Homoclinic Orbits and Chaos in Discretized Perturbed NLS Systems: Part II. Symbolic Dynamics

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1 J. Noninear Sci. Vo. 7: pp ) 997 Springer-Verag New York Inc. Homocinic Orbits and Chaos in Discretized Perturbed NLS Systems: Part II. Symboic Dynamics Y. Li and S. Wiggins 2 Department of Mathematics, University of Caifornia at Los Angees, Los Angees, CA 90024, USA Present Address: Department of Mathematics, 2-336, Massachusetts Institute of Technoogy, Cambridge, MA 0239, USA 2 Appied Mechanics, and Contro and Dynamica Systems 6-8, Caifornia Institute of Technoogy, Pasadena, CA 925, USA Received September, 995; revised manuscript accepted for pubication September, 996 Communicated by Jerrod Marsden Summary. In Part I [9], this journa), Li and McLaughin proved the existence of homocinic orbits in certain discrete NLS systems. In this paper, we wi construct Smae horseshoes based on the existence of homocinic orbits in these systems. First, we wi construct Smae horseshoes for a genera high dimensiona dynamica system. As a resut, a certain compact, invariant Cantor set is constructed. The Poincaré map on induced by the fow is shown to be topoogicay conjugate to the shift automorphism on two symbos, 0 and. This gives rise to deterministic chaos. We appy the genera theory to the discrete NLS systems as concrete exampes. Of particuar interest is the fact that the discrete NLS systems possess a symmetric pair of homocinic orbits. The Smae horseshoes and chaos created by the pair of homocinic orbits are aso studied using the genera theory. As a consequence we can interpret certain numerica experiments on the discrete NLS systems as chaotic center-wing jumping. Key words. discretized noninear Schroedinger equations, Smae horseshoes, chaos MSC numbers. 58F07, 58F3, 70K50. Introduction In Part I of this study [9], this journa), Li and McLaughin estabished the existence of homocinic orbits for the foowing N-partice any 2 < N < ) finite-difference dynamica system: iq n = µ [q 2 n+ 2q n + q n ] + q n 2 q n+ +q n ) 2ω 2 q n [ +iɛ αq n + β ] µ 2q n+ 2q n +q n )+Ɣ,.)

2 36 Y. Li and S. Wiggins where i =, q n s are compex variabes, and q n+n = q n q N n = q n periodic condition), even condition); therefore, this is a 2M + ) dimensiona system, where M = N/2 N even), M = N )/2 N odd), with µ = N, N tan π N <ω<ntan 2π, N for N > 3, 3 tan π <ω<, 3 for N = 3. ɛ [0,ɛ ), α > 0), β > 0), Ɣ > 0) are constants. This system is a finite-difference discretization of the foowing perturbed NLS system: iq t = q xx +2 [ q 2 ω 2] q + iɛ[ αq +βq xx +Ɣ],.2) where qx + ) = qx), q x) = qx); π<ω<2π,ɛ [0,ɛ ),α>0),β>0), and Ɣ>0)are constants. The motivation for this study comes from the numerica experiments showing the existence of chaos in near-integrabe systems as summarized in []. These numerica experiments show that the perturbed noninear Schroedinger equations.2) possess soutions which consist of beautifu reguar spatia patterns that evove irreguary chaoticay) in time. See Figure.) These numerica studies aso correate this chaotic behavior in the perturbed system with the presence of a hyperboic structure, and homocinic manifods connecting this hyperboic structure, in the unperturbed ɛ = 0) integrabe NLS equation see Li [8], Li and McLaughin [0]). In Part I [9], this journa), Li and McLaughin proved the existence of homocinic orbits for.), and we now state the main resut from that paper. We denote the externa parameter space by N N 7) where N = ω,α,β,ɣ) ω Ntan π N, N tan 2π ), N Ɣ 0, ), α 0,α 0 ), β 0,β 0 ), } where α 0 and β 0 are any positive numbers < ). Theorem. Li and McLaughin [9]). For any N 7 N < ), there exists a positive number ɛ 0, such that for any ɛ 0,ɛ 0 ), there exists a codimension submanifod E ɛ in N. The submanifod E ɛ is in an Oɛ ν ) neighborhood of the hyperpane β = κα, where κ = κω; N) is shown in Figure 2, ν = /2 δ 0, 0 <δ 0 /2. For any externa parameters ω, α, β, Ɣ) one ɛ, there exists a homocinic orbit asymptotic to a fixed point q ɛ.

3 Symboic Dynamics 37 Fig.. Soution of the noninear Schroedinger equation in the chaotic regime. The waveform randomy switches from a form ocaized at the center of the interva to a form ocaized at the ends or wings ) of the interva with an intermediate passage through a spatiay uniform or fat state. Remark.. In the cases 3 N 6), κ is aways negative as shown in Figure 2. Since we require both dissipation parameters α and β to be positive, the reation β = κα shows that the existence of homocinic orbits vioates this positivity. When N is even and > 7, there is in fact a pair of homocinic orbits asymptotic to a fixed point q ɛ ; since if q n = f n, t) soves.), then q n = f n + N/2, t) aso soves.). In this paper, we show how Smae horseshoes can be constructed near these homocinic orbits and, most importanty, how the geometry associated with the horseshoes gives a mechanism for chaotic center-wing jumping as described in []. We wi first construct Smae horseshoes for a genera system, then cast the discrete NLS system.) into the specia form of the genera system. Horseshoes, and the associated symboic dynamics, have been constructed for the same types of systems by Sinikov [8] and Deng [4]. However, they do not treat the case of two homocinic orbits which arise as a resut of a symmetry. Moreover, our construction uses different methods. We use n-dimensiona versions of the Coney-Moser conditions see Moser [3] and Wiggins [9]). This method of proof offers some advantages over that of Sinikov and Deng.. The Coney-Moser conditions give rise to a geometrica description of chaotic dynamics in phase space that is particuar to the probem being studied. Using our horseshoe construction we are abe to interpret the numerica observation in Figure using

4 38 Y. Li and S. Wiggins a) b) c) Fig. 2. The sope κ, as a function of ω, of the hyperpane in the parameter space near which homocinic orbits occur. a) The sope for different vaues of N showing convergence to the PDE curve corresponding to N =. b) Sopes for N N c = 7. c) Sopes for N < N c = 7.

