Numerical continuation of families of homoclinic connections of periodic orbits in the RTBP

Transcription

1 Numerical continuation of families of homoclinic connections of periodic orbits in the RTBP E Barrabés, J M Mondelo 2 and M Ollé 3 Dept. Informàtica i Matemàtica Aplicada, Universitat de Girona, Avda. Lluís Santaló s/n, 77 Girona, Spain 2 IEEC & Dept. Matemàtiques. Universitat Autònoma de Barcelona, Edifici C, 893 Bellaterra, Spain. 3 Dept. de Matemàtica Aplicada I. Universitat Politècnica de Cataluna, Diagonal 647, 828 Barcelona, Spain. barrabes@ima.udg.edu, jmm@mat.uab.cat, merce.olle@upc.edu Abstract. The goal of this paper is the numerical computation and continuation of homoclinic connections of the Lapunov families of periodic orbits (p.o.) associated with the collinear equilibrium points, L, L 2, and L 3, of the planar circular Restricted Three Bod Problem (RTBP). We describe the method used that allows to follow individual families of homoclinic connections b numerical continuation of a sstem of (nonlinear) equations that has as unknowns the initial condition of the p.o., the linear approimation of its stable and unstable manifolds, and a point in a given Poincaré section in which the unstable and stable manifolds match. For the L 3 case, some comments are made on the geometr of the manifold tubes and the possibilit of obtaining trajectories with prescribed itineraries. AMS classification scheme numbers: 7F7, 7F5, 7H2, 7H33, 7K44 Submitted to: Nonlinearit. Introduction Homoclinic and heteroclinic connections of hperbolic invariant sets pla an important role in the stud of dnamical sstems from a global point of view. Of special interest is their application to the design of space missions using libration point dnamics. Among the libration point missions up to present [], Genesis [8] has been the first one to make use of a heteroclinic connection. The use of homoclinic and heteroclinic phenomena allows to envisage more comple missions, like low energ transfers to the Moon [26] and the Petit Grand Tour to the moons of Jupiter [4]. Having as a goal the design of such comple missions in an automatic wa, it is desirable to construct maps of homoclinic and heteroclinic connections in several primar secondar sstems. The methodolog developed in this paper is mostl aimed at the construction of such maps.

2 Numerical continuation of homoclinic connections 2 For the design of such missions, the circular Restricted Three Bod Problem (RTBP) is the natural problem to start with. In the literature, attention is mostl focused on L and L 2 because of its suitabilit to place permanent observatories of the Sun or of the whole celestial sphere []. The L 3 point has not been considered for astrodnamical applications, but horseshoe tpe motion has drawn some attention from the astronomical point of view, as it is performed b co-orbital satellites of Saturn like Janus and Epimetheus [22], or b near Earth asteroids [9]. In [5], it is seen that the manifolds of both L 3 and its corresponding famil of Lapunov periodic orbits (LPO) pla a role in this kind of motion. Analtical proofs of the eistence of homoclinic orbits to LPO around the L and L 2 points have been given in [2, 23]. More recentl, and b means of a computed assisted proof approach [28, 29], proofs of the eistence of particularl shaped homoclinic and heteroclinic connections of Lapunov orbits around L and L 2 have been given. From a numerical point of view, families of homoclinic and heteroclinic connections of both periodic orbits (p.o.) and 2D invariant tori in the L and L 2 neighborhoods have been computed in the literature [6, 4, 5, 6, 2] b means of the use of semi-analtical techniques (asmptotic epansions with coefficients computed in finite precision arithmetic). In these references, individual connections are found b matching the corresponding manifolds on a surface of section. Families are described b manuall repeating this matching process for several values of a parameter, which is often the energ (or, equivalentl, the Jacobi constant). This paper is aimed at the numerical computation of families of homoclinic connections of p.o. The automation of the process just described was one of its main motivations. Another point we wanted to address is to overcome the (effective) convergence restrictions of semi analtical procedures, which give ver good approimations close to the libration points, but become unusable awa from them. A final motivation for this paper is the eploration of the neighborhood of L 3, for which semi analtical procedures do not give useful approimations. We have used a method for the continuation of homoclinic connections that consists of raising a (nonlinear) sstem of equations whose solution is a curve that corresponds to a famil of homoclinic connections. This sstem includes the equations of a p.o. in a Poincaré section, the eigenvalue/eigenvector equations for the linear approimation of the invariant manifolds of the p.o., and matching conditions for the manifolds on a second Poincaré section. The sstem of equations is numericall continued b a standard predictor-corrector method [2]. Both the sstem of equations and its differential with respect to the unknowns are evaluated b direct numerical integration of the differential equations, together with its first and second variational equations. For this, a variable step Runge Kutta Fehlberg method of orders 7 and 8 has been used with standard double precision arithmetic. The instabilit due to the hperbolic character of all the p.o. considered is coped with b a multiple shooting strateg. The method just described has been applied to the continuation of families of transversal homoclinic connections of planar Lapunov p.o. associated to the collinear

