A computer-assisted proof of the existence of Smale horseshoe for the folded-towel map
AA COMPUTER-ASSISTED PROOF OF THE EXISTENCE OF SMALEHORSESHOE FOR THE FOLDED-TOWEL MAP
ANNA GIERZKIEWICZ
Abstract.
The paper contains a rigorous proof of existence of symbolic dynamics chaosin the generalized H´enon map’s 4th iterate H , which was conjectured in the paper A3D Smale Horseshoe in a Hyperchaotic Discrete-Time System of Li and Yang, 2007. Weprove also the uniform hyperbolicity of the invariant set with symbolic dynamics. Theproofs are computer-assisted with the use of C++ library
CAPD for interval arithmetic,differentiation and integration. Introduction
The H´enon map [8] is a well known example of chaotic diffeomorphism on R , and hasbeen widely developed and generalized to many different contexts. R¨ossler in 1979 [14]introduced a similar R map to study hyper-chaos, which he described as ‘a higher formof chaos with two directions of hyperbolic instability on the attractor’. The system gainedthe name of ‘folded-towel map’, thanks to its attractor’s shape. Further generalization to R n and its study via Lyapunov characteristic exponents may be found in [1]. For otherstudies of 3D H´enon-like maps by estimating their two maximal Lyapunov exponents see[6], also [13].Our motivation is the article [12], where the 3D case with fixed parameters is consid-ered:(1) H ( x, y, z ) = (1 . − y − . z, x, y ) . The system (1) is also investigated, implemented as an electronic circuit, in [7]. Its ‘foldedtowel’ attractor is depicted on Fig. 1. The authors of [12] observe that the hyper-chaosmay be also studied as containing a 3D generalization of Smale horseshoe dynamics ona compact subset of R . They show explicitly two cuboids a , b , which most probablycontain a Smale horseshoe with two expanding directions for the fourth iterate of themap (see Fig. 1, to the right).The images of a , b via H have properties that can be described intuitively as follows: • each image H ( a ) and H ( b ) intersects both a and b ; • each image is compressed in the direction along the shortest edge of a or b . Thisdirection is ‘locally normal’ to the attractor; • each image is expanded in two other directions, almost along the other edges of a and b .Fig. 2 shows the way the images H ( a ) and H ( b ) intersect a and b .This is a well known method for establishing (hyper-)chaos by computer assisted proof,which can be implemented with interval arithmetic (see, for example, [15, 2]). Our paperpresents such a proof of the fact that the set a ∪ b is indeed a topological horseshoe, whichimplies the existence of symbolic dynamics for the system (1) (Theorem 4). Additionallywe prove that the invariant set contained in a ∪ b is uniformly hyperbolic (Theorem 6).The proofs of Theorems 4, 6 are computer-assisted. It means that their essential partsare C++ codes available on-line [4]. We are aware that such codes get outdated quickly a r X i v : . [ m a t h . D S ] J a n igure 1. The ‘folded towel’ attractor for H´enon 3D map H defined byEq. (1).To the right: The location of the sets a , b on the attractor. Figure 2.
The location of the images H ( a ), H ( b ) with respect to sets a and b — three-quarter view to the left and side view to the right. The exitsets and their images are marked in red (for a ) and green (for b ).and after a few years they may not compile with the current C++ compilers any more.We hope, however, that outlines of the proofs will help to understand the idea and thegeneral construction of the codes. . Topological covering and periodic orbits
For full description of h-sets and their covering relations see [17]. In [5] one may alsofind a 2D simplified version, which may be useful for understanding the general idea.Here we introduce a short collection of necessary notions.We use the standard notation for the closure, interior, and boundary of a topologicalset A ⊂ R k , which are A , int A , and ∂A , respectively. The k -dimensional open unit ballcentred at the origin is denoted by B k . We use the balls in the maximum norm (cubes),for they are easily interpreted in the interval algebra as interval vectors: B k = { x ∈ R k : (cid:107) x (cid:107) ∞ < } . H-sets.
The basic object we work on is
Definition 1 ([17], Def. 3.1) . An h-set is a quadruple N = ( | N | , u ( N ) , s ( N ) , C N ), where | N | is a compact subset of R n , which we will call the support of an h-set and(1) two numbers u ( N ), s ( N ) ∈ N ∪ { } complement the dimension of space: u ( N ) + s ( N ) = n ;we will call them the exit and entry dimension , or unstable and stable dimension ,respectively;(2) the homeomorphism C N : R n → R n = R u ( N ) × R s ( N ) is such that C N ( | N | ) = B u ( N ) × B s ( N ) .The support of an h-set is sometimes called simply an h-set, if it does not lead to confusionor its structure is unimportant. We also often use the notation ‘ f ( N )’ interchangeablywith ‘ f ( | N | )’ to simplify formulas.Let us set also some useful notations:dim N = n , N c = B u ( N ) × B s ( N ) , N − c = ∂ B u ( N ) × B s ( N ) , N + c = B u ( N ) × ∂ B s ( N ) , N − = C − N ( N − c ), N + = C − N ( N + c ).Therefore, we can assume that an h-set is a product of two unitary balls moved tosome coordinate system with the exit set N − and entrance set N + distinguished. Thenotions with the subscript c refer to the ‘straight’ coordinate system in the image of C N .2.2. Covering relation.
