Distribution of Maps with Transversal Homoclinic Orbits in a Continuous Map Space

Abstract

This paper is concerned with distribution of maps with transversal homoclinic orbits in a continuous map space, which consists of continuous maps defined in a closed and bounded set of a Banach space. By the transversal homoclinic theorem, it is shown that the map space contains a dense set of maps that have transversal homoclinic orbits and are chaotic in the sense of both Li-Yorke and Devaney with positive topological entropy.

1. Introduction

Distribution of a set of maps with some dynamical properties in some continuous map space is a very interesting topic. In the 1960s, Smale [1] studied density of hyperbolicity. Some scholars believed that hyperbolic systems are dense in spaces of all dimensions, but it was shown that the conjecture is false in the late 1960s for diffeomorphisms on manifolds of dimension ≥2. The problem whether hyperbolic systems are dense in the one-dimension case was studied by many scholars. It was solved in the 𝐶¹ topology by Jakobson [2], a partial solution was given in the 𝐶² topology by Blokh and Misiurewicz [3], and 𝐶² density was finally proved by Shen [4]. In 2007, Kozlovki et al. got the result in 𝐶^𝑘 topology; that is, hyperbolic (i.e., Axiom A) maps are dense in the space of 𝐶^𝑘 maps defined in a compact interval or circle, 𝑘=1,2,…,∞,𝜔 [5]. At the same time, some other scholars considered the distribution of hyperbolic diffeomorphisms in Diff(𝑀), where 𝑀 is a manifold. Just like the work of Smale, Palis [6, 7] gave the following conjecture: (1) any 𝑓∈Diff(𝑀) can be approximated by a hyperbolic diffeomorphism or by a diffeomorphism exhibiting a homoclinic bifurcation (tangency or cycle), (2) any diffeomorphism can be 𝐶^𝑟 approximated by a Morse-Smale one or by one exhibiting transversal homoclinic orbit. Later, it was shown that the conjecture (1) holds for 𝐶¹ diffeomorphisms of surfaces [8]. And some good results have been obtained, such as any diffeomorphism can be 𝐶¹ approximated by a Morse-Smale one or by one displaying a transversal homoclinic orbit [9], any diffeomorphism can be 𝐶¹ approximated by one that exhibits either a homoclinic tangency or a heterodimensional cycle or by one that is essentially hyperbolic [10].

In 1963, Smale gave the well-known Smale-Birkhoff homoclinic theorem for diffeomorphisms [11], from which one can easily know that if a diffeomorphism 𝐹 on a manifold 𝑀 has a transversal homoclinic orbit, then there exists an integer 𝑘>0 such that 𝐹^𝑘 is chaotic in the sense of both Li-Yorke and Devaney. Later, in 1986, Hale and Lin introduced a generalized definition of transversal homoclinic orbit for continuous maps and got the generalized Smale-Birkhoff homoclinic theorem, that is, a transversal homoclinic orbit implies chaos in the sense of both Li-Yorke and Devaney for continuous maps in Banach spaces [12]. In the meanwhile, some scholars studied the density of maps which are chaotic in the sense of Li-Yorke or Devaney. In particular, some results have been obtained in one-dimensional maps (cf. [13–15]).

Since 2004, Shi, Chen, and Yu extended the result about turbulent maps for one-dimensional maps introduced by Block and Coppel in 1992 [16] to maps in metric spaces. This map is termed by a new terminology: coupled-expanding map. Under certain conditions, the authors showed that a strictly coupled-expanding map is chaotic [17, 18]. Applying this coupled-expansion theory, they extended the criterion of chaos induced by snap-back repellers for finite-dimensional maps, introduced by Marotto in 1978 [19], to maps in metric spaces [17, 20, 21]. Recently, we studied the distribution of chaotic maps in continuous map spaces, in which maps are defined in general Banach spaces and finite-dimensional normed spaces, and obtained that the following several types of chaotic maps are dense in some continuous map spaces: (1) maps that are chaotic in the sense of both Li-Yorke and Devaney; (2) maps that are strictly coupled-expanding; (3) maps that have nondegenerated and regular snap-back repeller; (4) maps that have nondegenerate and regular homoclinic orbit to a repeller [22].

In the present paper, we will construct a set of continuous chaotic maps with generalized transversal homoclinic orbits, and show that the set is dense in the continuous map space. The method used in the present paper is motivated by the idea in [22].

This paper is organized as follows. In Section 2, we first introduce some notations and basic concepts including the Li-Yorke and Devaney chaos, hyperbolic fixed point, and transversal homoclinic orbit, and then give a useful lemma. In Section 3, we pay our attention to distribution of maps with generalized transversal homoclinic orbits in a continuous map space, in which every map is defined in a closed, bounded, and convex set or a closed bounded set in a general Banach space. Constructing a continuous map with a generalized transversal homoclinic orbit, we simultaneously show the density of maps which are chaotic in the sense of both Li-Yorke and Devaney in the map space.

