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2 Available online at Chaos, Solitons and Fractals 36 (2008) Tubes in dynamical systems Y. Charles Li Department of Mathematics, University of Missouri, Columbia, MO 65211, United States Accepted 20 July 2006 Abstract Important concepts concerning tubes in dynamical systems are defined in details. In the cases of a homoclinic tube and a heteroclinically tubular cycle in autonomous systems, existence of tubular chaos is established. The main goal of this article is to stress the importance of tubes in high dimensional dynamical systems. Ó 2006 Elsevier Ltd. All rights reserved. 1. Introduction The basic element of a dynamical system is the orbit. Some orbits have important implications. For instance, a homoclinic orbit often implies the existence of chaos in its neighborhood, so does a heteroclinic cycle formed by two heteroclinic orbits. By looking at a group of orbits simultaneously, one gets a flow tube. Some flow tubes may have intricated topology, and dynamics within them can be rather complicated. If the flow tubes form homoclinic tubes or heteroclinically tubular cycles, tubular chaos can be created by treating flow tubes as the basic elements. That is, on the level of tubes, dynamics of (not within) tubes can be chaotic. Such chaos is on a larger scale than the orbital chaos. Of course, dynamics within the tubes can be chaotic too. This is the smaller scale orbital chaos. One can repeat the process to obtain a chain of chaos of descending scales a chaos cascade. The impact of the concept of chaos cascade in understanding fluid turbulence has not been investigated. Starting from a neighborhood, one can generate a flow tube. Treating it as a basic element, the flow tube can be viewed as a tubular average of orbits. Such averages can be chaotic too. In such cases, averaging is a bad tool. I personally believe that fluid turbulence contains such chaos cascades. This chaos cascade is different from the well-known energy cascade in fluid turbulence. It is also different from the so-called coherent structure in fluid turbulence. It has some spirit of the island chain structure inside a stochastic layer in Hamiltonian systems, but it is a different phenomenon. In some simpler cases, such chaos cascades can be rigorously established [1,2]. These are the non-autonomous sine-gordon systems. For autonomous systems, less perfect homoclinic tubes can be established, but establishing chaos cascades is a challenging open problem [3,4]. In this article, we shall establish tubular chaos around a homoclinic tube or a heteroclinically tubular cycle in autonomous systems in the general setting by amplifying the techniques in [5,6]. In a sense, invariant manifolds are flow tubes which have unique characters. Like periodic orbits or limit cycles, periodic tubes are also interesting to study. For instance, often inside a periodic tube, there is a periodic orbit. For low address: cli@math.missouri.edu /$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi: /j.chaos

3 1216 Y. Charles Li / Chaos, Solitons and Fractals 36 (2008) dimensional systems, fixed points and orbits are the main features. With the increase of the dimensions of the systems, fixed points turn into equilibrium manifolds, flow tubes are more distinct than orbits. The article is organized as follows: In Section 2, we present some general concepts. Section 3 deals with tubular chaos around a homoclinic tube in autonomous systems. Section 4 deals with tubular chaos around a heteroclinically tubular cycle in autonomous systems. 2. General concepts A dynamical system has two components, a phase space S and a flow F t where t can be either continuous or discrete time. We list below a few concepts of tubes in dynamical systems. These concepts have potentially important implications. Definition 2.1 (Flow tube). Let G be a set, then S t2i F t ðgþ is a flow tube, where I is a time interval, 0 2 I, I R or Z. G is called a generator. G is a minimal generator if no proper subset of G can generate the same flow tube over the same time interval I. See Fig. 1 for an illustration. S Definition 2.2 (Periodic tube). Let G be a set, if there is a T such that F T (G) =G and G is a minimal generator for 06t6T F t ðgþ, then S 06t6T F t ðgþ is a periodic tube. The smallest such T is called the period. See Fig. 2 for an illustration. Theorem 2.3. If the minimal generator of a periodic tube is convex and compact, then there is at least one periodic orbit inside the periodic tube. Proof. Let G be the minimal generator of the periodic tube F T ðgþ ¼G: Then by the Schauder fixed point theorem [7], the map F T : G 7! G has at least one fixed point. h Definition 2.4 (Invariant tube). Let G be a set, if for all t such that F t (G)=G, then G is called an invariant tube. Corollary 2.5. If an invariant tube is convex and compact, then for any t there is a periodic orbit of period t inside the invariant tube. Fig. 1. An illustration of a flow tube. The generator is the circle on the left end. Starting from the generator, the flow generates this flow tube over a time interval. Fig. 2. An illustration of a periodic tube. The generator is the small circle at the bottom. Starting from the generator, the flow generates this periodic tube (a 2-torus) over a time interval.

