := sup{Lx : x ∈ X with x = 1}.

Chaotification of Discrete Dynamical Systems in Banach Spaces

For a linear map L : X → X, introduce the following notation: L0 := inf{Lx : x ∈ X with x = 1}. If a bounded linear map L : X → X has a bounded linear inverse map, then L is called an invertible linear map. Lemma 2.2. Let (X,  · ) be a Banach space and

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let a map F : X → X be differentiable in an open subset A ⊂ X. If there exists a constant λ > 0 such that F (x) − F (y) ≥ λx − y,

∀ x, y ∈ A,

(6)

then DF (x)0 ≥ λ,

∀ x ∈ A.

(7)

By definition of the Frech´et derivative, for any fixed x ∈ A,

Proof.

F (y) = F (x) + DF (x)(y − x) + α(y)

(8)

with limy→x α(y)/y − x = 0. Hence, for any positive constant ε < λ, there exists a constant rε > 0 such that α(y) ≤ εy − x,

∀ y ∈ Brε (x),

which, together with (6) and (8), implies that DF (x)(y − x) ≥ (λ − ε)y − x,

∀ y ∈ Brε (x),

so that DF (x)0 ≥ λ − ε, and consequently (7) holds. This completes the proof.
Lemma 2.3. Let (X,  · ) be a Banach space and

let a map F : X → X be continuous in Br (x0 ) and continuously differentiable in Br (x0 ) for some x0 ∈ X and some r > 0. If there exists a constant L > 0 such that DF (x) ≤ L for all x ∈ Br (x0 ), then F (x) − F (y) ≤ Lx − y,

∀ x, y ∈ Br (x0 ).

(9)

Since DF (x) is continuous in Br (x0 ), it follows from Theorem 9.2 in [Pugachev & Sinitsyn, 1999] that
1 DF (y + t(x − y))(x − y)dt, F (x) − F (y) = Proof.

0

∀ x, y ∈ Br (x0 ),

which implies that F (x) − F (y) ≤ Lx − y,

∀ x, y ∈ Br (x0 ).

So, (9) follows from the above relation and the continuity of F in Br (x0 ). This completes the proof.


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Let F : X → X be a map and (X,  · ) be a Banach space. Assume that z is a snap-back repeller of F in Br0 (z) with x0 and m as specified in (ii) of Definition 2.5. If F is continuously differentiable in Br0 (z) and DF (x) is an invertible linear map for all x ∈ Br0 (z), then x0 is homoclinic to z and its backward orbit {F −j (x0 )}∞ j=1 is contained in Br0 (z). Lemma 2.4.

Since F is continuously differentiable in Br0 (z) and DF (x) is invertible for all x ∈ Br0 (z), F (x) is an interior point of F (Br0 (z)) for any x ∈ Br0 (z) by Theorem 10.39 in [Rudin, 1973]. So, F (Br0 (z)) is open and F (Br (z)) is also open for any positive constant r < r0 . For any fixed positive constant r < r0 , it is be shown that F (Br (z)) is closed. Suppose that {yn }∞ n=1 is any convergent sequence of F (Br (z)) and y0 is its limit. Then, there exists xn ∈ Br (z) such that yn = F (xn ) for each n ≥ 1. Since z is a snap-back repeller of F in Br0 (z), there exists a constant λ > 1 such that Proof.

ym − yn  = F (xm ) − F (xn ) ≥ λxm − xn , ∀ m, n ≥ 1, which implies that {xn }∞ n=1 is a fundamental sequence. By the completeness of X, {xn }∞ n=1 is convergent. Let limn→∞ xn = x0 . Then x0 ∈ Br (z) and y0 = limn→∞ yn = limn→∞ F (xn ) = F (x0 ) by the continuity of F in Br0 (z). Hence, y0 ∈ F (Br (z)) and consequently F (Br (z)) is a closed set. With a similar argument to that in the last part of the proof of [Shi & Chen, 2004a, Lemma 2.1], it can be concluded that for any positive constant r < r0 , F (Br (z)) ⊃ Br (z),

F (Br (z)) ⊃ Br (z).

So, F −j (x0 ) ∈ Br0 (z) exists for each j ≥ 1. Moreover, F −j (x0 ) − z ≤ λ−j x0 − z,

for j ≥ 1,

where λ > 1 is an expanding coefficient of F in Br0 (z). Hence, F −j (x0 ) → z as j → ∞. This implies u (z) and is homoclinic to z. Thus, the that x0 ∈ Wloc proof is complete.
The following two criteria of chaos for system (5) were established in [Shi & Chen, 2004a, 2004b], which play an important role in the rest of this paper. Lemma 2.5 [Shi & Chen, 2004a, Theorem 3.1]. Let (X, d) be a complete metric space and V1 , V2 be

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nonempty, closed, and bounded subsets of X with d(V1 , V2 ) > 0. If a continuous map F : V1 ∪ V2 → X satisfies

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(i) F is expanding in set on V1 and V2 , respectively, and furthermore F (Vj ) ⊃ V1 ∪ V2 for j = 1, 2; (ii) F is expanding in distance on V1 and V2 , respectively; (iii) there exists a constant γ > 0 such that d(F (x), F (y)) ≤ γd(x, y), ∀ x, y ∈ V1 and ∀ x, y ∈ V2 ; then there exists a Cantor set Λ ⊂ V1 ∪ V2 such that F : Λ → Λ is topologically + to the symbolic  conjugate → dynamical system σ : + 2 2 . Consequently, F is chaotic on Λ in the sense of Devaney. The following result is a direct consequence of Lemmas 2.1 and 2.5. Corollary 2.1. Assume that all the assumptions listed in Lemma 2.5 hold. Then F is chaotic in the sense of Li–Yorke. Lemma 2.6 [Shi & Chen, 2004b, Theorem 3.1]. Let (X,  · ) be a Banach space and F : X → X be a map with a fixed point z ∈ X. Assume that

(i) F is continuously differentiable in a neighborhood of z, and DF (z) is an invertible linear map satisfying DF (z)0 > 1; (ii) F has a homoclinic orbit Γ to z and is continuously differentiable in a neighborhood of each point x on Γ, and DF (x) is an invertible linear map satisfying DF (x)0 > 0. Then, for any neighborhood U of z, there exist a positive integer n and a Cantor set Λ ⊂ U such conjugate that F n : Λ → Λ is topologically  +to + → the symbolic dynamical system σ : 2 2. Consequently, F n is chaotic on Λ in the sense of Devaney. Corollary 2.2. Assume that all the assumptions listed in Lemma 2.6 hold. Then there exists a compact and perfect invariant set D ⊂ X, containing a Cantor set, such that F is chaotic in the sense of Devaney on D as well as in the sense of Li–Yorke, and has a dense orbit in D.

Fix a neighborhood U of the fixed point z. By Lemma 2.6 and by the proofs of [Shi & Chen, 2004b, Theorem 3.1] and [Shi & Chen, 2004a, Theorem 4.1], there exist a Cantor set Λ ⊂ U and a positive integer n such that F n : Λ → Λ is topologically conjugate one-sided symbolic dynamical + to the + system σ : 2 → 2 , and F j is continuous in Λ  for 1 ≤ j ≤ n. Since (σ, + 2 ) has a dense set of periodic points and a dense orbit [Devaney, 1989, Part 1, Proposition 6.6], F n also has a dense set of periodic points and a dense orbit in Λ. Set Proof.

D = {F j (x) : x ∈ Λ, 0 ≤ j ≤ n − 1}. Then D ⊃ Λ and F is continuous in D. Since σ is surjective, F n is surjective in Λ. So F is surjective in D. It can be verified that D is a compact and perfect invariant set of F by using the continuity of F , the compactness and perfectness of the Cantor set Λ, and the invariance of Λ under F n . In addition, D is an uncountable set since Λ is a Cantor set. Next, consider the denseness of the set of periodic points of F in D. For any point x ∈ D and any neighborhood V of x, there exist y ∈ Λ and an integer j, 0 ≤ j ≤ n − 1, such that F j (y) = x. By the continuity of F , (F j )−1 (V ) is a neighborhood of y. So, there exists a k-periodic point p ∈ Λ of F n such that p ∈ (F j )−1 (V ). This implies that F j (p) ∈ V and F nk (F j (p)) = F j (F nk (p)) = F j (p). Consequently, F j (p) ∈ D is a periodic point of F . Therefore, the set of periodic points of F is dense in D. Finally, we show that F has a dense orbit in D. n Suppose that {(F n )i (y0 )}∞ i=0 is a dense orbit of F i ∞ in Λ. One can easily prove that {F (y0 )}i=0 is dense in D by using an argument similar to that used in the above discussion. Therefore, F has a dense orbit in D and consequently is topologically transitive in D. Hence, F is chaotic in the sense of Devaney on D as well as in the sense of Li–Yorke by Lemma 2.1. This completes the proof.
The following result is directly derived from [Shi & Chen, 2004b, Lemma 2.3], Lemmas 2.2, 2.4 and 2.6, and Corollary 2.2. Theorem 2.1. Let (X,  · ) be a Banach space and