3 Numerical continuation of homoclinic connections 3 points of the planar, circular RTBP. For the L, L 2 case, the computations presented here are done for the Earth Moon mass ratio, and can be considered an etension of the results of [6], both in the computation of new families and the continuation to higher energ levels. The computations for the L 3 case are new, and have been done for the Sun Jupiter mass ratio in order to relate the results presented to previous works on horseshoe motion [4, 5]. Although the method presented here has onl been applied to homoclinic connections of planar LPO of the RTBP, it can be used without modification to follow homoclinic connections of an famil of p.o. of an Hamiltonian sstem. Minor modifications would allow to follow heteroclinic connections of p.o. as well. It ma also be generalized naturall to the continuation of connections of invariant tori, which will be the subject of future work. A different approach to the continuation of homoclinic connections of p.o. can be found in [], where the equations to solve are written in terms of boundar value problems, in order to use the continuation package AUTO [2]. In the former reference, the continuation method proposed is applied to the the detection and continuation of a ccle-to-ccle connecting orbit in a food chain model. The paper is structured as follows. Sect. 2 briefl recalls the circular, planar RTBP and describes the numerical method used for the continuation of families of homoclinic connections of p.o. Sect. 3 is devoted to numerical results, for L, L 2, in Sect. 3., and for L 3, in Sect For all three points, the main families of homoclinic connections are first sstematicall detected, in terms of the behavior of the manifold branches of the p.o., and then continued. For brevit, onl primar homoclinics are considered in the L, L 2 cases. In the L 3 case, some comments are made on higher order homoclinics and the possibilit of generating trajectories with prescribed itineraries, using transit and non transit trajectories as defined b Conle [8]. 2. Homoclinic connections of Lapunov p.o. of the RTBP 2.. The circular, planar RTBP The circular RTBP describes the motion of a particle of infinitesimal mass moving under the gravitational influence of two massive bodies, called primaries, that describe circular orbits around their common center of mass. We will consider the planar problem, in which the motion of the third bod is contained in the plane of motion of the primaries. Taking a coordinate sstem that rotates with the primaries, with origin placed at their center of mass, and suitable units, we can assume that the primaries have masses µ and µ, µ (, /2], their positions are fied at (µ, ) and (µ, ), and the period of their motions is 2π. With these assumptions, and introducing momenta p = ẋ and p = ẏ+, the equations of motion of the third bod ma be described b a Hamiltonian Actuall, the method could work in an sstem in which a famil of homoclinic connections eist. In our implementation we have assumed the eistence of a Hamiltonian first integral.

4 Numerical continuation of homoclinic connections 4 sstem with associated Hamiltonian function (see e.g. [27]) H(,, p, p ) = 2 (p2 + p 2 ) p + p µ r µ r 2, () where r = ( µ) and r 2 = ( µ + ) From now on, the value of the Hamiltonian on each orbit will be referred as the energ, denoted b h. We recall that the sstem of differential equations satisfies the well known smmetr (t,,, p, p ) ( t,,, p, p ). (2) This implies that, for each solution, there also eists another one, which is seen as smmetric with respect to = in configuration space. We also recall that the RTBP has five equilibrium points: the collinear points, L, L 2, and L 3, situated on the line containing the primaries at Li, i =, 2, 3, with L2 µ L µ L3, and the equilateral ones, L 4 and L 5, both forming an equilateral triangle with the two primaries. We denote h i the value of the energ at the equilibrium point L i, i =,...,5. We focus our attention on the dnamics of the RTBP around the collinear equilibrium points. It is well known [27] that, if we write the differential equations associated to () as ż = X H (z), then Spec DX H (L i ) = {±iω, ±λ}, with ω, λ >, so the equilibrium point L i, i =, 2, 3, is a center saddle point and Lapunov s center theorem applies (see e.g. [24]). Thus, each equilibrium point gives rise to a one parametric famil of periodic orbits, spanning a 2D manifold tangent to real and imaginar parts of the eigenvectors of eigenvalues ±iω at the equilibrium point. This famil is known as Lapunov famil of periodic orbits (LPO), and, close to the equilibrium point, it can be parametrized b the energ. Since the equilibrium point is hperbolic, the Lapunov famil inherits hperbolicit and therefore the periodic orbits have unstable and stable manifolds. These manifolds are immersed clinders R S. Geometricall, the can be viewed as two dimensional tubes approaching (forwards and backwards in time) the periodic orbit (see, for eample, Fig. 3 where the projection of the branches of the manifolds in the configuration space is shown). These unstable and stable manifolds can intersect, giving rise to homoclinic connections of the LPO Numerical methodolog A simple, general strateg to state the computation of a homoclinic connection as a zero finding problem is the following. Let X be a hperbolic p.o. with stable and unstable directions given b one dimensional real eigenspaces of its monodrom matri (it could also have central part or additional hperbolic directions). Assume we have parametrizations of the manifolds ψ s (θ, ξ), ψ u (θ, ξ), where θ is an angle and ξ R is such that the parametrizations describe the p.o. for ξ = and the different branches of the manifolds for ξ > and ξ < (actual epressions are given below). Let Σ = {g(z) = }

