# Persistence of Heteroclinic Cycles Connecting Repellers in Banach Spaces

Authors: Zongcheng Li
Journal: Journal of Mathematics
 (2022)

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## Abstract

This paper is concerned with persistence of heteroclinic cycles connecting repellers in Banach spaces. It is proved that if a map with a regular and nondegenerate heteroclinic cycle connecting repellers undergoes a small perturbation, then the perturbed map can still have a regular and nondegenerate heteroclinic cycle connecting repellers. The perturbation rang is given by an explicit positive constant according to the properties of the original map. Hence, the perturbed map and the original map are simultaneously chaotic in the sense of both Devaney and Li-Yorke. Especially, the persistence of heteroclinic cycles connecting repellers is also discussed in the Euclidean space, where the repellers can expand in different norms. Finally, three examples are provided to illustrate the validity of the theoretical results.


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## Body

## 1. Introduction

Chaos is a very important kind of dynamical behaviors in nonlinear systems and chaos problems have attracted a lot of attention from many scientists and sgd mathematicians. In 1975, the first mathematical definition of chaos and a famous result that “period three implies chaos” were given by Li and Yorke [1] in studying continuous interval maps. After that, different definitions of chaos from different points of view were proposed by researchers, one can see [2–4] for some related definitions of chaos. Among these mathematical definitions, chaos in the sense of Li-Yorke, Devaney or Wiggins are often used in the literature, see [5–8] for discussions of their relationships. Then, there appeared many works to study chaotic behaviors of multidimensional maps. A very famous work that “a snap-back repeller implies chaos” in the sense of Li-Yorke was proposed by Marotto [9], which is a generalization of Li and Yorke’s result from one-dimensional maps to multidimensional maps. This result shows great power in studying chaos of finite dimensional maps. However, it is clear that there are many systems should be studied in infinite dimensional spaces, such as Banach spaces and metric spaces. Then, a lot of works have been done on chaotic behaviors of infinite dimensional maps. Some of these important results were given by Shi and her cooperators. In 2004, Shi and Chen [10] extended the concept of snap-back repeller to metric spaces and obtained several criteria of chaos. Later, Shi and her cooperators developed the coupled-expansion theory and used it to study chaos, see [11–14] and references therein.Structural stability of chaotic maps is a very important and interesting question, and many results have been achieved. Marotto first studied perturbations of maps with snap-back repellers in [15, 16], and proved that if a scalar system with a snap-back repeller undergoes a small perturbation, then the perturbation system will have a transversal homoclinic point and thus has chaotic behaviors. Later, there appeared several results about multidimensional perturbations of chaotic systems, see [17–19]. In 2009, Li and Lyu [20] proved that if a map with a snap-back repeller in Rn undergoes a mall C1 perturbation, then the perturbed map still has a snap-back repeller and consequently is chaotic in the sense of Li-Yorke. However, all the above perturbations of chaotic systems were made in finite dimensional spaces. In 2011, Chen et al. [21] studied the persistence of snap-back repellers under small C1 perturbations in Banach spaces. In 2012, Zhang et al. [22] used a different method to study the persistence of snap-back repellers under small Lipschitz perturbations in Banach spaces. Moreover, Zhang and Shi [23] studied the persistence of coupled-expansion for time-varying systems under small time-varying perturbations in Banach spaces, and showed the persistence of snap-back repellers.In 2006, Lin and Chen [24] gave a result that heteroclinical repellers imply chaos in the sense of Li-Yorke in Rn. In their definition of heteroclinical repellers, there were some conditions given by the Jacobian matrices of a map. However, a map in a metric space may not have derivatives in general. In 2008, based on their work, Li et al. [25] grasped the essential meanings of the definition of heteroclinical repellers to extend it to general metric spaces without needing the continuity or continuous differentiability. For more intuitive to reflect the relationships of the repellers, they redefined it as a heteroclinic cycle connecting repellers and obtained several criteria of chaos. Later, they studied chaos induced by heteroclinic cycles connecting repellers in general Banach spaces [26], and used these results to study existence of chaos or chaotification problems [27]. This shows that the heteroclinic cycle connecting repellers has significant effects on chaos studying. Hence, it is worth studying whether a heteroclinic cycle connecting repellers has the persistence under small perturbations as that for a snap-back repeller. Recently, in 2020, Chen and Wu [28] studied the persistence of heteroclinic repellers in Rn for C1 maps under small C1 perturbations. Chen et al. [29] studied the persistence of heteroclinic repellers in Banach spaces for C1 maps under small C1 perturbations. It is noted that the definitions of heteroclinic repellers in [28, 29] both needed the differentiability of a map as that definition in [24]. In 2021, Wu [30] extended the concept of heteroclinic repellers in [24] to heteroclinic cycle connecting expanding periodic points in Rn and studied the persistence of it for C1 maps under C1 perturbations, where the maps needed to be continuously differentiable in the whole space. More recently, Chen and Luo [31] studied the persistence of regular nondegenerate snap-back repellers and heteroclinic cycles for continuous maps under small Lipschitz perturbations, where the maps were continuous in the whole Banach space. On the one hand, it should be pointed out that all the above results needed the maps to be continuous or continuously differentiable in the whole space. However, there are a lot of maps that may not be continuous or continuously differentiable in the whole space. On the other hand, it should be pointed out that all the above results needed the perturbations to be small enough and did not give a relatively explicit expression for the range of small perturbations, which is convenient and useful in applications to quickly check out whether the persistence is maintained. So, it is meaningful to study persistence of heteroclinic cycles connecting repellers for maps which are only continuous or continuously differentiable in some domains of the whole space, and it is also meaningful to study the explicit expression for the range of small perturbations.The fixed point theory has become an essential tool to resolve some problems in nonlinear analysis, including fractional calculus, see [32, 33] and references therein for more details about this theory. Here, we will apply the Banach contractive mapping principle and the ideas used in [22, 23] to study the persistence of regular and nondegenerate heteroclinic cycles connecting repellers in Banach spaces, where the original maps are only continuous or continuously differentiable in some neighborhoods of points. An important result is that an explicit expression for the range of perturbations is given. It will be proved that if a map with a regular and nondegenerate heteroclinic cycle connecting repellers undergoes a small Lipschitz perturbation, then the perturbed map can still have a regular and nondegenerate heteroclinic cycle connecting repellers. So, the perturbed map and the original map are simultaneously chaotic in the sense of both Devaney and Li-Yorke. Particularly, the persistence of heteroclinic cycles connecting repellers is also discussed in Rn. The significant difference between our result and those obtained in [28, 30] is that the repellers in our result expand in different norms, while the repellers in the latter expand in the single Euclidean norm. It is clear that different fixed points can expand in different norms in Rn. So, our result is more general in practice.The rest of the paper is organized as follows. Some concepts and lemmas are given in Section2. Several theorems about perturbations of maps with heteroclinic cycles connecting repellers in general Banach spaces or the Euclidean space are given in Section 3. Three examples are provided to illustrate the validity the theoretical results in Section 4. Finally, conclusions are made in Section 5.