F : X → X be a map with a fixed point z ∈ X. Assume that (i) F is continuously differentiable in Br0 (z) for some r0 > 0 and DF (z) is an invertible linear

Chaotification of Discrete Dynamical Systems in Banach Spaces

map satisfying

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DF (z)0 > 1; which is equivalent to that there exists a positive constant r ≤ r0 such that z is a regular expanding fixed point of F in Br (z); (ii) z is a snap-back repeller of F with F m (x0 ) = z for some x0 ∈ Br (z), x0 = z, and for some positive integer m. Furthermore, F is continuously differentiable in some neighborhoods of x1 , . . . , xm−1 , respectively, satisfying that DF (x) is an invertible linear map for all x ∈ Br (z) and for x = xj (1 ≤ j ≤ m − 1), and DF (xj )0 > 0 for 1 ≤ j ≤ m − 1, where xj = F (xj−1 ), 1 ≤ j ≤ m − 1. Then, for any neighborhood U of z, there exist an integer n > m and a Cantor set Λ ⊂ U such conjugate that F n : Λ → Λ is topologically  +to + → the symbolic dynamical system σ : 2 2. Consequently, there exists a compact and perfect invariant set D ⊂ X, containing a Cantor set, such that system (5) is chaotic on D in the sense of Devaney as well as in the sense of Li–Yorke, and has a dense orbit in D. Remark 2.2. Several remarks on Theorem 2.1 are in

Devaney and Li–Yorke for a map in a Banach space”. We refer to [Shi & Chen, 2004a, Theorem 4.1] for a similar result that a regular and nondegenerate snap-back repeller implies chaos for a map in a complete metric space. It can be similarly proved that the chaos is also in the sense of both Devaney and Li–Yorke.

3. Chaotification in General Banach Spaces In this section, the chaotification problem for system (1) with a feedback controller un given by (3) and (4), respectively, is studied, where (X,  · ) is a general Banach space. Without loss of generality and for simplicity, assume that the fixed point x∗ of the map f is the origin, i.e. f (0) = 0, throughout the rest of the paper. Theorem 3.1. Consider the controlled system (2)

with controller (3). Assume that (i) x∗ = 0 is a fixed point of f and there exist positive constants r and L such that f is continuous in Br (0) and continuously differentiable in Br (0), satisfying Df (x) ≤ L,

order. (i) An equivalence in assumption (i): suppose that F is continuously differentiable in Br0 (z) for some r0 > 0 and DF (z) is an invertible linear map. Then DF (z)0 > 1 is equivalent to that z is a regular expanding fixed point of F in Br (z) for some positive constant r ≤ r0 by [Shi & Chen, 2004b, Lemma 2.3]. (ii) Theorem 2.1 extends and improves the Marotto theorem in some way. It is noted that there are some differences between these two theorems. First, in the Marotto theorem the map has points of any period k ≥ N for some positive integer N . Although Theorem 2.1 cannot guarantee that the preceding result holds, it concludes that the set of periodic points of the map is dense in D. Secondly, by the Marotto theorem it can be only concluded that the map is chaotic in the sense of Li–Yorke, but by Theorem 2.1 it can conclude that the system is chaotic not only in the sense of Devaney on D but also in the sense of Li–Yorke. (iii) Theorem 2.1 can be restated by one simple sentence, “A regular and nondegenerate snapback repeller implies chaos in the sense of both

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∀ x ∈ Br (0);

(10)

(ii) g satisfies the following conditions: (iia) g is continuous in Br (0) ∪ Ω and continuously differentiable in Br (0) ∪ Ω, where Ω = {x ∈ X : a < x < b} with r < a < b; (iib) x∗ = 0 is a fixed point of g and there exists a point ξ ∈ Ω such that g(ξ) = 0; (iic) Dg(x) is an invertible linear operator for each x ∈ Br (0) ∪ Ω and there exists a positive constant N such that g(x) − g(y) ≥ N x − y, ∀ x, y ∈ Br (0) and ∀ x, y ∈ Ω.

(11)

Then, for any constant µ satisfying µ > µ0

  Lb Lb b Lr + b , , := max , r N r N (ξ − a) N (b − ξ) (12)

and for any neighborhood U of x∗ = 0, there exist a positive integer n > 2 and a Cantor set Λ ⊂ U such that Fµn : Λ → Λ is topologically conjugate to the +  symbolic dynamical system σ : + 2 → 2 , where Fµ (x) = f (x) + g(µx). Consequently, there exists a compact and perfect invariant set D ⊂ X containing

Y. Shi et al.

2622

a Cantor set such that the controlled system (2) with controller (3) is chaotic on D in the sense of both Devaney and Li–Yorke, and has a dense orbit in D.

and continuously differentiable in Ωµ . Further, by assumption (iib),

Theorem 2.1 is used to prove this theorem. The proof is divided into three parts. As assumed in the statement of the theorem, let µ > µ0 throughout the proof. (i) It is clear that x∗ = 0 is also a fixed point of Fµ . From (10) and Lemma 2.3, it follows that

and, by assumption (iic),

Proof.

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f (x) − f (y) ≤ Lx − y,

∀ x, y ∈ Br (0). (13)

Set

Gµ (µ−1 ξ) = g(ξ) − ξ = −ξ

Gµ (x) − Gµ (y) = g(µx) − g(µy) ≥ µN x − y, ∀ x, y ∈ Ωµ . Hence, Gµ : Ωµ → Gµ (Ωµ ) is invertible, and its : Gµ (Ωµ ) → Ωµ is continuous and inverse G−1 µ satisfies −1 −1 G−1 µ (x) − Gµ (y) ≤ (µN ) x − y,

∀ x, y ∈ Gµ (Ωµ ).

Ωµ := {x ∈ X : µ−1 a < x < µ−1 b}. Then, Bµ−1 r (0) ∪ Ωµ ⊂ Br (0), Fµ is continuous in Bµ−1 r (0) ∪ Ωµ and continuously differentiable in Bµ−1 r (0) ∪ Ωµ . Obviously, DFµ (x) = Df (x) + µDg(µx) for x ∈ Bµ−1 r (0) ∪ Ωµ . It follows from (11) and Lemma 2.2 that Dg(x)0 ≥ N,

∀ x ∈ Bµ−1 r (0) ∪ Ωµ .

So, for any fixed x ∈ Bµ−1 r (0) ∪ Ωµ and for each y ∈ X, DFµ (x)y ≥ µDg(µx)y − Df (x)y ≥ (µDg(µx)0 − Df (x))y ≥ (µN − L)y, (14) where (10) has been used. This implies that for any x ∈ Bµ−1 r (0) ∪ Ωµ , DFµ (x)0 ≥ µN − L > 1.

(15)

On the other hand, it follows from (11) and (13) that for all x, y ∈ Bµ−1 r (0), Fµ (x) − Fµ (y) ≥ g(µx) − g(µy) − f (x) − f (y) ≥ (µN − L)x − y, which implies that x∗ = 0 is an expanding fixed point of Fµ in Bµ−1 r (0). (ii) Now, it is to be proved that x∗ is a snapback repeller of Fµ in Bµ−1 r (0). Specifically, it is to show that there exists a point x0 ∈ Bµ−1 r (0) such that Fµ2 (x0 ) = 0. To achieve this, consider the equation Fµ (x) = 0

(19)

Therefore, Eq. (17), i.e. (16), can be rewritten as G−1 µ (−(f (x) + ξ)) = x.

(20)

In the following, it is to show that −(f (x) + ξ) ∈ Gµ (Ωµ ),

∀ x ∈ Ωµ .

(21)

By the definition of Gµ , it suffices to prove that for all x ∈ Ωµ , −f (x) ∈ g(µΩµ ) := {g(µx) : x ∈ Ωµ }.

(22)

In fact, it follows from (13) that for all x ∈ Ωµ , f (x) ≤ Lx ≤ µ−1 Lb. µ−1 ξ

(23)

g(µµ−1 ξ)

∈ Ωµ and = g(ξ) = 0. Obviously, So, 0 ∈ g(µΩµ ). Consequently, for each x ∈ ∂Ωµ , it follows from (11) that g(µx) = g(µx) − g(ξ) ≥ Nµx − µ−1 ξ ≥ N min{ξ − a, b − ξ} > µ−1 Lb. (24) In addition, Dg(µx) is invertible at each point x ∈ Ωµ by assumption (iic). So, g(µΩµ ) is open due to [Rudin, 1973, Theorem 10.39]. This, together with (23) and (24), implies that (22) holds and consequently (21) is true. To this end, set h1 (x) := G−1 µ (−(f (x) + ξ)), which is well defined on Ωµ , and discuss the properties of h1 (x) in Ωµ . It follows from (18) that for each x ∈ Ωµ , −1 −1 h1 (x) = G−1 µ (−(f (x) + ξ)) − Gµ (−ξ) + µ ξ, (25)

(16)

which, with the aid of (19) and (23), implies that for each x ∈ Ωµ ,

(17)

h1 (x) −1 ≥ µ−1 ξ − G−1 µ (−(f (x) + ξ)) − Gµ (−ξ)

on Ωµ , which can be written as −(f (x) + ξ) = g(µx) − ξ,

(18)

where ξ is specified as in assumption (iib). Let Gµ (x) := g(µx) − ξ. Then Gµ is continuous in Ωµ

≥ µ−1 (ξ − N −1 f (x)) ≥ µ−1 (ξ − (µN )−1 Lb) > µ−1 a.

