Abrupt bifurcations in chaotic scattering: view from the anti-integrable limit

Bleher, Ott and Grebogi found numerically an interesting chaotic phenomenon in 1989 for the scattering of a particle in a plane from a potential field with several peaks of equal height. They claimed that when the energy E of the particle is slightly less than the peak height Ec there is a hyperbolic suspension of a topological Markov chain from which chaotic scattering occurs, whereas for E > Ec there are no bounded orbits. They called the bifurcation at E = Ec an abrupt bifurcation to chaotic scattering. The aim of this paper is to establish a rigorous mathematical explanation for how chaotic orbits occur via the bifurcation, from the viewpoint of the anti-integrable limit, and to do so for a general range of chaotic scattering problems.


Introduction
Bleher et al [7] found numerically an interesting chaotic phenomenon in 1989 when investigating the motion of a particle scattered by a smooth planar potential field V : R 2 → R of the form V (x, y) = x 2 y 2 exp(−(x 2 + y 2 )). (1) When the total energy E of the particle is slightly less than the height E c := max (x,y)∈R 2 V (x, y) of the peaks of the potential, their study suggested that there exists a bounded hyperbolic invariant set of the form of a suspension of a Cantor set in the energy level, on which chaotic scattering occurs, whereas there are no bounded orbits when E > E c . If the energy is close to E c with E < E c , they proposed that the Lyapunov exponents of orbits of the return map to a cross-section of the invariant set are of order ln(E c − E) −1 , while the box-counting dimension of a cross-section to the invariant set is asymptotically proportional to 1/ ln(E c − E) −1 . They called the bifurcation of dynamics at E = E c an abrupt bifurcation [7]; it is related to a change in the topology of the energy surface. Potential (1) has four peaks and π/2 rotation symmetry. In [6], a potential of the form V (x, y) = (x 2 + y 2 ) 2 sin 2 3 2 tan −1 y x − π 4 exp(−(x 2 + y 2 )), = r 2 (r 3 + y 3 − 3x 2 y)e −r 2 , where r = x 2 + y 2 , which has three peaks and 2π/3 rotation symmetry, was also studied and much the same bifurcation behaviour was reported. Chaotic scattering from a similar three-hill example had been reported earlier in [26]. The topological entropy of the return map was observed to jump down to 0 as E increases through E c [23].
In order to explain the aforementioned phenomenon, an argument that invokes [18,33] on heteroclinic intersection of stable and unstable manifolds of periodic orbits was presented in [6]. There, they numerically verified a sufficient condition that guarantees the intersection. Their analysis, however, falls a long way short of proving the full range of claimed behaviour. In particular their argument does not produce the full topological Markov chain for E < E c , only an unspecified subset, nor does it prove hyperbolicity. Although suggestive, their argument for no bounded orbits when E > E c is not a proof. In the three-hill case (2) the hills are elliptic, not circular as they assumed, and the analysis of the elliptic case in their section 7 and the later paper [36], although again suggestive, is not complete.
In our paper we develop a general approach to proving chaotic scattering, in particular constructing hyperbolic suspensions of topological Markov chains. Unfortunately, it still does not prove the full extent of the claims of [6], for (1) because of the right angles between the heteroclinic orbits along the edges of the square (a problem already recognized in [6]), and for (2) because the hills are elliptic with their short axes pointing towards the centre whereas we would need the long axes towards the centre to deduce abrupt formation of a topological Markov chain. Nevertheless, we apply our method to deduce a large hyperbolic topological Markov chain for (1) and for two modified three-hill examples.
As already made clear in [6], the dynamics in the vicinity of the peaks of the potential play a crucial role in the bifurcation. Let O be a non-degenerate maximum point of V : R 2 → R. The potential is said to be elliptic around O if D 2 V (O) has distinct eigenvalues, otherwise it is said to be circular around O. The main difference between the dynamics around an elliptic maximum point and a circular one is that trajectories can approach a circular peak in any direction, but can approach an elliptic peak along the long and short axes only, and all but two do so along the long axis. The discussion [6] was primarily for the case where the potential is circular around the peaks, though the elliptic case was considered briefly. For a special setting of potential with large enough non-circularity around a peak, a geometrical sufficient condition for abrupt bifurcation was provided in [36] by analysing the dynamics near the peak, but we shall go well beyond this. For a review on chaotic scattering and its applications, see [30].
The point of our paper is to show that the limit E → E c can be interpreted as a nondegenerate anti-integrable (AI) limit (for this concept, see e.g. [1,2,9,15,17]) and that a rigorous analysis is possible in general. It is based on ideas of Turaev and Shilnikov from 1989 [38] for the elliptic case and on [9] for the circular case. Turaev and Shilnikov studied Hamiltonian systems with a hyperbolic equilibrium O m with a simple pair of eigenvalues of smallest real part and several homoclinic orbits to O m . One can describe the union of homoclinic orbits as a bouquet. They considered the generic case in which the homoclinics leave and return tangent to the slowest eigenvectors of O m . They showed that for energy just below (or above, respectively) that of O m there is a subshift of trajectories continuing all concatenations of homoclinics in the bouquet for which at each visit to O m the direction reverses (stays the same, respectively). Although it is clear they had a strategy in mind, their paper does not contain a proof. A variational proof was provided in [11]. An implicit function theorem version of the result was obtained for the case of circular hills in [9], where the second derivative at the equilibrium has rotation symmetry.
The principal point of our paper is to extend the above to an implicit function theorem proof valid for potentials with more than one local maximum, whether circular or elliptic, provided they all have the same height. Note that, by structural stability of the resulting subshifts, one can perturb the heights by a small amount proportional to the difference of the energy from their mean and still keep the chaotic sets, but cannot expect their creation to be abrupt as the energy is varied.
To complete the analysis one would like to find conditions under which there are no other bounded orbits. This can be done in some circumstances, though even to determine the complete set of homoclinics and heteroclinics to the equilibria is not obvious. We give in this paper two examples, examples 12 and 14 in section 3, for which we are able to prove that there are no bounded orbits when E > E c , the only bounded orbits are the heteroclinics when E = E c , and that there is a (bounded) hyperbolic set of chaotic orbits for all E ∈ (E c − δ, E c ) for some δ > 0.
The rest of this paper is organized as follows. In the next section, we specify the setting, define AI trajectories, and state the main results. In section 3, we recall the non-degeneracy condition for homoclinic and heteroclinic trajectories from [9], and examine their applicability in some examples. In section 4, existence, uniqueness and smooth dependence on the energy of trajectories inside a small neighbourhood of each equilibrium point are studied. In section 5, we define a variational functional which depends on = E − E c , and stationary points of which give rise to trajectories of energy E for = 0. In particular, when = 0 all stationary points of the functional are non-degenerate and correspond to the AI-trajectories. Then we apply the implicit function theorem to continue the AI-trajectories to true trajectories for = 0.

