Order of Appearance of Homoclinic Points for the Hénon Map

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1 1029 Progress of Theoretical Physics, Vol. 116, No. 6, December 2006 Order of Appearance of Homoclinic Points for the Hénon Map Yoshihiro Yamaguchi 1 and Kiyotaka Tanikawa 2 1 Teikyo Heisei University, Ichihara , Japan 2 National Astronomical Observatory, Mitaka , Japan (Received February 15, 2006) After the completion of the two-fold horseshoe in the area- and orientation-preserving Hénon map, every homoclinic point is characterized by the symbol sequence. A procedure for deriving their dynamical ordering is introduced, and a lower bound for topological entropy is evaluated. 1. Introduction For the area- and orientation-preserving Hénon map, 1) we previously derived a generalized dynamical ordering of the symmetric periodic orbits appearing through saddle-node bifurcations. 2) The procedure consists of, first, fixing the homoclinic tangency of the stable and unstable manifolds of a saddle fixed point and, then, deriving dynamical order relations for the symmetric periodic orbits associated with this homoclinic tangency. Geometrically, the interpretation of the procedure is simple: Iterates of symmetry axes accumulate at stable and unstable manifolds, and therefore before and/or after the tangency of stable and unstable manifolds, an infinite number of arcs of the symmetry axes become tangent to and/or intersect each other. This implies that the tangency of the stable and unstable manifolds is the key factor determining the dynamical order relations of the symmetric periodic orbits. If the order of appearance of the homoclinic points is determined, we have a great deal of information about the dynamical ordering for the symmetric periodic orbits and the complexity of the system. There have been several attempts tried to determine the forcing relations among homoclinic orbits for the horseshoe map. 3) 5) However, the complete set of order relations has not been derived. The purpose of the present work is to obtain as many kinds of dynamical orders as possible among homoclinic orbits for the Hénon map. We use the area- and orientation-preserving Hénon map T of the plane represented by the symplectic form 1),6) y n+1 = y n + f(x n ), (1) x n+1 = x n + y n+1, (2) where f(x) =a(x x 2 )anda 0 is a parameter. There are two fixed points, P =(0, 0) and Q =(1, 0), with P being a saddle. It is noted that a horseshoe exists for a a hs = In 2, we prepare the basic tools used in later sections. In 3, the properties of the homoclinic orbits are discussed, and their symbol sequences are determined. In 4, the dynamical orderings for the symmetric homoclinic orbits are obtained.

2 1030 Y. Yamaguchi and K. Tanikawa From these dynamical orderings, the order relations for the symbol sequences are also determined. In 5, the local dynamical orderings for the homoclinic orbits are obtained. Using their symbol sequences, we evaluate a lower bound for topological entropy. 2. Basic tools 2.1. Symmetry axes The map T can be expressed as a product of two involutions, h and g, as T = h g, (3) where h and g are defined by ( ) ( x x y h = y y ) and ( x g y ) ( = x y f(x) ), (4) with det h =det g = 1. The sets of fixed points of h and g are called the symmetry axes. 7) The symmetry axes are given by Points on the symmetry axes are said to be symmetric. S 1 : y =0forh, (5) S 2 : y = f(x)/2 forg. (6) 2.2. The stable and unstable manifolds and homoclinic orbits Let W u = W u (P ) be a branch of the unstable manifold starting at P and extending in the upper-right direction, and let W s = W s (P ) be a branch of the stable manifold extending in the lower-right direction. The stable manifold W s transversely intersects S 1 at v and S 2 at u, in this order, and the unstable manifold W u transversely intersects S 2 at u and S 1 at v, in this order (see Fig. 1). It is known that W s and W u intersect transversely at the homoclinic points u and v. 8) Let us denote by (α, β) Γ an open arc of the one-dimensional manifold Γ with α, β Γ. If Γ has an orientation, we use the convention that the upstream point be the left terminal of the interval. A closed arc [α, β] Γ is defined similarly. We sometimes write α Γ β if α is upstream of β along Γ. We define the arcs γ u = [u, v] Wu, γ s = [u, v] Ws, Γ u = [v, Tu] Wu and Γ s = [v, Tu] Ws. Further, we define Z as the closed region bounded by [P, u] Wu, γ s, Γ u and [Tu,P] Ws. Note that all homoclinic orbits (i.e., orbits consisting of homoclinic points) are contained in Z. In the situation that T 1 γ s is tangent to Γ u, TΓ u is also tangent to γ s. This situation marks the completion of the horseshoe (see Fig. 2). Bedford and Smillie 9) proved that the topological entropy is ln 2 in this situation (a = a hs ). Letusdenotetheorbitofapoints by O(s) ={...,T 1 s,s,ts,...}. We define ho(s) byh{...,t 1 s,s,ts,...} = {...,ht 1 s,hs,hts,...}. The orbit go(s) is defined similarly.