4 Definition 2.6 (Homoclinic tube). Let G and G 0 be two sets, if F t ðgþ!g 0 as jtj!1; then G generates a homoclinic tube asymptotic to G 0. Y. Charles Li / Chaos, Solitons and Fractals 36 (2008) Remark 2.7. Here the convergence in a metric space is in the sense of C 0 graph distance over G 0. More conditions can be posed on this convergence as in the setup later. Definition 2.8 (Heteroclinic tube). Let G and G ± be three sets, if F t ðgþ!g as t!1; then G generates a heteroclinic tube asymptotic to G ±. Definition 2.9 (Heteroclinically tubular cycle). Let G 1, G 2 and G ± be four sets, if F t ðg 1 Þ!G and F t ðg 2 Þ!G as t!1; then G 1 and G 2 generate a heteroclinically tubular cycle asymptotic to G ±. 3. Tubular chaos around a transversal homoclinic tube in autonomous systems A transversal homoclinic tube in an autonomous system is a homoclinic tube asymptotic to a periodic tube of the same dimension. See Fig. 3 for an illustration. Of course, a transversal homoclinic orbit in an autonomous system is a homoclinic orbit asymptotic to a limit cycle. The open problem on the existence of chaos around a transversal homoclinic orbit in an infinite dimensional autonomous system was solved recently [5]. The problem on the existence of tubular chaos around a transversal homoclinic tube under a map (a discrete flow, e.g. the period map of a periodic system) was also solved recently [6]. The machinery of these works consists of a stronger version of the shadowing lemmas, a stronger version of the k-lemmas, and the strategy of persistence of invariant manifolds. Here we shall amplify the above two works to establish the existence of tubular chaos around a transversal homoclinic tube in autonomous systems The setup The setup is as follows: Let B be a Banach space on which an autonomous flow F t is defined. The norm of the Banach space is denoted by k k. F t is C 0 in t for t 2 ( 1,+1). For any fixed t 2 ( 1,+1), F t is a C 4 -diffeomorphism. Fig. 3. An illustration of a transversal homoclinic tube in an autonomous system. The lower tube is a periodic tube. The upper tube is the transversal homoclinic tube which is asymptotic to the periodic tube as jtj!1.