## 2. Preliminaries

Some definitions and lemmas are given in this section.Two usually used definitions of chaos in the sense of Li-Yoke or Devaney are first introduced. Then, the concept of a heteroclinic cycle connecting repellers is introduced.Definition 1 (see [1]).
LetX,d be a metric space, f:X⟶X be a map, and S be a set of X with at least two distinct points. Then, S is called a scrambled set of f if for any two distinct points x,y∈S,(1)liminfn⟶∞dfnx,fny=0,limsupn⟶∞dfnx,fny>0.
The mapf is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scrambled set S of f.Remark 1.
There are three conditions in the original characterization of chaos in Li-Yorke’s theorem [1]. Since the third one is not essential, it is removed in Definition 1 in most literature.Example 1.
Consider the following Baker’s equation(2)xn+1=2xn,for0≤xn≤12,21−xn,for12<xn≤1,which models the mixing of a dye spot on a strip of dough that is repeatedly stretched and folded over on itself. The iterative scheme (2) maps the interval 0,1 into itself. It is easy to check that system (2) has a cycle of period three and hence is chaotic in the sense of Li-Yorke by the Li-Yorke theorem in [1]. This equation has been extensively discussed in the literature [7, 34] and references cited therein.Definition 2 (see [4]).
LetX,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 set of the periodic points off is dense in V(ii)
f is topologically transitive in V(iii)
f has sensitive dependence on initial conditions in VRemark 2.
In 1992, Banks et al. [5] proved that conditions (i) and (ii) together imply condition (iii) if f is continuous in V. So, condition (iii) is redundant in the above definition in this case. It has been proved by [6] that chaos in the sense of Devaney is stronger than chaos in the sense of Li-Yorke under some conditions.Example 2.
Let(3)∑2+:=s=s0,s1,s2,…:sj=0or1,with the distance(4)ρs,t:=∑i=0∞si−ti2i,where s=s0,s1,s2,… and t=t0,t1,t2,…. Then ∑2+ρ is a complete metric space and a Cantor set, see Lemma 2.5 in [10]. The shift map σ:Σ2+⟶Σ2+ defined by σs0,s1,s2,…=s1,s2,… is continuous. The dynamical system defined by σ is called a one-sided symbolic dynamical system. It follows from [[4], Part 1, Proposition 6.6] that σ has the following properties:(i)
CardPernσ=2n(ii)
Perσ is dense in Σ2+(iii)
there exists a dense orbit ofσ in Σ2+
Here,CardPernσ denotes the number of periodic points of period n for σ. It is clear that property (iii ) implies that σ is transitive. Therefore, the symbolic dynamical system is chaotic in the sense of Devaney. See [3, 4] for more discussions about this symbolic dynamical system.Definition 3 (see [26], Definition 2.5).
LetX,d be a metric space and f:X⟶X be a map with k≥2 fixed points z1,…,zk∈X.(I)
Suppose that, for eachi1≤i≤k, zi is an expanding fixed point of f in B¯rizi, and there exist a point xi0∈Brizi, xi0≠zi, and a positive integer mi≥1 such that fmixi0=zti, and zi is the limit for the backward orbit of xi0, where B¯rizi and B¯rizi are the closed and open balls of radius ri centered at zi, ti=imodk+1. Then all the points xi01≤i≤k, together with their backward and forward orbits consist of a set, which is called a k-heteroclinic cycle connecting repellers z1,…,zk.(II)
Suppose thatf has a k-heteroclinic cycle connecting repellers z1,…,zk. For each point x0 on the cycle, if there exists a positive constant r0 such that for each positive constant r≤r0, fx0 is an interior point of fBrx0, then the cycle is called regular; if there exist positive constants r1 and μ such that dfx,fy≥μdx,y,∀x,y∈B¯r1x0, then the cycle is called nondegenerate.Remark 3.
It is pointed out that the necessary and sufficient condition for a heteroclinic cycle connecting repellers is used to give the definition (I) for simplicity, see (1) of Remark 2.2 in [26]. In addition, it does not need the continuity or continuous differentiability in this definition, while some similar definitions need them, see [24–26, 28–31] for more details about this concept.
For convenience, some notations are given in the following. The continuously differentiable maps in a setU of a Banach space X are denoted by C1U,X. The derivative of a map f at a point x∈X is denoted by Dfx. In addition, for a linear map L:X⟶X, denote(5)L:=supLx:x∈X,x=1,L0:=infLx:x∈X,x=1.
If a bounded linear mapL has a bounded inverse, then L is said to be an invertible linear map, see Definition 4.17 in [35]. The following four lemmas will be used in the paper.Lemma 1 (see [22], Lemma 2.4).
LetX,⋅ be a Banach space, z∈X, and f:B¯rz⟶fB¯rz be a continuous map. Assume that fBrz is an open set of X and(6)fx−fy≥μx−y,∀x,y∈B¯rz,for some constant μ>0, then(7)Bμ−LrFz⊂FBrz,where F=f+g and g is a Lipschitz map in B¯rz with Lipschitz constant L<μ.Lemma 2 (see [25], Theorem 3.4).
LetX,d be a complete metric space and f:X⟶X be a map. Assume that(i)
f has a regular and nondegenerate k-heteroclinic cycle connecting repellers z1,…,zk∈X, k≥2(ii)
f is continuous in some neighborhood of each point on the cycle
Then there exists an uncountable, perfect, bounded, and closed setV such that fV=V and f is chaotic on V in the sense of Devaney as well as in the sense of Li-Yorke.Lemma 3 (see [26], Lemma 2.2; [22], Lemma 2.3).
LetX,⋅ be a Banach space and f:X⟶X be a map. Assume that f has a heteroclinic cycle connecting repellers z1,…,zk∈X, k≥2, and for each point x0 on the cycle f is continuously differentiable in some neighborhood of x0 and satisfies that Dfx0 is an invertible linear map, then the cycle is regular and nondegenerate.Lemma 4 (see [11], Lemma 2.2).
LetX,⋅ be a Banach space. Suppose that a map f:X⟶X is continuously differentiable in Br0x0 for some x0∈X and some r0>0, and satisfies that λ0=Dfx00>0, then for each ε∈0,λ0, there exists a positive constant r1<r0 such that(8)fx−fy≥λ0−εx−y,∀x,y∈B¯r1x0.