3 Order of Appearance of Homoclinic Points for the Hénon Map 1031 Fig. 1. The stable and unstable manifolds W s and W u, two homoclinic intersection points u and v, the segments of the stable manifolds γ s and Γ s, the segments of the unstable manifolds γ u and Γ u, and the two symmetry axes S 1 (x-axis) and S 2 are displayed. The figure displays the first tangency situation of TΓ u and T 1 γ s in S 1 at a = Proposition 1. For any point s in the plane, we have the following: (i) O(hs) =ho(s) =go(s) =O(gs), and (ii) O(s) =ho(s) =go(s) if and only if O(s) has a symmetric point. Proof. (i) O(hs) ={...,T 1 hs,hs,ths,...} = {...,hts,hs,ht 1 s,...} = ho(s). The other equalities are obtained in a similar manner. (ii) ( ) Thereisapointr and an integer k such that hr = T k r. If k is even, then T k/2 r = T k/2 hr = ht k/2 r, which implies that T k/2 r is on S 1.Ifk =2m +1, then T m r = T m 1 hr = gt m r, which implies that T m r is on S 2. ( ) Suppose r = hr O(s). Then, O(s) = O(r) = O(hr) = ho(r) = ho(s) = go(s). Q.E.D. The orbit specified by Proposition 1(ii) is called symmetric. By Proposition 1(ii), a symmetric orbit contains a symmetric point, and conversely, the orbit of a symmetric point is symmetric. If s is a homoclinic point, then O(s) is called a homoclinic orbit. If, in addition, O(s) is symmetric, then O(s) iscalledasymmetric homoclinic orbit. 3. Coding of the homoclinic point and the maximum eigenvalue Now, let us fix the parameter value to a hs, at which the horseshoe is just completed. At this parameter value, TΓ u is tangent to γ s,say,atβ, and correspondingly, T 1 γ s is tangent to Γ u,say,atα (see Fig. 2). We define four regions, V 0, V 1, H 0 and H 1, as displayed in Fig. 2. The region V 0 is the closed region bounded by the arcs [P, T 1 u] Wu,[T 1 u, α] Ws,[α, T u] Wu and [Tu,P] Ws, and the region V 1 is the closed region bounded by the arcs [T 1 v, u] Wu,[u, v] Ws,[v, α] Wu and [α, T 1 v] Ws.

4 1032 Y. Yamaguchi and K. Tanikawa Fig. 2. Definition of the regions V 0, V 1, H 0 and H 1 in the situation that the horseshoe is completed in a = a hs.hereαand β are the tangencies satisfying the relation β = Tα. The regions H 0 and H 1 are defined as The following relations also hold: H 0 = hv 0, (7) H 1 = hv 1. (8) V 0 = gv 0, (9) V 1 = gv 1. (10) If a point in the horseshoe is in V 0 (resp., V 1 ), we assign the value 0 (resp., 1) to this point. Then, it is well known that to the orbit of each point in the horseshoe, there corresponds a symbol sequence of the two symbols 0 and 1. We denote by 0 n and 1 n n consecutive repetitions of 0 and 1, respectively, in the symbol sequence. The notation 0 represents the infinite repetition of the symbol 0. The symbol sequence of a homoclinic orbit has the form 0 1s 0 s 1 s n 1 10, (11) where s i represents the symbol 0 or 1. We refer to the middle part, s 0 s 1 s n 1,as the core. It is to be noted that the definition of the core is different from that in Carvalho and Hall, 4) but it is identical with that in Sterling et al. 10) The presence of the symbol 1 located to the left of the core means that the homoclinic orbit has apointint 1 Γ u (= gγ s ), whereas that of the symbol 1 at the opposite side means that the homoclinic orbit has a point in γ s. A homoclinic orbit is characterized by its core. We refer to a core consisting of n words as an n-core. The reason we consider homoclinic orbits with a core is that we want to exclude from consideration two primary homoclinic orbits O(v) ando(u), which provide no information on the strength of the chaos exhibited by the map.

5 Order of Appearance of Homoclinic Points for the Hénon Map 1033 Fig. 3. An arc T 2 γ s intersects Γ u at the four points A 1, A 2, A 3 and A 4 at a =5.5. If TΓ u (resp., T 1 γ s ) (transversely) intersects γ s (resp., Γ u ), the two-fold horseshoe is completed. Therefore, there are two types of 1-cores, namely, 0 and 1. In this situation, T 2 γ s intersects Γ u at four points, A 1,A 2,A 3 and A 4 (see Fig. 3). Tracing the orbits starting at these points, we determine the 2-cores of these orbits to be A 1 :00, (12) A 2 :01, (13) A 3 :11, (14) A 4 :10. (15) From these symbol sequences, we see that the coding of the homoclinic points in Γ u is the same as that of the tent map represented by x n+1 =2x n for 0 x<1/2, (16) =2 2x n for 1/2 x<1, (17) where the symbol for x n is 0 (resp., 1) if x n exists in 0 x n < 1/2 (resp.,1/2 x n < 1). Applying the software Trains3, provided by Hall, 11) to the symbol sequences of homoclinic orbits, we can evaluate a lower bound for topological entropy h top (= ln λ max ). In Table I, we list the values of λ max for the homoclinic orbits whose cores have lengths less than or equal to 5. Before closing this section, we discuss an application of Table I. Let us consider the forced Duffing equations and study an appropriate surface of section and surface mapping T. Suppose that T n (n 1) possesses a two-fold horseshoe for a a hs and a secondary homoclinic point is found at a = a <a hs. We increase the parameter value up to a hs and determine the core of the orbit of the homoclinic point. As an example, consider the core A lower bound for topological entropy of the system for a a is (log 1.638)/n. In this way, we can determine a lower bound for entropy of a given system if the secondary homoclinic points are found.