5 Symboic Dynamics 39 symboic dynamics on four symbos 2, ;, 2). We show that geometricay the chaotic dynamics can be interpreted as the chaotic center-wing jumping that is seen in the numerica experiments. 2. The Coney-Moser conditions aow one to easiy concude structura stabiity of the invariant Cantor set. 3. Hyperboicity of the invariant Cantor set is an easy consequence. This aows one to bring in a variety of statistica techniques from dynamica systems theory. 4. The topoogica argument of Coney and Moser has some advantages in deaing with nonhyperboic fixed points. 5. The Coney-Moser topoogica approach can aso be used for deaing with transient chaos see, e.g., [7]). There is another important point to be made. The conditions of Coney and Moser are genera conditions that do not rey on being near a homocinic orbit for their appication. Rather, they are sufficient conditions for a dynamica system to possess an invariant set on which the dynamics is topoogicay conjugate to a subshift of finite type. The work of Coney and Moser was originay for two-dimensiona maps. In this ow-dimensiona case one can rather easiy get a hande on the geometry of the image and preimage of seected regions of the domain of a map. In higher dimensions this becomes more probematic. Consequenty, there are reativey few appications of this technique in higher dimensions, despite the obvious advantages. This is one of the contributions of this paper. We show how generaizations of the Coney-Moser conditions can be appied for this genera cass of vector fieds having a homocinic orbit. We do this by using the oca geometry of the fixed points of the Poincaré map. This construction pays a key roe in our equivariant construction of the symboic dynamics which is crucia for the interpretation of the center-wing jumping. The structura stabiity of the Coney-Moser construction is usefu in another area. In order to describe the dynamics near the hyperboic fixed point we use a smooth inearization argument. For an n-dimensiona system this requires a countabe number of nonresonance conditions on the eigenvaues of the matrix associated with the vector fied inearized at the fixed point. In [9] it is shown that a homocinic orbit exists on a codimension one surface in the parameter space. Each nonresonance condition aso defines a codimension one surface in the parameter space, and it intersects the surface on which the homocinic orbits exist in a set of measure zero. There are a countabe number of such resonance conditions, so the set of parameter vaues on this surface where nonresonance fais is sti a set of measure zero. Hence, on a codimension one surface in parameter space there is a set of fu measure where the nonresonance conditions hod. Now since the horseshoes constructed using the Coney-Moser construction are structuray stabe, they aso persist for this set of measure zero where the nonresonance conditions break down, since the set of fu measure where nonresonance hods is dense. This paper is organized as foows. Sections deveop the genera setting aowing us to construct the Poincaré map in a neighborhood of the homocinic orbit for the genera cass of systems under consideration. In Section 2.6 we compute fixed points of the Poincaré map. In Section 2.7 we deveop the n-dimensiona versions of the Coney- Moser conditions. These are sight generaizations of those given in [9], and possiby

6 320 Y. Li and S. Wiggins more easy to appy in high dimensiona probems. We then use them to construct Smae horseshoes near the homocinic orbit. In Section 3 we appy these resuts to homocinic orbits in the discrete NLS system. In the case where there are two homocinic orbits we show how the horseshoe chaos can be interpreted as chaotic center-wing jumping. 2. Genera Theory 2.. Genera Setting We study the foowing 2m + n)-dimensiona system: x j = ɛ j α j x j β j y j + X j x, y, z), ẏ j = β j x j + ɛ j α j y j + Y j x, y, z), z k = δ k γ k z k + Z k x, y, z), j =,...,m;k =,...,n), 2.) where ɛ j = δ k =, j m,, m < j m., k n,, n < k n, x x,...,x m ) T, y y,...,y m ) T, z z,...,z n ) T, X j 0,0,0)=Y j 0,0,0)=0, grad X j 0, 0, 0) = grad Y j 0, 0, 0) = 0, j =,...,m), Z k 0, 0, 0) = grad Z k 0, 0, 0) = 0, k =,...,n). Moreover, X j x, y, z), Y j x, y, z), and Z k x, y, z) are C functions in a neighborhood of 0, 0, 0), and α j, β j, and γ k are positive constants. Therefore, 0, 0, 0) is a sadde point. We assume that this system 2.) has the foowing two properties:. There is a homocinic orbit h asymptotic to 0, 0, 0). 2. a) α <α j, for any 2 j m ; α <γ k, for any n < k n.

7 Symboic Dynamics 32 b) γ <γ k, for any k n ; γ <α j, for any m < j m. c) α <γ. Property 2 says that α is the smaest attracting rate, and γ is the smaest repeing rate; moreover, α is smaer than γ. We aso assume that the soution operator F0 t, F0 t : x0), y0), z0)) xt), yt), zt)), is C in t, and a C 2 diffeomorphism for fixed finite t. The motivation for this study comes from the study of the discrete NLS system [9], in which the discrete NLS system can be normaized into a specia form of the above system 2.) see Section 3). We are going to define two Poincaré sections 0 and in a sma neighborhood of 0, 0, 0). The Poincaré map P: U 0 0 induced by the fow can be expressed as the composition of two maps where P = P 0 P 0, P 0 : U 0 0, P 0 : U 0, are aso induced by the fow. In particuar, P0 is induced by the fow in a neighborhood of 0, 0, 0), and P 0 is induced by the goba fow outside a neighborhood of 0, 0, 0). We define 0 and in such a way that they sit in a tubuar neighborhood of the homocinic orbit h. Moreover, the fight time for orbits starting from points on to reach 0 is bounded. These facts enabe us to approximate P 0 by a inear transformation. The fight time for orbits starting from points on 0 to reach is unbounded. Nevertheess, P0 is induced by the oca fow. We wi construct Smae horseshoes on 0 under the map P. As a resut, restricted to some compact Cantor subset of 0, the dynamics of P is topoogicay conjugate to the shift automorphism on symbos. It is much more difficut to construct horseshoes in high-dimensiona systems than in ow-dimensiona systems. Beow we ist the main difficut points:. Since there are many different attracting and repeing rates in 2.) at the sadde point 0, 0, 0), inearized dynamics cannot approximate the fu noninear dynamics in the neighborhood of 0, 0, 0). A simpe iustration of this fact is given in the Appendix. 2. The affine transformation, as an approximation to P 0, has the representation as a nonsinguar arge-matrix transformation. It is very hard to track objects on after this arge-matrix transformation. 3. In high dimensiona systems, verification of the so-caed Coney-Moser conditions becomes much harder. Point wi be overcome through smooth norma form reduction [6]. Point 2 wi be overcome through identifying the fixed points of P and using the oca structure of the

8 322 Y. Li and S. Wiggins fixed points. Simiary, point 3 wi be overcome through certain generic considerations and the oca geometric structure near fixed points of P. In this way this paper sets up a genera program for constructing Smae horseshoes in a arge cass of high-dimensiona dynamica systems Smooth Norma Form Reduction The reference for this section is [6]. Consider the inear part of the system 2.), denote by a j the eigenvaues, δ j γ j, a j = ɛ k α k +iβ k, ɛ k α k iβ k, j =,...,2m+n) j =,...,n, j =2k +n, k =,...,m, j =2k+n, k =,...,m, where i = and, k m, ɛ k =, m < k m,, j n, δ j =, n < j n. We assume the foowing nonresonance condition: G) a j 2m+n k= k a k, mod[i2π], 2.2) for a j =,...,2m+n; and a sets of nonnegative integers,..., 2m+n such that 2 Then, there exists a C 2 diffeomorphism 2m+n k= k <. R: x, y, z) x, y, z ), which is the identity transformation outside a neighborhood of 0, 0, 0); moreover, R0, 0, 0) = 0, 0, 0), grad R0, 0, 0) = identity matrix.

9 Symboic Dynamics 323 More importanty, in a sma neighborhood of x, y, z ) = 0, 0, 0), system 2.) is reduced to the inear system [6], ẋ j = ɛ j α j x j β j y j, ẏ j = β j x j + ɛ jα j y j, 2.3) ż k = δ kγ k z k, j =,...,m;k =,...,n). Denote the soution operator of system 2.3) by L t. The soution operator of system 2.) is F0 t. Let F t RF0 t R, then, F t is the soution operator of the transformed system of 2.) under R. Moreover, in, F t = L t. From now on, we are mainy concerned with the transformed system, ẋ j = ɛ j α j x j β j y j + X j x, y, z), ẏ j = β j x j + ɛ jα j y j + Y j x, y, z), 2.4) ż k = δ kγ k z k + Z k x, y, z), j =,...,m;k =,...,n), where X j, Y j, and Z k vanish identicay inside. Since F t 0 is a C2 diffeomorphism for fixed t,soisf t. Moreover, since R is independent of t, F t 0 is C in t, and so is F t. From now on, we sha drop the primes in systems 2.3;2.4). Remark 2.. The requirement that X j, Y j, and Z k in 2.) are of cass C can be repaced by requiring that they are of cass C N, where [6] N = N2m + n;a j }, j =,...,2m+n) Some Definitions In this section we wi define two Poincaré sections 0 and and two Poincaré maps, P 0 : U 0 0, P 0 : U 0. We wi denote the stabe and unstabe manifods of 0, 0, 0) by W s and W u, respectivey. We know that h W s W u. We denote the components of W s and W u containing the origin by W s oc and W u oc,