(26)

Chaotification of Discrete Dynamical Systems in Banach Spaces

On the other hand, applying (19), (23) and (25) yields that for each x ∈ Ωµ , h1 (x) −1 ≤ µ−1 ξ + G−1 µ (−(f (x) + ξ)) − Gµ (−ξ) ≤ µ−1 (ξ + N −1 f (x)) (27) ≤ µ−1 (ξ + (µN )−1 Lb) < µ−1 b.

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It is easy to see, from (26) and (27), that h1 maps Ωµ into itself. In addition, from (19) and (13), it follows that h1 (x) − h1 (y) ≤ (µN )−1 f (x) − f (y) ≤ (µN )−1 Lx − y, ∀ x, y ∈ Ωµ , which implies that h1 is contractive on Ωµ since (µN )−1 L < 1. By the Banach contractive mapping principle, there exists a unique point x1 ∈ Ωµ such that h1 (x1 ) = x1 . It follows from (26) and (27) that x1 ∈ Ωµ . Therefore, x1 solves Eq. (16). Next, consider the following equation: Fµ (x) = x1

(28)

(29)

By assumption (ii), it can be easily verified that g : Br (0) → g(Br (0)) is invertible, and that its inverse g−1 : g(Br (0)) → Br (0) is continuous and satisfies g

−1

(x)−g

−1

(y) ≤ N

−1

x−y, ∀ x, y ∈ g(Br (0)).

So, Eq. (29), i.e. (28), can be rewritten as µ−1 g−1 (−f (x) + x1 ) = x.

this, for any fixed x ∈ Bµ−1 r (0) ∪ Ωµ and for any z ∈ X, consider the following equation: DFµ (x)y = z,

(31)

which can be written as µ−1 (−Df (x)y + z) = Dg(µx)y.

(32)

By assumption (iic), Dg(µx) is invertible. So, Eq. (32), i.e. (31), can be rewritten as µ−1 (Dg(µx))−1 (−Df (x)y + z) = y.

(33)

Let h3 (y) := µ−1 (Dg(µx))−1 (−Df (x)y + z). From (11) and by Lemma 2.2, one knows that Dg(µx)0 ≥ N , which implies that (Dg(µx))−1  ≤ N −1 . Set K = z(µN − L)−1 . Then, for each y ∈ BK (0), h3 (y) ≤ µ−1 (Dg(µx))−1 (Df (x)y + z) ≤ (µN )−1 (Ly + z) ≤ K, and for any y1 , y2 ∈ BK (0), by (10),

in Bµ−1 r (0), which can be written as −f (x) + x1 = g(µx).

2623

(30)

Let h2 (x) := µ−1 g−1 (−f (x) + x1 ). Similar to the discussion on h1 , one can easily show that h2 is well defined on Bµ−1 r (0), maps Bµ−1 r (0) into itself, and is contractive on Bµ−1 r (0) for µ > µ0 . This implies that there exists a unique point x0 ∈ Bµ−1 r (0) such that h2 (x0 ) = x0 , again by the Banach contractive mapping principle. One can also show that x0 ∈ Bµ−1 r (0). So, x0 solves Eq. (28). Obviously, x0 = 0 from (28). Based on the above discussions, one concludes that there exists x0 ∈ Bµ−1 r (0), x0 = 0, such that Fµ2 (x0 ) = 0 and x1 = Fµ (x0 ) ∈ Ωµ . Hence, x∗ = 0 is a snap-back repeller of Fµ . (iii) Now, it is shown that DFµ (x) is invertible for each x ∈ Bµ−1 r (0) ∪ Ωµ . With assumption (iic), it follows from (10) and (14) that DFµ (x) = Df (x) + µDg(µx) is bounded and injective for each x ∈ Bµ−1 r (0) ∪ Ωµ . It is also surjective. To show

h3 (y1 ) − h3 (y2 ) ≤ µ−1 (Dg(µx))−1 Df (x)y1 − y2  ≤ (µN )−1 Ly1 − y2 . Hence, h3 maps BK (0) into itself and is contractive on BK (0). Consequently, by the Banach contractive mapping principle, there exists a unique point y ∈ BK (0) such that h3 (y) = y. This implies that DFµ (x) is surjective. So, DFµ (x) has an inverse (DFµ (x))−1 . Thus, it follows from (14) that (DFµ (x))−1 is a bounded linear operator, so that DFµ (x) is invertible for each x ∈ Bµ−1 r (0) ∪ Ωµ . In summary, x∗ = 0 is a regular expanding fixed point of Fµ in Bµ−1 r (0) by [Shi & Chen, 2004b, Lemma 2.3] and is a snap-back repeller of Fµ with Fµ2 (x0 ) = 0 for some x0 ∈ Bµ−1 r (0), x0 = 0; Fµ is continuously differentiable in Bµ−1 r (0) and also in a neighborhood of x1 ; and DFµ (x) is invertible for all x ∈ Bµ−1 r (0) and for x = x1 satisfying (15). Hence, all the assumptions in Theorem 2.1 are satisfied for each µ > µ0 , so that the theorem follows from Theorem 2.1. This completes the proof.
Remark 3.1. The controller (3) can be easily designed such that the corresponding map g is simple and satisfies assumption (ii) in Theorem 3.1. For example, g can be taken as one of the following four

Y. Shi et al.

2624

4. Chaotification in Some Special Banach Spaces

simple functions:  if x ≤ r  ±x, g1 (x) = arbitrary, if r < x < a   ±(x − ξ), if a ≤ x ≤ b,

(34)

where 0 < r < a < b and ξ ∈ X can be any point satisfying a < ξ < b. Theorem 3.2. Consider the controlled system (2) Int. J. Bifurcation Chaos 2006.16:2615-2636. Downloaded from www.worldscientific.com by UNIVERSITY OF WESTERN ONTARIO WESTERN LIBRARIES on 07/24/12. For personal use only.

with controller (4). Assume that (i) assumption (i) in Theorem 3.1 holds; (ii) g satisfies the following conditions:  (iia) g is continuous in Ba (0) ∪ Ω and continuously differentiable in Ba (0) ∪ Ω , where Ω = {x ∈ X : b < x < r} with 0 < a < b < r; (iib) x∗ = 0 is a fixed point of g and there exists a point ξ ∈ Ω such that g(ξ) = 0; (iic) Dg(x) is an invertible linear operator for each x ∈ Ba (0) ∪ Ω and there exists a positive constant N such that g(x) − g(y) ≥ N x − y, ∀ x, y ∈ Ba (0)

and



∀ x, y ∈ Ω .

Then, for each constant µ satisfying µ > µ0

  Lr Lr La + r , , , := max N a N (ξ − b) N (r − ξ)

all the results in Theorem 3.1 hold for Fµ (x) = f (x) + µg(x) therein. The proof is similar to that of Theorem 3.1 with Bµ−1 r (0) and Ωµ replaced by Ba (0) and Ω , respectively. So, details are omitted.
Proof.

Remark 3.2. The controller (4) can be easily designed such that the corresponding map g is simple and satisfies assumption (ii) in Theorem 3.2. For example, similarly to the function g1 given in (34), here g can be taken as one of the following four simple functions:  if x ≤ a  ±x, (35) g2 (x) = arbitrary, if a < x < b  ±(x − ξ), if b ≤ x ≤ r,

where 0 < a < b < r and ξ ∈ X can be any point satisfying b < ξ < r.

In this section, chaotification in some special Banach spaces is studied. For convenience, let j k denote “m ≤ k for k < ∞ and j < ∞ for k = ∞”. Let Rk = {x = {xj }kj=1 : xj ∈ R for 1 ≤ j k} with 1 ≤ k ≤ ∞, and Yk = {x ∈ Rk : xk < ∞} with the norm xk = sup{|xj | : 1 ≤ j k}. Then, (Yk ,  · k ) is a Banach space. Clearly, in the special case of k < ∞, Yk is the classical k-dimensional real space Rk and its norm  · k is the sup-norm, while in the special case of k = ∞, Yk = l∞ and the norm  · k is the usual norm of l∞ . In general, of course, chaotification of system (1) in (Yk ,  · k ) can be achieved by using controller (3) or (4) with the map g satisfying assumption (ii) in Theorems 3.1 and 3.2, respectively. In this section, we consider the chaotification of system (1) in the space (Yk ,  · k ) with a certain special feedback controller un given in the form of (3) or (4), and show that the controlled system is chaotic in the sense of both Devaney and Li–Yorke. As a consequence, the controlled system (28) with controller (29) in [Wang & Chen, 2000] is proved to be chaotic in the sense of Devaney. Further, we extend the Chen–Lai anti-control algorithm with mod-operation in a finite-dimensional real space proposed in [Chen & Lai, 1998] to Y∞ and show that the controlled system is chaotic in the sense of both Devaney and Li–Yorke for 1 ≤ k ≤ ∞, by which the controlled system (21) with controller (22) in [Chen & Lai, 1998] is proved to be chaotic in the sense of Devaney, as well as in the sense of both Li–Yorke and Wiggins. For convenience, denote I k := {x = {xj }kj=1 : xj ∈ I for 1 ≤ j k}, where I is a bounded subset of R. Clearly, I k ⊂ Yk .