6 1034 Y. Yamaguchi and K. Tanikawa Table I. Maximum eigenvalues. Core λ max Core λ max Core λ max Core λ max Dynamical ordering for the homoclinic orbits with symmetric cores 4.1. Order relations We consider homoclinic orbits characterized by symmetric cores. These appear through the tangency of the stable and unstable manifolds at a point of the symmetry axis. A homoclinic orbit with a symmetric core containing an even number of symbols (abbreviated as a symmetric even-core ) appears through the tangency between T n Γ u and S 1, whereas a homoclinic orbit with a symmetric odd-core appears through the tangency between T n Γ u and S 2. In this section, we separately study the orders of appearance of symmetric even-cores and symmetric odd-cores. Let us consider the situation for the parameter value at which the horseshoe is completed. The folded arc T n Γ u (n 1) starting at the foot-points (two terminal points) on (Tu,P) Ws extends and wraps round the fixed point Q. The arc intersects the symmetry axes multiple times. Let us name these intersections. The first pair of intersection points of T n Γ u (n 1) and S 1 is born between P and Q and is named 1 h n, the second pair of intersection points is born between Q and v and is named 2 h n, and so on. (Here, the counting starts at the foot-points in the direction of the extension.) We thus have 2 n 1 pairs of symmetric homoclinic points from 1 h n to 2 n 1 h n. Examples are displayed in Fig. 4 for small n. In this situation, the horseshoe is completed. It is conjectured that the k h n (1 k 2 n 1 )arewell defined. For the odd-cores, we use the similar notation k g n (n 0, 1 k 2 n ). These appear through the tangency of T n Γ u and S 2. Now, we consider the parameter region in which the horseshoe is not completed. The intersection points k h n appear through the tangency between T n Γ u and S 1, but in the parameter region between this tangency and the completion of horseshoe, we cannot predict the behavior of k h n. For example, we cannot prove that there is no bifurcation of k h n, such as a bifurcation of cubic type. If such a bifurcation occurs,

7 Order of Appearance of Homoclinic Points for the Hénon Map 1035 Fig. 4. Naming of the symmetric even-cores at the completion of the horseshoe. one homoclinic point bifurcates into three points. There is another possibility. The deformation of T n Γ u other than extension may give rise to additional intersections of T n Γ u (n 1) and S 1. Then k h n (1 k 2 n 1 ) may not be unique. Indeed, we may have a sequence k h 1 n, k h 2 n,... It is difficult to select a point among these points that survives until the completion of the horseshoe. In order to avoid the ambiguity due to the non-uniqueness of k h n, we introduce k h min n as one of k h 1 n, k h 2 n,...which bifurcates for the smallest a. Then, although k h min n may not be the points which survive until a = a hs, the uniqueness is recovered. We hereafter denote k h min n simply by k h n. In Fig. 5(a), (not necessarily dynamical) order relations of the tree structure for i h n on (P, v) S1 are displayed. We refer to the top of the binary tree as the first floor, the next two points as the second floor, and so on. The suffix n of i h n corresponds to the ordinal number of the floor and the iteration number of Γ u. In Fig. 5(b), the order relations of the symmetric odd-cores on (P, u) S2 are displayed. We refer to the top of the binary tree as the zeroth floor. The appearance of 1 g 0 means the appearance of the horseshoe. In fact, 1 g 0 is the situation displayed in Fig. 2. The meanings of L and R in Fig. 5 are explained in 4.2. We found the orders of appearance for the symmetric even-cores given in Eqs. (18) (26). The notation A B used there means that if the homoclinic intersection points or the tangency named A exists, then the homoclinic intersection points named B also exist. k h n k h n+1, ((k =1,n 1), (k =2,n 2)) (18)

8 1036 Y. Yamaguchi and K. Tanikawa Fig. 5. (a) The symmetric even cores in (P, v) S1.Then-th floor includes the 2 n 1 cores. (b) The symmetric odd-cores in (P, u) S2. The n-th floor includes the 2 n cores. The symbols L and R represent the coding operators. 2 h n+1 1 h n, (n 1) (19) 2 n 1 h n 8 h 4 4 h 3 2 h 2 1 h 1, (20) 3 h n 5 h n+1 (2 k+1 +1) h n+k, (n 3,k 0) (21) 6 h n+1 10 h n+2 (2 k+1 +2) h n+k, (22) 6 h n+1 3 h n, (23) (2 k+1 +2) h n+k (2 k+1 +1) h n+k, (k 1) (24) (2 n 1 1) h n 15 h 5 7 h 4 3 h 3, (25) 2 n+1 h n (2 n 1) h n+1. (n 3) (26) In the following, we give comments on these relations, except for Eqs. (18) and (19). These relations are trivial. Comment on Eq. (20). The last order relation is included in Eq. (19), and therefore we first prove the second-to-last relation. The inverse image of 4 h 3 is