10 324 Y. Li and S. Wiggins respectivey. We know that Woc s and W oc u coincide with the corresponding stabe and unstabe subspaces, respectivey. In particuar, they have the foowing representations: Woc s x, y, z) } xj = 0, y j = 0, m < j m; z k = 0, k n, Woc u x, y, z) xj = 0, y j = 0, j m ; z k = 0, n < k n }. We refer to h Woc s h+ as the the forward time segment and h Woc u h as the backward time segment, respectivey. By assumption 2 on system 2.), α is the smaest attracting rate, γ is the smaest repeing rate. Therefore, genericay, G2) h + is tangent to the x, y )-pane at 0, 0, 0), h is tangent to the positive z -axis at 0, 0, 0). Specificay, h + and h can be parametrized as foows: x + j t) = e α j t x + j 0) cos β j t y + j 0) sin β j t), y + j t) = e α j t x + j 0) sin β j t + y + j 0) cos β j t); j m ), 2.5) z + k t) = e γkt z + k 0) n < k n), 0 t <, and x j t) = e α j t x j 0) cos β j t y j 0) sin β j t), y j t) = e α j t x j 0) sin β j t + y j 0) cos β j t); z k t) = eγkt z k 0) k n ), m < j m), 2.6) < t 0, respectivey, where x + j 0), y + j 0), z + k 0)) and x j 0), y j 0), z k 0)) represent any points on h + Woc s and on h Woc u, respectivey. The genericity corresponds to x + 0)) 2 + y + 0)) 2 0, 2.7) z 0) ) Let η be a sma parameter, and et η be another sma parameter, such that η exp 2πα /β } <η <η. This constraint is necessary in order for P 0 to be, as we show in Lemma 2.2. Definition. The Poincaré section 0 is defined by the constraints, y = 0, η < x <η, 0<z <η, xj 2 +yj 2 <η 2, j=2,...,m, z k <η, k=2,...,n.

11 Symboic Dynamics 325 Definition 2. The Poincaré section is defined by the constraints, z = η, z k <η, k=2,...,n, x 2 j +y 2 j <η 2, j=,...,m. Definition 3. The Poincaré map P 0 from 0 to is defined as P 0 : U 0 0, q U 0, P 0 q) = F t q), where t = t q) >0 is the smaest time such that F t q). Definition 4. The Poincaré map P 0 from to ) is defined as P 0 : U 0, q U, P 0 q) = F T q) q) 0, where T q) >0 is the smaest time such that F T q) q) 0. If η is sufficienty sma, 0,. By the representation 2.5) of h +, we can choose η, such that h + intersects the z = 0) boundary of 0 at where q + has the coordinates q + h + 0, x = x +, y = 0, x j = x + j, y j = y + j, 2 j m ), z k = z + k, n < k n), x j = 0, y j = 0, m < j m), z k = 0, k n ), with η < x + <η, and x + j, y + j, and z + k satisfy the corresponding inequaities in the definition of 0. Simiary, h intersects at q h,

12 326 Y. Li and S. Wiggins where q has the coordinates z = η, x j = x j, y j = y j m < j m), z k = z k 2 k n ), x j = 0, y j = 0, j m ), z k = 0, n < k n), and x j, y j, and z k satisfy the corresponding inequaities in the definition of. Finay, we have the crucia fact, P 0 q ) = q ) Lemma 2.. If η is sufficienty sma, then the Poincaré sections 0 and are transversa to the vector fied. Proof. First we wi show that 0 is transversa to the vector fied at the point q +. Let n 0 be the unit norma vector to 0 ; then n 0 has the coordinate representation as y =, and a other coordinates are zeros. The vector fied v + at q + can be obtained through differentiating 2.5) with respect to t. Notice that for the cacuation of the inner product n 0,v +, we ony need to know the y -coordinate of v +, which is where t + satisfies the equation x + 0) cos β t + y + 0) sin β t +, x + 0) sin β t + + y + 0) cos β t + = ) Assume that n 0,v + =0; then, x + 0) cos β t + y + 0) sin β t + = 0. 2.) Eqs. 2.0), 2.) impy that x + 0) = y+ 0) = 0. This contradicts the generic condition 2.7). Thus, n 0,v + 0. This, together with the smoothness of the vector fied, impies that for sufficienty sma η, 0 is transversa to the vector fied at any point of 0. Simiary for. This competes the proof of the emma.

13 Symboic Dynamics 327 Lemma 2.2. If η is sufficienty sma, then the Poincaré map P0 is -; moreover, if in addition U is inside a sufficienty sma neighborhood of q, then P 0 is -. Proof. The caim that P 0 is - foows immediatey from the fact that F t is C in t and a C 2 diffeomorphism for fixed t. For detais, see Section 2.5. Next, we show that P0 is -. Let p, p 2 be two different points in U 0 0, and assume that P 0 p ) = P 0 p 2). 2.2) We wi ony need to study the x, y, z ) coordinates of points p, p 2 : P 0 p ) and P 0 p 2). Let t and t 2 be the time for orbits starting from p and p 2 to reach P 0 p ) and P 0 p 2), respectivey. Then, t = γ n η/z p, t 2 = γ n η/z p2. We introduce the notation z p to denote the z coordinate of p. Since z p > 0, z p2 > 0, both t and t 2 are finite. The reations ead to x P 0 p ) = x P 0 p 2), y P 0 p ) = y P 0 p 2), x p e α t cos β t = x p2 e α t 2 cos β t 2, 2.3) x p e α t sin β t = x p2 e α t 2 sin β t 2, 2.4) since both t and t 2 are finite; moreover, both x p and x p2 satisfy the constraint, η < x <η, 2.5) where η >ηexp 2πα /β }. Then from 2.3;2.4), we have and thus, tan β t = tan β t 2, β t = β t 2 + jπ, for some j Z. If j is odd, then neither 2.3) nor 2.4) wi hod. Therefore, j is even = 2 j 0 ). Finay, from 2.3;2.4), we have x p x p2 = exp2α j 0 π/β }. This reation contradicts the constraint 2.5), except for the case j 0 = 0. But, if j 0 = 0, then t = t 2. In this case, the assumption 2.2) contradicts the fact that F t is a diffeomorphism. Thus, in any case, the assumption 2.2) is not vaid, which shows that P0 is -. This competes the proof of the emma.

14 328 Y. Li and S. Wiggins 2.4. The Poincaré Map P 0 Denote the coordinates on 0 by x 0 ;xj 0,y0 j },j=2,...,m; z0 k,k =,...,n). Denote the coordinates on by x j, yj }, j =,...,m; z k,k =2,...,n). In, the soution operator F t = L t has the representation x j t) = e ɛ j α j t x j 0) cos β j t y j 0) sin β j t ), y j t) = e ɛ j α j t x j 0) sin β j t + y j 0) cos β j t ), z k t) = e δ kγ k t z k 0) j =,...,m; k =,...,n). Let t be the fight time for an orbit starting from a point on 0 to reach. Then, t = γ n η/z ) Using this expression for the soution operator and the fight time, P 0 is given by x = e α t x 0 cos β t, y = e α t x 0 sin β t, x j = e α j t xj 0 cos β j t yj 0 sin β j t ), yj = e α j t xj 0 sin β j t + yj 0 cos β j t ) 2 j m ), z k = e γ kt z 0 k n < k n), 2.7) x j = e α j t x 0 j cos β j t y 0 j sin β j t ), y j = e α j t x 0 j sin β j t + y 0 j cos β j t ) m < j m), z k = eγ kt z 0 k k n ). 2.8) 2.5. The Poincaré Map P 0 Let ˆ 0 be the enargement of 0 specified by the condition η <z <η. The Poincaré map P 0 can be naturay extended to ˆP 0, and we denote the extension by ˆP 0 : Û ˆ 0.

