On the vertical families of two-dimensional tori near the triangular points of the Bicircular problem

This paper focuses on some aspects of the motion of a small particle moving near the Lagrangian points of the Earth–Moon system. The model for the motion of the particle is the so-called bicircular problem (BCP), that includes the effect of Earth and Moon as in the spatial restricted three body problem (RTBP), plus the effect of the Sun as a periodic time-dependent perturbation of the RTBP. Due to this periodic forcing coming from the Sun, the Lagrangian points are no longer equilibrium solutions for the BCP. On the other hand, the BCP has three periodic orbits (with the same period as the forcing) that can be seen as the dynamical equivalent of the Lagrangian points. In this work, we first discuss some numerical methods for the accurate computation of quasi-periodic solutions, and then we apply them to the BCP to obtain families of 2-D tori in an extended neighbourhood of the Lagrangian points. These families start on the three periodic orbits mentioned above and they are continued in the vertical (z and ż) direction up to a high distance. These (Cantor) families can be seen as the continuation, into the BCP, of the Lyapunov family of periodic orbits of the Lagrangian points that goes in the (z, ż) direction. These results are used in a forthcoming work [9] to find regions where trajectories remain confined for a very long time. It is remarkable that these regions seem to persist in the real system. This revised version was published online in July 2006 with corrections to the Cover Date.     

Extension of fast periodic transfer orbits from the Earth–Moon RTBP to the Sun–Earth–Moon Quasi-Bicircular Problem

Starting from 80 families of low-energy fast periodic transfer orbits in the Earth–Moon planar circular Restricted Three Body Problem (RTBP), we obtain by analytical continuation 11 periodic orbits and 25 periodic arcs with similar properties in the Sun–Earth–Moon Quasi-Bicircular Problem (QBCP). A novel and very simple procedure is introduced giving the solar phases at which to attempt continuation. Detailed numerical results for each periodic orbit and arc found are given, including their stability parameters and minimal distances to the Earth and Moon. The periods of these orbits are between 2.5 and 5 synodic months, their energies are among the lowest possible to achieve an Earth–Moon transfer, and they show a diversity of circumlunar trajectories, making them good candidates for missions requiring repeated passages around the Earth and the Moon with close approaches to the last.     

Multiple Poincaré sections method for finding the quasiperiodic orbits of the restricted three body problem

A new fully numerical method is presented which employs multiple Poincaré sections to find quasiperiodic orbits of the Restricted Three-Body Problem (RTBP). The main advantages of this method are the small overhead cost of programming and very fast execution times, robust behavior near chaotic regions that leads to full convergence for given family of quasiperiodic orbits and the minimal memory required to store these orbits. This method reduces the calculations required for searching two-dimensional invariant tori to a search for closed orbits, which are the intersection of the invariant tori with the Poincaré sections. Truncated Fourier series are employed to represent these closed orbits. The flow of the differential equation on the invariant tori is reduced to maps between the consecutive Poincaré maps. A Newton iteration scheme utilizes the invariance of the circles of the maps on these Poincaré sections in order to find the Fourier coefficients that define the circles to any given accuracy. A continuation procedure that uses the incremental behavior of the Fourier coefficients between close quasiperiodic orbits is utilized to extend the results from a single orbit to a family of orbits. Quasi-halo and Lissajous families of the Sun–Earth RTBP around the L2 libration point are obtained via this method. Results are compared with the existing literature. A numerical method to transform these orbits from the RTBP model to the real ephemeris model of the Solar System is introduced and applied.     

Fast Periodic Transfer Orbits in the Sun–Earth–Moon Quasi-Bicircular Problem

Starting from the identification and classification of a family of fast periodic transfer orbits in the Earth–Moon planar circular Restricted Three Body Problem (RTBP), and using analytic continuation techniques, we find two unstable periodic orbits in the Sun–Earth–Moon Quasi-Bicircular Problem (QBCP). The orbits found perform periodic Earth–Moon transfers with a period of approximately 29.5 days.     