5 Numerical continuation of homoclinic connections 5 be a hpersurface known to be intersected b the manifold tubes, and consider two associated Poincaré maps: P + Σ, which propagates the flow forward in time until the net intersection with Σ, and P Σ, which does the same backward in time. Choose values ξ u, ξs, with ξu, ξs small, and consider the function F(θ u, θ s ) = P + Σ (ψu (θ u, ξ u )) P Σ (ψs (θ s, ξ s )). (3) The values of (θ u, θ s ) for which F(θ u, θ s ) is zero correspond to a homoclinic connection. The function F and its differential DF can be numericall evaluated b numerical integration of ż = X H (z) and its first variational equations, so Newton s method can be used to find roots of F. Initial conditions for the Newton iteration can be found graphicall b plotting the section of the tubes with Σ, that is, {P + Σ (ψu (θ, ξ))} u θ [,2π) and {P Σ (ψs (θ, ξ s))} θ [,2π), as in Fig. 4. Each common point of these two sets corresponds to a zero of F. Theoreticall, the section Σ that defines the P + Σ and P Σ maps is defined locall, in a neighborhood of the intersection of the manifold tubes. In practice, it is more convenient to work with global sections defined b an implicit equation {g(z) = } (actuall, in all the computations of Sect. 3 we have used just hperplanes of the form coordinate = constant). It ma happen then, that the first cut with the section is not the one we are interested in. For instance, in Fig. 6, in order to match in Σ = { = } the first cut of the unstable manifold with the stable manifold, we need to consider the sith cut of the stable manifold with Σ. In Sect. 3, notation will be introduced in order to deal with this issue. The actual formula we have used for ψ u (θ, ξ) is ψ u (θ, ξ) = φ θ 2π T(z ) + ξ(λ u ) θ/(2π) Dφ θ 2π T(z )v u, (4) where φ t (z) is the time t flow of the RTBP, z is an initial condition of the p.o., T is its period (φ T (z ) = z ), v u is an eigenvector of the monodrom matri Dφ T (z ) corresponding to the unstable manifold, and Λ u is the corresponding eigenvalue. An analogous epression has been used for the stable manifold. Note that we are using the linear approimation. A first-order Talor epansion shows that, for bounded t, φ t (ψ(θ, ξ)) = ψ(θ + tω, e tλ ξ) + O(ξ 2 ), (5) for ψ = ψ u/s, ω = 2π/T, λ = (ω ln Λ)/(2π). Therefore, the manifold parametrized b ψ(θ, ξ) is invariant b the flow ecept for a quadratic term in ξ. Since we use double precision, in all the computations presented in Sect. 3 we have taken ξ of the order of 6. The automatic continuation of families of homoclinic connections of p.o. b a predictor corrector method [2] could be done letting z free in F(θ u, θ s ) = and considering v u = v u (z ), Λ u = Λ u (z ), etc. However, it is difficult to evaluate the derivatives of eigenvalues and eigenvectors with respect to z when the are computed For Λ <, a real Floquet change is not possible. In such a case, we can consider the p.o. 2T periodic and work with Dφ T (z ) 2 = Dφ 2T (z ), which will have Λ 2 > as corresponding eigenvalue. This has never happened in the computations of Sect. 3.

