# Heteroclinic Cycles Imply Chaos and Are Structurally Stable

Authors: Xiaoying Wu
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## 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 maph has heteroclinic cycles in ℝn, then g has heteroclinic cycles with h−gC1 being sufficiently small. The results demonstrate C1 structural stability of heteroclinic cycles. In the end, two examples are given to illustrate our theoretical results and applications.


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## Body

## 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 f:ℝn⟶ℝn has positive topological entropy if f 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 ℝn [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 C1 structural stability of snapback repellers in ℝn. 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 C1 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:(1)xn+1=fxn,n=1,2,…,where f is a continuous map from ℝn 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 ℝn; 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.
Leth be a continuous differential map from ℝn into itself. If h has heteroclinic cycles connecting expanding periodic points, then h is chaotic in the sense of Devaney.Theorem 2.
Letg,h be continuous differential maps from ℝn into itself. If h has heteroclinic cycles connecting expanding periodic points and g−hC1 is sufficiently small, where g−hC1=maxsupx∈Xgx−hx,supx∈XDgx−Dhx,g 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 Section2, 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.
Letf:ℝn⟶ℝn be a continuous differentiable map. f is expanding at a point x0 if the derivative operator Dfx0 is invertible and the norm of each of its eigenvalues is larger than 1. f is expanding in a nonempty subset U of ℝn if there exists positive constant λ>1 such that, for any x,y∈U, fx−fy≥λx−y.It is well known that iff is differentiable, then f is expanding at a point x0 which is equivalent to the fact that there exist a norm ⋅ in ℝn and a constant λ>1 such that ∀x∈ℝn,Dfx0x≥λx [3]. The following heteroclinic repellers in ℝn were presented by Lin and Chen in [17].where ti≡imodk.Definition 2.
(see [17]). Let f:ℝn⟶ℝn be a continuous differentiable map and the integer k≥2 be fixed. The fixed points x1,x2,…,xk of f are called heteroclinic repellers if for any integer 1≤i≤k, there are(1)
f is expanding at the point xi(2)
f has a forward orbit Γi=zi−nn≥0 of xti+1 such that ∀n∈ℕ∩1,+∞,fzi−n=zi−n+1 and zi0=xti+1,limn⟶+∞zi−n=xi(3)
f is continuously differentiable in a neighborhood of any point z on Γi and Dfz is invertible,Based on Definition2, we introduce the following new concept.Definition 3.
Letf be a continuous differentiable map from ℝn into itself and k≥1 be fixed integer. Let f have k different periodic points x1,x2,…,xk with periods n1,n2,…,nk, respectively. For any 1≤m≤ni and i≠j, fmxi≠xj. For any 1≤i≤k, suppose that(1)
f is expanding at every point x which is in the periodic orbits of xi.(2)
f has a forward orbit Γi=zi−nn≥0 of xti+1 connecting periodic points xi and xti+1. That is, it satisfies that ∀n∈ℕ∩1,+∞,fzi−n=zi−n+1, zi0=xti+1 and there exists a positive integer mi such that limn⟶+∞zi−nni−mi=xi.(3)
For any pointz on Γi, the linear operator Dfz:ℝn⟶ℝn is invertible.
Then, the set∪i=1kΓi is called a heteroclinic cycle connecting expanding periodic points x1,x2,…,xk.Remark 1.
By Definition3, for any point z in heteroclinic cycles connecting expanding periodic points x1,x2,…,xk, Dhz is invertible and the norm of each of its eigenvalues is larger than 1. Thus, x1,x2,…,xk are heteroclinic repellers when every periodic point is fixed point in Definition 3 and k≥2. Example 1 shows that heteroclinic cycles connecting expanding periodic points are different from heteroclinic repellers.To prove theC1 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 X,⋅ and Y,⋅ be Banach spaces, Λ,d be a metric space, and U be an open set of Λ×X. Suppose that F:U⟶Y is a continuous differentiable map and there exists a point λ0,x0∈U satisfying the following conditions:(1)
DFxλ,x, the Fr e´ chet partial derivative of Fλ,x with respect to x, is continuous with respect to x in some neighborhood of λ0,x0(2)
DFxλ0,x0:X⟶Y is an invertible linear operator(3)
Fλ0,x0=0,
Then, there exist open ballBr1x0=x|x−x0<r1 and Bδλ0=λ|dλ,λ0<δ, where r1>0,δ>0, such that for any λ∈Bδλ0, Fλ,x=0, the unique continuous solution x=hλ∈Br1x0 exists and x0=hλ0 is satisfied.Lemma 2.