2. Preliminaries

In this section, some notations and basic concepts are first introduced, including Li-Yorke and Devaney chaos, hyperbolic fixed point, and transversal homoclinic orbit. And then a useful lemma is given.

First, we give two definitions of chaos which will be used in the paper.

Definition 2.1 (see [23]). Let (𝑋,𝑑) be a metric space, 𝑓∶𝑋→𝑋 a map, and 𝑆 a set of 𝑋 with at least two points. Then, 𝑆 is called a scrambled set of 𝑓 if, for any two distinct points 𝑥,𝑦∈𝑆, liminf_𝑛→∞𝑑𝑓^𝑛(𝑥),𝑓^𝑛𝑦=0,limsup_𝑛→∞𝑑𝑓^𝑛(𝑥),𝑓^𝑛𝑦>0. (2.1) The map 𝑓 is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scrambled set 𝑆 of 𝑓.

Definition 2.2 (see [24]). Let (𝑋,𝑑) be a metric space. A map 𝑓∶𝑉⊂𝑋→𝑉 is said to be chaotic on 𝑉 in the sense of Devaney if (i) the periodic points of 𝑓 in 𝑉 are dense in 𝑉; (ii)𝑓 is topologically transitive in 𝑉; (iii)𝑓 has sensitive dependence on initial conditions in 𝑉. Now, we give the definition of hyperbolic fixed point.

Definition 2.3 (see [25]). Let 𝑋 be a Banach space, 𝑈⊂𝑋 be a set, and 𝐹∶𝑈⊂𝑋→𝑋 be a map. Assume that 𝑧 is a fixed point of 𝐹, 𝐹 is continuously differentiable in some neighborhood of 𝑧, and denote 𝐷𝐹(𝑧) the Fréchet derivative of 𝐹 at 𝑧. The fixed point 𝑧 is called hyperbolic if 𝜎(𝐷𝐹(𝑧))∩{𝜆∈𝐶∶|𝜆|=1}=∅, where 𝜎(𝐴) denotes the spectrum of a linear operator 𝐴. The hyperbolic fixed point 𝑧 is called a saddle point if 𝜎(𝐷𝐹(𝑧))∩{𝜆∈𝐶∶|𝜆|>1}≠∅ and 𝜎(𝐷𝐹(𝑧))∩{𝜆∈𝐶∶|𝜆|<1}≠∅.
If 𝐹 is invertible, then for any 𝑝₀∈𝑋, the set 𝑂⁺_𝐹(𝑝₀)∶={𝐹^𝑘(𝑝₀)∶𝑘≥0} is said to be the forward orbit of 𝐹 from 𝑝₀, the set 𝑂⁻_𝐹(𝑝₀)∶={𝐹^𝑘(𝑝₀)∶𝑘≤0} is said to be the backward orbit of 𝐹 from 𝑝₀. Since it is not required that 𝐹 is invertible in this paper, a backward orbit of 𝑝₀ is a set 𝑂⁻_𝐹(𝑝₀)={𝑝_𝑗∶𝑗≤0} with 𝑝_𝑗+1=𝐹(𝑝_𝑗),𝑗≤−1, which may not exist, or exist but may not be unique. A whole orbit of 𝑝₀ is the union 𝑂⁺_𝐹(𝑝₀)∪𝑂⁻_𝐹(𝑝₀), denoted by 𝑂_𝐹(𝑝₀) in the case that it has a backward orbit 𝑂⁻_𝐹(𝑝₀). The stable set 𝑊^𝑠(𝑧,𝐹) and the unstable set 𝑊^𝑢(𝑧,𝐹) of a hyperbolic fixed point 𝑧 of 𝐹 are defined by 𝑊^𝑠(𝑧,𝐹)∶=𝑝₀∈𝑋:theforwardorbit𝑝_𝑗^+∞_𝑗=0of𝐹from𝑝₀𝑝suchthat,_𝑗,𝑊⟶𝑧as𝑗⟶+∞_𝑢(𝑧,𝐹)∶=𝑝₀∈𝑋:thereexistsabackwardorbit𝑝_−𝑗^+∞_𝑗=0of𝐹from𝑝₀suchthat𝑝_−𝑗⟶𝑧as𝑗⟶+∞, (2.2) respectively. The local stable and unstable sets are defined by 𝑊^𝑠_loc(𝑧,𝑈,𝐹)∶=𝑊^𝑠(𝑧,𝐹)∩𝑈,𝑊^𝑢_loc(𝑧,𝑈,𝐹)∶=𝑊^𝑢(𝑧,𝐹)∩𝑈, (2.3) respectively, where 𝑈 is some neighborhood of 𝑧. If 𝑈=𝐵(𝑧,𝑟) or 𝐵(𝑧,𝑟) for some 𝑟>0, then the corresponding local stable and unstable sets of 𝐹 are denoted by 𝑊^𝑠_loc(𝑧,𝑟,𝐹) and 𝑊^𝑢_loc(𝑧,𝑟,𝐹), respectively. By the Stable Manifold Theorem [26], if 𝐹 is continuously differentiable in some neighborhood of a saddle point 𝑧, then there exists a neighborhood 𝑈 of 𝑧 such that the corresponding local stable and unstable set of 𝑧 are submanifolds of 𝑋, respectively.
In the following, we first give the definition that two manifolds intersect transversally and then give the definition of transversal homoclinic orbit for continuous maps.