We define the topological covering:
Definition 2 ([17], Def. 3.4, simplified) . Let two h-sets M , N be such that u ( M ) = u ( N ) = u and s ( M ) = s ( N ) = s . For a continuous map f : | M | → R n denote f c = C N ◦ f ◦ C − M : M c → R u × R s .We say that M f -covers the h-set N (denoted shortly as M f = ⇒ N ) if there exists acontinuous homotopy h : [0 , × M c → R u × R s , such that:(1) h = f c ,(2) h ([0 , , M − c ) ∩ N c = ∅ (the exit condition), (3) h ([0 , , M c ) ∩ N + c = ∅ (the entry condition).
4) Additionally, if u >
0, then there exists a linear map A : R u → R u such that h ( x, y ) = ( A ( x ) ,
0) for x ∈ B u and y ∈ B s , A ( ∂ B u ) ⊂ R u \ B u . For some geometrical intuition of covering in low-dimensional cases, see Figs. 3, 4.
Figure 3.
To the left: an example of topological self-covering N f = ⇒ N in R . To the right: f ( N ) is homotopy equivalent to the image of N via alinear map A with certain properties (see Definition 2). The exit sets andtheir images are marked in red. Figure 4.
Examples of topological self-covering N f = ⇒ N in R : with oneexit direction (left) and two exit directions (right). The exit sets and theirimages are marked in red.Topological covering has a property of tracking orbits [10]. In other words, for a chaincovering N f = ⇒ N f = ⇒ N , one can find a point in N that is mapped into N andthen to N . Moreover, one can prove the existence of a periodic orbit related to a closedsequence of covering relations: Theorem 1 ([17], Theorem 3.6, simplified) . Suppose there exists a sequence of h-sets N ,. . . , N n = N , such that N f = ⇒ N f = ⇒ . . . f = ⇒ N n = N ,then there exists a point x ∈ int | N | , such that f k ( x ) ∈ int | N k | for k = 0 , , . . . , n and f n ( x ) = x . n particular, if N f = ⇒ N , then there exists a stationary point for the map f , containedin N . 3. Detecting symbolic dynamics via covering relations
We shall prove that the map (1) is chaotic in the sense of symbolic dynamics for H .First let us recall this notion:3.1. Symbolic dynamics.
Let Σ = { , } Z be the set of bi-infinite sequences of twosymbols understood as a compact space with the metricfor l = { l n } n ∈ Z , l (cid:48) = { l (cid:48) n } n ∈ Z , dist ( l, l (cid:48) ) = + ∞ (cid:88) n = −∞ | l n − l (cid:48) n | | n | ,which induces the product topology. The homeomorphism σ : Σ → Σ , given by( σ ( l )) n = l n +1 ,is called the shift map . It has many interesting topological properties, in particular: theexistence of dense orbits, or the density of periodic orbits’ set in the whole space.In our study, by the chaotic behaviour of a discrete dynamical system induced by ahomeomorphism f : X → X we understand the existence of a compact set I ⊂ X invariant for f (or sometimes its higher iterate) such that f | I is semi-conjugate to σ , i.e. there exists a continuous surjection g : I → Σ such that g ◦ f | I = σ ◦ g. In other words, f admits on I at least as rich dynamics as σ on Σ . It means, inparticular, that the topological entropy of f is greater or equal log 2. The system (Σ , σ )or any system semi-conjugate to it is often called in literature a symbolic dynamics system [11].Symbolic dynamics is sometimes used as one of the definitions of chaotic dynamics,because the discrete system (Σ , σ ) defined above evinces all the typical chaotic phenom-ena as transitivity, density of periodic orbits set or sensitivity to initial conditions. Also,there exists a periodic orbit of any prescribed period [11].3.2. Topological horseshoe.
In general, to isolate a set with symbolic dynamics doesnot seem to be an easy task. It occurs, however, that some sets which fulfil certaincovering relations must contain symbolic dynamics. An important example is:
Definition 3 (Topological horseshoe) . Let N , N ⊂ R n be two disjoint h-sets. We saythat a continuous map f : R n → R n is a topological horseshoe for N , N if (see Fig. 5)(2) N f = ⇒ N , N f = ⇒ N , N f = ⇒ N , N f = ⇒ N . It can be shown that for any topological horseshoe we obtain symbolic dynamics.