(see [21]). Let Y,⋅ and Z,⋅ be two Banach spaces and V0=Bry0 be r-open ball neighborhood of y0. Assume that h is a Cp map from V0 into Z such that hy0=z0p≥1 and Dhy0 is an invertible linear operator from Y into Z. Then, there are constants r0>0,δ0>0,λ0>0 and a map f from Vh,λ0×Bδ0z0 into Br0y0 satisfying the following conditions:(1)
For anyφ∈Vh,λ0, φ is one to one on Br0y0, and Dφy is invertible for every y∈Br0y0(2)
For anyφ∈Vh,λ0 and z∈Bδ0z0, there is a unique fφ,z∈Br0y0 satisfying φfφ,z=z(3)
The mapf is a continuous map from Vh,λ0×Bδ0z0 into Br0y0(4)
For everyφ∈Vh,λ0, fφ,⋅:Bδ0z0⟶Br0y0 is a p-order continuous differentiable map, and Dzfφ,z=Dφfφ,z−1,where(2)Vh,λ0=φ|φ−hC1<λ0,φ∈CpV0,Z,denotes the λ0−C1-open ball neighborhood of h and CpV0,Z is a p-order continuous differentiable map set.Lemma 3.
Leth be a continuous differentiable map from ℝn into itself. Let h have k≥2 periodic points x1,x2,…,xk with periods n1,n2,…,nk, respectively. If h is expanding at every point x which is in the periodic orbit of xi, then for all 1≤i≤k, there exist positive constants ri and λ>0 such that φni is homeomorphism on closed ball B¯ri0xi for every φ∈Vh,λ and ri0<ri, φni is expanding on B¯ri0xi and(3)Bri0xi⊂φniBri0xi,B¯ri0xi⊂φniB¯ri0xi.Proof.
For any1≤i≤k, because h is expanding at every point x which is in the periodic orbit of xi, by the continuity of h, there exists some closed neighborhood B¯rixi of xi such that h is expanding in hmB¯rixi for 0≤m≤ni−1. Therefore, there exists a positive constant λi>1 such that, for all 0≤m≤ni−1 and for any x,y∈hmB¯rixi, there is(4)hx−hy≥λix−y.
Letλi1∈1,λi. Thus, for every φ∈Vh,λi−λi1 and x,y∈Brixi, there is(5)φnix−φniy≥hφni−1x−hφni−1y−φ−hφni−1x−φ−hφni−1y>λi−λi−λi1φni−1x−φni−1y=λi1φni−1x−φni−1y>λi1nix−y.
Becausehnixi=xi and Dhnixi:ℝn⟶ℝn is invertible, by Lemma 2, there are constants 0<λi′<λi−λi1,0<ri0<ri such that for every φ∈Vh,λi′, φni:Bri0xi⟶φniBri0xi is a differentiable homeomorphism. Therefore, for any 1≤m≤ni, φ:φm−1Bri0xi⟶φmBri0xi is a differentiable homeomorphism and(6)φmB¯ri0xi⊃φmB¯ri0xi¯,φm∂Bri0xi⊃∂φmBri0xi,where ∂Bri0xi denotes the boundary of the set Bri0xi and φmBri0xi¯ is the closure of the set φmBri0xi. Take λi2∈maxλi1,λi+ri0/1+ri0,λi. For every φ∈Vh,λi−λi2/2,x∈∂Bri0xi, there is(7)φx−φxi≥hx−hxi−φx−hx−φxi+hxi>λi2ri0−λi−λi2>ri0.
Hence,φBri0xi⊃Bri0φxi,φB¯ri0xi⊃B¯ri0φxi. Inductively, for every 0≤m≤ni−1, there are(8)φBri0φmxi⊃Bri0φmxi,φB¯ri0φmxi⊃B¯ri0φmxi.
Letλ=minλi′,λi−λi2|1≤i≤k, and for every φ∈Vh,λ, there are(9)Bri0xi⊂φniBri0xi,B¯ri0xi⊂φniB¯ri0xi.
The 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.
Leth be a continuous differential map from ℝn into itself. If h has heteroclinic cycles connecting expanding periodic points, h is chaotic in the sense of Devaney.Proof.
Letx1,x2,…,xk be k≥1 different periodic points with periods n1,n2,…,ni, respectively, and h be expanding at x1,x2,…,xk, respectively. Let ∪i=1i=kΓi=zi−nn≥0 be heteroclinic cycles connecting periodic points x1,x2,…,xk.