Setting and main results
The systems considered in this paper are governed by a Lagrangian of the form , where the scalar potential V is C 4 in x and | . | denotes the norm associated to an inner product , on R 2 . In a future paper we will consider the effects of adding a vector potential. We could allow more general kinetic energy of the form 1 2 v, M(x)v with M everywhere positive definite, as arises for general constrained Lagrangian systems, e.g. double pendulum, but in two degrees of freedom, as here, Birkhoff proved this could be reduced to the Euclidean case by change of coordinates and rate of time [5].
The Euler-Lagrange equation is We assume the system has properties (V1)-(V3) as follows: , and the associated Hamilton's equations arė The Hamiltonian is preserved under the Hamiltonian flow, so we have the energy integral j the eigenvalues of the Hessian of V at O. Without loss of generality, we take 0 < µ 1 µ 2 . The equilibrium is a saddle-saddle equilibrium.
Devaney [20] proved that if an analytic Hamiltonian system possesses a transverse homoclinic orbit to a saddle-focus equilibrium, then for any integer N 2 the system admits on the zero energy level a subsystem that is a suspension of the horseshoe with N symbols. In the presence of a gyroscopic force, Buffoni and Séré [13] gave a variational version of Devaney's result using a non-degeneracy condition.
The situation for saddle-saddle equilibria is more subtle, for there are examples where the existence of several transverse homoclinic orbits to a saddle-saddle equilibrium does not imply chaos. See [21] and example 11 to come in section 3. To ensure the occurrence of chaos, additional conditions have to be imposed. For instance, assuming that there are two transverse homoclinic orbits to an elliptic maximum point and some conditions on these orbits, Holmes [29] showed that there is an invariant set on the zero energy level on which the flow is a suspension of a shift automorphism and being hyperbolic it persists to all nearby energies. This contrasts with the scenario of abrupt bifurcation of chaos proposed by [7]. A variational version of his result was given by Berti and Bolle [3]. On the other hand, [38] states that in situations which include that of [29] on the zero energy level, the only orbits which remain forever in a small neighbourhood of the union of the homoclinics are the homoclinics and the equilibrium. Indeed Holmes' chaos is bounded away from the homoclinics.
Nevertheless, [38] also states that chaos is formed for all small enough positive or negative energies, under suitable conditions on homoclinics to a saddle-saddle. This is the direction we follow. The key conditions are non-degeneracy (section 3) and admissibility (definition 1 below). Let us introduce some notation. Suppose x 1 , x 2 , x 3 are three points in configuration space for which x 1 is connected to x 2 by a trajectory γ 12 and x 2 is connected to x 3 by a trajectory γ 23 . These trajectories are represented by curves. If there is no ambiguity, these curves are still denoted by γ 12 and γ 23 . We use the product path γ 12 · γ 23 to represent the concatenation of trajectory curves connecting x 1 , x 2 and x 3 in such a way that it starts from x 1 following γ 12 to x 2 , then following γ 23 to x 3 . With this notation, an anti-integrable (AI) trajectory is defined to be a bi-infinite product of curves · · · · k i−1 · k i · k i+1 · · · · with k i ∈ K and subject to lim t→∞ k i (t) = lim t→−∞ k i+1 (t) for every i ∈ Z. Define