9 Order of Appearance of Homoclinic Points for the Hénon Map 1037 Fig. 6. Relation between 2 k+3 h k+4 and 2 k+2 h k+3. located below the x-axis, because 4 h 3 is to the left of Q. In the situation that 4 h 3 exists, 2 h 3 also exists on the right side of Q. The inverse image of 2 h 3 is located above the x axis. These facts imply that T 2 Γ u intersects the x axis. Therefore, 2 h 2 exists. Now let us first turn to the remaining relations. Let us first consider the negative iterates of (P, Q) S1, i.e., T k (P, Q) S1, k =1, 2,... Since there are no fixed points on (P, Q) S1, the two successive iterates T k (P, Q) S1 and T k 1 (P, Q) S1 have no common points. Hence, T k (P, Q) S1 are ordered according to k. There exists a sequence of arcs of T k (P, Q) S1 that accumulate at T 3 γ s as k. Then, the intersections T 3 Γ u T k (P, Q) S1 exist if T 3 Γ u T k 1 (P, Q) S1 (k 0) holds (see Fig. 6). As is easily confirmed, T 3 Γ u T k (P, Q) S1 are homoclinic points on the (k + 3)-rd floor and are 2 k+2 h k+3, since these are the last intersections counting from the foot-points. Similarly, T 3 Γ u T k 1 (P, Q) S1 are named 2 k+3 h k+4.thus, we have 2 k+3 h k+4 2 k+2 h k+3. Let a c ( i h n ) be the first critical parameter value at which i h n appears. For the odd-cores, we use the similar notation i g n and a c ( i g n ). From the accumulation property, we have the following relation: lim a c( 2 n 1 n h n )=a hs. (27) Comment on Eq. (21). First, we consider the cases n =2m +1 (m 1). We consider the situation in which T m Γ u and T m γ s have four intersection points. In this situation, T m Γ u and T m γ s have already experienced the first tangency and the first cubic-type tangency both in (P, Q) S1. Then T 2m Γ u and γ s have four intersection points; that is, T 2m Γ u is divided into three kinds of arcs by γ s : the first ones are closer to the foot-points and are contained in Z; the second ones are out of Z; and the third one contains the tip and is again in Z. Under an additional iteration of T, the last arc is mapped into Z, with its foot-points on Tγ s. Let us call this arc ξ u ( T n Γ u ) [see Fig. 7(a)].

10 1038 Y. Yamaguchi and K. Tanikawa (a) (b) <3 h> n P Q S 1 P Q ξ u ξ T n u Γ u <5 h> n +1 T -1 ( P, Q) S1 Fig. 7. (a) Definition of ξ u and (b) the relation between 3 h n and 5 h n+1 We consider the situation in which ξ u and (P, Q) S1 has the first tangency. This necessarily takes place at some a satisfying 0 <a<a hs because, as we know, these intersect each other at a = a hs. This corresponds to the appearance of 3 h n. The tangency is mapped in the region y>0. This implies that ξ u intersects T 1 (P, Q) S1 and the intersection points are 5 h n+1. The relation 3 h n 5 h n+1 is obtained [see Fig. 7(b)]. In general, if T k ξ u (P, Q) S1 holds, T k ξ u T 1 (P, Q) S1 also holds. This implies the order relation (2 k+1 +1) h n+k (2 k+2 +1) h n+k+1.thus, Eq. (21) is proved for odd n. Next, we consider the cases n =2m +2 (m 1). Suppose that T m Γ u and T m 1 γ s intersect each other at four points, two of them being in (P, Q) S2. There exists an arc ξ u of T n Γ u which has foot-points in Tγ and extends into Z. Byusing ξ u, Eq. (21) is proved for even n. We then use the fact that the tangency point or the intersection point z of ξ u and (P, Q) S2 is mapped into the region y>0. In fact, Tz = h(gz) =hz holds, and hz is located in the region y>0, because z is in the region y<0. Similarly, the relations for odd n are obtained. Comment on Eq. (22). We consider the situation represented by Eq. (21), that is, that in which the tip arc of T n Γ u, which is called ξ u, has its foot-points on Tγ s and extends into Z. We consider the situation in which ξ u and T 1 (Q, v) S1, represented by y = f(x)(y >0), have a tangency. In Fig. 8(a), T 1 (Q, v) S1 is illustrated by a thin line. The tangency is the inverse image of 6 h n+1. Using the fact that there exist branches of the backward images of (Q, v) S1 accumulating at Tγ s,ifξ u T k (Q, v) S1 holds for k 1, then ξ u T k 1 (Q, v) S1 also holds. This implies the order relation (2 k+1 +2) h n+k (2 k+2 +2) h n+k+1. Comment on Eqs. (23) and (24). Let us consider the parameter value at which 6 h n+1 just appears. The inverse image of 6 h n+1 is on T 1 (Q, v) S1. Obviously, ξ u intersects (Q, v) S1, and the intersection is 3 h n [see Fig. 8(b)]. This implies Eq. (23). Comparing the positions of T 1 (P, Q) S1 and T 1 (Q, v) S1,itiseasyto see that ξ u intersects T 1 (P, Q) S1 and T 1 (Q, v) S1, in this order. This implies the relation 6 h n+1 5 h n+1. In general, for k 1, ξ u intersects T k (P, Q) S1 and