Definition 2.4 (see [25]). Two submanifolds 𝑉 and 𝑊 in a manifold 𝑀 are transverse (in 𝑀) provided for any point 𝑞∈𝑉∩𝑊, we have that 𝑇_𝑞𝑉+𝑇_𝑞𝑊=𝑇_𝑞𝑀, where 𝑇_𝑞𝑉 and 𝑇_𝑞𝑊 denote the tangent spaces of 𝑉 and 𝑊 at 𝑞, respectively, and “+” means the sum of the two subspaces (this allows for the possibility that 𝑉∩𝑊=∅).

Remark 2.5. If 𝑀=𝐑^𝑛, then 𝑉 and 𝑊 in 𝑀 are transverse (in 𝑀) provided for any point 𝑞∈𝑉∩𝑊, we have that 𝑇_𝑞𝑉+𝑇_𝑞𝑊=𝐑^𝑛. Obviously, if dim𝑇_𝑞𝑉+dim𝑇_𝑞𝑊=𝑛, then the sum of the two subspaces 𝑇_𝑞𝑉 and 𝑇_𝑞𝑊 is a direct one, denoted by ⊕.

Definition 2.6 (see [12]). Let 𝑋 be a Banach space, 𝐹∶𝑋→𝑋 be a map, and 𝑧∈𝑋 be a saddle point of 𝐹. (i) An orbit 𝑂_𝐹(𝑝₀)={𝑝_𝑗}^+∞_𝑗=−∞ is said to be a homoclinic orbit (asymptotic) to 𝑧 if 𝑝₀≠𝑧 and lim_𝑗→+∞𝑝_𝑗=lim_{𝑗→−∞}𝑝_𝑗=𝑧. (ii)A homoclinic orbit 𝑂_𝐹(𝑝₀)={𝑝_𝑗}^+∞_𝑗=−∞ to 𝑧 is said to be transversal if there exists an open neighborhood 𝑈 of 𝑧 such that 𝑝_−𝑖∈𝑊^𝑢_loc(𝑧,𝑈,𝐹) and 𝑝_𝑗∈𝑊^𝑠_loc(𝑧,𝑈,𝐹) for any sufficiently large integers 𝑖,𝑗≥0, and 𝐹^𝑖+𝑗 sends a disc in 𝑊^𝑢_loc(𝑧,𝑈,𝐹) containing 𝑝_−𝑖 diffeomorphically onto its image that is transversal to 𝑊^𝑠_loc(𝑧,𝑈,𝐹) at 𝑝_𝑗.
The following lemma is taken from Theorems  3.1, and 5.2, Corollary  6.1, and the result in Section  7 of [12].

