Boundary-value problem formulations for computing invariant manifolds and connecting orbits in the circular restricted three body problem

We demonstrate the remarkable effectiveness of boundary value formulations coupled to numerical continuation for the computation of stable and unstable manifolds in systems of ordinary differential equations. Specifically, we consider the circular restricted three-body problem (CR3BP), which models the motion of a satellite in an Earth–Moon-like system. The CR3BP has many well-known families of periodic orbits, such as the planar Lyapunov orbits and the non-planar vertical and halo orbits. We compute the unstable manifolds of selected vertical and halo orbits, which in several cases leads to the detection of heteroclinic connections from such a periodic orbit to invariant tori. Subsequent continuation of these connecting orbits with a suitable end point condition and allowing the energy level to vary leads to the further detection of apparent homoclinic connections from the base periodic orbit to itself, or the detection of heteroclinic connections from the base periodic orbit to other periodic orbits. Some of these connecting orbits are of potential interest in space mission design.     

The existence of transversal homoclinic orbits in a planar circular restricted four-body problem

In this paper, we study the existence of transversal homoclinic orbits in a planar circular restricted four-body problem, based on the perturbation theory of integrable Hamiltonian systems. We start from a planar circular restricted four-body model and regard it as a perturbation of the two-body model. Then, in order to conveniently study unbounded orbits, we transform the infinite points to finite points by a non-canonical transformation, arriving at a non-Hamiltonian system with degenerate fixed points. According to the extended Melnikov method, we finally prove that there exist transversal homoclinic orbits in this four-body model.     

On Brown's conjecture

We examine the conjecture made by Brown (1911) that in the restricted three body problem, the long period family of periodic orbits aroundL ₄, ends on a homoclinic orbit toL ₃. By numerical integration we establish that for the mass ratio Sun-Jupiter such a homoclinic orbit toL ₃ does not exist but that there exists a family of homoclinic orbits to periodic orbits aroundL ₃.     

Families of Periodic Orbits Emanating From Homoclinic Orbits in the Restricted Problem of Three Bodies

We describe and comment the results of a numerical exploration on the evolution of the families of periodic orbits associated with homoclinic orbits emanating from the equilateral equilibria of the restricted three body problem for values of the mass ratio larger than μ ₁. This exploration is, in some sense, a continuation of the work reported in Henrard [Celes. Mech. Dyn. Astr. 2002, 83, 291]. Indeed it shows how, for values of μ. larger than μ ₁, the Trojan web described there is transformed into families of periodic orbits associated with homoclinic orbits. Also we describe how families of periodic orbits associated with homoclinic orbits can attach (or detach) themselves to (or from) the best known families of symmetric periodic orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

Homoclinic and heteroclinic solutions in the restricted three-body problem

In this work we have performed a systematic computation of the homoclinic and heteroclinic orbits associated with the triangular equilibrium points of the restricted three-body problem. Some analytical results are given, related to their number when the mass ratio varies.     

Transport orbits in an equilateral restricted four-body problem

In this paper we consider a restricted equilateral four-body problem where a particle of negligible mass is moving under the Newtonian gravitational attraction of three masses (called primaries) which move on circular orbits around their center of masses such that their configuration is always an equilateral triangle (Lagrangian configuration). We consider the case of two bodies of equal masses, which in adimensional units is the parameter of the problem. We study numerically the existence of families of unstable periodic orbits, whose invariant stable and unstable manifolds are responsible for the existence of homoclinic and heteroclinic connections, as well as of transit orbits traveling from and to different regions. We explore, for three different values of the mass parameter, what kind of transits and energy levels exist for which there are orbits with prescribed itineraries visiting the neighborhood of different primaries.     

Asymptotic Orbits and Terminations of Families in the Copenhagen Problem, II

We present families of periodic orbits of the restricted three-body problem terminating with homoclinic orbits asymptotic to equilibrium points or to periodic orbits, as opposed to heteroclinic orbits presented in part I. This revised version was published online in July 2006 with corrections to the Cover Date.     