5 1218 Y. Charles Li / Chaos, Solitons and Fractals 36 (2008) There is a normally (i.e. transversally to be defined below) hyperbolic invariant C 4 periodic tube S. Let W cu and W cs be C 4 center-unstable and center-stable manifolds of S. There exist a C 3 invariant family of C 4 unstable Fenichel fibers ff u ðqþ : q 2 Sg and a C 3 invariant family of C 4 stable Fenichel fibers ff s ðqþ : q 2 Sg inside W cu and W cs respectively, such that W cu ¼ [ F u ðqþ; q2s W cs ¼ [ F s ðqþ: q2s There are positive constants j, j 1 and C (j 1 j) such that in a neighborhood of S, kf t ðq Þ F t ðqþk 6 Ce jt kq qk; 8t P 0; 8q 2 S; 8q 2 F u ðqþ; kf t ðq þ Þ F t ðqþk 6 Ce jt kq þ qk; 8t P 0; 8q 2 S; 8q þ 2 F s ðqþ; kf t ðq 1 Þ F t ðq 2 Þk 6 Ce j1jtj kq 1 q 2 k; 8t 2 R; 8q 1 ; q 2 2 S: W cu and W cs intersect along an isolated transversal homoclinic tube n asymptotic to S. n [ S is compact. Restricted to n [ S, F t is C 4 in t. S ¼ [ F t ðgþ; t2½0;t Š where G is a minimal generator which is a C 4 connected submanifold with or without boundary, and T > 0 is the period of S. The vector field D t F t is transversal to G. In a neighborhood of S, let G + and G be any two connected pieces of n \ [! F s ðqþ and n \ [! F u ðqþ ; q2g q2g respectively. Denote the corresponding maps by u : G 7! G : G ± are C 4 minimal generators of n, and u ± are C 3 diffeomorphisms. Denote by u t u t ¼ F t u F t : F t ðgþ 7! F t ðg Þ; t 2 R: Let ku t idk C 1 ¼ sup fmaxfku t ðqþ qk; kdu t ðqþ idkgg q2f t ðgþ where id is the identity map and Du t denotes the differential of u t.ast! +1, ku þ t idk C 1! 0: As t! 1, ku t idk C 1! 0: the induced maps Transversality is defined in the following sense: For example, let # ¼ inf fminfku vk;kv wk;kw ukgju 2 T qf u ðq Þ;v 2 T q F s ðq þ Þ;q 2 S;w 2 T q n;kuk¼kvk¼kwk¼1g; q2n;u;v;w where T q denotes the tangent space at q, then the transversality of n implies that # is positive. Remark 3.1. In the theory of chaos in partial differential equations [8], often n [ S is compact, finite dimensional, and consists of classical solutions. In such cases, our assumption that F t is C 4 in t when restricted to n [ S can be realized. In fact, this assumption can be replaced by a weaker assumption that n [ S is C 4. But with this assumption, the presentation is a lot simpler. The Fenichel fiber setup is standard, see for instance [9]. The notion of transversality adopted here poses a stronger requirement than the classical notion of transversality in terms of dimensions. The current notion is needed for Banach space analysis where compactness is rare. For more details on the angle, see [10] Fenichel fiber coordinate and k-lemma We will introduce Fenichel fiber coordinates in a neighborhood of S. For any h 2 S, let E u ðhþ ¼T h F u ðhþ; E c ðhþ ¼T h S; E s ðhþ ¼T h F s ðhþ;

6 Y. Charles Li / Chaos, Solitons and Fractals 36 (2008) where T h denotes the tangent space at h. E u and E s provide a coordinate system for a neighborhood of S, ð~v s ; ~ h; ~v u Þ; where ~v s 2 E s ð ~ hþ; ~v u 2 E u ð ~ hþ; ~ h 2 S: Fenichel fibers provide another coordinate system for the neighborhood of S. For any h 2 S, the Fenichel fibers F s ðhþ and F u ðhþ have the expressions 8 8 >< ~v s ¼ v s ; >< ~v u ¼ v u ; ~h ¼ h þ H s ðv s ; hþ; and ~h ¼ h þ H u ðv u ; hþ; >: >: ~v u ¼ V s ðv s ; hþ; ~v s ¼ V u ðv u ; hþ; where v s and v u are the parameters parametrizing F s ðhþ and F u ðhþ, H z