For any integeri∈1,k and any integer j≥0, by Definition 3, Dhhjxi is invertible and the norm of each of its eigenvalues is larger than 1. Therefore, there exist constants ri>0,λi>1 such that, for any integer m∈0,ni−1 and x,y∈B¯rihmxi, hx−hy≥λix−y. There exists a positive integer mi such that zi−mi∈Bri/2xi and limn⟶+∞zi−nni−mi=xi. Without loss of generality, for any integer i,j∈1,k,m∈1,mi,l∈1,mj, let B¯rihmxi∩B¯rjhlxj=∅, if hmxi≠hlxj. Since Dhmizi−mi is invertible, there exist positive constants 0<δ˜i<min1/3xi−zi−mi,ri/2,μi>0 such that hmi is a homeomorphism on B¯δ˜izi−mi and(10)∀x,y∈B¯δ˜izi−mi,hmix−hmiy≥μimix−y.
By the continuity ofh, there exists a closed ball B¯δ1x1⊂B¯δ˜1x1 such that(11)h|B¯δkzk−mk−mkB¯δ1x1⊂B¯δ˜kzk−mk,B¯δ1x1∩B¯δ˜1z1−m1=∅,where h|B¯δ˜kzk−mk−mk is the inverse of hmk restricted to B¯δ˜kzk−mk. Similarly, for any integer i∈2,k, there exists B¯δixi⊂B¯δ˜ixi such that(12)h|B¯δ˜i−1zi−1−mi−1−mi−1B¯δixi⊂B¯δ˜i−1zi−1−mi−1.
Next we prove there exist two nonempty bounded closed subsetsV0andV1 with V0∩V1=∅ such that, for a positive integer n0, hn0Vj⊃V0∪V1,j=0,1 and hn0 restricted to V0 and V1, respectively, is expanding. Because hni is expanding on B¯rixi and xi is a fixed point of hni for every 1≤i≤k, there are(1)
There exists a positive integern^1 such that hn1|B¯r1x1−n^1B¯r1/2x1⊂B¯δ1x1.(2)
For every2≤i≤k, there exists a positive integer ni such that(13)hni|B¯rixi−n^iB¯δ˜izi−mi⊂B¯δixi.
Letλ=minλi|1≤i≤k; then, there is λ>1. There exists a positive integer n0>n^1n1+∑i=1kn^ini+mi such that n0−∑i=1kn^ini+mi≡0modn1 and λn0−∑i=1kmi∏i=1kμimi>1. Let(14)l≡n0modn1,V0=h|B¯r1hn1−lx1−lh|B¯r1x1−n0−lB¯r1/2x1,Vk=h|B¯rkxk−n^knkh|B¯δ˜kzk−mk−mkB¯δ1x1⊂B¯δkxk,Vi=h|B¯rixi−n^inih|B¯δ˜izi−mi−miVi+1,for all2≤i≤k−1,V1=h|B¯rixi−n0−∑i=1kn^ini+mih|B¯δ˜1z1−m1−m1V2.
Then,V0 and V1 are two nonempty bounded closed subsets of ℝn and(15)V0⊂B¯r1hn1−lx1,V1⊂B¯δ1x1.
Therefore,V0∩V1=∅. From definitions of V0andV1, we have(16)hn0V0=B¯r1/2x1⊃V0∪V1,hn0V1⊃B¯r1/2x1⊃V0∪V1.
In addition, we have that(17)hn0x−hn0y>λn0x−y,∀x,y∈V0,hn0x−hn0y>λn0−∑i=1kmi∏i=1kμimix−y∀x,yV1.∈V1.
We claim that(18)∀i0,i1,…,in,…∈Σ2+,#∩s=0∞h−n0sVis≤1,where #⋅ denotes the cardinality of a set.
Proceed the proof by contradiction. Supposex1≠x2 and x1,x2∈∩s=0∞h−n0sVis. Let i0,i1,…,in,…∈Σ2+; then, there exists a subsequence jnn∈ℕ of ℕ such that for any n∈ℕ,ijn=ijn+1. Without loss of generality, for any n∈ℕ,ijn=0. By (17), we have(19)hn0jnx1−hn0jnx2≥λn⋅λn0−∑i=1kmi∏i=1kμimijn−nx1−x2>λnx1−x2⟶+∞,which is contrary to the fact that hn0jnx1 and hn0jnx2 belong to the bounded set V0. Therefore, there exist an invariant set Λ0⊂V0 and a positive integer n0 such that Λ1,hn0 is topologically conjugate to the one-side symbolic system Σ2+,σ. That is, hn0 is chaotic in the sense of Devaney and so is h.From Theorem3, we obtain the following corollary.Corollary 1.