4.1. A generalized sawtooth function as the controller Introduce the following function in Yk for each k ≥ 1: Sawr (x) = {sawr (xj )}kj=1 ,

(36)

Chaotification of Discrete Dynamical Systems in Banach Spaces

where sawr is the classical sawtooth function, that is,

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sawr (x) = (−1)m (x − 2mr), (2m − 1)r ≤ x < (2m + 1)r, m ∈ Z, while Z denotes the integer set. Clearly, Sawr (x) is the sawtooth function sawr (x) in R when k = 1, and Sawr (x) is the sawtooth function sawr (x) in Rk when k < ∞, defined by (11) and (12) in [Wang & Chen, 2000], respectively. So, Sawr can be regarded as a generalization of the classical sawtooth function. Theorem 4.1. Consider the controlled system (2)

with controller (3). Assume that f (0) = 0 and there exists a positive constant r such that f is continuous in Br (0) and furthermore satisfies f (x) − f (y) ≤ Lx − y,

∀ x, y ∈ Br (0) (37)

for some constant L > 0. Let the map g given in (3) be taken as g(x) = Sawε (x), where ε > 0 is any given constant. Then, for each constant µ satisfying   5 −1 µ > µ0 := max r ε, 5(L + 1) , 2 there exists a Cantor set Λ ⊂ B 5 µ−1 ε (0) such that 2 Fµ (x) = f (x) + g(µx) : Λ → Λ is topologically con jugate to the symbolic dynamical system σ : + 2 → + 2 . Consequently, Fµ , i.e. the controlled system (2) with controller (3), is chaotic on Λ in the sense of both Devaney and Li–Yorke. Lemma 2.5 is used to prove the theorem. As assumed in the statement of the theorem, let µ > µ0 throughout the rest of the proof. Set

1 −1 1 −1 k V1 = − µ ε, µ ε , 2 2

3 −1 5 −1 k µ ε, µ ε . V2 = 2 2 Proof.

Then, V1 , V2 ⊂ Br (0) and consequently Fµ is continuous in V1 ∪ V2 . In addition, it is easy to verify that V1 and V2 are nonempty, closed, bounded, and d(V1 , V2 ) = inf{x − y : x ∈ V1 and y ∈ V2 } = µ−1 ε > 0. Next, it is shown that Fµ (Vi ) ⊃ V1 ∪ V2 for i = 1, 2. For each x ∈ V1 , Fµ,j (x) = fj (x)+sawε (µxj ) = fj (x) + µxj , where Fµ (x) = {Fµ,j (x)}kj=1 , f (x) =

2625

{fj (x)}kj=1 , and x = {xj }kj=1 . So, for each x ∈ V1 with xj = −(1/2)µ−1 ε, it follows from (37) that 1 1 Fµ,j (x) = fj (x) − ε ≤ Lx − ε 2 2 1 1 1 ≤ µ−1 Lε − ε ≤ − µ−1 ε 2 2 2

(38)

and for each x ∈ V1 with xj = (1/2)µ−1 ε, it follows again from (37) that 1 1 Fµ,j (x) = fj (x) + ε ≥ −Lx + ε 2 2 5 1 1 (39) ≥ − µ−1 Lε + ε ≥ µ−1 ε. 2 2 2 Since Fµ is continuous in V1 , by the intermediate value theorem it follows from (38) and (39) that Fµ (V1 ) ⊃ [−(1/2)µ−1 ε, (5/2)µ−1 ε]k , which implies that Fµ (V1 ) ⊃ V1 ∪ V2 . With a similar argument, one can show that Fµ (V2 ) ⊃ V1 ∪ V2 . Hence, Fµ is expanding in set on V1 and V2 , respectively. Since V1 , V2 ⊂ Br (0), it follows from (37) that f (x) − f (y) ≤ Lx − y, ∀ x, y ∈ V1 and ∀ x, y ∈ V2 .

(40)

On the other hand, it can be verified by the definition of g that g(µx) − g(µy) = µx − y, ∀ x, y ∈ V1 and ∀ x, y ∈ V2 .

(41)

Hence, it follows from (40) and (41) that for all x, y ∈ V1 and x, y ∈ V2 , (µ − L)x − y ≤ Fµ (x) − Fµ (y) ≤ (µ + L)x − y, which implies that Fµ is expanding in distance on V1 and V2 , respectively, due to µ − L > 5. The above discussions indicate that all the assumptions given in Lemma 2.5 are satisfied with the distance d(x, y) = x−y. Therefore, by Lemma 2.5 and Corollary 2.1, the conclusion of Theorem 4.1 is true. This completes the proof.
Remark 4.1. Wang and Chen [2000] considered the controlled system (2) with the controller

un = sawr (µx),

(42)

where x ∈ Rk with k < ∞, sawr (x) = (sawr (x1 ), sawr (x2 ), . . . , sawr (xk )), and µ > 0 is a parameter. They showed that the controlled system (2) with (42), i.e. (28) with (29) in [Wang & Chen, 2000], is chaotic in the sense of Li–Yorke by the Marotto

2626

Y. Shi et al.

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theorem [1978] for sufficiently large µ under the assumptions that f (0) = 0 and f is continuously differentiable in the closed ball B3µ−1 r (0), satisfying (43) Df (x)∞ ≤ L, ∀ x ∈ B3µ−1 r (0), k where C∞ = max{ j=1 |cij | : 1 ≤ i ≤ k} for a k × k real matrix C = (cij ). By the definition of the operator norm of bounded linear operators (defined in Sec. 2), it is easy to see that C = C∞ . Clearly, Sawr (x) = sawr (x) in this case. Note that since f is the map corresponding to the original system (1), it is independent of the parameter µ. So, the assumption for f to be restricted in B3µ−1 r (0) is not reasonable. Instead, B3µ−1 r (0) should be replaced by Br (0) for some constant r > 0. On the other hand, if f is continuous in Br (0), and continuously differentiable in Br (0) and satisfies (43) in Br (0), then f satisfies the Lipschitz condition (37) in Br (0) by Lemma 2.3. Hence, the following result can be directly obtained from Theorem 4.1, and thus the controlled system (28) with controller (29) in [Wang & Chen, 2000] is chaotic not only in the sense of Li–Yorke but also in the sense of Devaney for sufficiently large µ. Corollary 4.1. Consider the controlled system (2)

with controller (3). Assume that f (0) = 0 and there exist positive constants r and L such that f is continuous in Br (0), continuously differentiable in Br (0), and satisfies Df (x) ≤ L,

∀ x ∈ Br (0).

Let the map g in (3) be taken as g(x) = Sawε (x), where ε > 0 is any given constant. Then all the results in Theorem 4.1 hold. Remark 4.2. In Theorem 4.1 and Corollary 4.1, the

constant ε can be chosen very small so that the controllers can be arbitrarily small in norm. This means that the original system can be driven to be chaotic by using an arbitrarily small-amplitude state feedback control, which is very desirable for engineering applications. Theorem 4.2. Consider the controlled system (2)

with controller (4). Assume that there exist positive constants r and L such that f is continuous in Br (0) and satisfies (37). If the map g in (4) is taken as g(x) = Saw r3 (x),

then, for each constant µ satisfying µ > µ0 := max{1 + L, 5 + 6(L + f (0)r −1 )}, there exists a Cantor set Λ ⊂ Br (0) such that Fµ (x) = f (x) + µg(x) : Λ → Λ is topologically con jugate to the symbolic dynamical system σ : + 2 → + . Consequently, F , i.e. the controlled system (2) µ 2 with controller (4), is chaotic on Λ in the sense of both Devaney and Li–Yorke. Proof.

Set

1 1 k V1 = − r, r , 6 6

V2 =

1 5 r, r 2 6

k .

Then, V1 , V2 ⊂ Br (0) and consequently Fµ is continuous on V1 ∪V2 . Similar to the proof of Theorem 4.1, one can show that for each µ > µ0 , Fµ (Vj ) ⊃ V1 ∪V2 and for all x, y ∈ V1 and x, y ∈ V2 , (µ − L)x − y ≤ Fµ (x) − Fµ (y) ≤ (µ + L)x − y. Hence, all the assumptions given in Lemma 2.5 are satisfied for µ > µ0 with the distance d(x, y) = x−y. Therefore, by Lemma 2.5 and Corollary 2.1, the conclusion of the theorem is true. The proof is complete.
Similar to Corollary 4.1, the following is a consequence of Theorem 4.2. Corollary 4.2. Consider the controlled system (2)

with controller (4). Assume that there exist positive constants r and L such that f is continuous in Br (0), continuously differentiable in Br (0), and satisfies Df (x) ≤ L,

∀ x ∈ Br (0).

