Abstract
This paper is concerned with the chaos of discrete dynamical systems. A new concept of heteroclinic cycles connecting expanding periodic points is raised, and by a novel method, we prove an invariant subsystem is topologically conjugate to the one-side symbolic system. Thus, heteroclinic cycles imply chaos in the sense of Devaney. In addition, if a continuous differential map has heteroclinic cycles in , then has heteroclinic cycles with being sufficiently small. The results demonstrate structural stability of heteroclinic cycles. In the end, two examples are given to illustrate our theoretical results and applications.
1. Introduction
Since Li and Yorke first introduced the term “chaos” in 1975 [1], chaotic dynamics have been observed in various fields [2–10]. When chaotic theory was in its initial stage, Marotto generalized the results of Li and Yorke in interval mapping to multidimensional discrete systems and proved that a snapback repeller implies chaos [11]. In [2], Blanco demonstrated that has positive topological entropy if has snapback repellers. In 1989, Devaney gave an explicit chaotic definition which is called Devaney’s chaos in [12]. In order to understand the relationship between various kinds of chaos, Huang and Ye obtained that Devaney’s chaos implies Li–Yorke chaos in [13]. Based on the research of Marotto, Shi and Chen raised the concept of snapback repellers in Banach spaces and complete metric spaces [14, 15]. As is well known, some homoclinic and heteroclinic cycles imply chaos in dynamical systems [6, 16]. Lin and Chen introduced the new chaotic criteria of heteroclinic repellers in [17]. Recently, Li et al. generalized the definition of heteroclinic repellers to infinite dimensional dynamical systems [18].
From the view of application, we naturally ask whether a chaotic dynamical system still has chaotic behaviors under small perturbations. In [19], Marotto showed that the systems with snapback repellers under delayed perturbations still have chaotic behaviors. In [20], Li et al. showed the structural stability of snapback repellers in . Further, Chen et al. studied the structure stability of snapback repellers in Banach space [21]. Chen and Li provided a sufficient condition of a high-dimensional difference equation having symbolic embedding for enough small perturbations [22]. In 2020, Chen and Wu et al. showed that the system with heteroclinic repellers has the structural stability in [4, 16].
In this paper, we consider the following discrete dynamical systems:where is a continuous map from into itself. Based on the concept of heteroclinic repellers in discrete dynamical systems, we introduce a new concept of heteroclinic cycles connecting expanding periodic points in ; by a novel method, we can construct shift invariant sets and prove that heteroclinic cycles imply chaos in the sense of Devaney. It is very important to choose an appropriate recurrent time set in the proof of the theorem. In particular, it is easier to find the conditions of heteroclinic cycles than that of snap back repellers for the system in [3]. The following theorems are the main results.
Theorem 1. Let be a continuous differential map from into itself. If has heteroclinic cycles connecting expanding periodic points, then is chaotic in the sense of Devaney.
Theorem 2. Let be continuous differential maps from into itself. If has heteroclinic cycles connecting expanding periodic points and is sufficiently small, where also has heteroclinic cycles connecting expanding periodic points.
This paper is organized as follows: in Section 1, the relevant results of the previous studies are introduced. In Section 2, some definitions and necessary lemmas are given. A new concept of heteroclinic cycles in the sequel is raised. In Section 3, the main conclusions are drawn. At the end, some examples are presented to illustrate the main results and applications in Section 4.
2. Definitions and Lemmas
In this section, some definitions and lemmas will be introduced.
Definition 1. Let be a continuous differentiable map. is expanding at a point if the derivative operator is invertible and the norm of each of its eigenvalues is larger than 1. is expanding in a nonempty subset of if there exists positive constant such that, for any , .
It is well known that if is differentiable, then is expanding at a point which is equivalent to the fact that there exist a norm in and a constant such that [3]. The following heteroclinic repellers in were presented by Lin and Chen in [17].where .
Definition 2. (see [17]). Let be a continuous differentiable map and the integer be fixed. The fixed points of are called heteroclinic repellers if for any integer , there are(1) is expanding at the point (2) has a forward orbit of such that and (3) is continuously differentiable in a neighborhood of any point on and is invertible,
Based on Definition 2, we introduce the following new concept.