6 Numerical continuation of homoclinic connections 6 b a general purpose numerical method (as implemented in linear algebra packages such as LAPACK [3]). This difficult is avoided b adding the eigenvector condition to the sstem of equations, plus a normalization equation in order to have local uniqueness. In order to state the sstem of equations we have used, consider h R an energ level, z R 4 an initial condition of a p.o., T its period, Λ u, Λ s Spec Dφ T (z), with Λ u > and < Λ s <, v u, v s corresponding eigenvectors of Dφ T (z), θ u, θ s starting phases on the linear approimation of the unstable and stable manifolds, respectivel, and T u, T s R the integration times required to intersect the section Σ from the starting points ψ u (θ u, ξ u), ψs (θ s, ξ s ). Note that the introduction of these times avoids the need to take into account the number of cuts with the sections, which would have been necessar if we had considered Poincaré maps eplicitl, as in the function F previousl considered. Consider also g : R 4 R a function defining a Poincaré section for the periodic orbit, and g 2 : R 4 R a function defining the Poincaré section to match the manifolds, this is, Σ = {g 2 (z) = }. The sstem of equations we have used for the continuation of homoclinic connections of p.o. is H(z) h =, Dφ T (z)v s Λ s v s =, g (z) =, v s 2 =, ) φ T (z) z =, g 2 (φ T u(ψ u (θ u, ξ)) u =, ( ) Dφ T (z)v u Λ u v u =, g 2 φ T s(ψ s (θ s, ξ s)) =, (6) v u 2 =, φ T u(ψ u (θ u, ξ u)) φ T s(ψs (θ s, ξ s )) =, where ξ u, ξs are kept fied to a small value (usuall 6 ), and the unknowns are h, z, T, Λ u, v u, Λ s, v s, θ u, T u, θ s, T s. Here T, T u > and T s <. In our setting, the integration times T u, T s ma become large (this is the case in Sect. 3.2). In order to avoid loss of precision, we have also used a multiple shooting version of (6). For that, we have added new points as unknowns, along the p.o., z,...,z m, along the unstable branch, z, u...,zm u, and along the stable one, u z, s..., z m s. The corresponding matching equations have been added to sstem (6), namel φ Ti (z i ) z i+ =, i =,..., m 2, φ T u i (zi u ) zi+ u =, i =,..., m u 2, φ T s i (zi s ) zi+ s =, i =,..., m s 2, with z = z, z u = ψu (θ u, ξ u), zs = ψs (θ s, ξ s ). An adaptive strateg has been used, in the sense that the number of points, m, m u, m s, and integration times, {T i } i, {Ti u} i, {Ti s} i, have been recomputed at each continuation step in order to have Dφ t (z i ) < M, < t < T i, and analogous inequalities along the unstable and stable branches. M has been taken tpicall of the order of tens or hundreds.

7 Numerical continuation of homoclinic connections 7 In all the computations of the following Section, the sstem (6) (or its multiple shooting version) and their derivatives with respect to all the unknowns have been evaluated from numerical integration b a variable step Runge Kutta Felbergh method of orders 7 and 8 of the RTBP equations, together with its first and second variationals. For that, a routine evaluating the augmented sstem of the RTBP equations including st and 2nd variationals has been written. The tolerance of the RKF method used has been 4. The absolute tolerances used to stop Newton iterates in the solution of either sstem (6) or its multiple shooting version have ranged from to 2. Note also that, both in the single and multiple shooting cases, the sstem is over determined and rank deficient. For instance, one of the equations including g 2 can be eliminated (it follows from the other equation including g 2 and the last one). An additional redundanc is that the equation g (z) = allows to eliminate one of the equations in φ T (z) z =. The reason for these redundancies follows from numerical eperience, and its goal is to accelerate the convergence of Newton s method. Redundanc and rank deficienc can be coped with b using the minimum norm least squares (LS) solution for the linear sstem that gives the Newton correction. Although redundant equations could be eliminated before each Newton iteration, their detection would need a linear analsis which is done automaticall b the minimum norm LS strateg at negligible computational cost. In our implementation, this solution has been computed using QR decomposition with column pivoting. For more details, see [7, 25]. As mentioned in the Introduction, the sstem of equations in Eq. (6) is not specific to the RTBP but is valid for an Hamiltonian sstem. With minor modifications, it would allow to continue heteroclinic connections of p.o. 3. Numerical results This section is devoted to the presentation of numerical results on the computation of families of homoclinic connections of Lapunov periodic orbits around the collinear points of the planar RTBP with the methodolog presented in the previous section. In order to find homoclinic connections in the L,2 cases, onl the first intersection of the manifold tubes with a surface of section will be considered. We will speak of first cut homoclinics. Note that first cut homoclinics are alwas primar, but the converse is not necessaril true, as we shall see. In the L 3 case, some additional cuts will be taken into account. As shown in [8], + for the three collinear points, in the linear approimation of the Recent developments on general purpose Talor integrators using automatic differentiation [, 9] will allow to obtain arbitrar order derivatives of the flow just from the RTBP equations in the immediate future. The source code implementing the numerical methodolog of this section is available upon request to the authors. + Note that our convention for the ordering of the primaries on the ais is reversed with respect to this reference.