## 3. Persistence of Heteroclinic Cycles Connecting Repellers in Banach Spaces

In this section, we will study persistence of heteroclinic cycles connecting repellers in Banach spaces. Assume thatX,⋅ is a Banach space, f,g:X⟶X are two maps, and f has a regular and nondegenerate heteroclinic cycle connecting repellers and is continuous in some neighborhoods of interest points. Here, we study the following system:(9)xn+1=fxn+gxn,n≥0,where g is viewed as a mall perturbation. It is proved that there still has a regular and nondegenerate heteroclinic cycle connecting repellers in (9) when g satisfies some conditions. Consequently, the perturbed system (9) is chaotic in the sense of both Devaney and Li-Yorke.Theorem 1.
Suppose thatX,⋅ is a Banach space and f:X⟶X is a map with k≥2 different fixed points z1,…,zk∈X and satisfies the following:(i)
For eachi1≤i≤k, zi is a regular expanding fixed point of f in B¯rizi with expanding coefficient λi0 for some constant ri>0. Furthermore, there exist a point xi0∈Brizi, xi0≠zi, and a positive integer mi≥1 such that fmixi0=zti, where ti=imodk+1. Consequently, f has a heteroclinic cycle Γ connecting repellers z1,…,zk.(ii)
The heteroclinic cycleΓ connecting repellers is regular and nondegenerate, and f is continuous in B¯rizi and some neighborhood Uij of xij, where xij=fjxi0 for 1≤i≤k, 1≤j≤mi−1.Then, there exists a constantε0>0 such that for any Lipschitz map g in each set of B¯rizi and Uij, 1≤i≤k, 1≤j≤mi−1, with the Lipschitz constant L satisfying(10)maxL,gzi,gxij,1≤i≤k,0≤j≤mi−1<ε0.The perturbed system (9) also has a regular and nondegenerate heteroclinic cycle Γ′ connecting repellers, and consequently there exists an uncountable, perfect, bounded, and closed set V such that system (9) is chaotic on V in the sense of both Devaney and Li-Yorke.Proof.
Without loss of generality and for simplicity, we only show that Theorem1 is true for k=2. When k>2, one can use a similar method to prove it. For convenience, let Fx:=fx+gx in the rest of this paper and i=1 or 2 in the rest of this proof.
Without loss of generality, we can suppose thatB¯r1z1∩B¯r2z2=∅, and fxi0∉Brizi. Otherwise, one can see the third paragraph in the proof of Theorem 3.1 in [25].
Sincezi is a regular expanding fixed point of f in B¯rizi with an expanding coefficient λi0, we get that(11)fx−fy≥λi0x−y,∀x,y∈B¯rizi,f:B¯rizi⟶fB¯rizi is a homeomorphism and fBrizi is open, fD is open for any open set D⊂Brizi. Take a constant(12)δi0<ri−zi−xi02,such that B¯δi0xi0⊂Brizi. Then, it follows form (11) that f:Bδi0xi0⟶fBδi0xi0 is also a homeomorphism.
From assumption (ii), it follows that there exist positive constantsμij and δij such that(13)fx−fy≥μijx−y,∀x,y∈B¯δijxij,f:Bδijxij⟶fBδijxij is homeomorphic, and fBδijxij is open for 1≤j≤mi−1, where δij satisfies the following conditions(14)δi1<λi0δi0,δi,j+1<μijδij,for1≤j≤mi−2,B¯δijxij are disjoint subsets of Uij and B¯δijxij∩Brizi=∅ for fixed i and 1≤j≤mi−1.
In the following, we will show that the mapF satisfies the conditions in Lemma 2. It will be finished by the following three steps.Step 1.
It is to prove thatF has two regular expanding fixed points z1∗ and z2∗ when g satisfies some conditions.
For proving the existence ofz1∗, we take two positive constants δ2,m2 and ε1 such that(15)δ2,m2<minλ20δ20,r1−z1−x102−δ10,ifm2=1;minμ2,m2−1δ2,m2−1,r1−z1−x102−δ10,ifm2>1.ε1=λ10−1δ2,m21+δ2,m2.
Consider the following equation(16)Fx=x,x∈B¯δ2,m2z1,which is equivalent to the following equation:(17)fx=x−gx,x∈B¯δ2,m2z1.
It follows from the first relation of (15) that B¯δ2,m2z1⊂Br1z1. By assumption (i) and (11), we get that f:B¯δ2,m2z1⟶fB¯δ2,m2z1 is homeomorphic. Then we obtain that fBδ2,m2z1 is an open set and the inverse map f−1:fB¯δ2,m2z1⟶B¯δ2,m2z1 satisfies the following:(18)f−1x−f−1y≤λ10−1x−y,∀x,y∈fB¯δ2,m2z1.
Hence, equation (17) is translated into the following:(19)f−1x−gx=x.
Here, it should prove that(20)x−gx∈fB¯δ2,m2z1,∀x∈B¯δ2,m2z1.
On the one hand, for anyx∈B¯δ2,m2z1, we have(21)gx=gx−gz1+gz1≤gx−gz1+gz1≤Lδ2,m2+gz1.
Suppose that the mapg satisfies(22)maxL,gz1<ε1,then,(23)Lδ2,m2+gz1<ε1δ2,m2+ε1=λ10−1δ2,m2.
Therefore, it follows from (21) and (23) that, for any x∈B¯δ2,m2z1,(24)x−gx−z1≤gx+x−z1<λ10−1δ2,m2+δ2,m2=λ10δ2,m2.
On the other hand, for anyx∈∂Bδ2,m2z1,(25)fx−z1=fx−fz1≥λ10x−z1=λ10δ2,m2.
Sincez1∈fBδ2,m2z1 and fBδ2,m2z1 is an open set, it follows from (24) and (25) that (20) is true.
According to the above discussion, we can define a map(26)h1x=f−1x−gx,x∈B¯δ2,m2z1.
For anyx∈B¯δ2,m2z1, it follows from (18) and (24) that(27)h1x−z1=f−1x−gx−f−1z1≤λ10−1x−gx−z1<δ2,m2,which implies that h1 maps B¯δ2,m2z1 into itself. Moreover, for any x,y∈B¯δ2,m2z1, we get from (18) that(28)h1x−h1y=f−1x−gx−f−1y−gy≤λ10−1gx−gy+x−y≤λ10−1L+1x−y.
It follows from the second relation of (15) and (22) that(29)λ10>L+1,which together with (28) yields that h1 is contractive in B¯δ2,m2z1. It follows from the Banach contractive mapping principle and (27) that there exists a unique point z1∗∈Bδ2,m2z1 satisfying h1z1∗=z1∗. Consequently, Fz1∗=z1∗, that is, z1∗ is a fixed point of F in Bδ2,m2z1.
It should prove thatz1∗ is a regular expanding fixed point of F in some neighborhood of z1∗. To do this, take(30)r1∗=r1+z1−x102.
Then, it follows fromz1∗∈Bδ2,m2z1 and the first relation of (15) that(31)x10−z1∗=x10−z1+z1−z1∗≤x10−z1+z1−z1∗≤x10−z1+δ2,m2<x10−z1+r1−z1−x102−δ10=r1∗−δ10,which implies that B¯δ10x10⊂Br1∗z1∗. For any x∈Br1∗z1∗,(32)x−z1=x−z1∗+z1∗−z1≤x−z1∗+z1∗−z1<r1∗+δ2,m2<r1+z1−x102+r1−z1−x102−δ10=r1−δ10,which implies that B¯r1∗z1∗⊂Br1z1. Consequently, fBr1∗z1∗ is an open set. For any x,y∈B¯r1∗z1∗,(33)Fx−Fy=fx+gx−fy−gy≥fx−fy−gx−gy≥λ10−Lx−y.
Then, it follows from (29) and (33) that z1∗ is an expanding fixed point of F in B¯r1∗z1∗ with expanding coefficient λ10−L>1. Since fBr1∗z1∗ is an open set, it follows from Lemma 1 that(34)Bλ10−Lr1∗z1∗=Bλ10−Lr1∗Fz1∗⊂FBr1∗z1∗.which implies that z1∗ is an interior point of FBr1∗z1∗. Hence, z1∗ is a regular fixed point of F in B¯r1∗z1∗.
