Distribution of Maps with Transversal Homoclinic Orbits in a Continuous Map Space
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.
Distribution of a set of maps with some dynamical properties in some continuous map space is a very interesting topic. In the 1960s, Smale  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 topology by Jakobson , a partial solution was given in the topology by Blokh and Misiurewicz , and density was finally proved by Shen . In 2007, Kozlovki et al. got the result in topology; that is, hyperbolic (i.e., Axiom A) maps are dense in the space of maps defined in a compact interval or circle, . At the same time, some other scholars considered the distribution of hyperbolic diffeomorphisms in , where is a manifold. Just like the work of Smale, Palis [6, 7] gave the following conjecture: (1) any can be approximated by a hyperbolic diffeomorphism or by a diffeomorphism exhibiting a homoclinic bifurcation (tangency or cycle), (2) any diffeomorphism can be approximated by a Morse-Smale one or by one exhibiting transversal homoclinic orbit. Later, it was shown that the conjecture (1) holds for diffeomorphisms of surfaces . And some good results have been obtained, such as any diffeomorphism can be approximated by a Morse-Smale one or by one displaying a transversal homoclinic orbit , any diffeomorphism can be approximated by one that exhibits either a homoclinic tangency or a heterodimensional cycle or by one that is essentially hyperbolic .
In 1963, Smale gave the well-known Smale-Birkhoff homoclinic theorem for diffeomorphisms , from which one can easily know that if a diffeomorphism on a manifold has a transversal homoclinic orbit, then there exists an integer such that 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 . 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  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 , 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 .
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 .
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.
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 ). Let be a metric space, a map, and a set of with at least two points. Then, is called a scrambled set of if, for any two distinct points , The map is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scrambled set of .
Definition 2.2 (see ). Let be a metric space. A map is said to be chaotic on in the sense of Devaney if (i) the periodic points of in are dense in ; (ii) is topologically transitive in ; (iii) has sensitive dependence on initial conditions in . Now, we give the definition of hyperbolic fixed point.
Definition 2.3 (see ). Let be a Banach space, be a set, and be a map. Assume that is a fixed point of , is continuously differentiable in some neighborhood of , and denote the Fréchet derivative of at . The fixed point is called hyperbolic if , where denotes the spectrum of a linear operator . The hyperbolic fixed point is called a saddle point if and .
If is invertible, then for any , the set is said to be the forward orbit of from , the set is said to be the backward orbit of from . Since it is not required that is invertible in this paper, a backward orbit of is a set with , which may not exist, or exist but may not be unique. A whole orbit of is the union , denoted by in the case that it has a backward orbit . The stable set and the unstable set of a hyperbolic fixed point of are defined by respectively. The local stable and unstable sets are defined by respectively, where is some neighborhood of . If or for some , then the corresponding local stable and unstable sets of are denoted by and , respectively. By the Stable Manifold Theorem , if is continuously differentiable in some neighborhood of a saddle point , then there exists a neighborhood of such that the corresponding local stable and unstable set of are submanifolds of , 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 ). Two submanifolds and in a manifold are transverse (in ) provided for any point , we have that , where and denote the tangent spaces of and at , respectively, and “+” means the sum of the two subspaces (this allows for the possibility that ).
Remark 2.5. If , then and in are transverse (in ) provided for any point , we have that . Obviously, if , then the sum of the two subspaces and is a direct one, denoted by .
Definition 2.6 (see ). Let be a Banach space, be a map, and be a saddle point of . (i) An orbit is said to be a homoclinic orbit (asymptotic) to if and . (ii)A homoclinic orbit to is said to be transversal if there exists an open neighborhood of such that and for any sufficiently large integers , and sends a disc in containing diffeomorphically onto its image that is transversal to at .
The following lemma is taken from Theorems 3.1, and 5.2, Corollary 6.1, and the result in Section 7 of .
Lemma 2.7. Let be a map, where , and and are Banach spaces. (i) Let and be linear continuous maps in and , respectively, with the absolute values of the spectrum of less than 1 and the absolute values of the spectrum of larger than 1, and for some constant .(ii)Assume that is an open neighborhood of 0 in and are maps with . Further, assume that are uniformly continuous in , and satisfies that for some constants and , for all . (iii)Let be of the following form: and have the local stable and unstable manifolds and .(iv)Assume that is a homoclinic orbit of with as , and there exists an integer such that , , and(iv1) sends a disc centered at diffeomorphically onto containing ;(iv2) intersects transversally at . Then is a transversal homoclinic orbit of . Furthermore, there exists an integer and a subset in a neighborhood of such that on is topologically conjugate to the full shift map on the doubly infinite sequence of two symbols. Consequently, is chaotic in the sense of both Li-Yorke and Devaney, and its topological entropy .