ð0; hþ ¼ o ov H zð0; hþ ¼V z z ð0; hþ ¼ o ov V zð0; hþ ¼0; z ¼ u; s; z and H z (v z,h) and V z (v z,h) (z = u,s) are C 4 in v z and C 3 in h by the setup. The coordinate transformation from (v s,h,v u ) to (~v s ; ~ h; ~v u ) 8 >< ~v s ¼ v s þ V u ðv u ; hþ; ~h ¼ h þ H u ðv u ; hþþh s ðv s ; hþ; >: ~v u ¼ v u þ V s ðv s ; hþ is a C 3 diffeomorphism. In terms of the Fenichel coordinate (v s,h,v u ), the Fenichel fibers coincide with their tangent spaces. From now on, we always work with the Fenichel coordinate (v s,h,v u ). Next we present a k-lemma. For any q u,q s 2 n and q 2 S such that q z 2 F z ðqþ, (z = u,s). Let E z ðq z Þ¼T q zf z ðqþ; E c ðq z Þ¼T q zn; ðz ¼ u; sþ: Lemma 3.2 (k-lemma). For d small enough, there exists a m 0 > 0 (m 0 depends on d), such that for any q 2 S, q u 2 n \ F u ðqþ and q s 2 n \ F s ðqþ, (1) When kq s qk is less than Oðd m 0 Þ,E u (q s ) E c (q s )isd 3 -close to E u (q) E c (q) in the Lipschitz semi-norm of graphs over E u (q) E c (q). (2) When kq u qk is less than Oðd m 0 Þ,E s (q u ) E c (q u )isd 3 -close to E s (q) E c (q) in the Lipschitz semi-norm of graphs over E s (q) E c (q). Proof. Let q s 1 2 n \ Fs ðq 1 Þ (q 1 2 S) such that kq s 1 q 1kOðd 4 Þ. Notice that F s ðq 1 Þ¼E s ðq 1 Þ. Let v 1 2 E u ðq s 1 ÞEc ðq s 1 Þ, kv 1 k = 1. We represent v 1 in the frame (E s (q 1 ),E u (q 1 ) E c (q 1 )), v 1 ¼ðv s 1 ; vuc 1 Þ; where kv uc 1 k 6¼ 0, since n is transversal by the setup. Let k 1 ¼kv s 1 k=kvuc 1 k. The transversality of n implies that k 1 has an upper bound for all v 1 [10]. Since S is compact, k 1 has an upper bound for all q 1 2 S. For some m, let q s n ¼ F 4mT ðq s n 1 Þ and q n = F 4mT (q n 1 ) where T is the period of S. Let ð^v s 2 ; ^vuc 2 Þ¼DF 4mT ðq s 1 Þv 1 ¼ DF 4mT ðq s 1 Þðvs 1 ; 0ÞþDF 4mT ðq 1 Þð0; v uc 1 Þþ½DF 4mT ðq s 1 Þ DF 4mT ðq 1 ÞŠð0; v uc 1 Þ and ðr s ; r uc Þ¼½DF 4mT ðq s 1 Þ DF 4mT ðq 1 ÞŠð0; v uc 1 Þ; where the left hand sides are splittings in the frame (E s (q 2 ),E u (q 2 ) E c (q 2 )). Then ^v s 2 ¼ DF 4mT ðq s 1 Þvs 1 þ rs ; ^v uc 2 ¼ DF 4mT ðq 1 Þv uc 1 þ ruc : We choose m large enough such that DF 4mT ðq 1 Þv uc 1 P 2e 1 2 jmt kv uc 1 k: For such fixed m, choosing d small enough, we have kdf 4mT ðq s 1 Þ DF 4mT ðq 1 Þk 6 d 3 e jmt :

7 1220 Y. Charles Li / Chaos, Solitons and Fractals 36 (2008) Then Thus k^v s 2 k 6 kdf 4mT ðq 1 Þv s 1 kþk½df 4mT ðq s 1 Þ DF 4mT ðq 1 ÞŠv s 1 kþkrs k 6 e 3jmT kv s 1 kþd3 e jmt kv s 1 kþd3 e jmt kv uc 1 k 6 e jmt kv s 1 kþd3 e 1 2 jmt kv uc 1 k; k^v uc 2 k P 2e 1 2 jmt kv uc 1 k d3 e jmt kv uc 1 k P e 1 2 jmt kv uc 1 k: k 2 ¼k^v s 2 k=k^vuc 2 k 6 e 1 2 jmt k 1 þ d 3 : Iterating the argument, we obtain k N 6 e 1 2 jðn 1ÞmT k 1 þ d 3 XN 2 e 1 2 jlmt : l¼0 For such fixed d, when N is large enough, k N 6 4d 3 : There exists m 0 > 0 such that kf 4ðN 1ÞmT ðq s 1 Þ F 4ðN 1ÞmT ðq 1 Þk Oðd m 0 Þ; implies that N is large enough as required above. Similarly for the case of q u. The proof is completed. h The k-lemma is crucial in setting up the coordinate system in the proof of the shadowing lemma later Pseudo-flow-tube The building blocks of the pseudo-flow-tubes are what we call loop-0 and loop-1. Definition 3.3. Loop-0, denoted by g 0, is defined to be the m-times circulation of the periodic tube S, where m is large enough. One should view loop-0 as a time sequence over the time interval mt where T is the period of S. The flow tube to shadow loop-0 does