Planar Symmetric Periodic Orbits in Four Dipole Problem

We have extend Stormer’s problem considering four magnetic dipoles in motion trying to justify the phenomena of extreme “orderlines” such as the ones observed in the rings of Saturn; the aim is to account the strength of the Lorentz forces estimating that the Lorentz field, co-acting with the gravity field of the planet, will limit the motion of all charged particles and small size grains with surface charges inside a layer of about 200 m thickness as that which is observed in the rings of Saturn. For this purpose our interest feast in the motion of charged particles with neglected mass where only electromagnetic forces accounted in comparison to the weakness of the Newtonian fields. This study is particularly difficult because in the regions we investigate these motions there is enormous three dimensional instability. Following the Poincare’s hypothesis that periodic solutions are ‘dense’ in the set of all solutions in Hamiltonian systems we try to calculate many families of periodic solutions and to study their stability. In this work we prove that in this environment charged particles can trace planar symmetric periodic orbits. We discuss these orbits in details and we give their symplectic relations using the Hamiltonian formulation which is related to the symplectic matrix. We apply numerical procedures to find families of these orbits and to study their stability. Moreover we give the bifurcations of these families with families of planar asymmetric periodic orbits and families of three dimensional symmetric periodic orbits.     

Three dimensional periodic orbits in three dipole problem

In the three dipole problem where enormous electromagnetic forces obstruct the three dimensional movement of the charged particle we determined for the first time families of three dimensional asymmetric periodic orbits. We study how these families appear, branching from the planar motion and we develop the procedures we have followed to determine them numerically. Also we give their characteristics and the conical projections and plottings of some orbits.     

Symmetric and asymmetric librations in extrasolar planetary systems: a global view

We present a global view of the resonant structure of the phase space of a planetary system with two planets, moving in the same plane, as obtained from the set of the families of periodic orbits. An important tool to understand the topology of the phase space is to determine the position and the stability character of the families of periodic orbits. The region of the phase space close to a stable periodic orbit corresponds to stable, quasi periodic librations. In these regions it is possible for an extrasolar planetary system to exist, or to be trapped following a migration process due to dissipative forces. The mean motion resonances are associated with periodic orbits in a rotating frame, which means that the relative configuration is repeated in space. We start the study with the family of symmetric periodic orbits with nearly circular orbits of the two planets. Along this family the ratio of the periods of the two planets varies, and passes through rational values, which correspond to resonances. At these resonant points we have bifurcations of families of resonant elliptic periodic orbits. There are three topologically different resonances: (1) the resonances (n + 1):n, (2:1, 3:2, ...), (2) the resonances (2n + 1):(2n-1), (3:1, 5:3, ...) and (3) all other resonances. The topology at each one of the above three types of resonances is studied, for different values of the sum and of the ratio of the planetary masses. Both symmetric and asymmetric resonant elliptic periodic orbits exist. In general, the symmetric elliptic families bifurcate from the circular family, and the asymmetric elliptic families bifurcate from the symmetric elliptic families. The results are compared with the position of some observed extrasolar planetary systems. In some cases (e.g., Gliese 876) the observed system lies, with a very good accuracy, on the stable part of a family of resonant periodic orbits.     

Motion close to the Hopf Bifurcation Of the vertical family of periodic orbits of L4

The paper deals with different kinds of invariant motions (periodic orbits, 2D and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze the Hamiltonian direct Hopf bifurcation that takes place close to the Lyapunov vertical family of periodic orbits of the triangular equilibrium point L₄ in the 3D restricted three-body problem (RTBP) for the mass parameter, μ greater than (and close to) μ_R (Routh’s mass parameter). Consequences of such bifurcation, concerning the confinement of the motion close to the hyperbolic orbits and the 3D nearby tori are also described.     

Periodic solutions around the collinear langrangian points in the photo gravitational restricted three-body problem: Sun-Jupiter case

The evolution of the periodic orbits around the collinear equilibrium positions, belonging to the Strömgren families a, b and c, with the radiation pressure parameter of the more massive body is studied in the Sun-Jupiter system. These families are determined for a single value of the radiation pressure parameter and particularly when the radiation force of the more massive body is equal to one half of the gravitational attraction. Then the critical stability orbits of each family are transferred with the radiation parameter. The stability of each periodic solution is also studied.     