11 Order of Appearance of Homoclinic Points for the Hénon Map 1039 P T -2 v (a) -1 T v ξu Q -1 T v T -1 ( Q, v ) T -1 S (, v ) 1 ξ u <6 h > n +1 P (b) P S 1 <6 > h n +1 Q T -2 (, ) <10 > h n +2 Qv S1 <5 > h n +1 <3 > h n Fig. 8. (a) Relation between 6 h n+1 and 10 h n+2. T 1 (Q, v) S1 is illustrated by a thin line and T 2 (Q, v) S1 by a thick line. (b) Relations among 6 h n+1, 3 h n and 5 h n+1. T k (Q, v) S1, in this order. These imply Eq. (24). Comment on Eqs. (25) and (26). We consider the first stage introduced in Ref. 2), in which T 2 γ s is tangent to the arc of T 2 Γ u. This situation is displayed in Fig. 9, where the tangent point is marked by the small disk labeled α. By reversibility, the point labeled β = hα, represented by the small circle, is also the tangent point of T 2 γ s and T 2 Γ u. The two points T 2 α and T 2 β are located in γ s. The point T 2 β is the tip of T 4 Γ u. The intersection points of T 4 Γ u and (Q, v) S1 are located to the left of T 2 β. This implies the existence of 7 h 4 and the relation 4 h 3 7 h 4. Because we have T 3 β Tγ s, the tip of T 5 Γ u is located in the region y<0, which means that 7 h 4 exists if 15 h 5 exists. Thus, we have 15 h 5 7 h 4. We now consider the situation in which T 2 γ s is tangent to the arc of T 3 Γ u located above α in Fig. 9. This is the beginning of the second stage. Repeating the argument of the preceding paragraph, we obtain the following relations: 8 h 4 15 h 5, 31 h 6 15 h 5. We can repeat this treatment in the beginning of the k ( 1)-th stage, and this yields the following relations: 2 k+1 h k+2 (2 k+2 1) h k+3, (2 k+3 1) h k+4 (2 k+2 1) h k+3. Finally, we prove the relation 7 h 4 3 h 3. In the situation depicted in Fig. 9, 7 h 4 and 3 h 3 both exist. Consider the situation in which 3 h 3 exists. We know that 5 h 4 exists. The image of 3 h 3 is located in the region y>0. This implies that neither 6 h 4 nor 7 h 4 exists. Thus, the relation is proved. Combining all the relations above, Eqs. (25) and (26) are obtained.

12 1040 Y. Yamaguchi and K. Tanikawa Fig. 9. The beginning of the first stage, where T 2 Γ u at a = and T 2 γ s has the two tangency points α and β (= hα). The symbol 7 4 stands for 7 h 4. The diagonal line represents S 1. We have the following relation: lim a c( (2 n 1 1) h n )=a hs. (28) n We also find the following orders of appearance for the symmetric odd-cores: k g n k g n+1, ((k =1,n 0), (k =2,n 1)) (29) 2 g n+1 1 g n, (n 1) (30) 2 n 1 g n 1 16 h 4 8 h 3 4 h 2, (31) 3 g n 5 g n+1 (2 k+1 +1) g n+2, (n 2,k 0) (32) 6 g n+1 10 g n+2 (2 k+1 +2) g n+k, (33) (2 k+1 +2) g n+k (2 k+1 +1) g n+k. (k 1) (34) 4.2. Order relations for the cores Here, we introduce a convenient method for determining the core for any element in the two binary trees. Let us define two operators L and R by L 0ŝ0 =0 2 ŝ0 2, R 0ŝ0 =01ŝ10,

13 Order of Appearance of Homoclinic Points for the Hénon Map 1041 where ŝ is an arbitrary word including an empty one. Given a path to i h n or i g n in the tree, we can calculate the corresponding core. For example, the sequence of operators is RL from the top of the tree to 4 h 3. Thus, its core, , is determined as RLs, wheres =0 2 is the core of 1 h 1. The cores for the elements in Fig. 5(b) are also determined if the paths are given. The coding process begins at 1 g 1 with the core 0 3 or at 2 g 1 with the core 010. For example, the core of 4 g 3 is obtained as LRs = starting from s =0 3. There is a unique path from the top of the tree to any position of the tree. Correspondingly, there is a unique sequences of operators L and R organized from right to left when descending the tree. In fact, given any i 1andn (n 2forh, and n 1forg), we can find the path to i h n or i g n. The general algorithm is Algorithm-1 given in Appendix A. This algorithm also determines the sequence of operators L and R. Algorithm-2, given in Appendix A, determines the path to any position in the binary tree. Corresponding to Eqs. (18) to (29), we can derive the order relations Eqs. (35) (40) for cores, where the symbol is used to represent the forcing relation between two cores; that is, s 1 s 2 means that if the symmetric homoclinic orbit with core s 1 exists, then the symmetric homoclinic orbit with core s 2 exists: (35) (36) 0 n n 2 0 n n 2 0 n n 2 0 n n 2. (37) 0 n n 1 0 n n 1 0 n n 1 0 n n 1. (38) 010 2n (39) 010 n n (40) In the following, we give comments on these relations. For Eq. (35): Comparing the positions of S 1 and S 2 in the phase plane, we have the relation 1 h n 1 g n. (41) Combining this equation and Eqs. (18) and (29), we have Eq. (35). A lower bound for topological entropy in the situation characterized by the core 0 m (m 1) is given by ln λ max,whereλ max is the largest eigenvalue of the following equation: λ m+1 λ m 2=0. (42) This equation is derived using the trellis method. A detailed derivation is given in Refs. 12) and 2). In the limit m, the largest eigenvalues accumulate at 1, and thus the lower bound for topological entropy tends to 0. This situation is the integrable case with a =0. For Eq. (36):