Here, it is to show thatFBr1∗z1∗ is an open set. For each given point y∈FBr1∗z1∗, there is a point x∈Br1∗z1∗ satisfying Fx=y. Then, there is a constant r¯1>0 satisfying Br¯1x⊂Br1∗z1∗. From the third paragraph of the proof, it is easy to see that fBr¯1x is an open set because of Br1∗z1∗⊂Br1z1. It also follows from Lemma 1 again that(35)Bλ10−Lr¯1y=Bλ10−Lr¯1Fx⊂FBr¯1x⊂FBr1∗z1∗,which implies that y is an interior point of FBr1∗z1∗ and then FBr1∗z1∗ is an open set.
With a similar argument to the existence ofz1∗, we can obtain the following positive constants(36)δ1,m1<minλ10δ10,r2−z2−x202−δ20,ifm1=1;minμ1,m1−1δ1,m1−1,r2−z2−x202−δ20,ifm1>1.ε2=λ20−1δ1,m11+δ1,m1,r2∗=r2+z2−x202,such that when(37)maxL,gz2<ε2,there exists a point z2∗∈Bδ1,m1z2 satisfying that z2∗ is a regular expanding fixed point of F in B¯r2∗z2∗ with expanding coefficient λ20−L>1 and FBr2∗z2∗ is an open set.
A summary for this step is given as follows. When the following condition holds(38)maxL,gz1,gz2<minε1,ε2,the map F will have two regular expanding fixed points z1∗∈Br1∗z1∗ and z2∗∈Bδ1,m1z2, and FBri∗zi∗ is an open set for i=1,2.Step 2.
It is to show that for eachi1≤i≤2 there exists a point yi0 in Bri∗zi∗ such that Fmiyi0=zti∗, where ti=imod2+1.
We first prove that there existm1 points y1j∈Bδ1jx1j, 0≤j≤m1−1 such that(39)Fy1j=y1,j+1,for0≤j≤m1−2,Fy1,m1−1=z2∗.
That is, there exists a pointy10∈Bδ10x10⊂Br1∗z1∗ such that Fm1y10=z2∗.
In order to do that, we first prove the existence ofy1,m1−1 by solving the following equation:(40)Fx=z2∗,x∈B¯δ1,m1−1x1,m1−1,which can be translated into the following:(41)fx=z2∗−gx,x∈B¯δ1,m1−1x1,m1−1.
It follows from assumption (ii) and (13) that f:B¯δ1,m1−1x1,m1−1⟶fB¯δ1,m1−1x1,m1−1 is homeomorphic with the inverse map f−1:fB¯δ1,m1−1x1,m1−1⟶B¯δ1,m1−1x1,m1−1 satisfying(42)f−1x−f−1y≤μ1,m1−1−1x−y,∀x,y∈fB¯δ1,m1−1x1,m1−1.
Then, equation (41) can be translated into the following:(43)f−1z2∗−gx=x,x∈B¯δ1,m1−1x1,m1−1.
Here, it needs to prove that(44)z2∗−gx∈fB¯δ1,m1−1x1,m1−1,x∈B¯δ1,m1−1x1,m1−1.
Suppose thatg also satisfies(45)maxL,gx1j,0≤j≤m1−1<ε3,where(46)ε3=minε1,ε2,minλ10δ10−δ11,μ1jδ1j−δ1,j+1,for1≤j≤m1−11+maxδ1j,0≤j≤m1−1.
From (14) and (36), we get that ε3>0. On the one hand, for any x∈B¯δ1,m1−1x1,m1−1, it follows from (45) and (46) that(47)z2∗−gx−z2=gx1,m1−1−gx−gx1,m1−1+z2∗−z2≤Lx−x1,m1−1+gx1,m1−1+z2∗−z2<ε3δ1,m1−1+ε3+δ1,m1=1+δ1,m1−1ε3+δ1,m1≤1+δ1,m1−1μ1,m1−1δ1,m1−1−δ1,m11+δ1,m1−1+δ1,m1=μ1,m1−1δ1,m1−1.
On the other hand, for anyx∈∂Bδ1,m1−1x1,m1−1, it follows from (13) that(48)fx−z2=fx−fx1,m1−1≥μ1,m1−1x−x1,m1−1=μ1,m1−1δ1,m1−1.
Sincez2∈fBδ1,m1−1x1,m1−1 and fBδ1,m1−1x1,m1−1 is open, it follows from (47) and (48) that (44) is true. So, we can define a map(49)h2x=f−1z2∗−gx,x∈B¯δ1,m1−1x1,m1−1.
It follows from (45) and (46) that(50)L<ε3<μ1,m1−1δ1,m1−1−δ1,m11+δ1,m1−1<μ1,m1−1δ1,m1−11+δ1,m1−1<μ1,m1−1.
Then, for anyx∈B¯δ1,m1−1x1,m1−1, it follows from (42) and (47) that(51)h2x−x1,m1−1=f−1z2∗−gx−f−1z2≤μ1,m1−1−1z2∗−gx−z2<δ1,m1−1.
That is,h2 maps B¯δ1,m1−1x1,m1−1 into itself. Moreover, for any x,y∈B¯δ1,m1−1x1,m1−1, it follows from (42) that(52)h2x−h2y=f−1z2∗−gx−f−1z2∗−gy≤μ1,m1−1−1gx−gy≤Lμ1,m1−1−1x−y,which together with (50) implies that h2 is contractive in B¯δ1,m1−1x1,m1−1. It follows from the Banach contractive mapping principle and (51) that there exists a unique point y1,m1−1∈Bδ1,m1−1x1,m1−1 such that hy1,m1−1=z2∗. Consequently, Fy1,m1−1=z2∗, that is, equation (40) has a unique solution y1,m1−1∈Bδ1,m1−1x1,m1−1.
Using a similar method as above, we can prove that there existm1−1 unique points y1j∈Bδ1jx1j for 0≤j≤m1−2 such that Fy1j=y1,j+1. Then, we get that y10∈Bδ10x10⊂Br1∗z1∗ such that Fm1y10=z2∗.
Next, setg also to satisfy(53)maxL,gx2j,0≤j≤m2−1<ε4,where(54)ε4=minε1,ε2,minλ20δ20−δ21,μ2jδ2j−δ2,j+1,for1≤j≤m2−11+maxδ2j,0≤j≤m2−1.
It follows from (14) and (15) that ε4>0.
Repeating a similar discussion as above, we can get wheng satisfies (53), there exist m2 unique points y2j∈Bδ2jx2j, 0≤j≤m2−1, such that Fy2j=y2,j+1 for 0≤j≤m2−2, and Fy2,m2−1=z1∗. That is, there exists a point y20∈Bδ20x20⊂Br2∗z2∗ such that Fm2y20=z1∗.
Letε0=minεj,1≤j≤4. If g satisfies the following condition(55)maxL,gzi,gxij,1≤i≤2,0≤j≤mi−1<ε0,then the statements in Step 2 hold. Consequently, F has a heteroclinic cycle Γ′ connecting repellers z1∗ and z2∗.Step 3.
It is to show that the heteroclinic cycleΓ′ connecting repellers z1∗ and z2∗ of F is regular and nondegenerate.
When the mapg satisfies (55), it follows from the discussions in Step 2 that yij∈Bδijxij for 1≤i≤2, 0≤j≤mi−1. Hence, for 1≤i≤2, 0≤j≤mi−1, we can take positive constants δij′<δij such that(56)yij∈Bδij′yij⊂Bδijxij.
It follows from (11), (13), and (56) that for 1≤i≤2, 1≤j≤mi−1,(57)Fx−Fy≥fx−fy−gx−gy≥λi0−Lx−y,∀x,y∈B¯ri∗zi∗,(58)Fx−Fy≥fx−fy−gx−gy≥μij−Lx−y,∀x,y∈B¯δij′yij,where λi0>L+1 and μij>L can be derived from (45), (46), (53), and (54).
For eachi1≤i≤2, since zi∗ is a regular expanding fixed point of F and FBri∗zi∗ is an open set, the backward orbit of yi0 lies in Bri∗zi∗ and (57) holds in some neighborhood of each point on the backward orbit. The forward orbit of yi0 consists of yij for 1≤j≤mi−1 and (58) holds for each yij in B¯δij′yij. Therefore, the heteroclinic cycle Γ′ connecting repellers z1∗ and z2∗ is nondegenerate. In addition, it is clear that F is continuous in Bri∗zi∗ and Bδij′yij for 1≤i≤2, 1≤j≤mi−1. It follows from (3) of Remark 2.2 in [25] that if we prove that for each point y0 on the cycle Γ′, there exists a positive constant r0 such that Fy0 is an interior point of FBr0y0, then this cycle Γ′ is regular.
Firstly, for each pointy0 on Γ′ lying in Bri∗zi∗, there exists a constant r0 such that B¯r0y0⊂Bri∗zi∗. It follows from (57) and Lemma 1, by using F to replace f and making g=0, that(59)Bλi0−Lr0Fy0⊂FBr0y0,which implies that Fy0 is an interior point of FBr0y0.
Secondly, for each pointyij, 1≤j≤mi−1, on Γ′ lying out Bri∗zi∗, it follows from (58) and Lemma 1, by using F to replace f and making g=0 again, that(60)Bμij−Lδij′Fyij⊂FBδij′yij,which implies that Fyij is an interior point of FBδij′yij.
Hence, the cycleΓ′ is regular. That is, the heteroclinic cycle Γ′ connecting repellers z1∗ and z2∗ of F is regular and nondegenerate. Consequently, it follows form Lemma 2 that there exists an uncountable, perfect, bounded, and closed set V such that system (9) is chaotic on V in the sense of both Devaney and Li-Yorke. This completes the proof.Remark 4.