not need to have self-overlap. How large M has to be will be decided in the shadowing argument. To define loop-1, we focus in the neighborhood of the generator G of S, introduced in the setup. Let G u = F d (G) and G s = F d (G) where d > 0 is a small parameter. Let bg u ¼ F NT þd ðg Þ and bg s ¼ F NT d ðg þ Þ where G ± are introduced in the setup and T is the period of S. For any fixed m P 4, by the setup, when N is large enough, where ku NT þd idk C 1 6 dm and ku þ NT d idk C 1 6 dm ; u NT þd : G u 7! bg u and u þ NT d : G s 7! bg s are C 3 diffeomorphisms introduced in the setup. Inside the center-unstable manifold W cu, we are going to connect bg u and G in the following manner: In terms of the Fenichel coordinates, u NT þd can be expressed as v u ¼ v u ðhþ; v s ¼ 0; h 2 G u ; and u NT þt ¼ F t d u NT þd F d t (t 2 [0,d]) can be expressed as v u ¼ v u ðh; tþ; v s ¼ 0; h 2 G u ; t 2½0; dš; where v u (h,t) isc 3 in h and t. Let v(t) be a mollifier such that vðtþ : ½0; 1Š 7!½0; 1Š; vð0þ ¼0; vð1þ ¼1; v ðnþ ð0þ ¼v ðnþ ð1þ ¼0 ðn ¼ 1; 2; 3Þ; and v(t) isac 1 strictly monotonically increasing function. Then v u ¼ vðt=dþv u ðh; tþ; v s ¼ 0; h 2 G u ; t 2½0; dš

8 Y. Charles Li / Chaos, Solitons and Fractals 36 (2008) defines a tube f u connecting G u and G. Along G u and G, f u coincides with n and S to the third derivatives respectively. f u is C 3 diffeomorphic to the segment dgg u of S. Similarly, one can define such a tube f s inside W cs connecting G s and G. Let ^n be the portion of n, which connects bg u to bg s. Then ^n [ f s [ f u, dg s G [ f u and f s [ dgg u are all C 3. Definition 3.4. Loop-1, denoted by g 1, is defined by g 1 ¼ ^n [ f s [ f u. To define the pseudo-flow-tubes, we introduce the topological space of binary sequences. Definition 3.5. Let R be a set that consists of elements of the doubly infinite sequence form a ¼ða 2 a 1 a 0 ; a 1 a 2 Þ; where a k 2 {0,1}, k 2 Z. We introduce a topology in R by taking as neighborhood basis of a ¼ða 2 a 1 a 0 ; a 1 a 2 Þ; the set A j ¼fa2 Rja k ¼ a k ðjkj < jþg for j =1,2,... This makes R a topological space. The Bernoulli shift automorphism v is defined on R by v : R 7! R; 8a 2 R; vðaþ ¼b; where b k ¼ a kþ1 : The Bernoulli shift automorphism v exhibits sensitive dependence on initial conditions, which is a hallmark of chaos. Definition 3.6. To each a k 2 {0,1}, we associate the loop-a k, g ak. Then each doubly infinite sequence a ¼ða 2 a 1 a 0 ; a 1 a 2 Þ is associated with a d-pseudo-flow-tube g a ¼ðg a 2 g a 1 g a0 ; g a1 g a2 Þ: 3.4. Shadowing lemma and tubular chaos Lemma 3.7 (Shadowing lemma). When d is sufficiently small, there is a unique flow tube in the neighborhood of any d-pseudo-flow-tube. Proof. We provide a sketch of the proof here. For the detailed technical arguments, see [5]. The main difference from [5] is that here we are dealing with a periodic tube instead of the periodic orbit in [5]. Fortunately the available machinery on the persistence of invariant manifolds is powerful enough to handle such a general case. The d-pseudo-flow-tube g a is not invariant, it is nearly invariant, and the near invariance is measured by d. Thus g a can be regarded as flow tube of some perturbed flow of F t. The goal is to find a nearby flow tube of F t. Fenichel s persistence of invariant manifold idea goes along the converse direction [11], that is, starting from an invariant manifold of the