Note that it is not required that is a diffeomorphism, even may not be continuous on the whole space 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 and are Banach spaces, and and are bounded, convex, and open sets in and , respectively. It is evident that is a bounded, convex, and open set in , where the norm on is defined by , for any , where . Introduce the following map space: For any , let and for any , let Then is a metric space. It may not be complete because a limit of a sequence of maps in is continuous and bounded, but may not have a fixed point in . But in the special case that is finite-dimensional, 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 .
For convenience, by denote and , by Fix denote the set of all the fixed points of . For with , by denote the straight half-line connecting and :
Lemma 3.1 (see [22, Lemma 3.1]). For every map and any , there exists a map such that , , and is continuously differentiable in some neighborhood of some point .
Lemma 3.2. For every map and every , there exists a map with such that has a transversal homoclinic orbit in .
Proof. Fix any . By Lemma 3.1, it suffices to consider the case that has a fixed point with , and is continuously differentiable in some neighborhood of .
For any , there exists a positive constant with such that
Let , take two constants with and take two points , where .
The rest of the proof is divided into three steps.
Step 1. Construct a map that is locally controlled near .
Define where , , and and are real parameters and satisfy Note that since .
For any , is defined as follows. Let and be the intersection points of the straight line with and , respectively, (see Figure 1). Set where is determined as follows; It is noted that when continuously varies in , so do the intersection points and . Consequently, and then are continuous in .
Next, define for . Finally, for any , suppose that and are the intersection points of the straight line with and , respectively. Define as that in (3.8), where is determined by (3.9) with and replaced by and , respectively. Hence, is continuous in .
Obviously, is a saddle fixed point of , and Step 2. and satisfies that .
From the definition of , it is easy to know that is continuous on and has a fixed point , that is, .
Next, we will prove that . For , . For , it follows from (3.5) and (3.6) that For , it follows from (3.5), (3.6), and (3.8) that For , from (3.5) and (3.6), one has For , from (3.5), (3.6), and (3.8), one has Therefore, from the above discussion, .Step 3. has a transversal homoclinic orbit in .
It follows from (3.6) that and , where by (3.7). So, by (3.10) one has that , . Hence, satisfies that . Thus, and is a homoclinic orbit of .
Set a positive constant satisfying Then by (3.7), and consequently it follows from (3.10) that the disc Further, set the discs Then , , and (see Figure 2).
It is evident that intersects transversally at point . In addition, by the definition of in , one can get that is a diffeomorphism. Therefore, is a transversal homoclinic orbit asymptotic to of by Definition 2.6, where .
The entire proof is complete.
Theorem 3.3. Let and be Banach spaces, , and be a bounded, convex, and open set in . Then, for every map and for any , there exists a map satisfying (1); (2) has a transversal homoclinic orbit in ; (3) is chaotic in the sense of both Li-Yorke and Devaney; (4)the topological entropy .
Proof. Let be defined as in Lemma 3.2. Then (1) and (2) hold by Lemma 3.2.
Let , , , , and be specified in the proof of Lemma 3.2. Without loss of generality, suppose that the fixed point of is the origin.
Set , , . So and satisfy assumption (i) in Lemma 2.7, where . Take . Then for . Further, Hence, satisfies assumptions (ii) and (iii) in Lemma 2.7 with .
By the discussions in Step 3 in the proof of Lemma 3.2, satisfies assumption (iv) in Lemma 2.7, where , , , and are the same as those in the proof of Lemma 3.2 and . 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 into itself, the convexity of domain can be removed and all the corresponding results to Lemmas 3.1 and 3.2 and Theorem 3.3 still hold. In detail, let be a bounded open set in and Then is a metric space, where is defined the same as that in (3.3). The results of Lemma 3.2 and Theorem 3.3 hold, where is replaced by . Their proofs are similar.
Now, we only present the detailed result corresponding to Theorem 3.3.
Theorem 3.4. Let and be Banach spaces, , and be a bounded open set in . Then, for every map and for any , there exists a map satisfying the following(1); (2) has a transversal homoclinic orbit in ; (3) is chaotic in the sense of both Li-Yorke and Devaney; (4)the topological entropy .
Remark 3.5. A general Banach space may not be discomposed into a product of two Banach spaces with dimension greater than or equal to 1. However, it is true for with . So Theorem 3.3 holds for each -dimensional space with . In addition, if is a bounded and convex set in , every continuous map has a fixed point in by the Schauder fixed point theorem. In this case one has that
Remark 3.6. (1) As we all know, under perturbation, the hyperbolicity of a map is preserved. But it is obvious that the conclusion does not hold in the sense.
(2) In the 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 topology. For example, consider the map Clearly, and is a globally asymptotically stable fixed point of in . By Theorem 3.3, for each , there exists a map with such that is chaotic in the sense of both Li-Yorke and Devaney. But, in the topology, for each positive constant and for every map with is globally asymptotically stable in , and so is not chaotic in any sense.
This paper was supported by the RFDP of Higher Education of China (Grant 20100131110024) and the NNSF of China (Grant 11071143).
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