15 Symboic Dynamics 329 We know from 2.9) that ˆP 0 q ) = q +. Denote by T q) the fight time for the orbit starting from the point q on to reach ˆ 0. Then, F T q ) q ) = q +. Since F t is C in t and a C 2 diffeomorphism for fixed t, the impicit function theorem impies that there is a sma neighborhood U of q in in which T q) is a C function of q. Restricted to U, ˆP 0 has the representation, where ˆP 0 q + q) = q + + grad ˆP 0 q ), q +Nq, q), 2.9) Nq, q) o q ), as q 0;, and are the usua Cartesian inner product and norm; moreover, grad ˆP 0 q ) = t F T q ) q ) q T q ) + q F T q ) q ). Next, we introduce new coordinates on 0 and with q + and q as origins, respectivey. On 0, x 0 = x + + x0, x 0 j =x + j + x j 0, yj 0 =y + j +ỹj 0 2 j m ), z 0 k = z+ k + z0 k n <k n), x 0 j = x 0 j, y 0 j =ỹ 0 j m <j m), z 0 k = z0 k k n ). On, zk = z k + z k 2 k n ), x j = x j + x j, yj =y j +ỹj m <j m), z k = z k n <k n), x j = x j, yj =ỹj j m ).

16 330 Y. Li and S. Wiggins In terms of the new coordinates, Eq. 2.9) can be written as x 0 x ỹ 0 z 0 = A ỹ z + B, 2.20) where x 0 x 0,..., x0 m )T, ỹ 0 ỹ 0 2,...,ỹ0 m )T, z 0 z 0,..., z0 n )T, x x,..., x m )T, ỹ ỹ,...,ỹ m )T, z z 2,..., z n )T. Ais a 2m + n ) 2m + n ) constant matrix, B is a 2m + n ) coumn vector function of q and q; moreover, B o q ), as q 0, q= x ỹ z. A convenient notation for A is the foowing bock form: A x,x) jk A x,y) jk A x,z) jk A y,x) jk A y,y) jk A y,z) jk, 2.2) A z,x) jk A z,y) jk A z,z) jk where the index j in a bock runs through the dimension of the first superscript and the index k runs through the dimension of the second superscript. A simiar notation is used for the entries of B: B x) j, B y) k, etc Fixed Points of the Poincaré Map P P 0 P 0 The Poincaré map P is defined as P: U 0 0, P = P 0 P )

17 Symboic Dynamics 33 A fixed point of P corresponds to a periodic orbit of system 2.4).Suppose q U 0 is a fixed point of P, with coordinates x 0 ; x 0 j,ỹ0 j }, j =2,...,m; z0 k,k=,...,n). 2.23) Then, Pq) = q 2.24) gives 2m + n ) equations with 2m + n ) variabes 2.23). The task is to find soutions to the system 2.24). However, the set of variabes 2.23) is not the best for soving system 2.24). We wi next give a set of variabes that is more suited to this purpose The Sinikov Variabes. We consider a set of variabes coordinatizing a region S, and constructed from the variabes on 0 and,given by x 0 ; x0 j,ỹ0 j },2 j m ; x j,ỹ j },m < j m; ) t ; z k,2 k n ; z k 0,n <k n, which are defined by the transformation T : S 0, x 0 x0, x 0 j,ỹ0 j } x0 j,ỹ0 j }, 2 j m, with z 0 k z0 k, n < k n, x j x 0 j= e α jt [ x j + x j )cos β jt + y j +ỹ j )sin β jt ], m < j m, ỹ j ỹ 0 j= e α jt [ x j + x j )sin β jt + y j +ỹ j )cos β jt ], m < j m, z k z0 k = e γ kt z k + z k ), 2 k n, t z 0 = e γ t η, 2.25) T : 0 S, x 0 x0, x 0 j,ỹ0 j } x0 j,ỹ0 j }, 2 j m, z 0 k z0 k, n < k n,

18 332 Y. Li and S. Wiggins x 0 j x j= x j +e α jt [ x 0 j cos β j t ỹ 0 j sin β j t ], m < j m, ỹ 0 j ỹ j= y j +e α jt [ x 0 j sin β j t +ỹ 0 j cos β j t ], m < j m, z 0 k z k = z k +eγ kt z 0 k, 2 k n, z 0 t = og η z γ ) We refer to these variabes as the Sinikov variabes since they were first used by Sinikov [5], [6], [7], [8], and further deveoped by Deng [3]. These variabes wi be particuary usefu for finding fixed points of the Poincaré map. In particuar, we wi use them by computing the conjugate of P 0 P 0 with T, i.e., T P 0 P 0 T, 2.27) and seeking fixed points of this map. Ceary, these fixed points correspond to fixed points of P 0 P 0 under the map T. Using 2.7), 2.20), 2.25), and 2.26), we compute the components of the conjugated map. Mutipying each equation by e α t, and using the ordering of the eigenvaues given in Section 2., we obtain the foowing equations: ) e α t σ k 0 = x + + x0 ) A σ,x) k cos β t + A σ,y) k sin β t + m j=m + n + j=2 A σ,x) ) e α t x j + A σ,y) e α t ỹ j A σ,z) e α t z j +C σ ) k, 2.28) σ = x, < k m, σ = y, 2 < k m, σ = z, n < k n. ) 0 = x + + x0 ) A z,x) k cos β t + A z,y) k sin β t + m j=m + n + j=2 A z,x) ) e α t x j + A z,y) e α t ỹ j A z,z) e α t z j +C z) k, k n, 2.29) ) 0 = x + + x0 ) A x,x) k cos β t + A x,y) k sin β t cos β k t ) } A y,x) k cos β t + A y,y) k sin β t sin β k t

19 Symboic Dynamics m j=m + n + j=2 m j=m + + n j=2 A x,x) A x,y) e α t z j A y,x) A y,z) e α t z j ) e α t x j + A x,y) e α t ỹ j } cos β k t ) e α t x j + A y,y) e α t ỹ j } sin β k t +C x) k, m < k m, 2.30) 0 = x + + x0 ) + + m j=m + + n j=2 m j=m + + n j=2 ) A x,x) k cos β t + A x,y) k sin β t sin β k t ) } + A y,x) k cos β t + A y,y) k sin β t cos β k t A x,x) A x,y) e α t z j A y,x) A y,z) e α t z j ) e α t x j + A x,y) e α t ỹ j } sin β k t ) e α t x j + A y,y) e α t ỹ j } cos β k t +C y) k, m < k m, 2.3) where the primes on the variabes in 2.28) denote the images of the unprimed coordinates and the functions C σ ) k 0, as t +.The common factor of e α t makes it natura to consider these equations as functions of the foowing scaed coordinates: t ;ẑk eα t zk, 2 k n ; 2.32) ˆx j e α t x j, ŷ j e α t ỹj ), m < j m; ˆx 0 eα t x 0 ;ẑ0 k eα t zk 0, n <k n; 0 ˆxj e α t xj 0,ŷ0 j e α t ỹ j 0 ), 2 j m. 2.33) Note that the coefficient mutipying sin β k t in 2.30) is the same as the coefficient mutipying cos β k t in 2.3). Simiary, the coefficient mutipying cos β k t in 2.30) is