If the map g in (4) is taken as g(x) = Saw r3 (x), then all the results in Theorem 4.2 hold. Remark 4.3. The sawtooth function sawr (x) is unimodal interval-wise on R. In fact, any function, which has a similar geometric shape to the sawtooth function, may generate a function in Yk in the same way as in (36) and the generated function can be used as a controller such that the controlled system is chaotic. This fact was first observed by Chen [2003] for the one-dimensional case. For example, the classical sinusoidal function sin x has similar geometric properties to the sawtooth function

Chaotification of Discrete Dynamical Systems in Banach Spaces

interval-wise. Similar to Sawr (x) in (36), the following function is defined:

Int. J. Bifurcation Chaos 2006.16:2615-2636. Downloaded from www.worldscientific.com by UNIVERSITY OF WESTERN ONTARIO WESTERN LIBRARIES on 07/24/12. For personal use only.

sin x = {sin xj }kj=1 ,

x = {xj }kj=1 ∈ Yk .

If g(x) = sin(r −1 x), then, with a similar argument to the proof of Theorem 4.1, one can show that there exists a constant µ0 > 0 such that for any constant µ > µ0 , all the results in Theorem 4.1 and Corollary 4.1 hold. Similarly, if g(x) = sin(4r −1 x), then there exists a constant µ0 > 0 such that for any constant µ > µ0 , all the results in Theorem 4.2 and Corollary 4.2 hold. Remark 4.4. Here, it is not required that the map f , corresponding to the original system (1), has a fixed point in Theorem 4.2 and Corollary 4.2. This is weaker than many existing results.

Chen and Lai [1998] developed the first anti-control algorithm based on feedback control with modoperation, described by (mod 1)

(44)

with un = µxn ,

Theorem 4.3. Consider the chaotification of sys-

tem (1) in space Yk . Assume that f (0) = 0 and there exists a positive constant r such that f is continuous in Br (0) and furthermore satisfies f (x) − f (y) ≤ Lx − y,

∀ x, y ∈ Br (0) (47)

µ > µ0 := max{5(1 + L), 10L}, there exists a Cantor set Λ ⊂ B 5 µ−1 r (0) such that 2 Fµ (x) := f (x) + µx (mod r) : Λ → Λ is topologicallyconjugate +to the symbolic dynamical system + σ : 2 → 2 . Consequently, Fµ , i.e. the controlled system

(45)

where f (0) = 0, f is continuously differentiable, at least locally in a region containing x∗ = 0, in the usual k-dimensional real space Rk , and satisfies Df (x)1 ≤ L

chaos in the sense of Devaney is stronger than that in the sense of Wiggins and also stronger than that in the sense of Li–Yorke under some conditions (see Lemma 2.1). In this subsection, the chaotification algorithm with the mod-operation is extended to space Yk (defined at the beginning of this section) and the controlled system is proved to be chaotic in the sense of Devaney, Li–Yorke, and Wiggins. As a consequence, the controlled system (44)–(45) is shown to be chaotic in the sense of Devaney as well as in the sense of both Li–Yorke and Wiggins.

for some constant L > 0. Then, for each constant µ satisfying

4.2. Chaotification via the mod-operation

xn+1 = f (xn ) + un

2627

(46)

for all x ∈ Rk or x in a region containing x∗ = 0, and for some constant L > 0, with C1 being the spectral norm of a k × k matrix C = (cij ), µ = L + ec , and c > 0 is a parameter. First, it is noted that C1 ≤ C. System (44) with controller (45) here is the autonomous form of (21) with controller (22) given in [Chen & Lai, 1998]. It was shown in [Chen & Lai, 1998] that for c > 0, the controlled system (44) and (45) is chaotic in the sense of Devaney in the linear case f (x) = Ax, where A is a k × k real matrix, and is chaotic in the sense of Wiggins in the nonlinear case. Later, Wang and Chen [1999] showed that the controlled system (44)–(45) is chaotic in the sense of Li–Yorke by using the Marotto theorem [Marotto, 1978]. Since there is some error in the Marotto theorem (see the relative discussion in [Shi & Chen, 2004b]), the proof of the Wang–Chen result contains a relative error. On the other hand, it is known that

xn+1 = f (xn ) + µxn

(mod r),

(48)

is chaotic on Λ in the sense of Devaney, and also in the sense of both Li–Yorke and Wiggins, where the mod-operation is componentwise. The idea in the proof is similar to that in the proof of Theorem 4.1. But, since Fµ is not continuous in Br (0) for sufficiently large µ, the proof is somewhat more complicated. For completeness, details are given below. Set Gµ (x) := f (x) + µx. Then, Fµ (x) = Gµ (x) (mod r). It is evident that f is continuous on Br (0) from (47). Further, set Proof.

U1 = I1k ,

U2 = I2k ,

where I1 = [(3/4)µ−1 r, (3/2)µ−1 r] and I2 = [(7/4)µ−1 r, (5/2)µ−1 r]. Clearly, for any µ > µ0 , U1 , U2 ⊂ Br (0) and consequently Gµ is continuous on U1 ∪ U2 . In addition, it is easy to verify that d(U1 , U2 ) = (1/4)µ−1 r > 0. As assumed in the statement of the theorem, let µ > µ0 in the following discussions. First, construct a subset V1 of U1 such that Fµ satisfies all the assumptions in Lemma 2.5 on V1 . For any fixed j, 1 ≤ j ≤ k, and for any fixed xi ∈ I1

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Y. Shi et al.

for 1 ≤ i ≤ k, i = j, if xj = (3/4)µ−1 r, it follows from (47) that 3 3 3 Gµ,j (x) ≤ Lx + r ≤ µ−1 Lr + r < r; (49) 4 2 4 if xj = (3/2)µ−1 r, it follows again from (47) that 3 3 3 Gµ,j (x) ≥ −Lx + r ≥ − µ−1 Lr + r > r; 2 2 2 (50)

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and for all xj ∈ I1 , it follows again from (47) that 3 3 3 Gµ,j (x) ≤ Lx + r ≤ µ−1 Lr + r < 2r. (51) 2 2 2 By the continuity of Gµ,j in U1 and the inequalities in (49) and (50), there exists a unique point x0j ∈ ((3/4)µ−1 r, (3/2)µ−1 r) such that Gµ,j (x0 ) = r, where x0 = {xi }ki=1 with xj = x0j . Clearly, Gµ,j (x) > r for the fixed xi (1 ≤ i ≤ k, i = j) and xj ∈ (x0j , (3/2)µ−1 r]. It is evident that x0j is dependent on µ and xi for 1 ≤ i ≤ k, i = j. Denote x0j = hj (µ, x1 , . . . , xj−1 , xj+1 , . . . , xk ). Then, hj is a continuous function of xi in I1 for 1 ≤ i ≤ k, i = j. In fact, for any xi , yi ∈ I1 (1 ≤ i ≤ k, i = j), fj (x1 , . . . , xj−1 , x0j , xj+1 , . . . , xk ) + µx0j = r,

(52)

fj (y1 , . . . , yj−1 , yj0 , yj+1 , . . . , yxk ) + µyj0 = r,

where x0j = hj (µ, x1 , . . . , xj−1 , xj+1 , . . . , xk ) and yj0 = hj (µ, y1 , . . . , yj−1 , yj+1 , . . . , yk ). Since µ > L, Eqs. (52) with (47) together imply that |x0j − yj0 | ≤ µ−1 L max{max{|xi − yi | : 1 ≤ i ≤ k, i = j}, |x0j − yj0 |} ≤ µ−1 L max{|xi − yi | : 1 ≤ i ≤ k, i = j}. This implies that hj is continuous in xi for xi ∈ I1 (1 ≤ i ≤ k, i = j). Set 

3 V1 = x = {xj }kj=1 ∈ U1 : xj ∈ x0j , µ−1 r , 2  1≤j≤k . Then, V1 is a nonempty, closed, and bounded subset of U1 . Clearly, V1 is a bounded subset of U1 . Because x = {xj }kj=1 with xj = (3/2)µ−1 r (1 ≤ j ≤ k) is in V1 , V1 is nonempty. So, it suffices to show that V1 is closed. In fact, for any convergent sequence {x(n) } in V1 , suppose x(n) → x as n → ∞. For con(n) venience, denote x(n) = {xj }kj=1 and x = {xj }kj=1 . (n) Then, xj → xj as n → ∞ for 1 ≤ j ≤ k. By (n)

the continuity of hj and by noticing that xj

∈

(n)

(n)

(n)

(n)

[hj (µ, x1 , . . . , xj−1 , xj+1 , . . . , xk ), (3/2)µ−1 r], one has that xj ∈ [hj (µ, x1 , . . . , xj−1 , xj+1 , . . . , xk ), (3/2)(µ)−1 r] for 1 ≤ j ≤ k, which implies that x ∈ V1 . So, V1 is closed. On the other hand, it follows from (51) that for any x ∈ V1 , r ≤ Gµ,j (x) < 2r,

1 ≤ j ≤ k.