Definition 1 (Admissibility condition). An AI-trajectory
We say that a trajectory ϒ shadows the AI-trajectory · · · · k −1 · k 0 · k 1 · · · · if there exists a sequence · · · < a −1 The notion of non-degeneracy of a homo/heteroclinic trajectory will be recalled in section 3. We say that an AI-trajectory is non-degenerate if it is made up from non-degenerate homo/heteroclinic trajectories.  The two theorems above are a corollary of a more detailed result, theorem 23. By the admissibility condition, the collection of all AI-trajectories over { k } k∈K admissible for E < 0 (respectively, E > 0) determines a topological Markov graph (or subshift of finite type) G − K (respectively, G + K ) with vertices belonging to K. So a bi-infinite sequence { k i } i∈Z is admissible for E < 0 (respectively E > 0) if and only if the bi-infinite sequence {k i } i∈Z is admissible by the topological Markov graph G − K (respectively, G + K ). We shall see that theorem 23 implies that if is non-zero and sufficiently close to zero, then there is an invariant set on the energy level {E = } on which the dynamics is a suspension of the topological Markov chain G − K or G + K . To be precise, Theorem 6. There exists 1 > 0 such that for any 0 < < 1 there exists a cross-section N ∈ T R 2 on the energy level {E = } for which A := N ∩ is a uniformly hyperbolic invariant set with Lyapunov exponents of order ln | | −1 for the Poincaré map P . The restriction of P to A is topologically conjugate to the topological Markov chain G + K . We remark that, by the above two theorems and a result of Fathi [27], the upper boxcounting dimension of A is at most of order 1/ ln | | −1 .
To illustrate the admissibility condition we give two examples.
has two other homoclinic trajectories as in figure 2, 3 traversing clockwise and its time-reverse 4 . Also assume that O 0 has two homoclinic trajectories going out and coming back on opposite sides of the long axis, 5 traversing clockwise and its time-reverse 6 , as in figure 2. Let K = {0, 1, 2, 3, 4, 5, 6}. We obtain the indicated graphs G ± K of admissible transitions for energy above or below 0.
To apply our theorems to these sorts of example, we need also to verify non-degeneracy hypotheses on the homoclinic or heteroclinic trajectories. We will address this in the next section.   figure 4). Thus, there are 12 heteroclinic trajectories to the circular maximum points (±1, ±1) of the potential (circularity can be proved by direct calculation of the quadratic part; alternatively the sides of the square form orbits and so the reflection symmetry across the diagonal forces the quadratic part to be circular). We prove in example 17 that these heteroclinics are non-degenerate. Hence, by theorem 5, we are able to prove that, when E decreases through E c = e −2 , there is an abrupt bifurcation to a subshift on 4 symbols, say 0, 1, 2 and 3, representing the peaks.
We do not, however, obtain the whole of the subshift that Bleher et al claim [6], namely that with forbidden words {00}, {11}, {22} and {33}, call it BGO . This is because the possibility (c) ('90-degree turn') is not covered by our general approach (it was already perceived as problematic in [6] √ 2) (this is a relatively simple calculation, based on using symmetry to reduce the equation for the largest eigenvalue of the adjacency matrix for the graph to that for a 2 × 2 matrix). It is conceivable that the subshift BGO occurs as soon as E < E c , because the shape of the potential might be such that turning 90-degrees is always feasible, though it would be very weakly hyperbolic for E near E c . Alternatively, it is possible, indeed likely, that one does not obtain the whole of BGO straightaway. Instead, orbits are probably added gradually to as energy decreases, to make up BGO at some lower energy.
Potential (2) has a maximum point at (0, √ 2), and numerical computation shows that the eigenvalues of the Hessian matrix of the potential there are approximately −1.218 02 and −2.165 36 (indeed, it is diagonal and easy to show that V xx = V yy ). Hence, the maximum point is elliptic. Its short axis points towards the centre and numerically there are heteroclinics coming in along the long axes (look ahead to figure 8(b)). This means that we cannot deduce an abrupt bifurcation when the energy is lowered through the critical value E c . Instead the most one can hope for is gradual formation of a full chaotic set as the energy goes sufficiently below E c so that the dynamics does not notice that the axes are the wrong way round. We did not prove that there are heteroclinic connections, but their existence could be proved by minimizing the Jacobi length of absolutely continuous curves connecting the maximum points inside a bounded region, using compactness and showing the gradient vector ∇V points inwards on the boundary. Furthermore they will come out along the long axes (as in cases (f ), (k) in figure 3) because the short axes give only reflection symmetric orbits, which go to infinity. Presumably we could also show that they come in on opposite sides of the maxima as our numerical computation indicates in figure 8(b), but we did not invest time in this. Interestingly, what theorem 4 would give in this setting is two periodic orbits for energy slightly above E c , rotating in opposite directions round the triangle formed by the heteroclinic connections.
In examples 12 and 14, to come in section 3, we provide two potentials that do exhibit abrupt bifurcation. For these examples, we rigorously prove the existence and non-degeneracy of heteroclinic trajectories, and the absence of bounded trajectories for energy above the maximum. Other examples that could be studied using our results are those of [24,25].