Lemma 2.7. Let 𝐹∶𝑍→𝑍 be a map, where 𝑍=𝑋×𝑌, and 𝑋 and 𝑌 are Banach spaces. (i) Let 𝐴 and 𝐵 be linear continuous maps in 𝑋 and 𝑌, respectively, with the absolute values of the spectrum of 𝐴 less than 1 and the absolute values of the spectrum of 𝐵 larger than 1, and ‖𝐴‖,‖𝐵⁻¹‖≤𝜆₀ for some constant 0<𝜆₀<1.(ii)Assume that 𝑈 is an open neighborhood of 0 in 𝑍 and 𝑓₁∶𝑈→𝑋,𝑓₂∶𝑈→𝑌 are 𝐶^𝑘(𝑘≥1) maps with 𝑓_𝑖(0)=0,𝐷𝑓_𝑖(0)=0,𝑖=1,2. Further, assume that 𝐷𝑓₁,𝐷𝑓₂ are uniformly continuous in 𝑈, and satisfies that for some constants 0<𝜃<1−𝜆₀ and 𝛾>0, ‖𝐷𝑓₁(𝑥,𝑦)‖,‖𝐷𝑓₂(𝑥,𝑦)‖<𝜃 for all (𝑥,𝑦)∈𝐵(0,𝛾)⊂𝑈. (iii)Let 𝐹∶𝑈→𝑍 be of the following form: 𝐹𝑥,𝑦=𝐴𝑥+𝑓₁𝑥,𝑦,𝐵𝑦+𝑓₂𝑥,𝑦, (2.4) and have the local stable and unstable manifolds 𝑊^𝑠_loc(0,𝑈,𝐹)={(𝑥,𝑦)∣(𝑥,𝑦)∈𝑈,𝑦=0} and 𝑊^𝑢_loc(0,𝑈,𝐹)={(𝑥,𝑦)∣(𝑥,𝑦)∈𝑈,𝑥=0}≠{0}.(iv)Assume that 𝑂(𝑝₀)={𝑝_𝑖}^∞_𝑖=−∞ is a homoclinic orbit of 𝐹 with 𝑝_𝑖→0 as 𝑖→±∞, and there exists an integer 𝑁>0 such that 𝑝_−𝑁∈𝑊^𝑢_loc(0,𝑈,𝐹), 𝑝_𝑁∈𝑊^𝑠_loc(0,𝑈,𝐹), and(iv₁)𝐹^2𝑁 sends a disc 𝑂₁∩𝑊^𝑢_loc(0,𝑈,𝐹) centered at 𝑝_−𝑁 diffeomorphically onto 𝑂₂=𝐹^2𝑁(𝑂₁) containing 𝑝_𝑁;(iv₂)𝑂₂ intersects 𝑊^𝑠_loc(0,𝑈,𝐹) transversally at 𝑝_𝑁. Then 𝑂(𝑝₀) is a transversal homoclinic orbit of 𝐹. Furthermore, there exists an integer 𝑘>0 and a subset Λ in a neighborhood of 𝑂(𝑝₀) such that 𝐹^𝑘 on Λ is topologically conjugate to the full shift map on the doubly infinite sequence of two symbols. Consequently, 𝐹 is chaotic in the sense of both Li-Yorke and Devaney, and its topological entropy ℎ(𝐹)≥log2/𝑘.

Note that it is not required that 𝐹 is a diffeomorphism, even 𝐹 may not be continuous on the whole space 𝑍 in Lemma 2.7.

3. Distribution of Maps with Transversal Homoclinic Orbits

In this section, we first consider distribution of maps with transversal homoclinic orbits in a continuous self-map space, which consists of continuous maps that transform a closed, bounded, and convex set in a Banach space into itself. At the end of this section, we discuss distribution of chaotic maps in a continuous map space, in which a map may not transform its domain into itself.

Without special illustration, we always assume that (𝑋,‖⋅‖_𝑋) and (𝑌,‖⋅‖_𝑌) are Banach spaces, and 𝐷_𝑋 and 𝐷_𝑌 are bounded, convex, and open sets in 𝑋 and 𝑌, respectively. It is evident that 𝐷=𝐷_𝑋×𝐷_𝑌 is a bounded, convex, and open set in 𝑍=𝑋×𝑌, where the norm ‖⋅‖ on 𝑍 is defined by ‖(𝑥,𝑦)‖=max{‖𝑥‖_𝑋,‖𝑦‖_𝑌}, for any (𝑥,𝑦)∈𝑍, where 𝑥∈𝑋,𝑦∈𝑌. Introduce the following map space: 𝐶₀𝐷,𝐷∶=𝑓∶𝐷⟶𝐷iscontinuousandhasafixedpointin𝐷. (3.1) For any 𝑓∈𝐶₀(𝐷,𝐷), let 𝑓∶=sup𝑓(𝑥)∶𝑥∈𝐷, (3.2) and for any 𝑓,𝑔∈𝐶₀(𝐷,𝐷), let 𝑑𝑓,𝑔∶=𝑓−𝑔. (3.3) Then (𝐶₀(𝐷,𝐷),𝑑) is a metric space. It may not be complete because a limit of a sequence of maps in 𝐶₀(𝐷,𝐷) is continuous and bounded, but may not have a fixed point in 𝐷. But in the special case that 𝑍 is finite-dimensional, (𝐶₀(𝐷,𝐷),𝑑) is a complete metric space by the Schauder fixed point theorem.

In this section, we first study distribution of maps with transversal homoclinic orbits in 𝐶₀(𝐷,𝐷).