(53)

Hence, Fµ,j (x) = Gµ,j (x) − r for x ∈ V1 and, consequently, Fµ,j (x) is continuous on V1 and Fµ,j (x) = 0 for x ∈ V1 with xj = hj (µ, x1 , . . . , xj−1 , xj+1 , . . . , xk ). Furthermore, it follows from (50) that for x ∈ V1 with xj = (3/2)µ−1 r, 5 3 1 Fµ,j (x) ≥ − µ−1 Lr + r > µ−1 r. 2 2 2 Hence, by the intermediate value theorem, one has that Fµ (V1 ) ⊃ [0, (5/2)µ−1 r]k and consequently Fµ (V1 ) ⊃ U1 ∪ U2 . In addition, Fµ (x) − Fµ (y) = Gµ (x) − Gµ (y) and then from (47) one has that for all x, y ∈ V1 , (µ − L)x − y ≤ Fµ (x) − Fµ (y) ≤ (µ + L)x − y.

(54)

With similar arguments, one can show that there exists a nonempty, closed and bounded subset V2 of U2 such that Fµ,j (x) = Gµ,j (x) − 2r (1 ≤ j ≤ k) for all x ∈ V2 , Fµ is continuous in V2 , Fµ (V2 ) ⊃ U1 ∪ U2 , and (54) holds for all x, y ∈ V2 . Summarizing the above discussions, and also noting that d(V1 , V2 ) ≥ d(U1 , U2 ) = (1/4)µ−1 r > 0, it has been shown that for each µ > µ0 , Fµ satisfies all the assumptions given in Lemma 2.5 on V1 and V2 . Thus, the theorem follows from Lemma 2.5 and Corollary 2.1. The proof is complete.
Similar to Corollary 4.1, the following is a consequence of Theorem 4.3. Corollary 4.3. Consider the chaotification of system (1) in space Yk . Assume that f (0) = 0 and there exist positive constants r and L such that f is continuous in Br (0) and continuously differentiable in Br (0), satisfying

Df (x) ≤ L,

∀ x ∈ Br (0).

Then, all the results in Theorem 4.3 hold. Remark 4.5. Taking r = 1 and k < ∞ in Corollary

4.3, one can conclude that the controlled system (44) with controller (45) is chaotic on a Cantor set in the sense of Devaney, as well as in the sense of both Li–Yorke and Wiggins. So, after all, the Chen– Lai anti-control algorithm via feedback control with

Chaotification of Discrete Dynamical Systems in Banach Spaces

mod-operation actually leads to chaos in the sense of Devaney, Li–Yorke, and Wiggins.

5. An Application to First-Order Partial Difference Equations

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In this section, we illustrate how to apply the chaotification schemes established in the preceding sections to a class of first-order partial difference equations. Consider the following first-order partial difference equation: x(n + 1, m) = f (m, x(n, m), x(n, m + 1)),

(55)

where n ≥ 0 is the discrete time step, m is the lattice point with 0 ≤ m k ≤ ∞, k + 1 is the system size, and f : {(m, x, y) : 0 ≤ m k, (x, y) ∈ D} → R is a map with D ⊂ R2 . We first need to give a certain boundary condition for Eq. (55). In the case of k < ∞, the following periodic boundary condition is imposed on (55): x(n, 0) = x(n, k + 1),

n ≥ 0.

(56)

In the case of k = ∞, the boundary condition is that the solution {x(n, m)}∞ n,m=0 of Eq. (55) is bounded for each n ≥ 0, that is, {x(n, m)}∞ m=0 ∈ Y∞ (defined at the beginning of Sec. 4) for each n ≥ 0. Note that other boundary conditions can also be imposed. For any given initial condition of Eq. (55), in the case of k < ∞, x(0, m) = φ(m),

0 ≤ m ≤ k + 1,

where φ satisfies the boundary condition (56), Eq. (55) has a unique solution {x(n, m) : n ≥ 0, 0 ≤ m ≤ k} satisfying the initial condition and the boundary condition (56), which can be verified by successive iterations. This means that Eq. (55) for any given initial-boundary condition (56) has a unique solution. So this type of initial-boundary problems is well-defined. In the case of k = ∞, for any given initial condition x(0, m) = φ(m),

m ≥ 0,

the initial value problem has a unique solution {x(n, m)}∞ n,m=0 , again by successive iterations. It is noted that {x(n, m)}∞ m=0 may not be bounded for all n ≥ 0 even if the initial-valued function φ = {φ(m)}∞ m=0 is bounded. The solution {x(n, m) : n ≥ 0, 0 ≤ m k} is also called an orbit of Eq. (55) (with (56) in the case of k < ∞) starting at φ. Remark 5.1. Equation (55) can be regarded as a discretization of the following first-order partial

differential equation: ut (t, s) = fˆ(s, u(t, s), us (t, s)),

2629

(57)

where t ≥ 0 is the time variable, s ∈ [0, s0 ] (or [0, s0 ) for s0 = ∞ ) is the spatial variable with 0 < s0 ≤ ∞, ˆ → R is a and fˆ : {(s, x, y) : 0 ≤ s s0 , (x, y) ∈ D} ˆ ⊂ R2 . The boundary condition for function with D the above partial differential equation in the case of s0 < ∞ is u(t, 0) = u(t, s0 ),

t ∈ [0, ∞),

(58)

and u(t, s) is bounded on [0, ∞) for any t ∈ [0, ∞) in the case of s0 = ∞. For discussions on chaos for linear partial differential equations, we refer to [Chen et al., 1998; Desch et al., 1997; Dyson et al., 2000] and some references therein. It is evident that studying the dynamical behavior of Eq. (55) with the time evolution will be helpful for understanding dynamical behavior of Eq. (57) with the time evolution. When f is independent of m and k = ∞, Chen et al. [2005] studied the stability and chaos problems and gave some fundamental concepts for Eq. (55) by reformulating (55) into a certain discrete dynamical system of the form (1). The objective here is to design a control input sequence {u(n, m)} such that the output of the controlled system x(n + 1, m) = f (m, x(n, m), x(n, m + 1))+u(n, m) (59) is chaotic in the sense of Devaney or Li–Yorke (see Definition 5.2 below). Similar to (3) and (4), the controller can be designed in the form of u(n, m) = g(µx(n, m))

(60)

u(n, m) = µg(x(n, m)),

(61)

or where µ is a positive parameter and g : J → R is a map, while J is an interval. We only discuss a special controller in this section, leaving further studies to our forthcoming papers. This section consists of four subsections. The first subsection gives a reformulation of Eq. (55) and some fundamental concepts. The second subsection studies the stability and expansion of fixed points of Eq. (55). In the third subsection, a chaotification scheme for Eq. (55) is designed as an application of the chaotification theory established in the above two sections. Finally, some specific examples are discussed and some simulations are given in the fourth subsection.

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5.1. Reformulation and some relative concepts Instead of studying spatiotemporal dynamics for Eq. (55), our attention is focused on the dynamical behavior of Eq. (55) with respect to the time evolution. We first reformulate Eq. (55) into a special discrete system of the form (1). By setting

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xn = (x(n, 0), x(n, 1), . . .)T ∈ Rk+1 ,

n ≥ 0,

Eq. (55) (with (56) in the case of k < ∞) can be written as xn+1 = F (xn ),

n ≥ 0,

(62)

where F (xn ) = (f (0, x(n, 0), x(n, 1)), f (1, x(n, 1), x(n, 2)), . . . , f (k, x(n, k), x(n, 0)))T in the case of k < ∞ and F (xn ) = (f (0, x(n, 0), x(n, 1)), f (1, x(n, 1), x(n, 2)), . . .)T in the case of k = ∞, and the controlled Eq. (59) with (60) or (61) (with (56) in the case of k < ∞) can be written as xn+1 = F (xn ) + G(xn , µ),

n ≥ 0,

(63)

where G(xn , µ) = (g(µx(n, 0)), g(µx(n, 1)), . . . , g(µx(n, k)))T or G(xn , µ) = µ(g(x(n, 0)), g(x(n, 1)), . . . , g(x(n, k)))T in the case of k < ∞; and G(xn , µ) = (g(µx(n, 0)), g(µx(n, 1)), . . .)T or G(xn , µ) = µ(g(x(n, 0)), g(x(n, 1)), . . .)T in the case of k = ∞. System (62) is said to be induced by Eq. (55) (with (56) in the case of k < ∞) in the Banach space Yk+1 , defined at the beginning of Sec. 4. It is evident that a solution {x(n, m) : n ≥ 0, 0 ≤ m k} of Eq. (55) (with (56) in the case of k < ∞) corresponds to a solution {xn }∞ n=0 of system (62) with xn = {x(n, m)}km=0 , which is called the solution of system (62) induced by the solution {x(n, m) : n ≥ 0, 0 ≤ m k} of Eq. (55) (with (56) in the case of k < ∞). In the case of k < ∞, for any induced solution {xn }∞ n=0 , each term xn is in Yk+1 , while in the case of k = ∞, the solution {x(n, m) : n ≥ 0, 0 ≤ m k} of Eq. (55) is bounded over 0 ≤ m < ∞ for each n ≥ 0 if and only if each term xn of its induced solution {xn }∞ n=0 is in Y∞ . Further, we have xn k = sup{|x(n, m)| : 0 ≤ m k},

n ≥ 0,

which is exactly the sup-norm of the initial function space, i.e. the phase space for Eq. (55). Therefore, the dynamical behavior of Eq. (55) (with (56) in the case of k < ∞) with respect to the time evolution in the phase space with the sup-norm defined above is the same as that of its induced system (62) in Yk+1 .