20 334 Y. Li and S. Wiggins the same as the coefficient mutipying sin β k t in 2.3). Hence, equivaent equations that the fixed points must satisfy are given by ) 0 = x + A σ,x) k cos β t + A σ,y) k sin β t + m j=m + A σ,x) ) ˆx j + A σ,y) ŷj n + j=2 σ = x, y, m < k m, σ = z, k n. ˆσ k 0 ) = x + A σ,x) k cos β t + A σ,y) k sin β t + m j=m + A σ,x) ) ˆx j + A σ,y) ŷj σ = x, k m, σ = y, 2 k m, σ = z, n < k n. n + j=2 A σ,z) ẑ j +C σ ) k, 2.34) A σ,z) ẑ j +C σ ) k, 2.35) C σ ) k = 0 Soutions. Next, we set C σ ) k = 0 in system 2.34;2.35), and sove the resuting system. In particuar, we first want to sove system 2.34) for t ;ẑ k, 2 k n, ˆx j,ŷ j ), m < j m. Then, we substitute these vaues into 2.35) to get ˆx 0 j, j m, ŷ 0 j, 2 j m, ẑ 0 k, n < k n. Before soving system 2.34), we want to show the foowing fact. Lemma 2.3. Let A be the [2m m ) + n ] [2m m ) + n ] matrix, ) rowa ) = A σ,x) km +)...Aσ,x) km Aσ,y) km +)...Aσ,y) km Aσ,z) k2...a σ,z) kn, σ = x, y, m < k m, If the generic condition, σ = z, k n. G3) dimt q +W s T q + W u }=,

21 Symboic Dynamics 335 is true, then ranka ) = 2m m ) + n. Proof. Let A 0 be the [2m + n ] [2m m ) + n ] matrix, ) rowa 0 ) = A σ,x) km +)...Aσ,x) km Aσ,y) km +)...Aσ,y) km Aσ,z) k2...a σ,z) kn, σ = x, k m, σ = y, 2 k m, σ = z, k n. Notice that ˆx j, ŷ j ), m < j m, ẑ k, 2 k n ; coordinatize Woc u ; moreover, Woc u is [2m m )+n ]-dimensiona. Since P 0 is a diffeomorphism, when restricted to the sma neighborhood U of q in, P 0W oc u ) is aso [2m m ) + n ]-dimensiona. Moreover, T q + P 0W oc u ) has the representation ˆx A 0 ŷ ẑ, 2.36) where ˆx = ˆx m +,..., ˆx m )T, ŷ = ŷ m +,...,ŷ m )T, ẑ = ẑ 2,...,ẑ n ) T. Therefore, ranka 0 ) = 2m m ) + n. 2.37) Assume ranka )<2m m )+n, then coumns of A are ineary dependent, and thus there exists a nonzero [2m m )+ n ] coumn vector a, such that A a = ) Moreover, by 2.37) A 0 a )

22 336 Y. Li and S. Wiggins By 2.38), By 2.36), Notice that then We aso know that A 0 a Woc s ) 0 = T q + W s oc 0. A 0 a T q + P 0 W u oc U ). P 0 W u oc U ) W u, A 0 a T q + W s T q + W u. T q +h T q +W s T q + W u. Moreover, T q +h is transversa to 0, whie A 0 a 0. By 2.39), T q +h and A 0 a are ineary independent, and then dimt q + W s T q +W u }=2. This contradicts the assumption in the emma, and the emma is proved. By this Lemma 2.3), without oss of generaity, we assume the [2m m ) + n ] [2m m ) + n ] matrix A 2, ) rowa 2 ) = A σ,x) km +)...Aσ,x) km Aσ,y) km +)...Aσ,y) km Aσ,z) k2...a σ,z) kn, σ = x, y, m < k m; σ = z, 2 k n, is nonsinguar. This can be used to provide an equation for the variabe t that we can easiy sove, which is seen as foows. Writing out 2.34) in matrix form gives A z,x) j A z,y) j A z,z) j A x,x) A y,x) A z,x) A x,y) A y,y) A z,y) A x,z) A y,z) A z,z) ˆx ŷ ẑ = x + A z,x) cos β t + A z,y) sin β t A x,x) k A y,x) k A z,x) k cos β t + A x,y) k sin β t cos β t + A y,y) k sin β t cos β t + A z,y) k where the matrix on the eft of this expression is A and the submatrix, A x,x) A y,x) A z,x) A x,y) A y,y) A z,y) A x,z) A y,z) A z,z), sin β t 2.40) 2.4)

23 Symboic Dynamics 337 is A 2. Then, by the nonsinguarity of A 2, there is a unique [2m m ) + n ] row vector b that satisfies the foowing equation: However, from 2.40), we have rowa ) σ =z, k= = ba ) rowa ) σ =z, k= = A z,x) cos β t + A z,y) sin β t. 2.43) Computing the right-hand side of 2.42) using 2.4) and equating the resut to the right-hand side of 2.43) gives or A z,x) cos β t + A z,y) sin β t = σ =x,y m <k m + σ =x,y m <k m b σ k Aσ,x) k + σ =z b σ k Aσ,y) k + σ =z σ bk Aσ,x) k 2 k n σ bk Aσ,y) k 2 k n cos β t sin β t, cos β t + 2 sin β t = 0, 2.44) where 2 = A z,x) = A z,y) σ =x,y;m <k m σ =x,y;m <k m bk σ Aσ,x) k bk σ Aσ,x) k, σ =z,2 k n bk σ Aσ,y) k bk σ Aσ,y) k, σ =z,2 k n which provides an equation for t. If the generic condition that G4) and 2 do not vanish simutaneousy, is true, then Eq. 2.44) has infinitey many soutions: t = β π ϕ), Z, 2.45) where ϕ = arctan 2 }. We substitute each t into equations σ = x, y, m < k m, σ = z, 2 k n,

24 338 Y. Li and S. Wiggins in system 2.34), and obtain the system A 2 ˆx ŷ ẑ = f, 2.46) where ˆx, ŷ, and ẑ are defined in 2.36), and f isa[2m m )+n ] coumn vector, ) entry f ) = x + A σ,x) k cos β t + Aσ,y) k cos β t, σ = x, y, m < k m, σ = z, 2 k n. Since A 2 is nonsinguar, system 2.46) has a unique soution: ˆx ŷ = A ẑ 2 f. 2.47) We substitute each soution 2.45;2.47) into system 2.35),and obtain the soution, ˆσ k = x + + A σ,x) k m j=m + cos β t + Aσ,y) k sin β t A σ,x) σ = x, k m, σ = y, 2 k m, σ = z, n < k n. ) ˆx j + Aσ,y) ŷj ) n + j=2 A σ,z) ẑ j, 2.48) Fixed Points. Starting from soutions obtained in the ast section, we want to sove system 2.34;2.35) in the imit t +. Since C σ ) k 0, as t +, by the impicit function theorem, we have the foowing theorem. Theorem 2.. There exists an integer 0, such that there are infinitey many soutions, abeed by 0 ), to system 2.34;2.35): t = T, ˆx 0 =ˆx0,), ) ˆx j 0 =ˆx 0,) j,ŷj 0 =ŷ 0,) j, 2 j m, ẑk 0 =ẑ0,) k, n <k n, ) ˆx j =ˆx,) j,ŷj =ŷ,) j, m <j m, ẑ k =ẑ,) k, 2 k n,

25 Symboic Dynamics 339 where, as, T = β π ϕ) + o), ˆx 0,) =ˆx +o), ˆx r,) j ŷ r,) j =ˆx j +o), =ŷj +o), r = 0, 2 j m, r =, m < j m, =ẑ k +o), r = 0, n < k n, r =, 2 k n, ẑ r,) k in which ˆx j, ŷ j, and ẑ k are given in Eqs. 2.47;2.48). Sketch of the proof. Let t = β [2sπ + τ], τ [0, 2π], s Z +, and et v denote the rest of the variabes in 2.34;2.35). Then, Eqs. 2.34;2.35) can be written as f τ, v) gτ, v) + Cs; τ,v) = 0, 2.49) where as s +,Cs;τ,v) 0. By the study in the ast subsection, there exist two soutions to gτ, v) = 0, which are denoted by τ,v ) and τ 2,v 2 ). Moreover, gτ i,v i ), i =, 2, 2.50) are inear diffeomorphisms. For τ [0, 2π], v in some bounded region D ; D [0, 2π] D, We know that sup Cs; τ,v) 0, as s, 2.5) τ,v) D sup τ,v) D Cs; τ,v) 0, as s. 2.52) f τ i,v i )=Cs;τ i,v i ).