Non-degeneracy conditions and examples
Define Hill's region) and let be the Maupertuis functional on the set W 1,1 of absolutely continuous curves [14] γ : [0, 1] → M E with fixed ends γ (0) and γ (1). The functional is independent of the parametrization of the curve, and with v =γ , if γ is parametrised in such a way that The Maupertuis principle says that if γ is a trajectory of (3) of energy E, then it is a critical point of J , and conversely, any critical point of J becomes a trajectory of energy E after reparametrizing as in (7). Alternatively, one could restrict to energy E by considering critical points of S = τ 0 (L + E) dt with free duration τ . This has the advantage of allowing a more general form of Lagrangian L. Nevertheless we do not need that generality here and furthermore the values of S and J are equal on trajectories of energy E (see section 4).
Suppose O e ∈ O max . In the phase space T * R 2 , the hyperbolic equilibrium (O e , 0) has two-dimensional local stable and unstable manifolds W ± loc [31]. Since W ± loc are Lagrangian manifolds and locally project diffeomorphically to the configuration plane [28], they are defined by C 2 generating functions S ± on a sufficiently small neighbourhood U e of O e : The functions S ± have a non-degenerate minimum at O e ; without loss of generality S ± (O e ) = 0.
For any x ∈ U e , there exists a unique trajectory ω + then S ± (x) = δ for all x ∈ ∂U e,δ . We write U e = U e,δ . Its boundary ∂U e is C 2 and for all x ∈ ∂U e , ω ± x intersect ∂U e orthogonally, by the first variation formula, see [8,12].
be the space of absolutely continuous curves with endpoints in ∂U e 0 and ∂U e 1 .
The condition that γ is non-degenerate can be expressed in the following equivalent ways.
(1) γ is a non-degenerate critical point of the Maupertuis functional J : → R on the subset of curves with the parametrization (7), where is the space of absolutely continuous curves with fixed boundary points O e 0 and O e 1 .
andγ is a non-degenerate critical point of the Maupertuis functional J : → R on the subset of curves with the parametrization (7). (1) and (ũ, τ ) is a non-degenerate critical point for G. We will find it convenient to use the following sufficient (but not necessary) condition for non-degeneracy. Equipped with the Jacobi metric √ 2(E − V (x)) |dx|, geodesics in Hill's region M E are trajectories of the Lagrangian system with potential V and energy E, and vice versa. The Jacobi metric is a Riemannian metric which degenerates on the boundary ∂M E .
where is the Laplacian operator, that is, V (x) means the trace of the Hessian of V . To show a heteroclinic γ is non-degenerate it suffices that the Gaussian curvature in a neighbourhood of {γ (t)| 0 < t < 1} be negative.
The following result, which will be useful later for proving absence of with plus sign if C is anti-clockwise, or minus sign otherwise, where D is the region bounded by C.
Proof. The curvature κ(x) of a trajectory is given by where ∇ ⊥ V (x) means ∇V (x) ·n withn the unit tangent vector perpendicular toẋ satisfyinĝ n ×ẋ = |ẋ|k (in the usual (î,ĵ,k)-Euclidean coordinates). Suppose C = γ 1 · γ 2 · · · γ m is a product of m pieces of trajectories, with t t 1 2 }, etc. Then by Green's theorem the angle turned by these trajectories along C is ±2π − m i=1 θ i : where dS is an element of surface area in D, and (10) takes the minus sign if the trajectory traverses C anti-clockwise, the plus sign otherwise.

Corollary 10.
Suppose K E is non-positive.
(i) Any trajectory of energy E cannot self-intersect.
(ii) Any two distinct trajectories of energy E can intersect at most once.

Example 11.
(a) For the system of two uncoupled pendula there are four non-degenerate homoclinic trajectories to (0, 0), along the sides of the unit square in both directions, and a continuum of degenerate homoclinic orbits from one corner to the diagonally opposite one, see [3,11]. Any pair of non-degenerate homoclinics joining at (0, 0) behave like case (c) or (e) in figure 3. (b) Similarly, all except four homoclinic trajectories to (0, 0) for the Lagrangian are degenerate. Any pair of non-degenerate homoclinic orbits joining at the origin behave like case (h), (i), (k) or (l) in figure 3.  which includes the lower boundary. So we can achieve a reduction in the Jacobi length by pushing the curve into the interior of the triangle. Because the eigenvectors of the smaller eigenvalue point to the centre of the triangle, the heteroclinics meet at any of the maximum points like case (g) in figure 3. The heteroclinic orbit from one maximum to another is unique by the hyperbolicity to be shown in (ii) and the constraint (v) that they must lie in the triangle. The six heteroclinics found numerically are illustrated in figure 6(a) (they occur in pairs in opposite directions). (ii) It suffices to show that the Gaussian curvature is negative in the interior of the equilateral triangle minus the origin (which the heteroclinics do not visit). Direct calculation shows that Thus, the zero set of the curvature consists of the three straight lines through the maxima plus the origin, and the curvature is negative inside the triangle, except at the centre. The curvature has a singularity at the maximum points. See figure 6(b). (iii) We show that for any E, any trajectory that leaves the equilateral triangle will go to infinity.
For the region {0 x − √ 3y, y −3/8}, a simple calculation shows that So the force has negative vertical component in the region. When a particle leaves the triangle from a point on the segment {0 x 3 √ 3/8, y = −3/8}, the vertical component of its velocity must be non-positive. Hence, the vertical component of the particle's velocity will always stay negative and bounded away from zero. Therefore, the particle goes to infinity. (For trajectories that cross the lines {x = 0} or {y = −x/ √ 3}, use the symmetry of the system.) (iv) We have shown that K 27/256 (x) < 0 when x is in the interior of the equilateral triangle minus the origin. It follows that K E (x) < 0 for all x in the triangle when E > E c = 27/256. To see this, K 27/256 (x) < 0 implies from (9) . Suppose there is a bounded trajectory γ . It has to lie within the triangle, so must have an accumulation point in phase space (since the part of the energy level that projects to the configuration space in the triangle is compact). The accumulation point cannot be an equilibrium because E > E c . This implies that there is µ 0 and t 1 < t 2 such that max{|γ (t 2 ) − γ (t 1 )|, |γ (t 2 ) −γ (t 1 )|} < µ (the velocity at γ (t 1 ) and γ (t 2 ) are in almost the same direction). If µ is zero, γ is a periodic trajectory, contradicting (i) of corollary 10. If µ is not zero, one can take µ sufficiently small so that there is a segment ν of another trajectory connecting γ (t 1 ) and γ (t 2 ). (This can be done since E > E c .) The two segments {γ (t)| t 1 t t 2 } and ν form a simple closed curve since γ cannot self-intersect (by (i) of corollary 10). But, this contradicts (ii) of the corollary. (Remark: alternatively, one can prove this assertion by proposition 13.) (v) The equilateral triangle can be decomposed into six right triangles of the same area and shape (up to translation and rotation). One of them is {0 x − √ 3y, − 3/8 y 0}.
In this one, we have with the last equality only when x = 0. And with the last equality only when x = 0 or x = 3 √ 3/8. This means that in the interior of the right triangle the force always has negative horizontal component, and implies that a particle in the interior must go to and cross the boundary of the right triangle unless it is asymptotic to the maximum point (3 √ 3/8, −3/8). If its trajectory is asymptotic to the maximum point, then consider it in the opposite time direction. If in the opposite time direction it is also asymptotic to a maximum point, it is a heteroclinic. It cannot be a homoclinic because then it would contain a loop, thus it, or another homoclinic by symmetry of the system, must have a point with zero horizontal velocity. (The case like 2 in figure 2(a) cannot happen here). Due to the negative horizontal force, the particle cannot go the maximum point. When the particle crosses the lower boundary {(x, y)| 0 x 3 √ 3/8, y = −3/8}, it has to go to infinity. When it crosses the side {(x, y)| x = 0, −3/8 y 0} or the hypotenuse to another right triangle, by (ii) of corollary 10, it cannot come back to the original right triangle. Hence, the particle will cross the segment {(x, y)| x = 0, −3/8 y 0} twice or cross the boundary of the equilateral triangle. The former case contradicts (ii) of corollary 10, while the latter results in an unbounded trajectory. Part (vi) is a consequence of theorem 3.