For convenience, by (𝑥,𝑦)∈𝑍 denote 𝑥∈𝑋 and 𝑦∈𝑌, by Fix(𝑓) denote the set of all the fixed points of 𝑓. For (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝑍 with (𝑥₁,𝑦₁)≠(𝑥₂,𝑦₂), by 𝑙((𝑥₁,𝑦₁),(𝑥₂,𝑦₂)) denote the straight half-line connecting (𝑥₁,𝑦₁) and (𝑥₂,𝑦₂): 𝑙𝑥₁,𝑦₁,𝑥₂,𝑦₂∶=𝑢=𝑥₁,𝑦₁+𝑡𝑥₂,𝑦₂−𝑥₁,𝑦₁∶𝑡≥0. (3.4)

Lemma 3.1 (see [22, Lemma  3.1]). For every map 𝑓∈𝐶₀(𝐷,𝐷) and any 𝜀>0, there exists a map 𝑔∈𝐶₀(𝐷,𝐷) such that 𝑑(𝑓,𝑔)<𝜀, Fix(𝑔)∩𝐷≠∅, and 𝑔 is continuously differentiable in some neighborhood of some point 𝑥^∗∈Fix(𝑔)∩𝐷.

Lemma 3.2. For every map 𝑓∈𝐶₀(𝐷,𝐷) and every 𝜀>0, there exists a map 𝐹∈𝐶₀(𝐷,𝐷) with 𝑑(𝑓,𝐹)<𝜀 such that 𝐹 has a transversal homoclinic orbit in 𝐷.