26 340 Y. Li and S. Wiggins We want to find τ i,v i ), such that f τ i + τ i,v i +v i ) fτ i,v i )= Cs;τ i,v i ). 2.53) When s is sufficienty arge, 2.53) is equivaent to where τ i,v i )= [ fτ i,v i )] [Cs;τ i,v i )+ Rs;τ i,v i ;τ i,v i )], 2.54) Rs; τ i,v i ;τ i,v i )= fτ i +τ i,v i+v i ) fτ i,v i ) f τ i,v i ) τ i,v i )=o τ i,v i ) ). Then, a fixed point argument [2][5] for 2.54) impies the theorem. This competes the proof of the theorem. Remark 2.2. By this theorem, there are infinitey many periodic orbits, in a neighborhoodof the homocinic orbit h. Moreover, by the asymptotic representations of the fixed points given in the theorem, this sequence of periodic orbits approaches the homocinic orbit h as Smae Horseshoes In this section, starting from Theorem 2., we construct Smae horseshoes. Our construction wi be geometrica in nature. We wi use n-dimensiona versions of the Coney- Moser conditions cf. Moser [3] and Wiggins [9]) Definition of Sabs. We begin by defining the notion of a sab. Definition 5. We define sabs S 2 0 )in 0 as foows: q 0 η exp γ T 2+) π/2)} z 0 q) ηexp γ T 2 π/2)}, S x 0 q) ηexp 2 α T 2 }, σ k P 0 q)) ηexp 2 α T 2 }, σ = x, y, m < k m; σ = z, 2 k n }. S is defined so that it incudes two fixed points of P see Theorem 2.). We denote these two fixed points by p + and p, where p + corresponds to T 2, and p corresponds to T 2+ in Theorem 2.. It foows that z 0 p+ )> z 0 p ). If is sufficienty arge, P0 S ) is incuded in a ba centered at q on, with radius of order O exp 2 }) α T 2.

27 Symboic Dynamics 34 Therefore, P0 S ) U. Thus, S U, where U is the domain of definition of P. See Eq. 2.22).) Then, since p + and p are fixed points, there exist respectivey neighborhoods of p + and p, V + and V, that are incuded in the intersection PS ) S S and P 0 S ). In this subsection we wi describe the geometry of the image of S under P 0. We denote coordinates on 0 and by x 0, z 0,ξ0 s,ξ0 u) and x, ỹ ),ξ s,ξ u, respectivey, where ξ τ s τ = 0, ) is the [2m ) + n n )] vector, entryξs τ ) = στ k, σ = x,y, 2 k m, σ = z, n < k n; ξ τ u τ = 0,) is the [2m m ) + n ] vector, entryξu τ ) = στ k, σ = x,y, m < k m, σ = z, 2 k n. Definition 6. We define the diameter of S aong the ξ 0 u directions as foows: where C stands for d u S ) sup C ξ 0 u q ) ξ 0 u q 2) }, C q, q 2 S ; x 0 q ) = x 0 q 2), z 0 q ) = z 0 q 2), ξ 0 s q ) = ξ 0 s q 2) }. We define the diameter of P 0 S ) aong the ξ s directions as foows: where C stands for d s P 0 S )) sup C ξ s q ) ξ s q 2) }, C q, q 2 P 0 S ); x q ) = x q 2), ỹ q ) =ỹ q 2), ξ u q ) = ξ u q 2) }.

28 342 Y. Li and S. Wiggins Fig. 3. a) Geometry of the sabs. b) Geometry of the image of a sab under P 0. By the definition of S and the representation 2.7) of P 0,wehave d u S ) oexp γ + α /2)T 2 }), as +, 2.55) d s P 0 S )) o exp α T 2 }), as ) S and P0 S ) have the product representations as shown in Figure 3. The width of the intersection of S with the x 0, z0 )-pane, aong the x 0 -direction, is 2η exp 2 } α T 2, 2.57) and aong z 0 -direction it is of the order Oexp γ T 2 }), as ) As shown in Figure 3, P0 S ) intersects the x, ỹ )-pane in the shape of an annuus which has width of order O exp 32 }) α T 2, as +, 2.59) and radius of order Oexp α T 2 }), as ) On the annuus, we have marked the reative coordinate-positions between P 0 p+ ) and P 0 p ).

29 Symboic Dynamics 343 Definition 7. Let S be the cosure of S in 0. The connected components of the stabe boundary of S are defined as + s S q S x 0 q) =+ηexp 2 }} α T 2, s S q S x 0 q) = ηexp 2 }} α T 2, σ s S q S σq) =η, for some σ = x 0 j )2 + y 0 j )2 2 j m ), } or some σ = zk 0 n < k n). The union of a the connected components of the stabe boundary of S is referred to as the stabe boundary of S, denoted by s S. Simiary, the connected components of the unstabe boundary of S are defined as + u S q S z 0 q) = η exp γ T 2 π/2)} }, u S q S z 0 q) = η exp γ T 2+) π/2)} }, σ u S q S σp 0 q)) =ηexp 2 } α T 2, for some σ = xj 0)2 + yj 0)2 m < j m), } or some σ = zk 0 k < n ). The union of a the connected components of the unstabe boundary of S is referred to as the unstabe boundary of S, denoted by u S. The stabe and unstabe boundaries of P 0 S ) and PS ) are defined, respectivey, as τ P 0 S ) P 0 τ S )τ=s,u), τ PS ) P τ S )τ=s,u). In Figure 3, we have marked two pieces of s S and u S by, 2) and 3, 4), respectivey. We aso marked the corresponding boundaries of P 0 S ) by the same etters. Note the contraction, expansion, and bending in the deformation process from the rectange in the x 0, z0 )-pane to the annuus in the x, ỹ )-pane.