Theorem1 gives a relatively explicit range of the Lipschitz perturbation g which is characterized by a constant ε0 determined by the properties of the original map f. From (10), we see that it only needs L and the values of g at zi, xij are less than ε0, and it does not need to compute all the values of g in some domains. Hence, the conditions about g in Theorem 1 are relatively easy to check out in practice. In addition, it only needs the original map f to be continuous near some points of interest without having to be continuous in the whole space.Remark 5.
From the proof of Theorem1, it is easy to see that the perturbed map F will have a regular and nondegenerate heteroclinic cycle Γ′ connecting repellers if the unperturbed map f with a regular and nondegenerate heteroclinic cycle Γ connecting repellers undergoes a small perturbation, and the cycle Γ′ is near to Γ. The perturbed range of g is characterized by ε0 determined in Theorem 1. Thus, this result can be viewed as persistence of regular and nondegenerate heteroclinic cycles connecting repellers in Banach spaces.
When the original mapf is continuously differentiable in some domains of interest, using a similar method to Theorem 1, we can get the following result.Theorem 2.
LetX,⋅ be a Banach space and f:X⟶X be a map with k≥2 different fixed points z1,…,zk∈X. Assume that(i)
For eachi1≤i≤k, f is continuously differentiable in Bri′zi for some constant ri′>0 and Dfzi is an invertible linear map satisfying Dfzi0>1, which is equivalent to that there exists a positive constant ri≤ri′ such that zi is a regular expanding fixed point of f in B¯rizi.(ii)
f has a heteroclinic cycle Γ connecting repellers z1,…,zk.(iii)
f is continuously differentiable in some neighborhood Ux0 of each point x0 on the cycle Γ, and Dfx0 is an invertible linear map.Then, there exists a constantε0>0 such that for any Lipschitz map g in each set of B¯rizi and Ux0 for x0∈Γ, with the Lipschitz constant L satisfying(61)maxL,gx0forx0∈Γ<ε0,the results of Theorem 1 hold.Proof.
It follows from the assumptions in Theorem2 and Lemma 3 that f has a regular and nondegenerate heteroclinic cycle Γ connecting repellers z1,…,zk. For each i1≤i≤k, since zi is a regular expanding fixed point of f, there exist a point xi0∈Brizi and a positive integer mi≥1 such that fxi0∉Brizi and fmixi0=zti, where ti=imodk+1. The rest of the proof is similar to that of Theorem 1, so it is omitted.
For a functionf∈C1U,X, the following norm is often used(62)fC1,U:=supfx,Dfx,x∈U⊂X.
Therefore, if the conditions in Theorems1 and 2 about g are replaced by those based on the above norm, then we can obtain two consequences of Theorems 1 and 2. For convenience, we list them as the following theorems.Theorem 3.
Suppose thatX,⋅ is a Banach space, f:X⟶X is a map with k≥2 different fixed points z1,…,zk∈X and satisfies the conditions (i) and (ii) in Theorem 1. Then, there exists a constant ε0>0 such that for any g∈C1U,X with gC1,U<ε0, the results of Theorem 1 hold, where U=Brizi∪∪j=1mi−1Uij,1≤i≤k.Theorem 4.
Suppose thatX,⋅ is a Banach space, f:X⟶X is a map with k≥2 different fixed points z1,…,zk∈X and satisfies the conditions (i)–(iii) in Theorem 2. Then, there exists a constant ε0>0 such that for any g∈C1U,X with gC1,U<ε0, the results of Theorem 1 hold, where U=∪x0∈ΓUx0.
At the last of this section, we discuss a usually used Banach spaceRn, which is the Euclidean space. As is well known, there are many different norms in Rn. A map in Rn can expand in different norms, see [11, 26] and references therein. It is natural to ask whether there is the persistence of heteroclinic cycles connecting repellers in Rn, where the repellers expand in different norms. The following Theorem 5 will answer this question.The usually used Euclidean norm is denoted by(63)x=∑j=1nxj21/2,x=x1,…,xnT∈Rn.In the following, we will use the neighborhood of a pointx∈Rn in different norms. For convenience, let B¯rx and Brx denote the closed and open balls of x with radius r in the Euclidean norm ⋅, let N¯rx and Nrz denote the closed and open balls of x with radius r in any other norm ⋅′.Now, we establish a result on persistence of heteroclinic cycles connecting repellers inRn, where the repellers expand in different norms.Theorem 5.
Suppose that a mapf:Rn⟶Rn has k≥2 different fixed points z1,…,zk∈Rn and satisfies the following conditions:(i)
for eachi1≤i≤k, f is continuously differentiable in some neighborhood of zi and all the eigenvalues of Dfzi have absolute values larger than 1, which implies that there exist a constant ri>0 and a norm ⋅i in Rn such that f is continuously differentiable in N¯rizi, and zi is a regular expanding fixed point of f in N¯rizi.(ii)
for eachi1≤i≤k, there exist a point xi0∈Nrizi, xi0≠zi, and a positive integer mi≥1 such that fmixi0=zti, where ti=imodk+1. Furthermore, f is continuously differentiable in some neighborhood Uij of xij and satisfies that detDfxij≠0, where xij=fjxi0 for 1≤j≤mi−1.Then, for any Lipschitz mapg with Lipschitz constant L in the Euclidean norm ⋅ in each set of N¯rizi and Uij, 1≤i≤k, 1≤j≤mi−1, there exists a constant ε0>0 satisfying(64)maxL,gzi,gxij,1≤i≤k,0≤j≤mi−1<ε0,such that the perturbed system (9) is chaotic in the sense of both Devaney and Li-Yorke on a compact and perfect set which contains a Cantor set.Proof.
Without loss of generality and for simplicity, we also only show that Theorem5 holds for k=2.
For convenience, leti=1 or 2 in the rest of proof. As pointed in the second paragraph of the proof in Theorem 1, we can also suppose that Nr1z1∩Nr2z2=∅ and fxi0∉Nrizi.
Since all the norms onRn are equivalent by Corollary 3.14 of Chapter II in [36], there exist positive constants b11, b12, ci1 and ci2 such that(65)b11⋅1≤⋅2≤b12⋅1,ci1⋅i≤⋅≤ci2⋅i.
Sinceg is a Lipschitz map with Lipschitz constant L in the Euclidean norm ⋅ in N¯rizi and Uij, for any x,y∈N¯rizi and any x,y∈Uij, 1≤j≤mi−1, it follows from (65) that(66)gx−gyi≤ci1−1gx−gy≤ci1−1Lx−y≤ci1−1ci2Lx−yi≤L′x−yi,where(67)L′=maxci1−1ci2L,i=1,2.
Then it follows from (66) that g is also a Lipschitz map with Lipschitz constant L′ in the norm ⋅i in N¯rizi and Uij.