Proof. Fix any 𝑓∈𝐶₀(𝐷,𝐷). By Lemma 3.1, it suffices to consider the case that 𝑓 has a fixed point 𝑧=(𝑧₁,𝑧₂)∈𝐷 with 𝑧₁∈𝑋,𝑧₂∈𝑌, and is continuously differentiable in some neighborhood of 𝑧.
For any 𝜀>0, there exists a positive constant 𝑟₀<𝜀/4 with 𝐵(𝑧,𝑟₀)⊂𝐷 such that 𝑓<𝜀𝑥,𝑦−𝑧4,𝑥,𝑦∈𝐵𝑧,𝑟₀. (3.5)
Let 𝑟=𝑟₀/4, take two constants 𝑎,𝑏 with 𝑟<𝑎<𝑏<𝑟₀/3 and take two points 𝑝₁=(𝑥₀,𝑧₂)∈𝐵(𝑧,𝑟),𝑝₀=(𝑧₁,𝑦₀)∈𝐵(𝑧,𝑏)⧵𝐵(𝑧,𝑎), where 𝑥₀∈𝑋,𝑦₀∈𝑌.
The rest of the proof is divided into three steps.
Step 1. Construct a map 𝐹 that is locally controlled near 𝑧.
Define𝐹=𝜆𝑥,𝑦𝑥−𝑧₁+𝑧₁,𝜇𝑦−𝑧₂+𝑧₂,𝑥,𝑦∈𝐵(),𝑥𝑧,𝑟𝑥+₀−𝑧₁,𝑦−𝑦₀−𝑧₂,𝑥,𝑦∈𝐵(𝑧,𝑏)⧵𝐵(𝑧,𝑎), (3.6) where 𝑥∈𝑋, 𝑦∈𝑌, and 𝜆 and 𝜇 are real parameters and satisfy |𝜆|𝑏<1,𝑟<𝜇<𝑟₀𝑟. (3.7) Note that |𝜇|>1 since 𝑟<𝑏.
For any (𝑥,𝑦)∈𝐵(𝑧,𝑎)⧵𝐵(𝑧,𝑟), 𝐹(𝑥,𝑦) is defined as follows. Let (𝑥^′₁,𝑦^′₁) and (𝑥^′₂,𝑦^′₂) be the intersection points of the straight line 𝑙(𝑧,(𝑥,𝑦)) with 𝜕𝐵(𝑧,𝑟) and 𝜕𝐵(𝑧,𝑎), respectively, (see Figure 1). Set 𝐹𝑥,𝑦=𝐹𝑥^′₁,𝑦^′₁+𝑡𝑥,𝑦𝐹𝑥^′₂,𝑦^′₂−𝐹𝑥^′₁,𝑦^′₁, (3.8) where 𝑡(𝑥,𝑦)∈(0,1) is determined as follows; 𝑥,𝑦=𝑥^′₁,𝑦^′₁+𝑡𝑥,𝑦𝑥^′₂,𝑦^′₂−𝑥^′₁,𝑦^′₁. (3.9) It is noted that when (𝑥,𝑦) continuously varies in 𝐵(𝑧,𝑎)⧵𝐵(𝑧,𝑟), so do the intersection points (𝑥^′₁,𝑦^′₁) and (𝑥^′₂,𝑦^′₂). Consequently, 𝑡(𝑥,𝑦) and then 𝐹(𝑥,𝑦) are continuous in 𝐵(𝑧,𝑎)⧵𝐵(𝑧,𝑟).
Next, define 𝐹(𝑥,𝑦)=𝑓(𝑥,𝑦) for (𝑥,𝑦)∈𝐷⧵𝐵(𝑧,𝑟₀). Finally, for any (𝑥,𝑦)∈𝐵(𝑧,𝑟₀)⧵𝐵(𝑧,𝑏), suppose that (𝑥₁^′′,𝑦₁^′′) and (𝑥₂^′′,𝑦₂^′′) are the intersection points of the straight line 𝑙(𝑧,(𝑥,𝑦)) with 𝜕𝐵(𝑧,𝑏) and 𝜕𝐵(𝑧,𝑟₀), respectively. Define 𝐹(𝑥,𝑦) as that in (3.8), where 𝑡(𝑥,𝑦) is determined by (3.9) with (𝑥^′₁,𝑦^′₁) and (𝑥^′₂,𝑦^′₂) replaced by (𝑥₁^′′,𝑦₁^′′) and (𝑥₂^′′,𝑦₂^′′), respectively. Hence, 𝐹(𝑥,𝑦) is continuous in 𝐵(𝑧,𝑟₀)⧵𝐵(𝑧,𝑏).
Obviously, 𝑧 is a saddle fixed point of 𝐹, and 𝑥,𝑦∶𝑥=𝑧₁,𝑦−𝑧_2𝑌<𝑏⊂𝑊^𝑢_loc(𝑧,𝑏,𝐹),𝑥,𝑦∶𝑦=𝑧₂,𝑥−𝑧_1𝑋<𝑟⊂𝑊^𝑠_loc(𝑧,𝑟,𝐹). (3.10)Step 2. 𝐹∈𝐶₀(𝐷,𝐷) and satisfies that 𝑑(𝑓,𝐹)<𝜀.