original flow, one finds a persistent invariant manifold for a perturbed flow. The main condition for such persistence is the normal hyperbolicity of the invariant manifold. The main tool to establish the persistence is graph transform. By the k-lemma 3.2, we can set up a C 2 transversal bundle [5] be ¼fðq; E u ðqþ; E s ðqþþjq 2 g a g; which serves as a coordinate system around g a. Then we define the space of sections of be equipped with the C 0 norm. Using the flow F t, we can define a graph transform of the sections. A contraction mapping argument leads to the existence of a unique fixed point of the graph transform [5]. The graph of the fixed point is the flow tube that shadows the d- pseudo-flow-tube g a. h Let N be a transversal codimension 1 section to the periodic tube S, N \ S = F s (G) where G is the minimal generator introduced in the setup, and s is some small time. For any d-pseudo-flow-tube g a, denote by h a0 the portion of the shadowing flow tube, that shadows the portion g a0. Let G a be the first intersection of h a0 with N. Let K be the set consisting of the submanifolds G a for all doubly infinite sequences a 2 R. These G a s are graphs over F s (G). The C 0 graph norm introduces the topology on K. We define a map P:K # K as follows: For any G a 2 K, P(G a )=G v(a). Then we have the theorem.

9 1222 Y. Charles Li / Chaos, Solitons and Fractals 36 (2008) Theorem 3.8 (Tubular chaos theorem). The set K is invariant under the map P. The action of P on K is topologically conjugate to the action of the Bernoulli shift automorphism v on R. That is, there exists a homeomorphism /:R # K such that the following diagram commutes: Proof. The invariance of K under P follows from the definitions of K and P. We define /:R # K as follows: For any a 2 R, /(a)=g a. It is straightforward to show that / is a homeomorphism, and P and v are topologically conjugate. h Since the return map is not defined in the entire neighborhood of S, here we define it only for the shadowing invariant tubes. The P defined here is a counterpart of the iterated Poincaré period map in periodic systems [6]. In fact, for an invariant tube shadowing loop-0 (a 0 = 0), we only record its first intersection with N. For such a return map P, we have the above chaos theorem. In general, replacing the periodic tube S specifically defined in the setup by a general normally hyperbolic invariant manifold, we do not have the above chaos theorem. It is possible to deal with this general case, but it seems very challenging. Again since the return map is not defined in the entire neighborhood of S, characterization of K, for instance hyperbolicity, is unclear. The concept of flow tube is very general a generalization of orbit. To gain nontrivial results, one needs to restrict the setup to special flow tubes, like periodic tubes, homoclinic tubes etc. The concept of flow tube is not directly related to the concepts of isolated invariant sets and isolating blocks [12], although isolated invariant sets are certainly special invariant tubes. 