Proposition 13. For E > E c , if there is a bounded trajectory γ in the configuration space there is one with a self-intersection.
Proof. Let be the phase space orbit corresponding to γ . is bounded so take a limit point A of and its orbit . Then consists of limit points of , so is bounded. If it has no self-intersections, take a limit point B. Denote the projections to configuration space of all the above by the corresponding lower case letters. The velocity in configuration space is non-zero at b because E > E c . Take a local transverse section σ through b. Then γ cuts σ repeatedly and these intersections are themselves limit points of γ . As in the proof of the Poincaré-Bendixson theorem, successive intersections of a non-self-intersecting trajectory with σ are monotonic and thus γ forms a closed curve, which contradicts its having no self-intersections. Proof.
(o) To find the maxima is simple calculus. The eigenvalues of the Hessian of the potential at the maximum points are double, equal to −9/2 1/3 , thus the peaks are circular. (ii) On the energy level {E = 1/4}, trajectories of the Lagrangian system are geodesics of the Jacobi metric √ 2(E − V (x, y)) |d(x, y)|. Let z = x + iy. Then in the complex z-plane the metric can be written as ρ(z)|dz| with Note that the metric degenerates only at the hilltops z 1 = 2 −1/3 e iπ/2 , z 2 = 2 −1/3 e i7π/6 and z 3 = 2 −1/3 e i11π/6 . Furthermore, the metric ρ(z)|dz| corresponds to the Euclidean metric |dw| in the w-plane where Since all geodesics in the w-plane with the metric |dw| are Euclidean straight lines, all trajectories for the Lagrangian system are non-degenerate. (i) Note that heteroclinics correspond to the straight lines joining the three points w(z 1 ), w(z 2 ) and w(z 3 ), thus can be obtained by solving the quartic equation (11) for z and choosing the right solution. (See figure 8(a)). Points z 1 , z 2 and z 3 are located at the vertices of an equilateral triangle. The mapping z → w has 2π/3 rotation symmetry, thus w(z 1 ), w(z 2 ) and w(z 3 ) are also located at the vertices of an equilateral triangle. Taking Taylor expansion around z 1 , z 2 and z 3 , we see that the heteroclinics meet at an angle of π/6 at z 1 , z 2 or z 3 . (iii) Note that ∂ ∂y V (x, y) = −3x 2 + 3y 2 − 6y(x 2 + y 2 ) 2 , The latter is negative except when x is at a maximum point. The second partial derivative is negative when y is negative. Hence − ∂ ∂y V (x, y) in the region {(x, y)| y −2 −4/3 } is negative except at a maximum point. In particular, it is negative in the region {(x, y)| 0 x − √ 3y, y −2 −4/3 } except at a maximum point. The rest of proof is similar to that of (iii) of example 12. (iv) It can be verified that the Gaussian curvature K 1/4 is everywhere zero except at the maximum points, where it has a singularity. Thus, from (9), (E c − V ) V = −|∇V | 2 everywhere. Since V < E c except at the maxima, we deduce that V < 0 except at the critical points of V (the maxima and the origin), and so K E is everywhere negative when E > 1/4 except at the origin. Then the proof goes the same as that of (iv) of example 12. (v) All trajectories can be obtained from straight lines in the complex w-plane. (vi) is a consequence of theorem 3.

Remark 15.
The logarithm of the absolute value of an analytic function is automatically harmonic, but it is easily checked explicitly, as follows. Write ρ(z) = ρ(x + iy) asρ(z,z). It is easy to verify that the Gaussian curvature K(z) =K(z,z) = − lnρ(z,z) ρ 2 (z,z) of any point other than z 1 , z 2 and z 3 in the complex z-plane with the metric ρ(z) |dz| is zero, since lnρ(z,z) = 4 ∂ 2 ∂z∂z lnρ(z,z) Remark 16. One thing we did not prove in examples 12 and 14 is that the set of bounded orbits for E ∈ (E c − δ, E c ) is not more than the constructed subshift. This could probably be done with a bit more work, see [22]. Proof. Along the trajectory (x * (t), y * (t)) from (−1, 1) to (1, 1), the orthogonal component of the linearized Euler-Lagrange equation is The coefficient in the right hand side of the equality is positive except when x * passes through zero. Thus there are no bounded solutions other than δy = 0. For the trajectory (x * (t), y * (t)) from (−1, The coefficient on the right hand side is positive except when u * passes through zero. Thus there are no bounded solutions other than zero. The non-degeneracy of all other heteroclinics follows from the symmetry of the system.