Definition 3. Let be a continuous differentiable map from into itself and be fixed integer. Let have different periodic points with periods , respectively. For any and , . For any , suppose that(1) is expanding at every point which is in the periodic orbits of .(2) has a forward orbit of connecting periodic points and . That is, it satisfies that , and there exists a positive integer such that .(3)For any point on , the linear operator is invertible.Then, the set is called a heteroclinic cycle connecting expanding periodic points .
Remark 1. By Definition 3, for any point in heteroclinic cycles connecting expanding periodic points , is invertible and the norm of each of its eigenvalues is larger than 1. Thus, are heteroclinic repellers when every periodic point is fixed point in Definition 3 and . Example 1 shows that heteroclinic cycles connecting expanding periodic points are different from heteroclinic repellers.
To prove the structural stability of heteroclinic cycles connecting expanding periodic points, we need the following implicit function theorem with parametric variables and continuous dependence theorem of inverse mapping.
Lemma 1. (see [21]). Let and be Banach spaces, be a metric space, and be an open set of . Suppose that is a continuous differentiable map and there exists a point satisfying the following conditions:(1), the Fr chet partial derivative of with respect to , is continuous with respect to in some neighborhood of (2) is an invertible linear operator(3),Then, there exist open ball and , where , such that for any , , the unique continuous solution exists and is satisfied.
Lemma 2. (see [21]). Let and be two Banach spaces and be -open ball neighborhood of . Assume that is a map from into such that and is an invertible linear operator from into . Then, there are constants and a map from into satisfying the following conditions:(1)For any , is one to one on , and is invertible for every (2)For any and , there is a unique satisfying (3)The map is a continuous map from into (4)For every , is a -order continuous differentiable map, and ,wheredenotes the -open ball neighborhood of and is a p-order continuous differentiable map set.
Lemma 3. Let be a continuous differentiable map from into itself. Let have periodic points with periods , respectively. If is expanding at every point which is in the periodic orbit of , then for all , there exist positive constants and such that is homeomorphism on closed ball for every and , is expanding on and
Proof. For any , because is expanding at every point which is in the periodic orbit of , by the continuity of , there exists some closed neighborhood of such that is expanding in for . Therefore, there exists a positive constant such that, for all and for any , there isLet . Thus, for every and , there isBecause and is invertible, by Lemma 2, there are constants such that for every , is a differentiable homeomorphism. Therefore, for any , is a differentiable homeomorphism andwhere denotes the boundary of the set and is the closure of the set . Take . For every , there isHence, . Inductively, for every , there areLet , and for every , there areThe proof of the lemma is done.
3. Heteroclinic Cycles Connecting Expanding Periodic Points Imply Chaos and Are Structurally Stable
In this section, we are going to consider that heteroclinic cycles imply Devaney’s chaos and are structurally stable.
Theorem 3. Let be a continuous differential map from into itself. If has heteroclinic cycles connecting expanding periodic points, is chaotic in the sense of Devaney.
Proof. Let be different periodic points with periods , respectively, and be expanding at , respectively. Let be heteroclinic cycles connecting periodic points .
For any integer and any integer , by Definition 3, is invertible and the norm of each of its eigenvalues is larger than 1. Therefore, there exist constants such that, for any integer and , . There exists a positive integer such that and . Without loss of generality, for any integer , let , if . Since is invertible, there exist positive constants such that is a homeomorphism on andBy the continuity of , there exists a closed ball such thatwhere is the inverse of restricted to . Similarly, for any integer , there exists such thatNext we prove there exist two nonempty bounded closed subsets with such that, for a positive integer , and restricted to and , respectively, is expanding. Because is expanding on and is a fixed point of for every , there are(1)There exists a positive integer such that .(2)For every , there exists a positive integer such thatLet ; then, there is . There exists a positive integer such that and . LetThen, and are two nonempty bounded closed subsets of andTherefore, . From definitions of , we haveIn addition, we have thatWe claim thatwhere denotes the cardinality of a set.