It follows from assumption (i) that there exists a constantλi0>1 such that(68)fx−fyi≥λi0x−yi,∀x,y∈N¯rizi,f:N¯rizi⟶fN¯rizi is a homeomorphism and fNrizi is open, fD is open for any open set D⊂Nrizi. Take a constant(69)δi0<ri−zi−xi0i2such that N¯δi0xi0⊂Nrizi. Then, it follows form (68) that f:Nδi0xi0⟶fNδi0xi0 is also a homeomorphism.
In addition, it follows fromdetDfxij≠0, 1≤j≤mi−1, that none of the eigenvalues of Dfxij is 0. Therefore, DfxijTDfxij is positive definite. Then,(70)‖Dfxij‖0=inf‖x‖=1xTDfxijTDfxijx12>0,where x∈Rn. It follows from (65) and (70) that(71)Dfxiji0=infx≠0Dfxijxixi≥ci1ci2−1infx≠0Dfxijxx=ci1ci2−1Dfxij0>0,
Hence, it follows from (71) and Lemma 4 that there exist positive constants μij and δij such that(72)fx−fyi≥μijx−yi,∀x,y∈N¯δijxij,which implies that f:Nδijxij⟶fNδijxij is homeomorphic, and fNδijxij is open for 1≤j≤mi−1, where δij satisfies the following conditions:(73)δi1<λi0δi0,δi,j+1<μijδij,for1≤j≤mi−2,N¯δijxij are disjoint subsets of Uij and N¯δijxij∩Nrizi=∅ for fixed i and 1≤j≤mi−1.
The rest of the proof is almost exactly the same to Steps1–3 in the proof of Theorem 1 except for three aspects. One is that L is replaced by L′ and the domains in the norm ⋅ are replaced by those in the norms ⋅1 or ⋅2, respectively. In brief, in the representations of the domains, the alphabet B is replaced by the alphabet N through the proof of Theorem 1. The second is that some values in the norm ⋅ are replaced by those in the norms ⋅1 or ⋅2, respectively. It is pointed out that (18) and (42) take the values in the norm ⋅1, the remainders follow the following rule: if the independent variables of functions are taken from Nrizi or Nδijxij, then the values in the norm ⋅ are replaced by those in the norm ⋅i. The third is that some related constants used in the proof are slightly modified since the norm ⋅ is replaced by the norms ⋅1 or ⋅2. For convenience, we list them as follows.(74)δ1,m1<minb11λ10δ10,r2−z2−x2022−δ20,ifm1=1;minb11μ1,m1−1δ1,m1−1,r2−z2−x2022−δ20,ifm1>1,δ2,m2<minb12−1λ20δ20,r1−z1−x1012−δ10,ifm2=1;minb12−1μ2,m2−1δ2,m2−1,r1−z1−x1012−δ10,ifm2>1,r1∗=r1+z1−x1012,r2∗=r1+z2−x2022,ε1=λ10−1δ2,m21+δ2,m2,ε2=λ20−1δ1,m11+δ1,m1,ε3=minε1,ε2,minλ10δ10−δ11,μ1,m1−1δ1,m1−1−b11−1δ1,m1,μ1jδ1j−δ1,j+1,for1≤j≤m1−21+maxδ1j,0≤j≤m1−1,ε4=minε1,ε2,minλ20δ20−δ21,μ2,m2−1δ2,m2−1−b12δ2,m2,μ2jδ2j−δ2,j+1,for1≤j≤m2−21+maxδ2j,0≤j≤m2−1.
It follows from (73) and the above conditions that εj>0 for 1≤j≤4. Here, it should explain how to take the values of the numerators in terms of m1 and m2 in ε3 and ε4. We only explain the values in terms of m1, while that for m2 is similar. For the numerator of the fraction at the right side of ε3, when m1=1, it only takes the first term; when m1=2, it only takes the first two terms; when m1≥3, it takes all of the terms.
Set(75)ε0′=minεj,1≤j≤4,ε0=minε0′maxc11−1c12,c21−1c22,c11ε0′,c21ε0′.
Ifg satisfies condition (64) in the Euclidean norm ⋅, that is the following:(76)maxL,gzi,gxij,1≤i≤2,0≤j≤mi−1<ε0,then it follows from (65), (75) and (76) that g also satisfies the following condition in the norms ⋅1 and ⋅2(77)maxL′,gzii,gxiji,1≤i≤2,0≤j≤mi−1<ε0′,
Therefore, repeating the Steps1–3 in the proof of Theorem 1, we can get that if g satisfies (76), consequently (77), then F has a regular and nondegenerate heteroclinic cycle Γ′ connecting repellers z1∗ and z2∗ in different norms ⋅1 and ⋅2. It follows from the proof of Theorem 4.1 in [26] that there exists a positive integer p such that Fp has a heteroclinic cycle Γ∗ connecting repellers z1∗ and z2∗ in the unified Euclidean norm ⋅. Since all the points on the cycle Γ∗ of Fp also lie on the cycle Γ′ of F, it is easy to prove that the cycle Γ∗ of Fp is also regular and nondegenerate. Consequently, F is chaotic on a compact and perfect set which contains a Cantor set in the sense of both Devaney and Li-Yorke. This completes the proof.Remark 6.
From the proof of Theorem5, we obtain that Fp has a regular and nondegenerate heteroclinic cycle connecting repellers in the unified Euclidean norm ⋅ for some positive integer p. Hence, Theorem 5 can also be regarded as the persistence of a regular and nondegenerate heteroclinic cycle connecting repellers in Rn. In the special case that all the norms ⋅i, 1≤i≤k, in assumption (i) become a unified norm, such as the Euclidean norm ⋅, then the positive integer p becomes 1. Hence, this special case of Theorem 5 is consistent with Theorem 1.
The following result is a direct consequence of Theorem5.Theorem 6.
Suppose that a mapf:Rn⟶Rn has k≥2 different fixed points z1,…,zk∈Rn and satisfies the following conditions(i)
For eachi1≤i≤k, zi is an expanding fixed point of f in some norm ⋅i;(ii)
f has a k-heteroclinic cycle Γ connecting fixed points z1,…,zk and is continuously differentiable in some neighborhood Ux0 of each point x0∈Γ satisfying detDfx0≠0.Then, there exists a constantε0>0 such that for any Lipschitz map g in each set of ∪x0∈ΓUx0 with Lipchitz constant L in the Euclidean norm ⋅ satisfying(78)maxL,gx0,forx0∈Γ<ε0,the results in Theorem 5 hold.Remark 7.
[28] studied the persistence of heteroclinic repellers in Rn for C1 maps with C1 perturbations, where the maps needed to be continuously differentiable in the whole space. Here, it only needs the maps to be continuously differentiable in some neighborhoods of points. The main differences between the above two theorems and the result in [28] are as follows. One is that Theorems 5 and 6 studied the Lipschitz perturbations, while the latter considered the C1 perturbations. The second is that Theorems 5 and 6 give an explicit expression for the range of perturbations, which are determined by the properties of the original maps, while the latter did not give such a range for perturbations. The third is that Theorems 5 and 6 use different norms for expansions of fixed points which are more general in practice, while the latter only used a single norm for expansions of fixed points.Remark 8.
Just as Theorems3 and 4, if the perturbed term g is continuously differentiable, then the conditions about g in Theorems 5 or 6 can be replaced by that g∈C1U,X with gC1,U<ε0, where U is taken the corresponding domains used in Theorems 5 or 6, respectively, then the results in Theorems 5 or 6 hold.