Boundary value problem
This section is devoted to the dynamics of a particle moving in a small neighbourhood of a non-degenerate local maximum O e ∈ O max of the potential V . We choose the neighbourhood to be U e = U e,δ defined in (8). In this section all results are local, so we omit the subscript e in the rest of this section. As in [4,9,11,12,32], lemmata 18, 19, 20 and proposition 21 stated below characterize the main features of the dynamics. For the trajectories ω ± x (t), let z ± x (t) ∈ W ± loc be the corresponding orbits in the phase space. Recall that for critical points of J (5), parametrized to achieve (7), J has the value Proof. From the parametrization (7),

Lemma 18.
There exists T > 0 large, such that for any x, y ∈ U and any a < b ∈ R with b − a T , the Lagrangian system (3) has a unique trajectory q(t) satisfying q(a) = x, q(b) = y, and q(t) ∈ U for a t b. Furthermore, • if q is written as a function of (x, y, a, b, t), then it is a C 2 function; in fact, it has the phase space representation where φ is a uniformly bounded C 2 function to T * R 2 ∼ = R 4 . • the action R(x, y, a, b) ∞ and E, depending on x, y, a, b, is the energy of q.
The representation (12) indicates that q(t) → ω + x (t −a) as b → ∞, that q(t) → ω − y (t −b) as a → −∞, and that q(t) → O as a → −∞ and b → ∞, all uniformly on compact time intervals. Since the cotangent planes T * x U and T * y U intersect transversally the invariant manifolds W ± loc of the hyperbolic equilibrium (O, 0), lemma 18 is a consequence of lemma 19 below by putting Y v = T * x U and Y u = T * y U . Lemma 19 can be derived from Shilnikov's lemma [34] or from the strong λ-lemma [19].

Lemma 19.
Let g t be the phase flow of the differential equationż = G(z), where G is a C 3 vector field on R n and g 0 ≡ id. Suppose that the origin 0 is a hyperbolic equilibrium with nonempty local stable and unstable manifolds W ± loc , and that µ = min |Re (Spec DG(0))|. Let Y u and Y v be C 2 manifolds intersecting transversally W − loc and W + loc , respectively, at points z u and z v . Then for sufficiently large T > 0 and any a < b with b − a > T , there exists a unique solution z(t) = g t (z(a)) such that • it has the representation where φ is a uniformly C 2 function to R n . If the manifolds Y u and Y v depend smoothly on a parameter taking values in a compact manifold, then φ is a uniformly bounded joint C 2 function with that parameter.
Proof. The proof follows closely that in [12]. Set z = (u, v) and use coordinates u ∈ W − loc , v ∈ W + loc in a neighbourhood N = W − loc × W + loc of the origin. Suppose Y u is the graph of a function: The strong λ-lemma [19] says that f t C 2 Ce µt , t 0, for some constant C. An analogous estimate h t C 2 Ce −µt also holds for g t (Y v . Thus, any point on the set {z(t)| a t b} satisfies u = f t−b (v) and v = h t−a (u). By the above estimates and the implicit function theorem, η(a, b, t) for a t b. The fact that C 2 e µ(a−b) and + a,b,t is quadratically small provided h t−a and f t−b are small: a−b) . A similar inequality holds for the derivative of η in a, b, t. The representation (13) is proved for the coordinate v. The proof is similar for the coordinate u. When Y u and Y v depend on a parameter, the estimate above will be uniform in such a parameter. Let Lemma 20. The energy h(x, y, a, b) of the trajectory q(t) is a C 2 function and has the form h 1 (x, y, a, b) , y, a, b) converges to zero in the C 2 topology as b − a → ∞.
Note that ζ + (x) and ζ − (y) are tangent vectors at O to the asymptotic trajectories ω + x and ω − y , respectively. In other words, write For equal eigenvalues µ 1 (O) = µ 2 (O), the scalar product ξ + (x), ξ − (y) may take any value in [−1, 1] for a pair x and y ∈ U . When µ 1 (O) = µ 2 (O), let W ++ loc ⊂ W + loc and W −− loc ⊂ W − loc be one-dimensional local strong stable and unstable manifolds of the hyperbolic equilibrium (O, 0) ∈ T * R 2 , namely the invariant manifolds tangent to the eigenvectors with eigenvalues ±µ 2 (the strong directions). The manifolds W ±± loc divide W ± loc into connected components: The projections π(W ±± ) to configuration space are identical and divide the neighbourhood U into connected components Proof of lemma 20. Since we are concerned with the dynamics only in a neighbourhood U of O, we may shift and rotate coordinates if necessary so that the origin is located at O and in the new coordinates (we still use x = (x 1 , x 2 ) ∈ R 2 and conjugate momentum p = (p 1 , p 2 ) ∈ R 2 ) the Hessian matrix of V at O is diagonal and the Hamiltonian takes the form where −ν 1 , −ν 2 are the eigenvalues of the Hessian matrix of V at O. If ν 1 = ν 2 the result was proved in [9]. If ν 1 = ν 2 , the eigenvalues of the Hamiltonian dynamics are ±µ 1 , ±µ 2 with There exists a symplectic transformation T : (x, p) → (u, v) given by such that W + loc = {u = 0}, W − loc = {v = 0}, and the Hamiltonian takes the form If µ 1 < µ 2 , then the coordinates can be chosen in such a way that W ++ loc = {u = 0, v 1 = 0} and W −− loc = {u 1 = 0, v = 0}. Hamilton's equations on W − loc have the forṁ where the right hand side of both equations is of class C 3 . By a C 2 change of variables , they can be transformed to linear equationsη 1 = µ 1 η 1 , η 2 = µ 2 η 2 unless µ 2 = 2µ 1 [35]. Hence if µ 2 = 2µ 1 the phase flow on W − loc takes the form where The representation given in (17) is called an exponential expansion in [19]. A similar representation holds for the flow on W + loc : where Put t = (a + b)/2 in (12), by (17) and (19), where h 2 = h 2 (x, y, a, b) and h 3 = h 3 (x, y, a, b) satisfying h 2 C 2 → 0 and h 3 C 2 → 0 as b − a → ∞. Passing to the variables x, p and using (21) and (22), we get lim t→∞ e µ 1 tω+ Putting together (24)-(26), we obtain the desired result: If µ 2 = 2µ 1 one can linearize just the first equation of (16) as in [11], and achieve the same result. In fact this argument works for any µ 2 > µ 1 .
Take small ν > 0 and let Then for (x, y) ∈ B − or B + the function h(x, y, a, b) is monotone in a−b (for b−a sufficiently large). Thus, solving the equation h(x, y, a, b) = for small yields a function τ (x, y), with b − a = τ . This together with lemmata 18 and 20 gives the following result.
where the function µ is uniformly bounded on B − (respectively, B + ) as → 0. y, ), where the function ζ is uniformly C 1 bounded as → 0.
• the action S(x, y, a, b) of the trajectory q x,y regarded as a function of (x, y) is C 2 on B − (respectively, B + ) and where r is uniformly C 2 bounded on B − (respectively, B + ) as → 0.