4. Tubular chaos around a transversal heteroclinically tubular cycle in autonomous systems Under the similar setup as in the homoclinic tube case, the above tubular chaos theorem can also be established around a transversal heteroclinically tubular cycle in autonomous systems (Fig. 4). The setup is as follows: Let B be a Banach space on which an autonomous flow F t is defined. The norm of the Banach space is denoted by k k. F t is C 0 in t for t 2 ( 1,+1). For any fixed t 2 ( 1,+1), F t is a C 4 -diffeomorphism. There are two normally (i.e. transversally to be defined below) hyperbolic invariant C 4 periodic tubes S ± which are C 4 diffeomorphic to each other. Let W cu cs and W be the C4 center-unstable and center-stable manifolds of S ±. The dimensions of W cu cs and W are independent of ±. There exist a C3 invariant family of C 4 unstable Fenichel fibers ff u ðqþ : q 2 S g and a C 3 invariant family of C 4 stable Fenichel fibers ff s ðqþ : q 2 S g inside W cu cs and W respectively, such that W cu ¼ [ F u ðqþ; W cs ¼ [ F s ðqþ: q2s q2s Fig. 4. An illustration of a transversal heteroclinically tubular cycle in an autonomous system. There are two periodic tubes (left and right). The top and the bottom tubes form the transversal heteroclinically tubular cycle. The top (bottom) tube is asymptotic to left and right (right and left) periodic tubes as t! +1 and t! 1respectively.

10 There are positive constants j and C such that kf t ðq 1 Þ F t ðqþk 6 Ce jt kq 1 qk; 8t P 0; 8q 2 S ; 8q 1 2 F u ðqþ; kf t ðq 1 Þ F t ðqþk 6 Ce jt kq 1 qk; 8t P 0; 8q 2 S ; 8q 1 2 F s ðqþ; kf t ðq 1 Þ F t ðq 2 Þk 6 Ce j1jtj kq 1 q 2 k; 8t 2 R; 8q 1 ; q 2 2 S ; where 0 6 j 1 j. W cu cs and W intersect along isolated transversal heteroclinic tubes n±. Together n + and n form a heteroclinically tubular cycle. n + [ n [ S + [ S is compact. Restricted to n + [ n [ S + [ S, F t is C 4 in t. S ¼ [ F t ðg Þ; t2½0;t Š where G ± are the minimal generators which are C 4 connected submanifolds with or without boundary, and T ± >0 are the periods of S ±. The vector field D t F t is transversal to G ±. In a neighborhood of S ±, let G þ and G be any two connected pieces of n \ [! F s ðqþ and n \ [! F u ðqþ ; q2g q2g respectively. Denote the corresponding maps by u þ : G 7! Gþ ; and u : G 7! G : G þ and G are C4 minimal generators of n ±, and u þ and u are C3 diffeomorphisms. Denote by u r ;t (r = ±) the induced maps u r ;t ¼ F t u r F t : F t ðg Þ 7! F t ðg r Þ; t 2 R; r ¼: Let ku r ;t idk C ¼ 1 sup q2f t ðg Þ fmaxfku r ;t ðqþ qk; kdur ;t ðqþ idkgg where id is the identity map and Du r ;t denotes the differential of u r ;t (r = ±). As t! +1, ku þ ;t idk C! 0: 1 As t! 1, ku ;t idk C! 0: 1 Transversality is defined in the following sense: For example, let # ¼ inf fminfku vk; kv wk; kw ukgju 2 T q F u ðq Þ; q2n ;u;v;w v 2 T q F s ðq Þ; q 2 S ; w 2 T q n; kuk ¼kvk ¼kwk ¼1g; where T q denotes the tangent space at q, then the transversality of n ± implies that # is positive. Loop-0, g 0, is defined to be the m-times circulation of the periodic tube S +, where m is large enough. By putting a junction (as before) on each of S ±, we get loop-1, g 1. Then each doubly infinite binary sequence a ¼ða 2 a 1 a 0 ; a 1 a 2 Þ is associated with a d-pseudo-flow-tube g a ¼ðg a 2 g a 1 g a0 ; g a1 g a2 Þ: Y. Charles Li / Chaos, Solitons and Fractals 36 (2008) Then similar arguments as before leads to shadowing lemma. Lemma 4.1 (Shadowing lemma). When d is sufficiently small, there is a unique flow tube in the neighborhood of any d- pseudo-flow-tube. Let N be a transversal codimension 1 section to the periodic tube S +, N \ S + = F s (G + ) where G is the minimal generator introduced in the setup, and s is some small time. For any d-pseudo-flow-tube g a, denote by h a0 the portion of the shadowing flow tube, that shadows the portion g a0. Let G a be the first intersection of h a0 with N. Let K be the set con-