Theorem 2 ([19], Theorem 18) . Let f be a topological horseshoe for N , N . Denoteby I = Inv( N ∪ N ) the invariant part of the set N ∪ N under f , and define a map g : I → Σ by g ( x ) k = j ∈ { , } iff f k ( x ) ∈ N j . Then g is a surjection satisfying g ◦ f | I = σ ◦ g and therefore f is semi-conjugate to theshift map σ on Σ . igure 5. Topological horseshoes in R and R : each N , covers itselfand the other set. The exit sets of N and N are marked in red and green,respectively. Compare also to Fig. 2.The conjugacy to the model space Σ may be understood as follows: for any sequenceof the symbols 0 and 1 there exists an orbit of the discrete system generated by f passingthrough the sets N and N in the order given by the sequence. Moreover, if the sequenceis k -periodic, then so is the orbit. Corollary 3.
Let f be a topological horseshoe for N , N . Then it follows from Theorem1 that for any finite sequence of zeros and ones ( l , l , . . . , l n − ) , l i ∈ { , } , there exists x ∈ N l such that f i ( x ) ∈ int N l i and f n ( x ) = x. The theorem on symbolic dynamics for H´enon 3D map.
In our case, weassume that h-sets N i are contained in R and the continuous map f = H : R → R issuch that N i have exit dimensions equal to 2, that is u ( N i ) = u = 2 and s ( N i ) = s = 1(as on the right part of Fig. 4).The sets a , b defined in [12] are parallelepipeds spanned by the sets of vertices (Fig. 1) • a : { (0 . , . , . , (1 . , . , . , (1 . , . , . , (0 . , . , . , (0 . , . , . , (1 . , . , . , (1 . , . , . , (0 . , . , . } ; • b : { (1 . , . , . , (1 . , . , . , (1 . , . , . , (1 . , . , . , (1 . , . , . , (1 . , . , . , (1 . , . , . , (1 . , . , . } .Define two h-sets N a , N b ⊂ R with supports a and b , respectively. The supports areimages of the cube B = ( N a ) c = ( N b ) c = B × B = [ − , by the following affinetransformations: | N a | = a = C − a ( B ) = . . . + . . − . . . . . − . − . · B ,(3) | N b | = b = C − b ( B ) = . . . + . . − . . . . . − . − . · B, (4) Theorem 4.
For h-sets N a , N b defined above the following chain of covering relationsoccurs: (5) N a H = ⇒ N a H = ⇒ N b H = ⇒ N b H = ⇒ N a ,which proves the existence of symbolic dynamics for the fourth iterate of map H , definedby (1) . roof. Using the CAPD library for C++ [3] one is able to calculate the (over-estimated)image of the h-set N through the map f and enclose it in a cuboid (an interval closureof f ( N ), denoted by [ f ( N )]).To prove each of four covering relations N f = ⇒ N , where f = H , N , N ∈ { N a , N b } (as in (5)), we check a sufficient condition, which is a conjunction of two:(I) [ f ( N )] is spanned across N , that is: • the image projected on any unstable coordinate (1 or 2) lies outside N : (cid:13)(cid:13) [ C N ◦ f ◦ C − N ( B )] , (cid:13)(cid:13) > • or else, the image projected on stable coordinate (3) lies between the twocomponents of N +1 : (cid:13)(cid:13) [ C N ◦ f ◦ C − N ( B )] (cid:13)(cid:13) < . (II) The estimated image of the exit set [ f ( N − )] lies outside N and it is homotopyequivalent to the estimated image of N − through a chosen linear map ( A, A satisfying the conditions from Def. 2 that we choose is A = D ( C N ◦ f ◦ C − N )(0) u . Fig. 6 compares, as an example, the images of the box B via maps C a ◦ H ◦ C − a and ( A = D ( C a ◦ H ◦ C − a )(0) u , Figure 6.
The images of the box B via maps C a ◦ H ◦ C − a and ( A = D ( C a ◦ H ◦ C − a )(0) u , N − ) , , if the interval hull of the union of its image through f and through ( A,
0) lies outside N , projected on the considered coordinate: (cid:13)(cid:13)(cid:13)(cid:2) [ C N ◦ f ◦ C − N ( B − (face) )] ∪ [ C N ◦ ( A, ◦ C − N ( B − (face) )] (cid:3) , (cid:13)(cid:13)(cid:13) > h : [0 , × B × B (cid:51) ( t, x, y ) (cid:55)−→ (1 − t ) f ( x, y ) + t ( A ( x ) , ∈ R connecting the images of the face through f and through ( A,
0) which does nottouch N . utline of the computer-assisted proof [4] . The program
03a Henon Towel Symbolic Dynamics.cpp consists of the following steps:(1) We define the interval map H and h-sets N a , N b .(2) We check each of four covering relations N f = ⇒ N , where f = H , N , N ∈{ N a , N b } by:(a) dividing N in 20 × ×
20 small h-sets and checking the condition (I) foreach part,(b) dividing each of two faces of ( N − ) in 10 ×
10 small parts and checking thecondition (II) for every part in unstable dimension 1,(c) dividing each of two faces of ( N − ) in 10 ×
10 small parts and checking thecondition (II) for every part in unstable dimension 2.(3) The above conditions are fulfilled, proving every relation N H = ⇒ N , where N , N ∈ { N a , N b } . Therefore, Theorem 4 is proved. (cid:3) Hyperbolicity
Uniform and strong hyperbolicity.