From the definition of 𝐹, it is easy to know that 𝐹 is continuous on 𝐷 and has a fixed point 𝑧∈𝐷, that is, 𝐹∈𝐶₀(𝐷,𝐷).
Next, we will prove that 𝑑(𝑓,𝐹)<𝜀. For (𝑥,𝑦)∈𝐷⧵𝐵(𝑧,𝑟₀), ‖𝐹(𝑥,𝑦)−𝑓(𝑥,𝑦)‖=0<𝜀. For (𝑥,𝑦)∈𝐵(𝑧,𝑟), it follows from (3.5) and (3.6) that 𝐹𝑥,𝑦−𝑓𝑥,𝑦≤𝐹𝑥,𝑦−𝑧+𝑓𝑥,𝑦−𝑧≤𝑟₀+𝜀4<𝜀. (3.11) For (𝑥,𝑦)∈𝐵(𝑧,𝑎)⧵𝐵(𝑧,𝑟), it follows from (3.5), (3.6), and (3.8) that 𝐹𝑥,𝑦−𝑓𝑥,𝑦=𝐹𝑥^′₁,𝑦^′₁+𝑡𝑥,𝑦𝐹𝑥^′₂,𝑦^′₂−𝐹𝑥^′₁,𝑦^′₁≤𝐹𝑥−𝑓𝑥,𝑦1−𝑡𝑥,𝑦^′₁,𝑦^′₁−𝑧+𝑡𝑥,𝑦𝐹𝑥^′₂,𝑦^′₂−𝑧+𝑓𝑥,𝑦−𝑧≤𝑟₀𝜀+2𝑏+4<𝜀. (3.12) For (𝑥,𝑦)∈𝐵(𝑧,𝑏)⧵𝐵(𝑧,𝑎), from (3.5) and (3.6), one has 𝐹≤𝐹+𝑓𝜀𝑥,𝑦−𝑓𝑥,𝑦𝑥,𝑦−𝑧𝑥,𝑦−𝑧<2𝑏+4<𝜀. (3.13) For (𝑥,𝑦)∈𝐵(𝑧,𝑟₀)⧵𝐵(𝑧,𝑏), from (3.5), (3.6), and (3.8), one has 𝐹𝑥,𝑦−𝑓𝑥,𝑦=𝐹𝑥₁^′′,𝑦₁^′′+𝑡𝑥,𝑦𝐹𝑥₂^′′,𝑦₂^′′−𝐹𝑥₁^′′,𝑦₁^′′≤𝐹𝑥−𝑓𝑥,𝑦1−𝑡𝑥,𝑦₁^′′,𝑦₁^′′−𝑧+𝑡𝑥,𝑦𝐹𝑥₂^′′,𝑦₂^′′−𝑧+𝑓𝑥,𝑦−𝑧≤𝑟₀+𝜀4+𝜀4<𝜀. (3.14) Therefore, from the above discussion, 𝑑(𝑓,𝐹)=‖𝐹−𝑓‖<𝜀.Step 3. 𝐹 has a transversal homoclinic orbit in 𝐷.
It follows from (3.6) that 𝐹(𝑝₋₁)=𝑝₀ and 𝐹(𝑝₀)=𝑝₁, where 𝑝₋₁=(𝑧₁,𝜇⁻¹(𝑦₀−𝑧₂)+𝑧₂)∈𝐵(𝑧,𝑟) by (3.7). So, by (3.10) one has that 𝑝₋₁∈𝑊^𝑢_loc(𝑧,𝑟,𝐹), 𝑝₁∈𝑊^𝑠_loc(𝑧,𝑟,𝐹). Hence, 𝑂(𝑝₀)={𝑝_𝑗}^∞_𝑗=−∞ satisfies that lim_𝑗→∞𝑝_𝑗=lim_{𝑗→−∞}𝑝_𝑗=𝑧. Thus, 𝑂(𝑝₀)⊂𝐷 and is a homoclinic orbit of 𝐹.
Set a positive constant 𝛿 satisfying 𝛿<min𝑏−𝑦₀−𝑧_2𝑌,𝑦₀−𝑧_2𝑌−𝑎,𝜇⁻¹𝑏,𝑟−𝜇⁻¹𝑏. (3.15) Then 0<𝛿<𝑟 by (3.7), and consequently it follows from (3.10) that the disc 𝑂₀∶=𝑥,𝑦∶𝑥=𝑧₁,𝑦−𝑦_0𝑌≤𝛿⊂𝐵(𝑧,𝑏)⧵𝐵(𝑧,𝑎)∩𝑊^𝑢_loc(𝑧,𝑏,𝐹). (3.16) Further, set the discs 𝑂₋₁∶=𝑥,𝑦∶𝑥=𝑧₁,𝑦−𝜇⁻¹𝑦₀−𝑧₂+𝑧_2𝑌≤𝜇⁻¹𝛿,𝑂₁∶=𝑥,𝑦∶𝑥=𝑥₀,𝑦−𝑧_2𝑌≤𝛿. (3.17) Then 𝑂₋₁⊂𝑊^𝑢_loc(𝑧,𝑟,𝐹), 𝑂₀=𝐹(𝑂₋₁), and 𝑂₁=𝐹(𝑂₀) (see Figure 2).
It is evident that 𝑂₁ intersects 𝑊^𝑠_loc(𝑧,𝑟,𝐹) transversally at point 𝑝₁. In addition, by the definition of 𝐹 in 𝐵(𝑧,𝑏), one can get that 𝐹²∶𝑂₋₁→𝑂₁ is a diffeomorphism. Therefore, 𝑂(𝑝₀) is a transversal homoclinic orbit asymptotic to 𝑧 of 𝐹 by Definition 2.6, where 𝑁=1.
The entire proof is complete.