14 1042 Y. Yamaguchi and K. Tanikawa In order to derive Eq. (36), we use the following relations, which are easily obtained: 2 g n 2 h n+1 for n 1, (43) 2 h n 2 g n for n 2. (44) Employing the trellis method, a lower bound for topological entropy in the situation characterized by the core 0 n 10 n for 2 g n and the core 0 n n for 2 h n are determined, respectively, by λ 3n+2 (λ 1) λ 2n+1 (λ +1) 2=0, (45) and λ 2n+3 (λ n+1 +1) 2 1=0. (46) In the limit n, the largest eigenvalues calculated from both Eqs. (45) and (46) accumulate at 1. For Eqs. (37) and (38): Relations (37) and (38) are determined from Eqs. (21) and (32), respectively. Let h(s) denote a lower bound for topological entropy of a homoclinic orbit with core s. Then we have lim k h(0n k n 2 )=h(0 n 1 ), (47) and lim k h(0n k n 1 )=h(0 n ). (48) Here 0 n k n 2 and 0 n k n 1 are the cores of (2k +3) h n+k (k 0) and (2k +3) g n+k (k 1), where 0 0 represents an empty word. It is noted that h(0 n 1 ) is determined by Eq. (42). For Eqs. (39) and (40): The order relations (39) and (40) are derived from Eqs. (20) and (25), respectively. In these cases, the following relations hold: lim n h(0102n 4 10) = ln 2, (49) lim n h(010n n 3 10) = ln 2. (50) The former relation comes from Eq. (27) and the latter from Eq. (28). 5. Local dynamical ordering There exists an accumulation sequence of arcs of the unstable manifold at T n Γ u, whereas there exists an accumulation sequence of arcs of the stable manifold at T n γ s. Consequently, before and after the tangency between T n Γ u and T n γ s, there appear infinitely many intersections between the unstable arcs accumulating at T n Γ u and the stable arcs accumulating at T n γ s. In this section, as typical

15 Order of Appearance of Homoclinic Points for the Hénon Map 1043 examples, we consider the first tangency of TΓ u and T 1 γ s, referred to as 1 h 1 in 4, and the tangency of Γ u and T 1 γ s, referred to as 1 g 0. The term local in the title of this section reflects the fact that the orders of appearance near 1 h 1 or 1 g 0 are considered Intervals in the stable and unstable manifolds LetusintroduceasequenceofintervalsonΓ u and γ s. We first consider the parameter value at which TΓ u and T 2 γ s have the first tangency point on (P, Q) S2. This tangency appears before the first tangency of TΓ u and T 1 γ s on (P, Q) S1 (see Fig. 1). Next, we consider the situation in which TΓ u and T 2 γ s have two intersection points. In this situation, TΓ u already intersects T 3 γ s. Wedefinethetwo arcs of TΓ u as [A, B] TΓu and [C, D] TΓu where the two points A and D are located in T 3 γ s and B and C in T 2 γ s (see Fig. 10). In this situation, T 3 γ s intersects Γ u. Then T 3 γ s Z has two components, the upstream and downstream arcs. TΓ u intersects both of them. Let A and D with A Ws D be the intersections with the downstream arc. We define I1 u = T 1 [A, B] TΓu T 1 [C, D] TΓu Γ u. Considering the fact that T 2 Γ u intersects both T 2 γ s and T 3 γ s, we similarly define the intersection points E, F, G and H and the intervals [E,F] T 2 Γ u and [G, H] T 2 Γ u, and we correspondingly define I2 u = T 2 [E,F] T 2 Γ T 2 [G, H] T 2 Γ u in Γ u. Using the same procedure, we then define the intervals Ii u Γ u for i 1. The following Property 1 is obtained. Property 1. (i) Ii u Iu j = for i, j 1andi j. (ii) For i 1, T i+5 Ii+1 u is located between T i+4 Ii u and TΓ u, and the following relation holds: lim T i+4 Ii u = TΓ u. (51) i Fig. 10. Definition of [A, B] TΓu,[C, D] TΓu,[E,F] T 2 Γ u and [G, H] T 2 Γ u at a =2.4 afterthefirst tangency of TΓ u and T 2 γ s.

16 1044 Y. Yamaguchi and K. Tanikawa The accumulation property stated in Eq. (51) is derived by application of the lambda lemma. 13) Similar relations hold for I s i (= hi u i ). Finally, we define the intervals L u i = T 1 I u i,i=1, 2,... in Ω u = T 1 Γ u Order relations We define two homoclinic orbits, H(k, k )andg(k, k ). Let H(k, k )bethe homoclinic orbit which has point(s) on T k Γ u T k γ s,withk + k 1, k 0and k 0, and let G(k, k ) be the homoclinic orbit which has point(s) on T k Ω u T k γ s, with k + k 2, k 0andk 0. Proposition 2. (i) H(k, k ) H(k,k)= for k k. Neither H(k, k )norh(k,k) is symmetric for k k. (ii) G(k, k ) G(k,k)= for k k. Neither G(k, k )norg(k,k) is symmetric for k k. Proof. A homoclinic orbit has at most one point in Γ u. In fact, if it has two points in Γ u, the unstable (or stable) manifold intersects itself at one of these points. The orbit H(k, k ) has a point in I u k, while H(k,k) has a point in I u k. Thus, the first statement in (i) is obtained. From Proposition 1, the second statement is derived. (ii) The proof is similar to that for (i) and thus is omitted. Q.E.D. The notation H(k, k ) H(m, m ) (52) expresses the forcing relation between H(k, k )andh(m, m ). We read it as H(k, k ) forces H(m, m ). The meaning is that if H(k, k ) exists, then H(m, m ) exists. The forcing relation G(k, k ) G(m, m ). (53) is also used. First, we consider the dynamical ordering forced by H(1, 1) (= 1 h 1 ). Theorem 1. Suppose that T 5 I u 1 T 5 I s 1 = when TΓ u and T 2 γ s has a first tangency point. Then, the dynamical ordering given in Table II holds. Remark 1. IntheHénon map, the first tangency between TΓ u and T 2 γ s occurs at Table II. Dynamical ordering. I1 s I2 s I3 s I1 u H(5, 5) H(5, 6) H(5, 7) H(5, 1) I2 u H(6, 5) H(6, 6) H(6, 7) H(6, 1) I3 u H(7, 5) H(7, 6) H(7, 7) H(7, 1) H(1, 5) H(1, 6) H(1, 7) H(1, 1)