Remark 22.
When µ 1 = µ 2 , a pair of points x and y is contained in B − or B + if x and y are sufficiently bounded away from the maximum point O and if ξ + (x), ξ − (y) is sufficiently bounded away from zero. When µ 1 < µ 2 , an additional condition is needed to guarantee that (x, y) is in B − or B + . The condition is that x must be sufficiently bounded away from π(W ++ loc ) and y must be sufficiently bounded away from π(W −− loc ). This can be seen from (18), (20), (25) and (26). As a matter of fact, [11,12,19,35].
This remark explains the definition of the admissibility condition in definition 1.

Shadowing hetero/homoclinic trajectories
The boundaries ∂U e,δ of U e,δ are C 2 .
Suppose that i−1,i = k i , k i ∈ K, is a hetero/homoclinic trajectory such that i−1, figure 9. Let W k be a small neighbourhood of k ((−∞, ∞)). By our assumption, the points y † i−1 and x † i are not conjugate to each other on the fixed energy level {E = 0}. Then by the implicit function theorem, there are neighbourhoods y i−1 of y † i−1 in ∂U e i−1 , x i of x † i in ∂U e i , and there are 0 < 0 < 1 such that for all 0 < < 1 any points y i−1 in y i−1 ⊂ ∂U e i−1 and x i in x i ⊂ ∂U e i can be connected by a unique trajectory q i−1,i : [0, T ] → W k i of energy satisfying q i−1,i (0) = y i−1 and q i−1,i (T ) = x i for some T = T (y i−1 , x i ) > 0. If q i−1,i and T are regarded as functions of (y i−1 , x i , ), then they depend C 2 on their variables. Theorem 23. There exists 0 < 0 (respectively, 1 > 0) such that for any 0 < < 0 (respectively, 0 < < 1 ) and any sequence of hetero/homoclinic trajectories { k i } i∈Z admissible for E < 0 (respectively, E > 0) the system admits a unique (up to time shift) trajectory ϒ : R → M ∩ ( k∈K W k ) of energy and a sequence · · · < a −1 < b −1 < a 0 < b 0 < · · · such that for all i ∈ Z More precisely, . Theorem 23 implies that there is an invariant set on the energy level {E = } on which the system is a suspension of the topological Markov chain G − K or G + K (i.e. theorems 5 and 6). To this end, we shall obtain a more detailed result first. We take the cross section N ∈ T R 2 described in theorem 5 to be the annuli whose projection to the configuration space is O e ∈O max ∂U e . That is, we require that if (x, v) ∈ N then x belongs to some ∂U e , |v| = √ 2( − V (x)), and the inner product of the velocity vector with the inner normal of ∂U e is positive. The Lagrangian flow induces a map (the Poincaré map) P on a subset of N : The meaning of P is that if x i , v i are the position and velocity with which a particle enters O e ∈O max U e then x i+1 , v i+1 are the next position and velocity with which it enters O e ∈O max U e . Suppose the corresponding orbits of the hetero/homoclinic trajectories { k } k∈K in the phase space T R 2 intersect N at a set A 0 : (We can take all U e sufficiently small so that N and ( k ,˙ k ) intersect at only one point for each k.) Define two topological Markov chains − A 0 and + A 0 , which are respectively topologically conjugate to the subshift with graphs G − K and G + K , by Let P ± 0 be the subshift ± A 0 → ± A 0 , Proposition 24. Let the sequence {(x † i , v † i ) ∈ A 0 } i∈Z be an orbit of the map P − 0 (respectively, P + 0 ). Then there are 0 < 0 < 1 so that for any ∈ ( 0 , 0) (respectively, (0, 1 )) there exists a unique sequence {(x * i ( ), v * i ( ))} i∈Z which is a hyperbolic orbit of the map P such that Define the following action function on y i−1 × x i : The function S is jointly C 2 in (y i−1 , x i ) and , and satisfies is an open set of i∈Z (∂U e i−1 × ∂U e i ) endowed with the supremum norm. The Jacobi length of a concatenation is the formal sum i∈Z S (y i−1 , x i ) + S(x i , y i , 0, τ (x i , y i )).
Rewrite this (which does not change the critical points) by adding and subtracting i∈Z S + (x i )+ S − (y i ) and subtracting the constant term i∈Z − log | | µ 1 (O e i ) , to obtain, using (27), the formal functional Note that since x i ∈ ∂U e i then S + (x i ) = δ, and similarly S − (y i ) = δ, but we write h more generally in case one might prefer to take different curves around O e i from ∂U e i . Further, we define a map where l ∞ is the subspace of (R 2 × R 2 ) Z with bounded supremum norm and F i = D z i W . (Remark: although the value of W may be infinite due to the summation over all i ∈ Z, the definition of F i makes sense because the derivative is taken with respect to z i (not with respect to z) and there are only finitely many terms having variable z i in the summation.) It is not difficult to see that F is a C 1 function. By our construction, z † is the only solution for F (z, 0) = 0 and corresponds to an admissible sequence of hetero/homoclinic trajectories (i.e. an AI-trajectory) associated with z † . Proposition 25. For slightly negative or positive , a point z * ( ) = z * = {(y * i−1 , x * i )} i∈Z which is a zero of F (·, ) corresponds to a trajectory ϒ of energy .
Proof. This is Maupertuis's principle. It can be checked by differentiation.