11 1224 Y. Charles Li / Chaos, Solitons and Fractals 36 (2008) sisting of the submanifolds G a for all doubly infinite sequences a 2 R. These G a s are graphs over F s (G + ). The C 0 graph norm introduces the topology on K. We define a map P:K # K as follows: For any G a 2 K, P(G a )=G v(a). Then we have the theorem. Theorem 4.2 (Tubular chaos theorem). The set K is invariant under the map P. The action of P on K is topologically conjugate to the action of the Bernoulli shift automorphism v on R. That is, there exists a homeomorphism /:R # K such that the following diagram commutes: This theorem shows that as in the setting of transversal homoclinic tubes, tubular chaos can be created in a neighborhood of a transversal heteroclinically tubular cycle. 5. Conclusion In this article, we have proved the existence of tubular chaos in a neighborhood of a transversal homoclinic tube or a transversal heteroclinically tubular cycle in the autonomous setting by extending our earlier works [6,1,2]. We believe that homoclinic tubes and heteroclinically tubular cycles are ubiquitous in high dimensional systems especially Hamiltonian systems, and play important roles in the global dynamics, e.g. large scale tubular chaos and large scale tubular diffusion, in the phase space. Since the dimension of the tubes can be large too, dynamics inside the tubes can be chaotic. This can lead to chaos cascade which we believe to be an important concept in high dimensional systems as discussed in the introduction. Chaos cascade reveals different scale chaos which we believe to be ubiquitous in high dimensional systems. So far, successful applications of our theory on chaos cascade have been conducted on chaotically driven sine- Gordon systems [1,2]. We also emphasized in general the importance of the concept of flow tube as a higher dimensional generalization of an orbit. References [1] Li Y. Homoclinic tubes and chaos in perturbed sine-gordon equation. Chaos, Solitons & Fractals 2004;20(4):791. [2] Li Y. Chaos and shadowing around a heteroclinically tubular cycle with an application to sine-gordon equation. Stud Appl Math 2006;116:145. [3] Li Y. Homoclinic tubes in nonlinear Schrödinger equation under Hamiltonian perturbations. Prog Theor Phys 1999;101(3):559. [4] Li Y. Homoclinic tubes in discrete nonlinear Schrödinger equation under Hamiltonian perturbations. Nonlinear Dynam 2003;31(4):393. [5] Li Y. Chaos and shadowing lemma for autonomous systems of infinite dimensions. J Dyn Diff Eq 2003;15(4):699. [6] Li Y. Chaos and shadowing around a homoclinic tube. Abstract Appl Anal 2003;2003(16):923. [7] Granas A, Dugundji J. Fixed point theory. New York: Springer-Verlag; [8] Li Y. Chaos in partial differential equations. International Press; [9] Li Y, Wiggins S. Invariant manifolds and vibrations for perturbed nonlinear Schrödinger equations. Applied Mathematical Sciences, vol Springer-Verlag; [10] Henry D. Exponential dichotomies, the shadowing lemma, and homoclinic orbits in Banach space. Resenhas IME-USP 1994;1(4):381. [11] Fenichel N. Persistence and smoothness of invariant manifolds for flows. Indiana Univ Math J 1971;21:193. [12] Conley C, Easton R. Isolating invariant sets and isolating blocks. Trans AMS 1971;158(1):35.