Proceed the proof by contradiction. Suppose and . Let ; then, there exists a subsequence of such that for any . Without loss of generality, for any . By (17), we havewhich is contrary to the fact that and belong to the bounded set . Therefore, there exist an invariant set and a positive integer such that is topologically conjugate to the one-side symbolic system . That is, is chaotic in the sense of Devaney and so is .
From Theorem 3, we obtain the following corollary.
Corollary 1. Let be a continuous differentiable map from into itself. If has heteroclinic repellers with , then for every neighborhood of , there exist an invariant subset of and a positive integer such that is topologically conjugate to the one-side symbolic system . Therefore, is chaotic in the sense of Devaney and so is .
Theorem 4. Let be continuous differential maps from into itself. If has heteroclinic cycles connecting expanding periodic points and is sufficiently small, then also has heteroclinic cycles connecting expanding periodic points.
Proof. Let be different periodic points with periods , respectively. Let be heteroclinic cycles connecting periodic points . By Definition 3, there exist constants such that, for any , . By Lemma 3, for every , there exists a positive constant , such that for any and ,and is expanding on . Since are heteroclinic cycles that connect periodic points , for any integer , there exists a positive integer such that . Note that for any point , is an invertible operator. Therefore, by Lemma 2, for every , there exist an open neighborhood of and a positive constant such that, for any , is a differentiable homeomorphism. Without loss of generality, suppose that . Thus, is expanding in .
Let . It is needed to prove that there exists a positive constant such that, for any , has heteroclinic cycles connecting expanding periodic points. Take with . Define a map asSince is expanding at , all eigenvalues of are greater than one in absolute value. Therefore, is invertible. Note that is invertible. Therefore, the diagonal block linear operatoris also invertible. Here we write, and if , then for any integer . By Lemma 1, there exist constants , and such that(1).(2)For every , is the unique solution to equations , , and , that is, .Therefore, for any integer , is expanding periodic point of , and if , for any . In particular, .
It is needed to show that for every , has a forward orbit of connecting periodic points and . First, for every , there isand is expanding on . Let denote the inverse of restricted to , and the -th iterate of denotes . For every integer , let . For any , becausethere isThis proves that .
For any integer and , letand for any integer , letTherefore, is a forward orbit of connecting and .
Because for every , is a differentiable homeomorphism, is an invertible linear operator. Moreover, for any , , so for . This implies that is an invertible linear operator for any integer . This proves that satisfies condition (3) in Definition 3. Let for any integer . Therefore, are heteroclinic cycles connecting expanding periodic points .
4. Examples
In the following, two examples are given which illustrate our results and applications.
Example 1. Consider the one-dimensional map has only one fixed point 0, and thus does not have heteroclinic repellers. Directing calculation, we have(1) has three 2-periodic points which are not in the same orbit .(2) is expanding in .(3).Therefore, has heteroclinic cycles connecting expanding periodic points and (see Figure 1).
Remark 2. Example 1 shows that does not have heteroclinic repellers but has heteroclinic cycles connecting expanding periodic points. So, heteroclinic cycles connecting expanding periodic points really contain heteroclinic repellers.
Example 2. Recall the wave equation with a van der Pol boundary condition [3], as follows:Letwhere is the unique real solution of the cubic equationLet . Then, the solution of system (30) is completely characterized by the interval maps and , between which there is topological conjugacy. If is chaotic on some invariant interval, we say that the gradient of system (30) is chaotic. Chen. et al. have analyzed the chaos caused by snapback repellers in system (30) (see [3]). For example, when and , has a snapback repeller. It can be checked that has heteroclinic cycles connecting expanding periodic points. In particular, it is easier to find the conditions that has heteroclinic cycles.
5. Conclusions
In this paper, a new criterion of chaos is established in . By constructing shift invariant sets, we prove that heteroclinic cycles imply chaos in the sense of Devaney. Heteroclinic cycles connecting expanding periodic points is structural stability.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was supported in part by the National Natural Science Foundation of China (11671410 and 61907010), the Natural Science Foundation of Guangdong Province (2018A0303130120), and the Foundation for Natural Science in Higher Education of Guangdong, China (2019KZDXM036).