17 Order of Appearance of Homoclinic Points for the Hénon Map 1045 a =2.35, and the first tangency between T 5 I1 u and T 5 I1 s Therefore, the assumption stated in Theorem 1 holds. occurs at a =3.1. Remark 2. The orbit H(k, 1) (resp., H(1,k)) for k 5 in the right (resp., bottom) box is the accumulation of the respective row (resp., column) sequence. In the limit k, H(k, 1) and H(1,k) accumulate at H(1, 1). The dynamical ordering appearing in Table II implies that H(1, 1) forces all other HOs. The homoclinic orbits H(k, k) (k 5) have tangency points (intersection points) on (P, Q) S1. Remark 3. Each of T k I u j and T k I s j consists of two arcs. The tangency between them occurs four times. Grouping these tangency situations into a unit, we call them the tangency between T k I u j and T k I s j. Proof. From Property 1, it is seen that T 5 I1 u, T6 I2 u, accumulate at TΓ u,and correspondingly, T 5 I1 s,t 6 I2 s, accumulate at T 1 γ s.ifwesitatt i+4 Ii u (i 1), we see that T (j+4) γ s (j 1) with increasing j approach and intersect T i+4 Ii u one by one. This gives the order relations of the i-th row. If we sit at T (j+4) Ij s (j 1), we see that T (i+4) γ s (i 1) with increasing i approach and intersect T (j+4) Ij s one by one. This gives us the order relations of the j-th column. Q.E.D. We determined the cores of the symbol sequence of the homoclinic orbits included in H(1, 1), H(k, 1), H(1,k)andH(k, k )(k, k 5) of Table II. The results are given in Table III. In Fig. 11, one of the orbits, H(1, 1), denoted by filled circles, starts from z 0 Γ u. Letusintroduceσ(z) as σ(z) =σ(gz) =i if z V i. (54) Then, we have σ(z 0 ) = 0. The next point, z 1, is the intersection point of TΓ u and T 1 γ s,andz 2 is located in γ s. FromFig.11,weseethatσ(z 1 ) = 0. Therefore, the core of this orbit is represented by 00. Next, we consider the orbit in H(5, 1), represented by gray disks. The initial point, z 0,islocatedinIu 1. The orbit moves near P and revolves around the fixed point Q once. It then comes back near z 0, represented by the filled circle. This point is z 4, and the next point, z 5,islocated Table III. The core for the homoclinic orbit in Table II. Orbit Core Core H(1, 1) H(k, 1) 00 k k k k H(1,k) 0010 k k k k 3 1 H(k, k ) 00 k k k k k k k k k k k k k k k k 3 1

18 1046 Y. Yamaguchi and K. Tanikawa Fig. 11. Three orbits of the homoclinic points after the completion of the horseshoe. above z 1 in T 1 γ s. Then the core of this orbit is obtained as The orbit denoted by filled rectangles in H(6, 1) passes in the vicinity of P andthenafterfour iterations reaches the neighborhood of z 3 for the orbit in H(5, 1). The core of this orbit is represented as Repeating the same procedure, we have the symbol sequence for the core of H(k, 1). The other representations are obtained similarly. By the reversibility with respect to h, the symbol sequences for the core of H(1,k) are also obtained. We next determine the core of the symmetric homoclinic orbit included in H(5, 5). The initial point, z 0, is located in I1 u, and z 5 is the intersection point of T 5 I1 u and T 5 I1 s,locatedins 1. The point z 5 exists in the vicinity of z 5 of the orbit in H(5, 1) mentioned above. Therefore, the symbol sequence of z 0 to z 5 is given by Using the relation hz 5 = z 5,wehave gz k = z 9 k. (55) From Eq. (9), we have σ(gz k )=σ(z 9 k ). (56) Using this relation, we can determine the symbol sequence from z 6 to z 9, and thus we obtain the expression Finally, we have the core represented by The other three symmetric homoclinic orbits of H(5, 5) are represented by , and There are four non-symmetric homoclinic orbits represented by , , and

19 Order of Appearance of Homoclinic Points for the Hénon Map 1047 Table IV. Maximum eigenvalues. Orbit λ max Orbit λ max Orbit λ max Orbit λ max H(1, 1) H(5, 5) H(6, 6) H(7, 7) H(5, 1) H(5, 6) H(6, 6) H(7, 8) H(6, 1) H(5, 7) H(6, 7) H(7, 9) H(7, 1) H(5, 8) H(6, 8) H(7, 10) H(8, 1) H(5, 9) H(6, 9) H(7, 11) Table V. Dynamical ordering. I1 s I2 s I3 s L u 1 G(4, 4) G(4, 5) G(4, 6) G(4, 1) L u 2 G(5, 4) G(5, 5) G(5, 6) G(5, 1) L u 3 G(6, 4) G(6, 5) G(6, 6) G(6, 1) G(1, 4) G(1, 5) G(1, 6) G(1, 1) These eight cores give the same value of the lower bound for topological entropy, ln Remarks. The relation H(k, k ) H(m, m ) does not necessarily imply h top (H(k, k )) >h top (H(m, m )), where h top (H(k, k )) denotes the lower bound for topological entropy of H(k, k ). We cannot obtain an order relation between H(4, 4) and H(5, 5), though we have the additional relation H(1, 1) H(4, 4). To obtain their order relation, we need more detailed information concerning the shapes of the stable and unstable manifolds. The following theorem gives the order relations of the HOs forced by G(1, 1) (= 1 g 0 ). The proof is similar to that of Theorem 1 and thus is omitted. Theorem 2. Suppose that T 4 L u 1 T 4 I s 1 = when TΓ u and T 1 γ s has the first tangency point. For the HOs forced by the symmetric homoclinic orbit G(1, 1), the dynamical ordering given in Table V holds. Remark. For the Hénon map, the first tangency between T 4 L u 1 and T 4 I s 1 occurs at a =4.69, and the first tangency of TΓ u and T 1 γ s occurs at a = Thus, the assumption in Theorem 2 holds. The cores for the homoclinic orbits in Table V are listed in Table VI, and their maximum eigenvalues are listed in Table VII: Acknowledgements The authors would like to thank the referees for useful comments.