## 4. Examples

In this section, three examples are given to illustrate the validity of the theoretical results.Example 3.
The original mapf is taken as the following map on R:(79)fx=2x,ifx∈−2,25x−9,ifx∈2,2.50.1x−0.2875,else.
The perturbed mapg is taken as gx=γx, where x∈R and γ is a positive real number. It is obvious that f is piecewise continuous on R, g is a Lipschitz map with a Lipschitz constant L=γ and it is not differentiable on R.
It is easy to see thatz1=0 and z2=2.25 are two regular expanding fixed points of f. Set x10=1.125∈−2,2, then fx10=z2. Set x20=2.375∈2,2.5, then x21=fx20=2.875 and fx21=z1, that is, f2x20=z1. So, f has a 2-heteroclinic cycle Γ connecting repellers z1 and z2. It is clear that the cycle is regular and nondegenerate, and assumptions (i) and (ii) in Theorem 1 holds with k=2, r1=2, r2=0.25, λ10=2, λ20=5, m1=1, m2=2, x10 and x20 as the above.
Some constants that appear in the proof of Theorem1 are taken as follows: μ21=0.1, δ10=0.43<r1−z1−x10/2=0.4375, δ20=0.06<r2−z2−x20/2=0.0625, δ11=0.002<minλ10δ10,r2−z2−x20/2−δ20=0.0025, δ21=0.28<λ20δ20=0.3, δ22=0.007<minμ21δ21,r1−z1−x10/2−δ10=0.0075, ε1=λ10−1δ22/1+δ22≈0.006951, ε2=λ20−1δ11/1+δ11≈0.007984, ε3=minε1,ε2,δ10λ10−δ11/1+δ10=0.006951, ε4=minε1,ε2,minλ20δ20−δ21,μ21δ21−δ22/1+maxδ20,δ21=0.006951, ε0=minεj,1≤j≤4=0.006951. It is easy to check that the perturbation g satisfies condition (10) for γ≤0.0024. Then, it follows from the result of Theorem 1 that the perturbed system F=f+g also has a regular and nondegenerate heteroclinic cycle Γ′ connecting repellers which is near to Γ. Consequently, F and f are chaotic in the sense of both Devaney and Li-Yorke. For illustrating the persistence of a heteroclinic cycle connecting repellers, we take γ=0.002 for example. It is easy to calculate the following results. The perturbed map F has two regular expanding fixed pints z1∗=0 and z2∗=1500/667≈2.248876. There exist two points x10∗=750000/667667≈1.123314 and x20∗=301375/127551≈2.362780 such that Fx10∗=z2∗, x21∗=Fx20∗=575/204≈2.818627 and Fx21∗=z1∗. Then, F has a 2-heteroclinic cycle Γ′ connecting repellers z1∗=0 and z2∗=0. It is clear that Γ′ is near to Γ. With the increase of γ, the heteroclinic cycle Γ′ will gradually run away from Γ until it breaks or disappears. Since f and F are chaotic on some intervals of R and the computer simulations of them are on intervals, we omit the computer simulations.Example 4.
The original mapf is taken as the following map on R2(80)fx,y=8x,y,ifx,y∈B¯10,0,2x−2,2y−2,ifx,y∈B¯40,0/B¯10,0,sinx−2−π2+y−2−π22,sinx−2−π22+y−2−π2,ifx,y∉B¯40,0.
This map is used as an example in [25] for illustrating chaos induced by a heteroclinic cycle connecting repellers. The perturbed map g is taken as gx,y=γx,y, where x,y∈R2 and γ is a real number. It is obvious that f is only continuously differentiable in some domains of R2, g is continuously differentiable in R2 and has a Lipschitz constant L=γ.
On the one hand, it is clear thatz1=0,0 and z2=2,2 are two fixed points of f, f is continuously differentiable in B¯1z1, B¯1.171z2, and satisfies that(81)Dfz1=8I2,Dfz2=2I2,where I2 is the identity matrix. So, the eigenvalues of Dfz1 and Dfz2 have absolute values larger than 1, which implies that z1 and z2 are two regular expanding expanding fixed points of f in B¯z1 and B¯1.171z2 in the Euclidean norm ⋅ with λ10=8, λ20=2, respectively. It is obvious that B¯1z1∩B¯1.171z2=∅ and both lie in B4z1. Set x10=1/4,1/4∈B1z1, then fx10=z2. Set x20=2+π/8,2+π/8∈B1.171z2, then x21=fx20=2+π/4,2+π/4∈B4z1\B1z1, x22=fx21=2+π/2,2+π/2∉B¯4z1 and fx22=z1, that is, f3x20=z1.
On the other hand, it is also obvious thatf is continuously differentiable in some neighborhoods of x10, x20, x21, and x22 and satisfies(82)Dfx10=8I2,Dfx20=Dfx21=2I2,Dfx22=I2.
Then, it follows from (82) that(83)Dfx100=8,Dfx200=Dfx210=2,Dfx220=1,which together with Lemma 4 imply that the cycle Γ is nondegenerate. Furthermore, it follows from (4) of Remark 2.2 in [25] that the cycle Γ is also regular. Consequently, f has a regular and nondegenerate 2-heteroclinic cycle Γ connecting the repellers z1 and z2.
Therefore, assumptions (i) and (ii) in Theorem5 hold with k=2, r1=1, r2=1.171, λ10=8, λ20=2, m1=1, m2=3, x10 and x20 as the above. Consequently, f has a 2-heteroclinic cycle Γ connecting repellers z1 and z2. As is pointed out in Remark 6, when the norms used in Theorem 5 become a unified norm, the special case of Theorem 5 is consistent with Theorem 1. So, we can take some constants that appear in the proof of Theorem 1 as follows:(84)μ21=2,μ22=1,δ10=0.2<r1−z1−x102≈0.323223,δ20=0.175<r2−z2−x202≈0.307820,δ11=0.1328<minλ10δ10,r2−z2−x202−δ20=0.132820,δ21=0.2<λ20δ20=0.35,δ22=0.2<μ21δ21=0.4,δ23=0.025<minμ22δ22,r1−z1−x102−δ10=0.123223,ε1=λ10−1δ231+δ23≈0.170732,ε2=λ20−1δ111+δ11≈0.117232,ε3=minε1,ε2,δ10λ10−δ111+δ10=0.117232,ε4=minε1,ε2,minλ20δ20−δ21,μ21δ21−δ22,μ22δ22−δ231+maxδ20,δ21,δ22=0.117232,ε0=minεj,1≤j≤4=0.117232. It is easy to check that the perturbation g satisfies condition (10) for γ≤0.0297. Then, it follows from the result of Theorem 5 that the perturbed system F=f+g also has a regular and nondegenerate heteroclinic cycle Γ′ connecting repellers which is near to Γ. Consequently, F and f are chaotic in the sense of both Devaney and Li-Yorke.
As is done in Example3, for a given γ, one can also directly calculate the heteroclinic cycle Γ′ of F to check whether it is near to Γ of f. However, it is not easy to directly calculate such a cycle for high-dimensional maps. If there is the persistence of a heteroclinic cycle connecting repellers, then the computer simulations of them will not change very much. The behaviors of the unperturbed map f with an initial point x,y=0.1,0.1 are illustrated in Figure 1. We do some computers simulations of F as γ increases from −0.0297 to 0 or from 0 to 0.0297, and find that all the simulations are similar with that of the original map f in Figure 1. Here, we give one simulation of F with an initial point x,y=0.1,0.1 for γ=0.0297, see Figure 2. We can see that Figure 2 is a small change to Figure 1, which shows that the heteroclinic cycle Γ′ of F is near to Γ of f. When we let γ continuous to increase, we find that the computer simulations gradually change until there is a big difference from that of the original map. This shows that the heteroclinic cycle Γ breaks or disappears. Are there new heteroclinic cycles connecting repellers not near Γ or new snap-back repellers to make the perturbed system still chaotic? It is an interesting question, while it is out of the scope of this paper and will be our further study.Figure 1
Complex behaviors of the original mapf in the x,y space, where the initial point is take as 0.1,0.1 and n=0,1,2,…,20000.Figure 2
Complex behaviors of the perturbed mapF in the x,y space, where γ=0.0297, the initial point is taken as 0.1,0.1 and n=0,1,2,…,20000.Example 5.
The original mapf is taken as the following map on R3(85)fx,y,z=6x,6y,6z,ifx,y,z∈B¯1O,4x−9,4y−9,4z−9,ifx,y,z∈B¯8O/B¯1O,sinx−5+y−52,siny−5+z−52,sinx−52+z−5,ifx,y,z∉B¯8O,where O=0,0,0 is the origin. The perturbed map g is taken as gx,y,z=γx,y,z, where x,y,z∈R3 and γ is a real number. It is obvious that f is only continuously differentiable in some domains of R3, g is continuously differentiable in R3 and has a Lipschitz constant L=γ.
Theorem5 is also used to verify the persistence of a heteroclinic cycle connecting repellers, and the process is similar to that of Example 4. So, we omit some details and only give some main results as follows. Assumptions (i) and (ii) in Theorem 5 hold with k=2, z1=0,0,0, z2=3,3,3, r1=1, r2=2.8, λ10=6, λ20=4, m1=1, m2=2, x10=0.5,0.5,0.5∈B1z1 and x20=3.5,3.5,3.5∈B2.8z2⊂B8O. The points z1 and z2 are two regular expanding fixed points of f in the Euclidean norm ⋅. In addition, fx10=z2, x21=fx20=5,5,5∉B¯8O and fx21=z1, that is, f2x20=z1. Then f has a regular and nondegenerate 2-heteroclinic cycle Γ connecting the repellers z1 and z2. Some constants that used to determine the range of perturbations are taken as follows: (86)μ21=1,δ10=0.03<r1−z1−x102≈0.066987,δ20=0.2<r2−z2−x202≈0.966987,δ11=0.04<minλ10δ10,r2−z2−x202−δ20=0.18,δ21=0.2<λ20δ20=0.8,δ22=0.03<minμ21δ21,r1−z1−x102−δ10=0.036987,ε1=λ10−1δ221+δ22≈0.145631,ε2=λ20−1δ111+δ11≈0.115385,ε3=minε1,ε2,δ10λ10−δ111+δ10=0.115385,ε4=minε1,ε2,minλ20δ20−δ21,μ21δ21−δ221+maxδ20,δ21=0.115385,ε0=minεj,1≤j≤4=0.115385. It is also easy to check that the perturbation g satisfies condition (10) for γ≤0.0133. Then, it follows from the result of Theorem 5 that the perturbed system F=f+g also has a regular and nondegenerate heteroclinic cycle Γ′ connecting repellers which is near to Γ. Consequently, F and f are chaotic in the sense of both Devaney and Li-Yorke.
The behaviors of the unperturbed mapf with an initial point x,y,z=0.1,0.1,0.1 are illustrated in Figure 3. We also do some computers simulations of F as γ increases from −0.0133 to 0 or from 0 to 0.0133, and find that all the simulations are also similar with that of the original map f in Figure 3. Here, we give one simulation of F with an initial point x,y,z=0.1,0.1,0.1 for γ=0.0133, see Figure 4. We can see that Figure 4 is also a small change to Figure 3, which shows that the heteroclinic cycle Γ′ of F is near to Γ of f. When we let γ continuous to increase, we also find that the computer simulations gradually change until there is a big difference from that of the original map. This shows that the heteroclinic cycle Γ breaks or disappears.Figure 3
Complex behaviors of the original mapf in the x,y,z space, where the initial point is take as 0.1,0.1,0.1 and n=0,1,2,…,20000.Figure 4
Complex behaviors of the perturbed mapF in the x,y,z space, where γ=0.0133, the initial point is taken as 0.1,0.1,0.1 and n=0,1,2,…,20000.Remark 9.
In the above examples, it only needs the Lipschitz constantL and the values of g at zi, xij for 1≤i≤2, 0≤j≤mi−1 to satisfy condition (10), and does not need to compute the values of g at any other points. This is very easy to check out and is very convenient in applications. Since there are few literature giving concrete methods to identify an exact expanding area of a fixed point, it is very hard to get the largest perturbation range. But we think that the results obtained in this paper are also useful in practice. Because when a perturbation range ε0 is determined as in the above examples, it can ensure that the persistence is maintained for a large range of parameters. The perturbation range obtained in these examples may not be the largest one for the persistence to be maintained. A more precise perturbed range is needed in practice and this will also be our further research.