Lemma 26.
For each i ∈ Z, the function F i (·, 0) on y i−1 × x i has a non-degenerate zero at z † i = (y † i−1 , x † i ).
Proof. is the concatenation ω − y i−1 · q y i−1 ,x i · ω + x i . Hence a zero of F i (·, 0) is a critical point of the action functional that is restricted to a finite-dimensional submanifold consisting of broken trajectories with break points y i−1 and x i connecting O e i−1 to O e i . Thus, the lemma follows from the assumption that i−1,i is a non-degenerate hetero/homoclinic trajectory.
Proof of theorem 23. Theorem 23 is a consequence of proposition 25 and lemma 26. The sequence of hetero/homoclinic trajectories { k i } i∈Z corresponds to a non-degenerate zero of F (·, 0) and the implicit function theorem gives rise to a locally unique continuation provided 0 < < 1 for some 0 < 0 < 1 . The values of 0 and 1 can be chosen independent of the sequence { k i } i∈Z because k i ∈ K and K is a finite set.
Proof of theorems 5 and 6. The set A 0 with the product topology is a Cantor set as soon as the corresponding graph is non-trivial (communicating with at least one branch point). The mapping composed of the following mappings is continuous in the product topologies because f 1 , and f 2 are all continuous (see, for example [16]). The projection i∈Z (x * i ( ), v * i ( )) π −→ (x * 0 ( ), v * 0 ( )) is also continuous. Let g := π • f 2 • • f 1 . Thence the two sets g ( A 0 ) := A and A 0 with the product topologies are homeomorphic to each other because all f 1 , , f 2 and π are injective. The homeomorphism g −1 is then the desired conjugacy: because by the time translation invariance of the system, the following diagram is commutative The proof of uniform hyperbolicity and the estimate of Lyapunov exponents are in a similar manner to that of [10]. We were able to define W (·, ) because any x i ∈ x i can be connected to any y i ∈ y i by a unique (up to time shift) trajectory of energy provided that the time needed to travel from x i to y i is sufficiently long, also any y i ∈ y i can be connected to any x i+1 ∈ x i+1 by a unique (up to time shift) trajectory of energy provided the trajectory is near i,i+1 . Actually, given a pair x i ∈ x i and x i+1 ∈ x i+1 , it determines a unique (up to time shift) trajectory of energy connecting them by gluing the aforementioned two unique trajectories because the aforementioned y i can be determined uniquely by x i and x i+1 and the resulting trajectory has no velocity discontinuity at y i . Indeed, our non-degeneracy condition implies that x i+1 is not conjugate to O e i , therefore det D 2 y i (w 0 (y i , x i+1 )) = 0. Then for small we can locally solve the equation D y i (w (y i , x i+1 ) + r(x i , y i , )) = 0 for y i = χ(x i , x i+1 , ). So we can define S (x i , x i+1 ) := w (χ (x i , x i+1 , ), x i+1 ) + r(x i , χ(x i , x i+1 , ), ). Therefore, we reduce W (·, ) to a functional of the sequence x = {x i } i∈Z only, Critical points of correspond to orbits of the symplectic map with the generating function S (x i , x i+1 ). It was proved by Aubry et al [2] that if the generating function has non-degenerate mixed second derivative (i.e. the twist condition) then the non-degeneracy of a critical point of is equivalent to uniform hyperbolicity of the corresponding orbit of the associated symplectic twist map.
The Poincaré map P : (x i , v i ) → (x i+1 , v i+1 ) defined in (28) is C 2 , and can be generated by D x i+1 S (x i , x i+1 ) = v i+1 , D x i S (x i , x i+1 ) = −v i . Consequently, non-degeneracy of D 2 x i x i+1 S (x i , x i+1 ) is equivalent to boundedness of the derivative of P : Proposition 21 implies that the time τ needed to connect x i and x i+1 is of order log | | −1 , the Poincaré map P , as a timeτ map induced from the Lagrangian flow, is thus uniformly C 1 bounded by c| | −1 , where τ = τ (x i , χ(x i , x i+1 , )) + T (χ (x i , x i+1 , ), x i+1 ). This gives an upper bound of the Lyapunov exponents. A lower bound of this order comes from the proof of proposition 1 of [2]. Note that the Lyapunov exponents of orbits of the topological Markov chain depend on the eigenvalues at the maxima, and the angles turned there if circular.