Leth be a continuous differentiable map from ℝn into itself. If h has heteroclinic repellers x1,x2,…,xk with k≥2, then for every neighborhood Ui of xi, there exist an invariant subset Λi of Ui and a positive integer ni such that Λi,hni is topologically conjugate to the one-side symbolic system Σ2+,σ. Therefore, hni is chaotic in the sense of Devaney and so is h.Theorem 4.
Letg,h be continuous differential maps from ℝn into itself. If h has heteroclinic cycles connecting expanding periodic points and g−hC1 is sufficiently small, then g also has heteroclinic cycles connecting expanding periodic points.Proof.
Letx1,x2,…,xk be k≥1 different periodic points with periods n1,n2,…,ni, respectively. Let ∪i=1i=kΓi=zi−nn≥0 be heteroclinic cycles connecting periodic points x1,x2,…,xk. By Definition 3, there exist constants ri>0,λi>1 such that, for any m∈0,ni−1∩ℕ,x,y∈B¯rihmxi, hx−hy≥λix−y. By Lemma 3, for every i∈1,k∩ℕ, there exists a positive constant λi1, such that for any g∈Vh,λi1 and m∈0,ni−1∩ℕ,(20)m∈0,ni−1∩ℕ,and g is expanding on Brigmxi. Since ∪i=1i=kΓi=zi−nn≥0 are heteroclinic cycles that connect periodic points x1,x2,…,xk, for any integer i∈1,k, there exists a positive integer mi such that zi−mi∈Brixi. Note that for any point z∈Γi, Dhz is an invertible operator. Therefore, by Lemma 2, for every m∈0,mi−1∩ℕ, there exist an open neighborhood Vim of hmzi−mi and a positive constant λi2 such that, for any g∈Vh,λi2, g:Vim⟶Vim+1 is a differentiable homeomorphism. Without loss of generality, suppose that Vi0⊂Brixi. Thus, g is expanding in Vi0.
Letλ=minλi1,λi2|1≤i≤k. It is needed to prove that there exists a positive constant λ0∈0,λ such that, for any g∈Vh,λ0, g has heteroclinic cycles connecting expanding periodic points. Take 0<r^i<ri with Br^ixi∩Vi0=∅. Define a map H:Vh,λ×Vk0×Br^1x1×V10×Br^2x2×V20×⋯×Vk−10×Br^kxk⟶ℝn×⋯×ℝn︷2k as(21)Hg,vk,u1,v1,…,uk=gmkvk−u1,gn1u1−u1,gm1v1−u2,gn1u2−u2,…,gnkuk−uk.
Sincehni is expanding at xi, all eigenvalues of Dhnixi are greater than one in absolute value. Therefore, Dhnixi−I is invertible. Note that Dhmizi−mi is invertible. Therefore, the diagonal block linear operator(22)DzHh,z0=∂Hg,z∂z|g=h,z0=zk−mk,x1,z1−m1,…,xk=Dhmkzk−mk−I00⋯00Dhn1x1−I00⋯0⋮⋮⋮⋮⋯⋮0000⋯Dhnkxk−I,is also invertible. Here we write(23)z=vk,u1,v1,…,uk∈Vk0×Br^1x1×V10×Br^2x2×V20×⋯×Vk−10×Br^kxk.
Hh,zk−mk,x1,z1−m1,…,xk=0, and if i≠j, then hmxi≠xj for any integer m∈1,ni. By Lemma 1, there exist constants 0<λ0<λ,0<ri0<r^i,0<δi, and φ:Vh,λ0⟶Bδkzk−mk×Br10x1×Bδ1z1−m1×⋯×Brk0xk such that(1)
Bδizi−mi⊂Vi0,Bδizi−mi∩Bri0xi=∅,∪m=1nihmBri0xi∩Brj0xj=∅.(2)
For everyg∈Vh,λ0, φg≜φ1g,φ2g,…,φ2kg is the unique solution to equations gmkvk=u1,gn1u1=u1, gm1v1=u2,gn2u2=u2,…,gmk−1vk−1=uk, and gnkuk=uk, that is, gmkφ1g=φ2g,gn1φ2g=φ2g,…,gnkφ2kg=φ2kg.
Therefore, for any integerj∈1,k, φ2jg is expanding periodic point of g, and if i≠j, gmφ2i≠φ2j for any m∈1,ni∩ℕ. In particular, φh=zk−mk,x1,z1−m1,…,xk.