8 Numerical continuation of homoclinic connections 8 flow around the point, and for a fied energ level, a branch of the unstable manifold departs to the { < Li, > } region, and the other one departs to the { > Li, < } region (see Fig. ). We will denote these branches as W u, W + u, respectivel. The application of smmetr (2) allows to obtain the branches W s, W + s, entering from the { < Li, < }, { > Li, > } regions, respectivel. Following [8], we will call transit orbits the trajectories crossing the bottleneck region determined b the zero velocit curves (see Figs., 2), going from the { < Li } half space to the { > Li } one or vice-versa, and non transit orbits the ones bouncing back to an of the half-spaces. A necessar and sufficient condition for a trajector approaching the LPO to be a transit orbit is to be inside one of the W± s tubes, and analogousl for an orbit departing from the LPO and the W± u tubes. Although all these considerations correspond to the linear flow, we have observed the same qualitative behavior for the full flow in all the families of homoclinics computed, even for large energies. W u W s + ais W s W u + Figure. Sketch in configuration space of the naming convention for the manifold branches of an LPO. The solid curves represent the LPO and the zero velocit curves (bounding the region of forbidden motion). The dotted lines represent the section with the plane of the manifold tubes. The arrows indicate direction of motion of trajectories following the manifold branches Figure 2. Hill s regions for h < h < h 2 (left), h 2 < h < h 3 (center) and h 3 < h < h 4 = h 5 (right). Inside the filled regions, motion is not possible. As mentioned before, in order to compute starting connections in each famil, we will need to consider the number of cuts with the section Σ used to match the manifolds.

9 Numerical continuation of homoclinic connections 9 In what follows, we will denote the j th cut of a branch of a manifold with a section Σ b W s/u ± Σ j. We will sa that there eists a homoclinic connection of tpe (j, k), j, k Z, if ( W u sign j Σ j ) ( Wsign s k Σ k ). (7) Observe that all the pairs (j, k ) such that sign j = sign j, sign k = sign k, j + k = j + k give the same homoclinic connection as the pair (j, k). Throughout this Section, in some figures where the projection of orbits or invariant manifolds in configuration space is shown, we have included the horizontal line = (dotted line in the plots) that contains the primaries for greater clarit. Blue and red indicate that the piece of orbit or the set of points shown belongs to the stable and unstable manifold respectivel. 3.. Homoclinic connections to Lapunov orbits around L and L 2 In this Section, the mass parameter is fied to the Earth Moon mass ratio, µ EM = B numericall propagating the manifolds of LPO, it is seen that In the L case, the W u/s + tubes go around the large primar (Earth), and the W u/s go around the small one (Moon). In the L 2 case, the W u/s + tubes go around the small primar, whereas the W u/s tubes eit the Hill region and go around the region of forbidden motion, surrounding both primaries. Due to the topolog of the region of forbidden motion (see Fig. 2 left, middle), first cut homoclinics will be found b matching either the W u/s + branches or the W u/s ones. We will present the corresponding results grouped in homoclinics surrounding the Moon and homoclinics surrounding either the Earth or both primaries Homoclinics to LPO surrounding the Moon From the behavior of the manifold branches, and using the section Σ = { = µ }, the candidates to be first cut homoclinics of LPO around the Moon are of tpe (, 2) for L (Fig. 3 left), and of tpe (+, +2), for L 2 (Fig. 3 right). Since L does not have a homoclinic connection of this tpe, neither do the corresponding LPO for energ levels close to h. As energ increases, the sets W u Σ, W s Σ2 become tangent and, after that, the intersect at two points giving rise to two homoclinic connections (Fig. 4 left). We have labeled these connections as Hm j, j =, 2, and we have followed them b the continuation procedure of Sect The corresponding characteristic curves are displaed in Fig. 5 left. The Hm 2 famil goes to collision with the Moon ver rapidl, whereas Hm survives up to energies over.5. The L 2 case is analogous: W+ u Σ, W+ s Σ2 are disjoint at first, the become tangent and, for increasing energies, the displa two intersection points (Fig. 4 right),

10 Numerical continuation of homoclinic connections h= h= W u -. W s W s - -. W u Figure 3. Projection in the configuration space of the invariant manifolds associated with a LPO around L (left, branches W u/s ), and L 2 (right, branches W u/s + ), up to a certain intersection with Σ = { = µ }. p u - h= s 2 - p h= s u p Figure 4. Left: for Σ = { = µ }, sets u = W u Σ and s 2 = W s Σ 2 in the (, p ) plane for a LPO around L. Right: sets u + = W u + Σ and s 2 + = W s + Σ 2 in the (p, p ) plane for a LPO around L 2 and the same section. giving rise to two families of homoclinics, which we have labeled as Hi j, j =, 2. We have also followed them (Fig. 5 right) Hi Hm 2 Hm Hi h h Figure 5. Characteristic curves in the (h, ) plane of families of homoclinic connections to LPO around L (left) and L 2 (right) that surround the Moon (the coordinates correspond to Σ = { = µ }).