So, this reformulation for Eq. (55) is natural in this sense of discretization. Now, we introduce some relative concepts for Eq. (55) (with (56) in the case of k < ∞). Definition 5.1

(i) A point x ∈ Yk+1 is called a k0 -periodic point of Eq. (55) (with (56) in the case of k < ∞) if x ∈ Yk+1 is a k0 -periodic point of its induced system (62), that is, F k0 (x) = x and F j (x) = x for 1 ≤ j ≤ k0 − 1. In the special case of k0 = 1, x is called a fixed point or a steady state of Eq. (55) (with (56) in the case of k < ∞). (ii) The concepts of stability, asymptotical stability, and instability of a fixed point or a periodic point for Eq. (55) (with (56) in the case of k < ∞) are defined similarly to those of the fixed point or the periodic point for its induced system (62) in Yk+1 ; and the concepts of density of periodic points, topological transitivity, sensitive dependence on initial conditions, and invariant set for Eq. (55) (with (56) in the case of k < ∞) are defined similarly to those for its induced system (62) in Yk+1 . Remark 5.2. From Definition 5.1, it follows that x = {x(m)}km=0 is a fixed point of Eq. (55) with (56) in the case of k < ∞ if and only if x ∈ Yk+1 satisfies x(m) = f (m, x(m), x(m + 1)), 0 ≤ m ≤ k − 1, x(k) = f (k, x(k), x(0)); (64)

and is a fixed point of Eq. (55) in the case of k = ∞ if and only if x ∈ Y∞ satisfies x(m) = f (m, x(m), x(m + 1)),

m ≥ 0.

(65)

These definitions of fixed and periodic points of Eq. (55) (with (56) in the case of k < ∞) come from those of steady-state and periodic solutions of the partial differential Eq. (57) (with (58) in the case of s0 < ∞), where only bounded solutions with respect to the spatial variables are concerned. It should be pointed out that the definition of a fixed point for Eq. (55) is different from that given in [Chen et al., 2005, Definition 3.2]. More specifically, it is defined in [Chen et al., 2005, Definition 3.2] that x ∈ R is a fixed point of Eq. (55) if and only if x satisfies x = f (x, x), where f is independent of the lattice point m. Therefore, all the components of this kind of fixed points are equal. This type of fixed points is special (although important) in the present definition.

Chaotification of Discrete Dynamical Systems in Banach Spaces

Now, we are ready to provide the following definition of chaos for Eq. (55) (with (56) in the case of k < ∞). Definition 5.2. Equation (55) (with (56) in the

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case of k < ∞) is chaotic in the sense of Devaney (or Li–Yorke) on V ⊂ Yk+1 if its induced system (62) is chaotic in the sense of Devaney (or Li–Yorke) on V ⊂ Yk+1 .

For simplicity, we assume that f in (55) is independent of the lattice point m, i.e. f (m, x, y) ≡ f (x, y), in the following discussion.

5.2. Stability and expansion of fixed points Suppose that f is differentiable at (0, 0) and f (0, 0) = 0. Then {x∗ (0, m) = 0 : 0 ≤ m k} is a fixed point of Eq. (55) (with (56) in the case of k < ∞), x∗ := 0 ∈ Yk+1 is a fixed point of the induced system (62), and F is differentiable at x∗ . Some relationship of stability of fixed points of Eq. (55) and its induced system (62) was discussed in [Chen et al., 2005, Theorems 4.1 and 4.2] and an exponential stablity of the fixed points of system (62) was concluded [Chen et al., 2005, Corollary 4.1] in the case of k = ∞. Since the stability and expansion of a map at a fixed point has a close relationship with the spectrum of its derivative operator when the map is differentiable at the fixed point, we now discuss the stability and expansion of this fixed point by applying the spectral theory for the two cases of k = ∞ and k < ∞, respectively. Let σ(DF (0)) denote the spectral set of DF (0) and r(σ(DF (0))) denote its spectral radius. In addition, let fx (x, y) and fy (x, y) denote the first partial derivatives of f with respect to the first and the second variables at the point (x, y), respectively. Further, denote a := fx (0, 0) and b := fy (0, 0). Case I : k < ∞. Since



a  0  DF (0) =  ... b

b a ... 0

0 b ... 0

... ... ... ...

 0 0   ,  ... a (k+1)×(k+1)

by a direct calculation, we can find that     2πj i :0≤j≤k , σ(DF (0)) = a + b exp k+1

2631

which implies that r(σ(DF (0)))        2πj   = max a + exp i b : 0 ≤ j ≤ k .   k+1 Therefore, we have the following result, which can be proved by [Zeidler, 1986, Theorem 4.C; Robinson, 1999, Theorem 10.14; Shi & Chen, 2004b, Theorem 4.3]. Theorem 5.1. Assume that k < ∞.

(i) If f is differentiable at (0, 0) and |fx (0, 0) + exp((2πj/(k + 1))i)fy (0, 0)| < 1 for 0 ≤ j ≤ k, then the fixed point {x∗ (0, m) = 0 : 0 ≤ m ≤ k} of Eq. (55) with (56) is asymptotically stable. (ii) If f is continuously differentiable in a neighborhood of (0, 0) and       2πj   i fy (0, 0) > 1 (66) fx (0, 0) + exp   k+1 for some j, 0 ≤ j ≤ k, then the fixed point {x∗ (0, m) = 0 : 0 ≤ m ≤ k} of Eq. (55) with (56) is unstable. (iii) If f is continuously differentiable in a neighborhood of (0, 0) and (66) holds for all j, 0 ≤ j ≤ k, then the fixed point {x∗ (0, m) = 0 : 0 ≤ m ≤ k} of Eq. (55) with (56) is a regular expanding fixed point in some norm. The following is a direct consequence of Theorem 5.1. Corollary 5.1. Assume that k < ∞.

(i) If f is differentiable at (0, 0) and |fx (0, 0)| + |fy (0, 0)| < 1,

(67)

then the fixed point {x∗ (n, m) = 0 : 0 ≤ m ≤ k} of Eq. (55) with (56) is asymptotically stable. (ii) If f is continuously differentiable in a neighborhood of (0, 0) and satisfies | |fx (0, 0)| − |fy (0, 0)| | > 1, then the fixed point {x∗ (0, m) = 0 : 0 ≤ m ≤ k} of Eq. (55) with (56) is a regular expanding fixed point in some norm. Case II : k = ∞. Since

DF (0)z = (az(0) + bz(1), az(1) + bz(2), . . .)T , (68)

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Y. Shi et al.

where z = {z(j)}∞ j=0 ∈ Y∞ , it is easy to verify that σ(DF (0)) = {λ ∈ C : |λ − a| ≤ |b|}, which is composed of eigenvalues, with r(σ(DF (0))) = |a| + |b|.

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Theorem 5.2. Assume that k = ∞.

(i) If f is differentiable at (0, 0) and (67) holds, then the fixed point {x∗ (0, m) = 0 : m ≥ 0} of Eq. (55) is asymptotically stable. (ii) If f is continuously differentiable in a neighborhood of (0, 0) and |fx (0, 0)| − |fy (0, 0)| > 1, then the fixed point {x∗ (0, m) = 0 : m ≥ 0} of Eq. (55) is a regular expanding fixed point.

only consider the controller (60) with the special function g(x) = sawε (x). Other controllers can be similarly studied. Suppose that f is continuously differentiable in [−r, r]2 for some r > 0. Denote L := max{|fx (x, y)| + |fy (x, y)| : x, y ∈ [−r, r]}. (69) Then, F is continuously differentiable in [−r, r]k+1 and DF (x)z = (fx (x(0), x(1))z(0) + fy (x(0), x(1))z(1), fx (x(1), x(2))z(1) + fy (x(1), x(2))z(2), . . .)T ,

The result (i) can be directly derived by using Theorem 4.C in [Zeidler, 1986]. We now prove result (ii) by applying Lemma 2.3 given in [Shi & Chen, 2004b]. For any given z = {z(j)}∞ j=0 ∈ Y∞ with z∞ = 1, it follows from (68) that

where x = {x(j)}kj=0 ∈ [−r, r]k+1 and z = {z(j)}kj=0 ∈ Yk+1 . This, together with (69), implies that

DF (0)z = sup{|az(j) + bz(j + 1)| : j ≥ 0}.

The following result is a direct consequence of Corollary 4.1.

Proof.