First, recall the notion of uniform hyperbol-icity. Let f : R n → R n be a diffeomorphism and I ⊂ R n – a compact invariant set for f . Definition 4 ([9]) . We say that f is uniformly hyperbolic on I if for every point x ∈ I the tangent space T x I is equal to a direct sum T x I = E ux ⊕ E sx such that Df ( x ) E ux = E uf ( x ) , Df ( x ) E sx = E sf ( x ) ,and for some constants c >
0, 0 < λ < x the inequalities ∀ v ∈ E sx (cid:107) Df k ( x ) v (cid:107) < cλ k (cid:107) v (cid:107) , ∀ v ∈ E ux (cid:107) Df − k ( x ) v (cid:107) < cλ k (cid:107) v (cid:107) hold for every k ≥ h-set with cones . In our case, we shall not need this definition except someauxiliary notations.Denote by Q the n × n = ( u + s ) × ( u + s ) block matrix Q = (cid:20) Id u − Id s (cid:21) ,where by Id k we mean the identity matrix of dimension k × k .Suppose now that we have a set M = (cid:83) Ni =1 M i , where M i ⊂ R n are compact and havepairwise disjoint interiors. Each M i is related to an affine coordinate system C i . Let I be the invariant part of M for the diffeomorphism f , that is I = Inv f ( M ) = { x ∈ M | ∀ n ∈ N f n ( x ) ∈ M ∧ f − n ( x ) ∈ M } . Denote also f ij = C j ◦ f ◦ C − i for i, j = 1 , . . . , N . Definition 5 ([16], Def. 2.2) . We say that f is strongly hyperbolic on M if for x ∈ M i and j = 1 , . . . , N such that f ( M i ) ∩ M j (cid:54) = ∅ (6) the matrix Df ij ( x ) T · Q · Df ij ( x ) − Q is positive definite. Theorem 5 ([16], Th. 2.3) . If f is strongly hyperbolic on M = (cid:83) Ni =1 M i , then f isuniformly hyperbolic on I = Inv f ( M ) . .2. The theorem on hyperbolicity.Theorem 6.
The map H is uniformly hyperbolic on Inv H ( a ∪ b ) .Proof. From the computer-assisted proof [4] we deduce that H is strongly hyperbolic on a ∪ b and therefore the thesis follows from Theorem 5. Outline of the computer-assisted proof [4] . The program
04 Henon Towel Hyperbolicity.cpp consists of the following steps:(1) We define four interval maps f aa = C a ◦ H ◦ C − a , and analogously f ab , f ba , f bb .We will check the condition (6) for x ∈ B = C a ( a ) = C b ( b ).(2) In our case u = 2, s = 1, so we also define Q = − .(3) We calculate the interval upper approximation of the matrices Df ( B ) T · Q · Df ( B ) − Q , f ∈ { f aa , f ab , f ba , f bb } , and check with Sylvester’s criterion if they are positivedefinite (the conditions are not fulfilled on the whole box B , for the images aretoo much over-estimated).(4) We divide the box B into (25 × ×
25) smaller boxes, and for each part B i andeach map f ∈ { f aa , f ab , f ba , f bb } we check if: • the estimated image [ f ( B i )] has an empty intersection with B : if so, then B i does not intersect I and we do not need to check the condition (6); • else, if the interval matrix [ Df ( B i )] T · Q · [ Df ( B i )] − Q is positive definite.(5) The above conditions are fulfilled for each map f ∈ { f aa , f ab , f ba , f bb } and everypoint in B . Therefore, Theorem 6 is proved. (cid:3) Conclusion
We proved the existence of symbolic dynamics in a ∪ b for the R map H defined by(1), and also the hyperbolicity of its invariant part Inv H ( a ∪ b ). We believe that themethods used in this paper can be easily applied also in other dimensions, especially formaps with two or more expanding directions. Acknowledgment
The author would like to thank Professors Piotr Zgliczy´nski and Daniel Wilczak for alltheir devoted time and insightful comments during common discussions.
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