11 Numerical continuation of homoclinic connections These computations are etensions of the computations in [6], where the Hm j, Hi j families are computed using Lindstedt Poincaré series, covering the energ range [.59475,.56865] for the L case, and [.58682, ] for the L 2 case Homoclinics to LPO surrounding the Earth From the behavior of the manifold branches, and using the section Σ = { = }, first cut homoclinics of LPO surrounding the Earth have to be of tpe (+, +6) for L (Fig. 6 left), and of tpe (, 2) for L 2 (Fig. 6 right). As before, L does not have a homoclinic connection of this tpe, so neither do the corresponding LPO for energ levels close to h. As the energ increases, W u + Σ, W u + Σ6 become tangent, two families of homoclinic connections appear, and, for larger energies, a new tangenc gives rise to two new families so the sets W u + Σ, W u + Σ6 have 4 common points (see Fig. 7 left). We have labeled the corresponding families He j, j =, 2, 3, 4, and we followed them. Their characteristic curves are displaed in Fig. 9 left. Some homoclinics of these families are displaed in the top row of Fig.. For large energies the approach both primaries, and thus become interesting for cargo transportation applications in the Earth Moon sstem [7]. h= W s h= W u - W u W s Figure 6. Invariant manifolds associated with a LPO around L (left, branches W u/s + ), and L 2 (right, branches W u/s ) up to a certain intersection with Σ = { = }. For the L 2 case, the situation is ver much as before and we have first 2 families of homoclinics, and then 4 (see Fig. 7 right). However, as energ increases, W s Σ2 spirals and generates new intersections with W u Σ giving rise to additional families (see Fig. 8). We have followed up to 2 families Ho i, i =,...,2, whose characteristic curves are displaed in Fig. 9 right. Some sample homoclinics of these families are displaed in the bottom row of Fig Homoclinic connections of Lapunov orbits around L 3 This subsection is devoted to families of homoclinics of LPO around L 3. As far as the authors know, these are the first computations of this kind done in the neighborhood of

12 Numerical continuation of homoclinic connections 2.8 h= h= p s 6 + u + p u - s p Figure 7. Left: for Σ = { = }, sets u + = W u + Σ and s 6 + = W s + Σ 6 in the (, p ) plane for a LPO around L. Right: sets u = W u Σ and s 2 = W s Σ 2 in the (p, p ) plane for a LPO around L 2 and the same section..8 h= h= h= u -.4 u -.4 u - p p p -.4 s s s p p p Figure 8. Curves u = W u Σ and s 2 = W s Σ 2 in the (p, p ) plane, for the Lapunov orbits around L 2 of the energies indicated and the section Σ = { = } He He 3 He 2 He h Ho 6 Ho Ho 3 Ho 4 Ho 2 Ho Ho 5 Ho 7 Ho 8Ho h Figure 9. Characteristic curves in the (h, ) plane of families of homoclinic orbits that surround either the Earth onl (left) or both the Earth and the Moon (right). The coordinates correspond to the section { = }. L 3. For this reason, the eposition will be a little more detailed, and we will finish with some comments on the construction of trajectories with prescribed itineraries following different homoclinics of LPO. All the computations of this section correspond to the Sun Jupiter mass parameter, µ = µ SJ = , in order to relate them to previous works on horseshoe motion [4, 5]. Locall, the branches of manifolds of LPO behave as in Fig.. When the go awa from the LPO, the surround the islands of the region of forbidden motion (see

13 Numerical continuation of homoclinic connections 3 h= h=-.558 h= h= h= h= Figure. Projection in configuration space of homoclinic orbits to LPO around L from the famil He (first row) and around L 2 from the famil Hi (second row). Fig. 2 right). For that reason, and using the section Σ = { = µ /2}, it seems reasonable to look for first cut homoclinics of tpe (+, 2) (which, b smmetr (2), give homoclinics of tpe ( 2, +)). These homoclinics have the shape of a half horseshoe (see Fig. left) Figure. Left: (, ) projection of a half horseshoe shaped homoclinic orbit of tpe (+, 2) with respect to Σ = { /2}. Right: full horseshoe shaped of tpe (+, +4) w.r.t. the same section. The situation is ver much like in the L,2 cases: the curves W u + Σ and W s Σ 2 do not intersect at first (Fig. 2 left) and, after becoming tangent at energ h t :=.54766, their intersection gives rise to two families of homoclinics (Fig. 2 right). We have labeled them as Hn i, i =, 2, and we have also followed them. Their characteristic curves are shown in Fig. 3 left. Before W u + Σ intersects W s Σ2, this is, for h < h t, all the trajectories in W u + Σ