It is needed to show that for everyj∈1,k∩ℕ, g has a forward orbit Γj of φ2tj+1g connecting periodic points φ2jg and φ2tj+1g. First, for every g∈Vh,λ0⊂∩j=1kVh,λj1, there is(24)Brjxj⊂gnjBrjxj,and gnj is expanding on Brjxj. Let g˜nj−1=gnj|Brjxj−1 denote the inverse of gnj restricted to Brjxj, and the l-th iterate of g˜nj−1 denotes g˜nj−1. For every integer l≥1, let z^j−mj=φ2tj+1g,z^j−lnj−mj=g˜nj−lz^j−mj. For any x,y∈Brjxj, because(25)g˜nj−1x−g˜nj−1y<λ^j−1x−y,λ^j>1,there is(26)z^j−lnj−mj−φ2jg=g˜nj−lz^j−mj−g˜nj−lφ2jg<λ^j−l‖φ2tj+1g−φ2jg‖≤λ^j−lrj.
This proves thatliml⟶∞z^j−lmj=φ2jg.
For any integerl≥1 and n∈l−1nj,lnj∩ℕ, let(27)z^j−n−mj=glnj−nz^j−lnj−mj,and for any integer n∈0,mj, let(28)z^j−n=gmj−nz^j−mj.
Therefore,z^j−nn∈ℕ is a forward orbit of φ2tj+1g connecting φ2jg and φ2tj+1g.
Because for every0≤m≤mi−1, g:Vim⟶Vim+1 is a differentiable homeomorphism, Dgmiz^j−mj is an invertible linear operator. Moreover, for any g∈Vh,λ0⊂Vh,λj1, Bδjz^j−mj⊂Vj0⊂Brjxj, so g˜−lnjz^j−mj∈Brjxj for l≥1. This implies that Dgnjg−lnjz^j−mj is an invertible linear operator for any integer l≥1. This proves that z^j−nn∈ℕ satisfies condition (3) in Definition 3. Let Γj=z^j−nn≥0 for any integer j∈1,k. Therefore, ∪j=1kΓj are heteroclinic cycles connecting expanding periodic points φ2g,φ4g,…,φ2kg.


## 4. Examples

In the following, two examples are given which illustrate our results and applications.Example 1.
Consider the one-dimensional map(29)fx=−4x1−x,x∈0,1,−4x1+x,x∈−1,0.
f has only one fixed point 0, and thus f does not have heteroclinic repellers. Directing calculation, we have(1)
f has three 2-periodic points which are not in the same orbit x1≈0.3454915,x2=0.75,x3≈0.9045084.(2)
f2 is expanding in 0.16325,0.4684,0.5315,0.8367,0.8682,1.(3)
f20.25=0.75,f20.6545085=0.3454915.
Therefore,f has heteroclinic cycles connecting expanding periodic points x1≈0.3454915 and x2=0.75(see Figure 1).Figure 1
Heteroclinic cycles connecting expanding periodic points.Remark 2.
Example1 shows that f does not have heteroclinic repellers but f 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:(30)wtt−wxx=0,x∈0,1,t>0,wx0,t=−ηwt0,t,η≠1,wx1,t=αwt1,t−βwt31,t,0<α<1,β>0,t>0,wx,0=w0x,wtx,0=w1x,0<x<1.
Let(31)Gηx=η+1η−1x,η≠1,Fαx=px+x,where y=px is the unique real solution of the cubic equation(32)βy3+1−αy+2x=0.
Letux,t=wxx,t+wtx,t/2,vx,t=wxx,t−wtx,t/2. Then, the solution u,v of system (30) is completely characterized by the interval maps Gη∘Fα⋅ and Fα∘Gη⋅, between which there is topological conjugacy. If Gη∘Fα⋅ is chaotic on some invariant interval, we say that the gradient w of system (30) is chaotic. Chen. et al. have analyzed the chaos caused by snapback repellers in system (30) (see [3]). For example, when α=0.6,β=1 and η=0.82, Gη∘Fα4⋅ has a snapback repeller. It can be checked that Gη∘Fα⋅ has heteroclinic cycles connecting expanding periodic points. In particular, it is easier to find the conditions that Gη∘Fα⋅ has heteroclinic cycles.


## 5. Conclusions

In this paper, a new criterion of chaos is established inℝn. By constructing shift invariant sets, we prove that heteroclinic cycles imply chaos in the sense of Devaney. Heteroclinic cycles connecting expanding periodic points is C1 structural stability.



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