30 344 Y. Li and S. Wiggins PS ). In this subsection we wi describe some features of the geometry of the image of S under P. The Poincaré map P 0 has the representation 2.20). Under the inear approximation of P 0, the coordinate frame x, ỹ,ξ s u),ξ on, is mapped into an affine coordinate frame x, ȳ, ξ s, ξ u ) 2.6) on 0 with origin at q +, where the x ȳ pane is the image of the x ỹ pane under the inear approximation of P 0. Notice that 0 is aready equipped with a Cartesian coordinate frame x 0, z 0 ),ξ0 s,ξ0 u, 2.62) aso with origin at q +. Since P 0 is a diffeomorphism, the representation 2.20) of P0 can be rewritten as x 0 x ỹ 0 z 0 = A + Ɣ ) ỹ z, 2.63) where δ x 0 δỹ 0 δ z 0 = A + Ɣ 2 ) Ɣ x ỹ z = B, δ x δỹ δ z, 2.64) and thus, Ɣ 0, as x 0, ỹ 0, and z 0. Based upon 2.63;2.64), we wi approximate PS ) by AP0 S )) in the ater construction. Then PS ) has the product representation as shown in Figure 4. In Figure 4 we aso marked corresponding pieces of s PS ) and u PS ), as in Figure 3. d sps )) can be defined simiary as for d sp 0 S )). d s PS )) o exp α T 2 }), as ) Since we are approximating P 0 by a inear transformation, and the x ȳ pane is the image of the x ỹ pane under this inear transformation, the annuus formed by the intersection of P0 S ) with the x ȳ pane is mapped to an annuus on the x ỹ pane. This annuus has width of order O exp 32 }) α T 2, as +, 2.66) and radius of order O exp α T 2 }), as )

31 Symboic Dynamics 345 Fig. 4. Geometry of the image of a sab under P Definition of Sices. We now define sices of sabs. Definition 8. A stabe sice V in S is a subset of S defined as the regionswept out through homeomorphicay moving and deforming s S in such a way that the part s S u S of s S ony moves and deforms inside u S. The new boundary obtained through such moving and deforming of s S is caed the stabe boundary of V, which is denoted by s V. The rest of the boundary of V is caed its unstabe boundary, which is denoted by u V. Unstabe sices of S, denoted by H, are defined simiary. By definition, u V u S, s H s S. In the coordinates x 0, z 0,ξ0 s,ξ0 u) on 0, et G be a 2m + n n )-dimensiona hyperpane in S, specified by the condition, z 0 = const., ξ 0 u = const.}. In genera, G V consists of severa singy-connected regions, K G V = G k. k= See Figure 5 for an iustration.

32 346 Y. Li and S. Wiggins Fig. 5. Components of G V, the intersection of a 2m +n n )-dimensiona hyperpane in S with a stabe sice. Definition 9. The diameter of the stabe sice V is defined as dv ) sup sup sup x 0 q ) x 0 q 2) + ξs 0 q ) ξs 0 q 2) } }}. G k q,q 2 G k The diameter of an unstabe sice H is defined simiary Generic Intersection and Horseshoes. Introducing poar coordinates r, θ ) on the x, ỹ )-pane, P 0 restricted to this pane has the foowing representation, r = e α t x + + x0 ), θ = β t. From Theorem 2., the time of fight of a fixed point from 0 to is given by T = β π ϕ) + o), as +. Consequenty, for any sma positive θ 0, there exists, such that, for any, θ P 0 p+ )) θ 0 θ P 0 p+ )) θ P 0 p+ )) + θ 0 s + ), 2.68) θ P 0 p )) θ 0 θ P 0 p )) θ P 0 p )) + θ 0 s ). 2.69) That is, the x, ỹ ) components of P 0 p+ ) and P 0 p ) are in the two sectors on the x, ỹ )-pane s + and s of anges 2θ 0, as shown in Figure 6. Under the restriction of P 0 to the x, ỹ )-pane, the sectors s + and s map to two sectors, s + = P 0 s +), 2.70) s = P 0 s ), 2.7) on the x, ȳ )-pane. See Figure 7. The boundaries, 2) incude parts of the boundaries

33 Symboic Dynamics 347 Fig. 6. The x, ỹ ) components of P 0 p+ ) and P0 p ) shown in the two sectors, denoted s + and s, on the x, ỹ )-pane. The anguar width of each sector is 2θ 0. _ p _ y _ s x P s S ) _ s + E u + u P P S )) E s + Fig. 7. The image of the sectors s + and s under the Poincaré map on the x, ȳ )-pane. p + of s + and s. For any, we introduce a system of curviinear coordinates ē u, ē s ) on the x, ȳ )-pane such that ē u = 0}, ē u = b u constant)}, ē s = 0}, ē s = b s constant)}, correspond to the boundaries 3, 4;, 2ofPS )restricted to the x, ȳ )-pane, respec-

34 348 Y. Li and S. Wiggins Fig. 8. Curviinear coordinates ē u, ē s )onthe x,ȳ )-pane. tivey. Cf. Figure 8. From now on, we restrict the coordinates ē u, ē s ) to the two sectors s + and s. Definition 0. Define two subsets of PS ) as foows: Let E u +, E s + ) be the tangent vectors We make the generic assumption that S + q PS ) ē u, ē s )q) s + }, S q PS ) ē u,ē s )q) s }. E + u T p + ē u, E + s T p + ē s. G5a) G5b) Span Span } e x 0, E + u, e ξs 0, e ξ u = 0, } e x 0, E + s, e ξs 0, e ξ u = 0, where e x 0, e ξ, etc., represent unit vectors corresponding to the respective coordinate u directions. The assumption says that, for exampe, genericay E u + is not parae to the codimension [2m m ) + n ] hyperpane Then, Span e x 0, e ξ 0 s }. } e x 0, E + u, e ξs 0 spans a codimension [2m m ) + n ] hyperpane. Simiary, for G5b).

35 Symboic Dynamics 349 Next, without oss of generaity, focusing attention at p +, we discuss the intersection PS ) S in the coordinates x 0, ē u,ξs 0, ξ u }. Definition. On S + and S we define the diameters of S and PS ) as foows: d u S ) sup ēu q ) ē u q 2 ) + ξ u q ) ξ u q 2) }, C where where C q, q 2 S ; x 0 q ) = x 0 q 2), ξs 0 q ) = ξs 0 q 2) }. d s PS )) sup x 0 q ) x 0 q 2) + ξs 0 q ) ξs 0 q 2) }, C C q, q 2 S + PS ); ē u q ) =ē u q 2 ), ξ u q ) = ξ u q 2) }. By assumption G5a) and Eqs. 2.55;2.58), By assumption G5a) and Eq. 2.65), d u S ) O exp γ T 2 }), as ) d s PS )) o exp α T 2 }), as ) Definition 2. Define two sections at p + as foows: s S ) q S ē u q) =ē u p + ), ξ u q) = ξ u p+ ) }, u PS )) q S + PS ) x 0 q) = x 0 p+ ), ξ 0 s q) = ξ 0 s p+ ) }. Then there exists an order O exp 2 }) α T 2, as +, 2.74) in the neighborhood of p + in s S ), and an order Oexp α T 2 }), as +, 2.75) in the neighborhood of p + in u PS )). Reca from 2.) that α <γ. 2.76) The geometry behind definitions 0 2 is iustrated in Figure 7. We now use the previousy deveoped geometry and estimates to prove the foowing proposition.

36 350 Y. Li and S. Wiggins Fig. 9. Geometry of the intersection of PS ) with S. Proposition. Under assumption G5a), there exists a sufficienty arge 0, such that for a 0,PS )intersects S in two disjointconnected components, V + and V.V + and V intersect both u + S and u S and they do not intersect s S. Moreover, V + and are stabe sices in S with V s V + s PS ), 2.77) and See Figure 9. s V s PS ). 2.78) Proof. We begin by showing that V + and V are disjoint. The proof is by contradiction. Assume they are not disjoint; then there exists a curve g connecting p + and p, such that g S, 2.79) and g PS ). 2.80) In the same coordinates x 0, ē u,ξs 0, ξ u }, define wg) sup ē u q ) ē u q 2 ) }, q,q 2 g

37 Symboic Dynamics 35 and then by 2.79), there is a constant D, such that wg) <D exp γ T 2 }, and, by 2.80) and the definitions of p + and p, there is a constant D 2, such that wg) >D 2 exp α T 2 }. By 2.76), for sufficienty arge, this is a contradiction. Thus V + and V are disjoint. For sufficienty arge, it foows from 2.73) and the definition of S Definition 5) that V + and V do not intersect s S. Simiary, for sufficienty arge, it foows from 2.72), 2.74), 2.75), 2.76), and the definition of S Definition 5) that V + and V intersect u + S and u S.2.77) and 2.78) foow from Definition 7 and the fact that V + and V intersect u + S and u S. We now et H + = P V + ), H = P V ). It foows from 2.77;2.78) and Definition 7 that H + unstabe boundaries of H + and H are and H are unstabe sices. The u H + = P u V + ), 2.8) u H = P u V ). 2.82) In summary, for sufficienty arge, on each S, we have defined two unstabe sices and two stabe sices H + V + and H, and V, such that V σ = PH σ ), s V σ = P s H σ ), 2.83) u V σ = P u H σ ), σ =+,. In the next subsection we wi estabish shift dynamics on each S. Consequenty, on each S, we have a Smae horseshoe. Thus, we have infinitey many Smae horseshoes abeed by on Symboic Dynamics In this section, we wi construct an invariant Cantor set in S and show that the Poincaré map P restricted to is topoogicay conjugate to the shift automorphism on two symbos 0 and.