Resonance transition periodic orbits in the circular restricted three-body problem

This work studies a special type of cislunar periodic orbits in the circular restricted three-body problem called resonance transition periodic orbits, which switch between different resonances and revolve about the secondary with multiple loops during one period. In the practical computation, families of multiple periodic orbits are identified first, and then the invariant manifolds emanating from the unstable multiple periodic orbits are taken to generate resonant homoclinic connections, which are used to determine the initial guesses for computing the desired periodic orbits by means of multiple-shooting scheme. The obtained periodic orbits have potential applications for the missions requiring long-term continuous observation of the secondary and tour missions in a multi-body environment.     

Bifurcation of Relative Equilibria in the Main Problem of Artificial Satellite Theory for a Prolate Body

In this paper, we study circular orbits of the J ₂ problem that are confined to constant-z planes. They correspond to fixed points of the dynamics in a meridian plane. It turns out that, in the case of a prolate body, such orbits can exist that are not equatorial and branch from the equatorial one through a saddle-center bifurcation. A closed-form parametrization of these branching solutions is given and the bifurcation is studied in detail. We show both theoretically and numerically that, close to the bifurcation point, quasi-periodic orbits are created, along with two families of reversible orbits that are homoclinic to each one of them.     

Homoclinic and heteroclinic orbits in the photogravitational restricted three-body problem

We study numerically the asymptotic homoclinic and heteroclinic orbits around the hyperbolic Lyapunov periodic orbits which emanate from Euler's critical points L ₁ and L ₂, in the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these Lyapunov orbits, are also presented. Poincaré surface of sections of these manifolds on appropriate planes and several homoclinic and heteroclinic orbits for the gravitational case as well as for varying radiation factor q ₁, are displayed. Homoclinic-homoclinic and homoclinic-heteroclinic-homoclinic chains which link the interior with the exterior Hill's regions, are illustrated. We adopt the Sun-Jupiter system and assume that only the larger primary radiates. It is found that for small deviations of its value from the gravitational case (q ₁ = 1), the radiation pressure exerts a significant impact on the Hill's regions and on these asymptotic orbits.     

Asymptotic Orbits at the Triangular Equilibria in the Photogravitational Restricted Three-Body Problem

We study numerically the asymptotic homoclinic and heteroclinic orbits associated with the triangular equilibrium points L ₄ and L ₅, in the gravitational and the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these critical points, are also presented. Hundreds of asymptotic orbits for equal mass of the primaries and for various values of the radiation pressure are computed and the most interesting of them are illustrated. In the Copenhagen case, which the problem is symmetric with respect to the x- and y-axis, we found and present non-symmetric heteroclinic asymptotic orbits. So pairs of heteroclinic connections (from L ₄ to L ₅ and vice versa) form non-symmetric heteroclinic cycles. The termination orbits (a combination of two asymptotic orbits) of all the simple families of symmetric periodic orbits, in the Copenhagen case, are illustrated.     

Qualitative study of the planar isosceles three-body problem

We consider the particular case of the planar three body problem obtained when the masses form an isosceles triangle for all time. Various authors [1, 2, 12, 8, 9, 13, 10] have contributed in the knowledge of the triple collision and of several families of periodic orbits in this problem. We study the flow on a fixed level of negative energy. First we obtain a topological representation of the energy manifold including the triple collision and infinity as boundaries of that manifold. The existence of orbits connecting the triple collision and infinity gives some homoclinic and heteroclinic orbits. Using these orbits and the homothetic solutions of the problem we can characterize orbits which pass near triple collision and near infinity by pairs of sequences. One of the sequences describes the regions visited by the orbit, the other refers to the behaviour of the orbit between two consecutive passages by a suitable surface of section. This symbolic dynamics which has a topological character is given in an abstract form and after it is applied to the isosceles problem. We try to keep globality as far as possible. This strongly relies on the fact that the intersection of some invariant manifolds with an equatorial plane (v=0) have nice spiraling properties. This can be proved by analytical means in some local cases. Numerical simulations given in Appendix A make clear that these properties hold globally.     

Distribution of periodic orbits and the homoclinic tangle

We study the distribution of regular and irregular periodic orbits on a Poincaré surface of section of a simple Hamiltonian system of 2 degrees of freedom. We explain the appearance of many lines of periodic orbits that form Farey trees. There are also lines that are very close to the asymptotic curves of the unstable periodic orbits. Some regular orbits, sometimes stable, are found inside the homoclinic tangle. We explain this phenomenon, which shows that the homoclinic tangle does not cover the whole area around an unstable orbit, but has gaps. Inside the lobes only irregular orbits appear, and some of them are stable. We conjecture that the opposite is also true, i.e. all irregular orbits are inside lobes.     