Figure 1 

Figure 2 

Theorem 3.3. Let 𝑋 and 𝑌 be Banach spaces, 𝑍=𝑋×𝑌, and 𝐷 be a bounded, convex, and open set in 𝑍. Then, for every map 𝑓∈𝐶₀(𝐷,𝐷) and for any 𝜀>0, there exists a map 𝐹∈𝐶₀(𝐷,𝐷) satisfying (1)𝑑(𝑓,𝐹)<𝜀; (2)𝐹 has a transversal homoclinic orbit in 𝐷; (3)𝐹 is chaotic in the sense of both Li-Yorke and Devaney; (4)the topological entropy ℎ(𝐹)>0.

Proof. Let 𝐹 be defined as in Lemma 3.2. Then (1) and (2) hold by Lemma 3.2.
Let 𝐹, 𝑟, 𝜆, 𝜇, and 𝑈 be specified in the proof of Lemma 3.2. Without loss of generality, suppose that the fixed point 𝑧 of 𝐹 is the origin.
Set 𝑈=𝐵(0,𝑟), 𝐴=𝜆𝐼,𝐵=𝜇𝐼, 𝑓₁|_𝑈=0,𝑓₂|_𝑈=0. So 𝐴 and 𝐵 satisfy assumption (i) in Lemma 2.7, where 𝜆₀=max{|𝜆|,|𝜇|⁻¹}<1. Take 𝜃=(1−𝜆₀)/2. Then ‖𝐷𝑓₁(𝑥,𝑦)‖=‖𝐷𝑓₂(𝑥,𝑦)‖=0<𝜃<1−𝜆₀ for (𝑥,𝑦)∈𝐵(0,𝑟). Further, 𝑊^𝑢_loc(0,𝑟,𝐹)=𝑥,𝑦∶𝑥=0,𝑦_𝑌,𝑊<𝑟_𝑠loc(0,𝑟,𝐹)=𝑥,𝑦∶𝑦=0,‖𝑥‖_𝑋<𝑟. (3.18) Hence, 𝐹 satisfies assumptions (ii) and (iii) in Lemma 2.7 with 𝛾=𝑟.
By the discussions in Step 3 in the proof of Lemma 3.2, 𝐹 satisfies assumption (iv) in Lemma 2.7, where 𝑂(𝑝₀), 𝑂₋₁, 𝑂₀, and 𝑂₁ are the same as those in the proof of Lemma 3.2 and 𝑁=1. So, all the assumptions in Lemma 2.7 are satisfied. Consequently, (3) and (4) hold by Lemma 2.7. The proof is complete.

When it is not required that a map transforms its domain 𝐷 into itself, the convexity of domain 𝐷 can be removed and all the corresponding results to Lemmas 3.1 and 3.2 and Theorem 3.3 still hold. In detail, let 𝑆 be a bounded open set in 𝑍 and 𝐶₀𝑆,𝑍∶=𝑓∶𝑆⟶𝑍iscontinuousandbounded,andhasafixedpointin𝑆. (3.19) Then (𝐶₀(𝑆,𝑍),𝑑) is a metric space, where 𝑑 is defined the same as that in (3.3). The results of Lemma 3.2 and Theorem 3.3 hold, where 𝐶₀(𝐷,𝐷) is replaced by 𝐶₀(𝑆,𝑍). Their proofs are similar.

Now, we only present the detailed result corresponding to Theorem 3.3.

Theorem 3.4. Let 𝑋 and 𝑌 be Banach spaces, 𝑍=𝑋×𝑌, and 𝑆 be a bounded open set in 𝑍. Then, for every map 𝑓∈𝐶₀(𝑆,𝑍) and for any 𝜀>0, there exists a map 𝐹∈𝐶₀(𝑆,𝑍) satisfying the following(1)𝑑(𝑓,𝐹)<𝜀; (2)𝐹 has a transversal homoclinic orbit in 𝑆; (3)𝐹 is chaotic in the sense of both Li-Yorke and Devaney; (4)the topological entropy ℎ(𝐹)>0.

Remark 3.5. A general Banach space 𝑍 may not be discomposed into a product of two Banach spaces with dimension greater than or equal to 1. However, it is true for 𝑍=𝐑^𝑛 with 𝑛≥2. So Theorem 3.3 holds for each 𝑛-dimensional space 𝐑^𝑛 with 𝑛≥2. In addition, if 𝐷 is a bounded and convex set in 𝐑^𝑛, every continuous map 𝑓∶𝐷→𝐷 has a fixed point in 𝐷 by the Schauder fixed point theorem. In this case one has that 𝐶₀𝐷,𝐷=𝐶𝐷,𝐷=𝑓∶𝐷⟶𝐷iscontinuous. (3.20)

Remark 3.6. (1) As we all know, under 𝐶¹ perturbation, the hyperbolicity of a map is preserved. But it is obvious that the conclusion does not hold in the 𝐶⁰ sense.
(2) In the 𝐶⁰ topology, Theorems 3.3 and 3.4 show the density of distributions of maps with transversal homoclinic orbits, and consequently in the sense of both Li-Yorke and Devaney. However, it is not true in the 𝐶¹ topology. For example, consider the map 𝑓=𝑥𝑥,𝑦2,𝑦2,𝑥,𝑦∈𝐼²=[0,1]×[0,1]. (3.21) Clearly, 𝑓∈𝐶₀(𝐼²,𝐼²) and 𝑧=0 is a globally asymptotically stable fixed point of 𝑓 in 𝐼². By Theorem 3.3, for each 𝜀>0, there exists a map 𝐹∈𝐶₀(𝐼²,𝐼²) with ‖𝐹−𝑓‖<𝜀 such that 𝐹 is chaotic in the sense of both Li-Yorke and Devaney. But, in the 𝐶¹ topology, for each positive constant 𝜀<1/2 and for every map 𝐹∈𝐶¹(𝐼²,𝐼²) with 𝐹−𝑓_𝐶1:=max𝐹−𝑓,𝐹^′−𝑓^′<𝜀, (3.22)𝐹 is globally asymptotically stable in 𝐼², and so is not chaotic in any sense.

Acknowledgments

This paper was supported by the RFDP of Higher Education of China (Grant 20100131110024) and the NNSF of China (Grant 11071143).