From z∞ = 1, either there exists a subsequence jn → ∞ as n → ∞ such that z(jn ) → α and z(jn + 1) → β as n → ∞ with |α| = 1 and |β| ≤ 1, or there exists a non-negative integer j0 such that |z(j0 )| = 1. In the second case, set α = z(j0 ) and β = z(j0 + 1). So, we have DF (0)z ≥ |aα + bβ| ≥ |a| − |b|, DF (0)0 ≥ |a| − |b| > 1. Further, it can be easily shown that DF (0) is invertible and its inverse, given by  ∞  (−1)j (a−1 b)j z(j), (DF (0))−1 z = a−1 ∞  (−1)j (a−1 b)j z(j + 1), . . .

x ∈ [−r, r]k+1 .

Theorem 5.3. Assume that f is continuously differentiable in [−r, r]2 for some r > 0 and f (0, 0) = 0. Then, for each constant µ satisfying   5 −1 µ > µ0 = max r ε, 5(L + 1) , 2

there exists a Cantor set Λ ⊂ B 5 µ−1 ε (0) ⊂ Yk+1 2 such that the controlled system

which implies that

j=0

DF (x) ≤ L,

T

x(n + 1, m) = f (x(n, m), x(n, m + 1)) + sawε (µx(n, m)), 0 ≤ m k (with (56) in the case of k < ∞) is chaotic on Λ in the sense of both Devaney and Li–Yorke, and has a dense orbit in Λ for any given ε > 0, where L is defined in (69). Remark 5.3. It is very interesting that the constant

,

µ0 is independent of the system size k.

j=0

is a bounded linear operator by the assumption of |a| − |b| > 1. Hence, DF (0) is an invertible linear map. By Lemma 2.3 in [Shi & Chen, 2004b], the point x∗ = 0 is a regular expanding fixed point of system (62). This completes the proof.


5.3. Chaotification for a partial difference equation In this subsection, the objective is to illustrate how to apply our chaotification schemes to the partial difference equation (55). Here, as an example, we

5.4. Examples and simulations Consider a special case of Eq. (55), given below: x(n + 1, m) − dx(n, m + 1) = cx(n, m)(1 − x(n, m)),

n ≥ 0,

0 ≤ m k, (70)

which is called a 2-D discrete logistic system, where c and d are real constants. This is a discretization of the nonlinear advection equation ˆ s (t, s) = cˆu(1 − u), ut (t, s) − du where cˆ and dˆ are real constants.

Chaotification of Discrete Dynamical Systems in Banach Spaces

Assume that the parameters in system (70) are inaccessible and what we can do is to measure its state. Then, in this case, feedback control design is more practical. It is evident that f (x, y) = cx(1 − x) + dy is continuously differentiable in R2 , f (0, 0) = 0, fx (x, y) = c(1 − 2x), and fy (x, y) = d. So, we have

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|fx (x, y)| + |fy (x, y)| ≤ 3|c| + |d|,

∀ x, y ∈ [−1, 1].

By Corollary 5.1 and Theorem 5.2, {x∗ (0, m) = 0 : 0 ≤ m k} is an asymptotically stable fixed point of Eq. (70) (with (56) in the case of k < ∞) if |c| + |d| < 1. On the other hand, by Theorem 5.3, for each constant µ satisfying   5 µ > µ0 := max ε, 5(3|c| + |d| + 1) , 2 there exists a Cantor set Λ ⊂ B 5 µ−1 ε (0) ⊂ Yk such 2 that the controlled system x(n + 1, m) = cx(n, m)(1 − x(n, m)) + dx(n, m + 1) + sawε (µx(n, m)), n ≥ 0, 0 ≤ m k (71) (with (56) in the case of k < ∞) is chaotic on Λ in the sense of both Devaney and Li–Yorke, and has a dense orbit in Λ for any given ε > 0. To end this section, we present some simulation results to verify the theoretical predictions. Simulation 1. Take c = d = 8−1 and k = 2. In this

-1.5

-1

-0.5

0

0.5

1

-1 1.5 -1.5

0 -0.5

0.5

1

1.5

Fig. 1. 3-D computer simulation result without control, showing convergence of trajectories to the origin in the (x(·, 0), x(·, 1), x(·, 2)) phase space for the system parameters: c = d = 0.125, k = 2.

µ = 15 > µ0 = 15/2. In this case, system (71) with (56) can be reformulated into the following threedimensional discrete dynamical system:   x(n, 0)(1 − x(n, 0)) + x(n, 1)   xn+1 = 8−1  x(n, 1)(1 − x(n, 1)) + x(n, 2)  x(n, 2)(1 − x(n, 2)) + x(n, 0)   saw1 (µx(n, 0))   +  saw1 (µx(n, 1)) . saw1 (µx(n, 2)) Figure 2 shows the chaotic behavior of the controlled system (71) in a rectangle box, centered at the fixed point {x∗ (n, m) = 0 : 0 ≤ m ≤ 2} that is originally asymptotically stable.

case, Eq. (70) with (56) can be reformulated into the following three-dimensional discrete dynamical system:   x(n + 1, 0)   xn+1 =  x(n + 1, 1)  x(n + 1, 2)   x(n, 0)(1 − x(n, 0)) + x(n, 1)   = 8−1  x(n, 1)(1 − x(n, 1)) + x(n, 2) . x(n, 2)(1 − x(n, 2)) + x(n, 0) The simulation result is shown in Fig. 1, indicating that without control the fixed point {x∗ (n, m) = 0 : 0 ≤ m ≤ 2} of Eq. (70) with (56) is asymptotically stable in the ball B1 (0) of the three-dimensional phase space. Simulation 2. Take c = d = 8−1 , and k = 2.

In the controlled system (71), we choose ε = 1 and

Fig. 2. 3-D computer simulation result with control, showing chaos in the (x(·, 0), x(·, 1), x(·, 2)) phase space for the system parameters: c = d = 0.125, k = 2, and the control parameters:
= 1, µ = 15.

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Simulation 3. Take c = d = 8−1 , and k = 400. In

the controlled system (71), we choose ε = 1 and µ = 15 > µ0 = 15/2. Figure 3 shows the chaotic behavior of the controlled system (71) with (56) in the (n, m, x) space. Note that this cannot be depicted in the phase space since the dimension of the phase space in this case is 401. Finally, we choose ε = 0.01 but keep other parameter values unchanged. Figure 4 shows that the chaotic behavior of the controlled system (71) with (56) is similar to the case of = 1, but is restricted to a very thin rectangle box, whose thickness is only (0.01/1) × 100% = 1% of that for the case of = 1,

Fig. 3. 3-D computer simulation result with control, showing chaos in the (n, m, x) space for the system parameters: c = d = 0.125, k = 400, and the control parameters:
= 1, µ = 15.

Fig. 4. 3-D computer simulation result with control, showing chaos in the (n, m, x) space for the system parameters: c = d = 0.125, k = 400, and the control parameters:
= 0.01, µ = 15.

as expected on the basis of the theoretical results in Theorem 5.3.

6. Conclusions Two simple feedback controllers have been designed for chaotification of discrete dynamical systems in general Banach spaces. All the controlled systems are proved to be chaotic in the sense of both Devaney and Li–Yorke. In particular, the controllers can be designed as a generalization of the classical sawtooth and sinusoidal functions in some special Banach spaces. As a consequence, the finitedimensional controlled system discussed in [Wang & Chen, 2000] has been proved to be chaotic not only in the sense of Li–Yorke but also in the sense of Devaney. In addition, the Chen–Lai anti-control algorithm via feedback control with mod-operation in a finite-dimensional real space developed in [Chen & Lai, 1998] has been further extended to a certain infinite-dimensional Banach space. Consequently, the controlled system discussed in [Chen & Lai, 1998] has been shown to be chaotic in the sense of Devaney, Li–Yorke and Wiggins. Moreover, it is not required that the map corresponding to the originally given system has a fixed point when the state space is Yk (defined in Sec. 4) and the controller is designed as in (4). This is different from and actually weaker than the conditions required in many existing papers. The amplitude of the feedback controller designed as in (3) can be arbitrarily small when the state space is Yk just like the finitedimensional case, which is very practical in engineering applications. A criterion of chaos in Banach spaces has been established. This criterion extends and improves the Marotto theorem. For illustration of how to use the chaotification schemes established in the paper, chaotification of a class of first-order partial difference equations has been studied. In view of the time evolution, the class of partial difference equations has first been reformulated into a specific class of finite-dimensional and infinite-dimensional discrete dynamical systems for the two cases: the system size is finite and infinite, respectively. Subsequently, some sufficient conditions for stability and expansion of fixed points of some first-order partial difference equations have been given and a chaotification scheme for partial difference equations has been established. Numerical simulations have been finally provided for partial difference equations with a finite system size, which verify the theoretical results obtained in the paper.

Chaotification of Discrete Dynamical Systems in Banach Spaces

Acknowledgments This research was supported by the NNSF of China (Grant 10471077), the NSERC of Canada (Grant R2686A02), and the Shandong Research Funds for Young Scientists (Grant 03BS094). The first author would like to thank Professor Xingfu Zou for some helpful discussions and providing some useful references.

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