14 Numerical continuation of homoclinic connections h= h= u + u + p p s 2 - s Figure 2. Projections in the (, p ) plane of the curves u + = W u + Σ (red) and s 2 = W s Σ 2 (blue) for a LPO around L 3 at the section Σ = { = µ /2}, for the values of the energies indicated Hs 6 Hs 3 - Hn Hn Hs Hs h h Figure 3. Left: characteristic curves of half horseshoe shaped homoclinics of tpe (+, 2) w.r.t. Σ = { = µ /2}. Right: characteristic curves of the full horseshoe shaped homoclinics of tpe (+, +4) w.r.t. the same section. These last homoclinics are of tpe (+2, +2) w.r.t. Σ = { = }, which has been used for their computation. The coordinates displaed correspond to Σ. are non transit. Therefore, after one turn around the island of the forbidden region in { < } (see Fig. 2 right), the avoid falling back to the LPO and perform a turn around the island in { > }, opening the possibilit for a smmetric, full horseshoe shaped homoclinic of tpe (+, +4) (Fig. right). It turns out that these kind of homoclinics appear before the (+, 2) do (see Fig. 4). When the (+, 2) homoclinics appear, the (+, +4) become secondar, so there must be an infinite number of them. This fact is reflected in that, as energ approaches h t, the W s + Σ4 section spirals and generates man new families, see Fig. 4. (Although not eplored, this phenomenon also takes place in the L,2 cases, as in Fig. 8.) In spite of this complicated behavior, the procedure of Sect. 2.2 is able to follow an of the (+, +4) families to energies past h t. Among these families, we have followed some of the smmetric ones, whose characteristic curves are displaed in Fig. 3 right. A remarkable difference between the dnamics around L, L 2 and that of L 3 comes from the geometr of the manifolds tubes. Close to the equilibrium point, the orbits can be approimated b a two-bod problem solution with period close to that of the small

15 Numerical continuation of homoclinic connections h= h= u u + p p s s Figure 4. Using the section Σ = { = µ /2}, (, p ) projection of the curves u + = W u + Σ (red) and s 4 + = W s + Σ 4 (blue), for the values of the energies indicated. primar. This slow dnamics causes the orbits to perform loops in the rotating frame, which increase as the LPO becomes larger (see Fig. 5 and [5] for an analsis of the presence of loops in horseshoe motion from the µ = case). Small loops straighten out along the manifold, like in Fig. 5 left, and present no problems. But, as the become bigger, the eventuall reach the surface of section (Fig. 5 right) and the (±i, ±j) classification of homoclinics stops making sense. This fact makes difficult to find the intersection sets defined in (7). This is not a problem for the continuation method of Sect. 2.2, because it does not make use of the number of cuts necessar in order to reach the section. However, in order to check for the appearance of new families of homoclinics, it is convenient to be able to have a clear representation of the section of the manifold tubes. h= h= Figure 5. Projection in the (, ) plane of two orbits of W u + up to its second intersection with Σ = { = µ /2}, plotted as a dotted line for clarit. As the energ increases, the loops reach a neighborhood of the section Σ. A wa to obtain these representations is b doing continuation of the equation g(ψ(θ, ξ)) = with respect to (θ, ξ), where ψ(θ, ξ) is a parametrization of the invariant manifold tube ( ξ not being necessaril small), and the section is given b Σ = {g(z) = }, for g : R 4 R. A simple wa to numericall evaluate such a parametrization is using the linear approimation and globalizing b numerical integration. Namel, for ξ not

16 Numerical continuation of homoclinic connections 6 necessaril small, we can take ( ) ψ(θ, ξ) = φ t ψ(θ t ω, e t λ ξ), for t such that e t λ ξ is small, ω, λ defined as in (5), and we can use (4) to evaluate ψ on the right hand side of the previous equation. -.2 h= h= p - u + p - u s 2 - s d d π/2 π θ 3π/2 2π π/2 π θ 3π/2 2π Figure 6. Top: for the energies indicated, (, p ) projection of curves W u + Σ (red) and W s Σ 2 (blue). Bottom: for the energies of the top row, distance d of each point of the curve W u + Σ to the curve W s Σ 2, which shows the number of intersections between them. The sections of the manifold tubes of the top row of Fig. 6 have been computed in this wa. The correspond to the (+, 2) homoclinics (Hn i families) introduced above. These plots are useful to check for the appearance of new families. With this, special care has to be taken because visual inspection can be misleading, as is illustrated in Fig. 6. For energ h = , a graph of the distance of each point of the curve W u + Σ to the curve W s Σ2 (Fig. 6 bottom left) shows that in Fig. 6 top left there are onl two intersections. For energ h = , the same kind of graph shows that in Fig. 6 top right there are 4 intersections, so two new families of homoclinics have appeared. We have not followed them The geometr of the manifold tubes In this subsection we would like to make some comments on the geometr of the manifold tubes for the L 3 case, in order to investigate the possible generation of trajectories with prescribed itineraries. We would also like to illustrate graphicall how the eistence of a