Hamiltonian Formulation and Perturbations for Dust Motion Around Cometary Nuclei

In this paper we analyze the dynamical behavior of large dust grains in the vicinity of a cometary nucleus. To this end we consider the gravitational field of the irregularly shaped body, as well as its electric and magnetic fields. Without considering the effect of gas friction and solar radiation, we find that there exist grains which are static relative to the cometary nucleus; the positions of these grains are the stable equilibria. There also exist grains in the stable periodic orbits close to the cometary nucleus. The grains in the stable equilibria or the stable periodic orbits won’t escape or impact on the surface of the cometary nucleus. The results are applicable for large charge dusts with small area-mass ratio which are near the cometary nucleus and far from the Solar. It is found that the resonant periodic orbit can be stable, and there exist stable non-resonant periodic orbits, stable resonant periodic orbits and unstable resonant periodic orbits in the potential field of cometary nuclei. The comet gravity force, solar gravity force, electric force, magnetic force, solar radiation pressure, as well as the gas drag force are all considered to analyze the order of magnitude of these forces acting on the grains with different parameters. Let the distance of the dust grain relative to the mass centre of the cometary nucleus, the charge and the mass of the dust grain vary, respectively, fix other parameters, we calculated the strengths of different forces. The motion of the dust grain depends on the area-mass ratio, the charge, and the distance relative to the comet’s mass center. For a large dust grain (> 1 mm) close to the cometary nucleus which has a small value of area-mass ratio, the comet gravity is the largest force acting on the dust grain. For a small dust grain (< 1 mm) close to the cometary nucleus with large value of area-mass ratio, both the solar radiation pressure and the comet gravity are two major forces. If the a small dust grain which is close to the cometary nucleus have the large value of charge, the magnetic force, the solar radiation pressure, and the electric force are all major forces. When the large dust grain is far away from the cometary nucleus, the solar gravity and solar radiation pressure are both major forces.     

Multiple Periodic Orbits in the Hill Problem with Oblate Secondary

We study the multiple periodic orbits of Hill’s problem with oblate secondary. In particular, the network of families of double and triple symmetric periodic orbits is determined numerically for an arbitrary value of the oblateness coefficient of the secondary. The stability of the families is computed and critical orbits are determined. Attention is paid to the critical orbits at which families of non-symmetric periodic orbits bifurcate from the families of symmetric periodic orbits. Six such bifurcations are found, one for double-periodic and five for triple-periodic orbits. Critical orbits at which families of sub-multiple symmetric periodic orbits bifurcate are also discussed. Finally, we present the full network of families of multiple periodic orbits (up to multiplicity 12) together with the parts of the space of initial conditions corresponding to escape and collision orbits, obtaining a global view of the orbital behavior of this model problem.     

Regular and irregular periodic orbits

We distinguish between regular orbits, that bifurcate from the main families of periodic orbits (those that exist also in the unperturbed case) and irregular periodic orbits, that are independent of the above. The genuine irregular families cannot be made to join the regular families by changing some parameters. We present evidence that all irregular families appear inside lobes formed by the asymptotic curves of the unstable periodic orbits. We study in particular a dynamical system of two degrees of freedom, that is symmetric with respect to the x-axis, and has also a triple resonance in its unperturbed form. The distribution of the periodic orbits (points on a Poincaré surface of section) shows some conspicuous lines composed of points of different multiplicities. The regular periodic orbits along these lines belong to Farey trees. But there are also lines composed mainly of irregular orbits. These are images of the x-axis in the map defined on the Poincaré surface of section. Higher order iterations of this map , close to the unstable triple periodic orbit, produce lines that are close to the asymptotic curves of this unstable orbit. The homoclinic tangle, formed by these asymptotic curves, contains many regular orbits, that were generated by bifurcation from the central orbit, but were trapped inside the tangle as the perturbation increased. We found some stable periodic orbits inside the homoclinic tangle, both regular and irregular. This proves that the homoclinic tangle is not completely chaotic, but contains gaps (islands of stability) filled with KAM curves.     

The manifolds of families of 3D periodic orbits associated to Sitnikov motions in the restricted three-body problem

This paper deals with the Sitnikov family of straight-line motions of the circular restricted three-body problem, viewed as generator of families of three-dimensional periodic orbits. We study the linear stability of the family, determine several new critical orbits at which families of three dimensional periodic orbits of the same or double period bifurcate and present an extensive numerical exploration of the bifurcating families. In the case of the same period bifurcations, 44 families are determined. All these families are computed for equal as well as for nearly equal primaries (μ = 0.5, μ = 0.4995). Some of the bifurcating families are determined for all values of the mass parameter μ for which they exist. Examples of families of three dimensional periodic orbits bifurcating from the Sitnikov family at double period bifurcations are also given. These are the only families of three-dimensional periodic orbits presented in the paper which do not terminate with coplanar orbits and some of them contain stable parts. By contrast, all families bifurcating at single-period bifurcations consist entirely of unstable orbits and terminate with coplanar orbits.     