## 5. Conclusions

In this paper, we studied persistence of heteroclinic cycles connecting repellers in Banach spaces. We proved that if a map with a regular and nondegenerate heteroclinic cycle connecting repellers undergoes a small Lipschitz perturbation, then the perturbed map still has a regular and nondegenerate heteroclinic cycle connecting repellers. Consequently, the perturbed map and the original map are simultaneously chaotic in the sense of both Devaney and Li-Yorke. We believe that the results obtained in the paper will be useful for studying the existence of chaos and will provide certain theoretical basis for practical applications of heteroclinic cycles of connecting repellers. Compared with some related papers, three major achievements on the persistence are summarized as follows. One is that the maps discussed in the paper only need to be continuous or continuously differentiable in some domains instead of the whole space. Since a lot of maps may not be continuous or continuously differentiable in the whole space, our results are more general in practice than those in some related papers. The second is that an explicit expression for the range of perturbations is given, while most related papers did not give such an expression. The expression is determined by some properties of the original maps. It only needs to check out some values of the perturbation map at certain points in practice. This is very convenient and has great potential in applications. The third is that different repellers are allowed to expand in different norms inRn, while some related papers only used the single Euclidean norm to do that. This is very meaningful since it is more general in practice for some fixed points to expand in different norms. To show the validity of the theoretical results, we give some illustrative examples. However, the range of perturbations obtained in this paper is only a sufficient condition for the persistence to be maintained, and it may not be the largest one. Since it is hard to determine the exact area of a fixed point and few researches have given concrete methods to do this, it is not easy to find the largest range of perturbations and this will be our further research.



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