Spiral Structures and Chaotic Scattering of Coorbital Satellites

The fractal nature of the transitions between two sets of orbits separated by heteroclinic or homoclinic orbits is well known. We analyze in detail this phenomenon in Hill's problem where one set of orbits corresponds to coorbital satellites exchanging semi-major axis after close encounter (horse-shoe orbits) and the other corresponds to orbits which do not exchange semi-major axis (passing-by orbits). With the help of a normalized approximation of the vicinity of unstable periodic orbits, we show that the fractal structure is intimately tied to a special spiral structure of the Poincaré maps. We show that each basin is composed of a few well behaved areas and of an infinity of intertwined tongues and subtongues winding around them. This behaviour is generic and is likely to be present in large classes of chaotic scattering problems.This revised version was published online in October 2005 with corrections to the Cover Date.     

Asymptotic and Periodic Orbits Around L 3 in the Photogravitational Restricted Three-Body Problem

Asymptotic motion near the collinear equilibrium points of the photogravitational restricted three-body problem is considered. In particular, non-symmetric homoclinic solutions are numerically explored. These orbits are connected with periodic ones. We have computed numerically the families containing these orbits and have found that they terminate at both ends by asymptotically approaching simple periodic solutions belonging to the Lyapunov family emanating from L₃.     

Preliminary Study on the Translunar Halo Orbits of the Real Earth–Moon System

The computation of translunar Halo orbits of the real Earth–Moon system (REMS) has been an open problem for a long time, but now, it is possible to compute Halo orbits of the REMS in a systematic way. In this paper, we describe the method used for the numerical computation of Halo orbits for a time span longer than 41 years. Halo orbits of the REMS are computed from quasi-periodic Halo orbits of the quasi-bicircular problem (QBCP). The QBCP is a model for the dynamics of a spacecraft in the Earth–Moon–Sun system. It is a Hamiltonian system with three degrees of freedom and depending periodically on time. In this model, Earth, Moon and Sun are moving in a self-consistent motion close to bicircular. The computed Halo orbits of the REMS are compared with the family of Halo orbits of the QBCP. The results show that the QBCP is a good model to understand the main features of the Halo family of the REMS.     

2/1 resonant periodic orbits in three dimensional planetary systems

We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic orbits of the system given in a suitable rotating frame. The stability of periodic orbits characterize the evolution of any planetary system with initial conditions in their vicinity. Stable periodic orbits are associated with long term regular evolution, while unstable periodic orbits are surrounded by regions of chaotic motion. We compute many families of symmetric periodic orbits by applying two schemes of analytical continuation. In the first scheme, we start from the 2/1 (or 1/2) resonant periodic orbits of the restricted problem and in the second scheme, we start from vertical critical periodic orbits of the general planar problem. Most of the periodic orbits are unstable, but many stable periodic orbits have been, also, found with mutual inclination up to 50^?–60^?, which may be related with the existence of real planetary systems.     

Numerical Determination of Homoclinic and Heteroclinic Orbits at Collinear Equilibria in the Restricted Three-Body Problem with Oblateness

Asymptotic motion to collinear equilibrium points of the restricted three-body problem with oblateness is considered. In particular, homoclinic and heteroclinic solutions to these points are computed. These solutions depart asymptotically from an equilibrium point and arrive asymptotically at the same or another equilibrium point and are important reference solutions. To compute an asymptotic orbit, we use a fourth order local analysis, numerical integration and standard differential corrections.     

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

We consider the planar restricted three-body problem and the collinear equilibrium point L ₃, as an example of a center × saddle equilibrium point in a Hamiltonian with two degrees of freedom. We explore numerically the existence of symmetric and non-symmetric homoclinic orbits to L ₃, when varying the mass parameter μ. Concerning the symmetric homoclinic orbits (SHO), we study the multi-round, m-round, SHO for m ≥ 2. More precisely, given a transversal value of μ for which there is a 1-round SHO, say μ ₁, we show that for any m ≥ 2, there are countable sets of values of μ, tending to μ ₁, corresponding to m-round SHO. Some comments on related analytical results are also made.