17 Numerical continuation of homoclinic connections 7 primar homoclinic connection leads to the appearance of an infinite number of higher order homoclinics. Similar studies for the L and L 2 points can be found in previous works [3, 4, 2] p.858 u 3 + u + p s s u - u Figure 7. For h = Left: (, p ) projection of the curves s 2 = W s Σ 2 (blue), u + = W u + Σ (black), and the component with < of the curve u 3 + = W u + Σ 3 (red). Right: (, p ) projection of the curves s 2 + = W s + Σ 2 (blue), u = W u Σ (black), and the component with > of the curve u 3 + = W u + Σ 3 (red). For the energ level h = and section Σ = { = µ /2}, consider the curves W+ u Σ and W s Σ 2, which are plotted in Fig. 7 left in black and blue colors respectivel. The points inside the black curve correspond to transit trajectories backward in time, meaning that, when propagated backwards, at the close approach with the LPO the eit through the W s branch and complete a turn around the island of forbidden motion in { < }. The points outside the black curve in Fig. 7 left are non transit trajectories backward in time, meaning that, when propagated backwards, at the close passage to the LPO the eit through the W+ s branch and perform a revolution around the island of forbidden motion in { > }. Eamples of such transit and non transit trajectories are given in Fig. 8. A similar description holds for transit and non transit trajectories forward in time with respect to the (blue) curve W s Σ2 in Fig. 7 left. As previousl mentioned, the proofs of these facts can be found in [8] Figure 8. For h = , eamples of a transit (left) and non transit (right) orbits in the neighborhood of L 3. The starting points have been taken in { = µ /2, < }.

18 Numerical continuation of homoclinic connections 8 Now consider the curve W u + Σ3 (red curve in Fig. 7 left), which is the second section with Σ forward in time of the points of W u + Σ (black curve). The curve W u + Σ is divided b the W s Σ2 (blue curve) in two segments: the one inside W s Σ2, which we denote as A, and the one outside, which we will call B. The points in the A segment correspond to transit orbits forward in time, so the sta in the { < } for the net 2 cuts with Σ, and give the red curve in Fig. 7 left (see also Fig. 8 left). The points in the B segment correspond to non transit orbits forward in time, so the have the net two cuts with Σ in { > }, giving the red curve in Fig. 7 right (see also Fig. 8 right). In this wa, W u + Σ3 has two components, that accumulate to the section of the manifold branch that the follow after the close passage to the LPO. Note that, in the first case, the accumulate inside W u + Σ, since the are transit orbits backward in time. In the second case the are non transit, so the accumulate outside W u Σ p Figure 9. For h = , (, p ) projection of curves W u + Σ (black), W u + Σ 3 (red), and W u + Σ 5 (blue). This scheme can be iterated. In Fig. 9, we have added W u + Σ 5 (in blue) to the curves in Fig. 7 left. All the points inside W u + Σ correspond to transit trajectories backward in time. In the second close passage to the LPO (backward in time), the ma be either transit, non transit, or belong to W u + Σ3, in which case the tend to the LPO. Therefore, W u + Σ3 separates transit from non transit points in the second LPO passage. Since the points of W u + Σ5 are transit in the second LPO passage, the determine the transit component. In this wa, the position of points with respect to the W u + Σ 3 curve determine their past histor up to two approaches to the LPO. A similar analsis can be done outside the curve W u + Σ, and also forward in time, with respect to the curve W s Σ2. The consideration of higher numbers of cuts of the manifolds would allow to determine the behavior of further close passages to the LPO, either forward or backward in time. A rigorous stud using smbolic dnamics would allow to show the eistence of orbits with prescribed itineraries. Studies of this kind have been done in [3, 4, 2] for LPO around the L and L 2 points. The difference between L 3 and the previous points is that, instead of prescribing itineraries between the interior and eterior regions of the zero velocit curve, here we would prescribe itineraries alternating between the { > } and { < } regions. Among these trajectories, we could find periodic orbits

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