Families of symmetric relative periodic orbits originating from the circular Euler solution in the isosceles three-body problem

We study symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches. We first prove that another family of symmetric relative periodic orbits is born from the circular Euler solution besides the elliptic Euler solutions. Previous studies also showed that there exist infinitely many families of symmetric relative periodic orbits which are born from heteroclinic connections between triple collisions as well as planar periodic orbits with binary collisions. We carry out numerical continuation analyses of symmetric relative periodic orbits, and observe abundant families of symmetric relative periodic orbits bifurcating from the two families born from the circular Euler solution. As the angular momentum tends to zero, many of the numerically observed families converge to heteroclinic connections between triple collisions or planar periodic orbits with binary collisions described in the previous results, while some of them converge to “previously unknown” periodic orbits in the planar problem.     

On the application of optimization methods to the determination of members of families of periodic solutions

The techniques used for the numerical computation of families of periodic orbits of dynamical systems rely on predictor-corrector algorithms. These algorithms usually depend on the solution of systems of approximate equations constructed from the periodicity conditions of these orbits. In this contribution we transform the root finding procedure to an optimization one which is applied on an objective function based on the exact periodicity conditions. Thus, the determination of periodic solutions and families of such orbits can be accomplished through unconstrained optimization. In this paper we apply and compare some well-known minimization methods for the solution of this problem. The obtained results are promising. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Self-resonant bifurcations of the Sitnikov family and the appearance of 3D isolas in the restricted three-body problem

The association of the Sitnikov family with families of multiple three-dimensional periodic orbits is studied. In particular, the families consisting of three-dimensional periodic orbits bifurcating from self-resonant orbits of the Sitnikov family at double, triple and quadruple period of the bifurcation orbit are considered. The branch families close upon themselves and remain 3D up to their terminations having two common members with the Sitnikov family. By varying the mass parameter we also study the evolution of some of the computed families and find that they become isolas and disappear gradually in three-dimensions by shrinking to point size.     

Vertical instability and inclination excitation during planetary migration

We consider a two-planet system migrating under the influence of dissipative forces that mimic the effects of gas-driven (Type II) migration. It has been shown that, in the planar case, migration leads to resonant capture after an evolution that forces the system to follow families of periodic orbits. Starting with planets that differ slightly from a coplanar configuration, capture can, also, occur and, additionally, excitation of planetary inclinations has been observed in some cases. We show that excitation of inclinations occurs, when the planar families of periodic orbits, which are followed during the initial stages of planetary migration, become vertically unstable. At these points, vertical critical orbits may give rise to generating stable families of \(3D\) periodic orbits, which drive the evolution of the migrating planets to non-coplanar motion. We have computed and present here the vertical critical orbits of the \(2/1\) and \(3/1\) resonances, for various values of the planetary mass ratio. Moreover, we determine the limiting values of eccentricity for which the “inclination resonance” occurs.     

The Structure of Periodic Solutions in the Restricted Problem of the Three Bodies II

In this work we consider four families of plane periodic orbits direct around the Sun which approach Jupiter but they are sufficiently far from it so as to be considered as predominantly two body orbits of the Sun-asteroid system. We study their horizontal and vertical stabilities and we give the exact orbits of bifurcations of these families with three-dimensional families of the same multiplicity or twice the multiplicity of the above families of plane symmetric periodic orbits. Moreover, we give the first segments of the three dimensional families of symmetric periodic orbits which emanate from these plane bifurcations and we study their stability relating it with the stability of the plane bifurcations.     

On the families of periodic orbits which bifurcate from the circular Sitnikov motions

In this paper we deal with the circular Sitnikov problem as a subsystem of the three-dimensional circular restricted three-body problem. It has a first analytical part where by using elliptic functions we give the analytical expressions for the solutions of the circular Sitnikov problem and for the period function of its family of periodic orbits. We also analyze the qualitative and quantitative behavior of the period function. In the second numerical part, we study the linear stability of the family of periodic orbits of the Sitnikov problem, and of the families of periodic orbits of the three-dimensional circular restricted three-body problem which bifurcate from them; and we follow these bifurcated families until they end in families of periodic orbits of the planar circular restricted three-body problem. We compare our results with the previous ones of other authors on this problem. Finally, the characteristic curves of some bifurcated families obtained for the mass parameter close to 1/2 are also described.