38 352 Y. Li and S. Wiggins The Shift Automorphism. Let be a set which consists of eements of the doube infinite sequence form, a =...a 2 a a 0,a a 2...), where a k = 0or,k Z. We introduce a topoogy in by taking as neighborhood basis of the set a =...a 2 a a 0,a a 2...), N j = a ak = a k k < j)}, for j =, 2,...This makes a topoogica space. The shift automorphism χ is defined on by b χa), b k = a k+. It is we-known that the shift automorphism has a countabe infinity of periodic orbits of a periods, an uncountabe infinity of nonperiodic orbits, and a dense orbit. Moreover, it aso exhibits sensitive dependence on initia conditions, which is a hamark of chaos Coney-Moser Conditions. The Coney-Moser conditions are sufficient conditions for estabishing the topoogica conjugacy between the Poincaré map P restricted to a Cantor set, and the shift automorphism on symbos; see [3] or [9]. Denote H +, H ; V +, V by respectivey. Then we have H 0, H ; V 0, V, Coney-Moser condition i): V j = PH j ), s V j = P s H j ), j = 0, ), u V j = P u H j ). Coney-Moser condition ii): There exists a constant 0 <ν<, such that, for any stabe sice V V j j = 0, ), where dṽ ) νdv ), Ṽ = PV H k ) k = 0,);

39 Symboic Dynamics 353 for any unstabe sice H H j j = 0, ), where d H) νdh), H = P H V k ) k = 0,). Remark 2.3. In Coney-Moser condition i), we have dropped the Lipschitz condition for the boundaries of the sices given in [3] [9]. In our case, a stabe sice V V j j = 0, ), and an unstabe sice, H H k k = 0, ), can possiby intersect into more than one point. In this case we ony choose one of them for the invariant set that we construct. The above Coney-Moser condition i) has been verified in the ast section cf. Reation 2.83)). We next discuss Coney-Moser condition ii). By the representation 2.7) of P 0 and the representation 2.63;2.64) of P0,wehave where where dṽ) ν dv), ν O exp α T 2 }), as + ; d H) ν 2 dh), ν 2 O exp γ T 2 }), as +. Lemma 2.4. If H k) H 2) H ) is an infinite sequence of unstabe sices, and, moreover, dh k) ) 0, as k, then H k) H ) k= isa2m +n n )-dimensiona connected surface; moreover, H ) s H ). Simiary, for stabe sices. Proof. By definition s H k) s H ) for each k. Hence H ) s H ). The dimension of H ) foows from the fact that dh k) ) 0, as k impies that the 2m m ) + n unstabe dimensions shrink to zero.

40 354 Y. Li and S. Wiggins Topoogica Conjugacy. Let a =...a 2 a a 0,a a 2...) be any eement of. Define inductivey for k 2 the stabe sices By the Coney-Moser condition ii), V a0 a = PH a ) H a0, V a0 a...a k = PV a...a k ) H a0. dv a0 a...a k ) ν dv a0 a...a k ) ) ν k dv a0 a ). By Lemma 2.4, V a) = k= V a0 a...a k defines a codimension 2m + n n ) connected surface; moreover, Simiary, define inductivey for k the unstabe sices By the Coney-Moser condition ii), By Lemma 2.4, V a) u S. 2.84) H a0 a = P H a V a0 ), H a0 a...a k = P H a...a k V a0 ). dh a0 a...a k ) ν 2 dh a0 a...a k ) ν k 2 dh a 0 ). Ha) = k=0 H a0 a...a k defines a 2m + n n )-dimensiona connected surface; moreover, By 2.84), 2.85), and the dimensions of Ha) and V a), Ha) s S. 2.85) V a) Ha) consists of points. Let p V a) Ha) be any point in the intersection set. Now we define the mapping φ : S, φa) = p.

41 Symboic Dynamics 355 By the above construction, Pp) = φχa)). That is, P φ = φ χ. Let φ ); then is a compact invariant under P) Cantor subset of S. Moreover, φ is a homeomorphism from to with the topoogy inherited from S ). For more detaied discussion, see [3] [9]. Thus we have the theorem: Theorem 2.2. There exists a compact invariant Cantor subset of S, such that P restricted to is topoogicay conjugate to the shift automorphism χ on two symbos 0 and. That is, there exists a homeomorphism φ:, such that the foowing diagram commutes. φ χ P φ 2.86) Remark 2.4. Topoogica conjugacy of P to the shift or subshift dynamics on many symbos can aso be estabished. But we omit that construction here. For a reevant discussion on such topics, see [9]. 3. Appication to Discrete NLS Systems In this section, we appy the theory deveoped in the ast section to the discretized perturbed NLS systems.) studied in [9]. 3.. Transformation of.) to the Form 2.) Here we state the resuts from [9] that are used in the transformation of.) to the form 2.). The phase space for the discrete NLS systems.) was defined as foows: ) q S q r = q, q=q r 0,q,...,q N ) T, q n+n =q n, q N n =q n }.

42 356 Y. Li and S. Wiggins It is easiy verified that.) has a sadde-type equiibrium point the approximate anaytica form can be found in [9]). The eigenvaues associated with the inearization of.) about q ɛ are given by ± 0 =±4ɛ /2 C /2 ω 3/2 + Oɛ), 3.) ± =±2 cos 2 k )/µ 2 + ω 2 ) ± j ω 2 N 2 tan 2 π ) + Oɛ), 3.2) N = ɛ [ α+2βµ 2 cos k j ) ] ± 2i E j F j j = 2,...,M), 3.3) where k j = 2 jπ/n and C, E j, and F j are constants which can be found in [9] the exact expressions are not important here, and so we omit them). From these formuae, for ɛ sufficienty sma, we see that 0 < + 0 < +, 0< Re + 2 }= Re 2 } < < Re + M }= Re M } < 0 <. Therefore, + 0 is the weakest growth rate, and Re + 2 } is the weakest decay rate. Moreover, we aso have Re + 2 } < ) Hence, the inearized system can be transformed to rea) Jordan canonica form. We then transform q ɛ in.) to the origin, and express.) in the coordinates which put the inear part in rea) Jordan canonica form. In this manner.) is reduced to the form of system 2.): x j = α j x j β j y j +X j x,y,z), ẏ j = β j x j α j y j + Y j x, y, z), 3.5) z k = δ k γ k z k + Z k x, y, z) j =,...,M,k =,...,4), where, k =, 2, δ k =, k = 3, 4, x x,...,x M ) T, y y,...,y M ) T, z z,...,z 4 ) T, X j 0,0,0)=Y j 0,0,0)=0, grad X j 0, 0, 0) = grad Y j 0, 0, 0) = 0 j =,...,M ), Z k 0, 0, 0) = grad Z k 0, 0, 0) = 0 k =,...,4),