20 1048 Y. Yamaguchi and K. Tanikawa Table VI. The cores for the homoclinic orbits in Table V. Orbit Core Core G(1, 1) 0 1 G(k, 1) 00 k k k k 3 11 G(1,k) 010 k k k k 3 1 G(k, k ) 00 k k k k k k k k k k k k k k k k 3 0 Table VII. Maximum eigenvalue. Orbit λ max Orbit λ max Orbit λ max Orbit λ max G(1, 1) 2 G(4, 4) G(5, 5) G(6, 6) G(4, 1) G(4, 5) G(5, 6) G(6, 7) G(5, 1) G(4, 6) G(5, 7) G(6, 8) G(6, 1) G(4, 7) G(5, 8) G(6, 9) G(7, 1) G(4, 8) G(5, 9) G(6, 10) Appendix A Algorithms For a given number n in a given floor of the binary tree, Algorithm-1 determines the path from the top of the tree to the number, and the sequence of L and R. Algorithm-2 determines for a given position in the tree the path from the top to it, and the sequence of the operators L and R. In the two algorithms, we use the following three properties of the binary tree. [P1] Any number A in the k-th (k 1) floor is a parent to two children in the (k + 1)-th floor. Any number A in the k-th floor (k 2) is a child to a parent in the (k 1)-th floor. A parent and a child are connected by a branch. [P2] If A is even (or odd), we call it the even- (or odd)-parent to its children. A right child is an even number, B, and a left child an odd number, C. If A is even, then B = A and C =2 k +1 A, whereasifa is odd, then B =2 k +1 A and C = A. [P3] L (or R) is attached to the right (or left) branch connecting the even-parent and the child. R (or L) is attached to the right (or left) branch connecting the odd-parent and the child. Algorithm-1. [1] Input a number n ( 2) and the floor number k ( k min = ln 2 n +1), where k min is the minimum number of the floor for which n exists. n is called a child. [2]Goto[3]ifk 2. If k =1,goto[6]. [3] Determine the pair child m of n as m =2 k 1 +1 n. [4] Using (4a) or (4b), the parent of m and n is determined, and the branch from the parent to the child is also determined. In (4a) and (4b), [P1] and [P2] are used.

21 Order of Appearance of Homoclinic Points for the Hénon Map 1049 (4a) The case n<m: The parent in the (k 1)-th floor is n. Ifn is even, the branch from the parent to the child is to the right. If n is odd, the branch from the parent to the child is to the left. (4b) The case n>m: The parent in the (k 1)-th floor is m. If m is even, the branch from the parent to the child is to the right. If m is odd, the branch from the parent to the child is to the left. [5] Redefine k as k 1andn as n or m, depending on whether (4a) or (4b) has been chosen. Return to [2]. [6] We arrive at the goal, that is, the top of the tree. At this point, we have a sequence of branches from the top to the initial number, n together with the corresponding sequence of the operators L and R. Algorithm-2. [1] Input a floor number k ( 2). There are 2 k 1 elements in the k-th floor. We name them 1, 2,, 2 n 1, from left to right. Input m (1 m 2 k 1 ) to specify the position of an element. [2] If k 2, go to [3]. If k =1,goto[5]. [3] The position of the parent is m/2, where m/2 is the smallest integer larger than or equal to m/2. If m is even, the branch from the parent to the child is to the right. If m is odd, the branch is to the left. [4] Redefine k as k 1andm as m/2. Return to [2]. [5] We arrive at the goal, that is, the top of the tree. At this point, we have a sequence of branches from the top to the initial position m, together with the corresponding sequence of the operators L and R. References 1) M. Hénon, Commun. Math. Phys. 50 (1976), 69. 2) Y. Yamaguchi and K. Tanikawa, Prog. Theor. Phys. 114 (2005), ) A. de Carvalho, Ergod. Th. & Dynam. Sys. 19 (1999), ) A. de Carvalho and T. Hall, Experimental Math. 11 (2002), ) P. Collins, Experimental Math. 14 (2005), 75. 6) Y. Yamaguchi and K. Tanikawa, Prog. Theor. Phys. 113 (2005), ) R. de Vogelaere, in Contributions to the Theory of Oscillations, Vol. IV, Ann. Math. Studies No. 41 (Princeton University Press, 1958). 8) R. Brown, Ergod. Th. & Dynam. Sys. 15 (1995), ) E. Bedford and J. Smillie, Ann. Math. 160 (2004), 1. 10) D. Sterling, H. R. Dullin and J. D. Meiss, Physica D 134 (1999), ) T. Hall, Trains3, Software available from SET/CONTENT/members/T Hall.html 12) P. Collins, Int. J. Bifurcation and Chaos 12 (2002), ) J. Palis, Topology 8 (1969), 385.