Infinite Feigenbaum sequences and spirals in the vicinity of the Lagrangian periodic solutions

We studied systematically cases of the families of non-symmetric periodic orbits in the planar restricted three-body problem. We took interesting information about the evolution, stability and termination of bifurcating families of various multiplicities. We found that the main families of simple non-symmetric periodic orbits present a similar dynamical structure and bifurcation pattern. As the Jacobi constant changes each branch of the characteristic of a main family spirals around a focal point-terminating point in x- at which the Jacobi constant is C _∞ = 3 and their periodic orbits terminate at the corotation (at the Lagrangian point L₄ or L₅). As the family approaches asymptotically its termination point infinite changes of stability to instability and vice versa occur along its characteristic. Thus, infinite bifurcation points appear and each one of them produces infinite inverse Feigenbaum sequences. That is, every bifurcating family of a Feigenbaum sequence produces the same phenomenon and so on. Therefore, infinite spiral characteristics appear and each one of them generates infinite new inner spirals and so on. Each member of these infinite sets of the spirals reproduces a basic bifurcation pattern. Therefore, we have in general large unstable regions that generate large chaotic regions near the corotation points L₄, L₅, which are unstable. As C varies along the spiral characteristic of every bifurcating family, which approaches its focal point, infinite loops, one inside the other, surrounding the unstable triangular points L₄ or L₅ are formed on their orbits. So, each terminating point corresponds to an asymptotic non-symmetric periodic orbit that spirals into the corotation points L₄, L₅ with infinite period. This is a new mechanism that produces very large degree of stochasticity. These conclusions help us to comprehend better the motions around the points L₄ and L₅ of Lagrange.     

The genealogy of periodic orbits in a plane rotating galaxy

We study the various families of periodic orbits in a dynamical system representing a plane rotating barred galaxy. One can have a general view of the main resonant types of orbits by considering the axisymmetric background. The introduction of a bar perturbation produces infinite gaps along the central familyx ₁ (the family of circular orbits in the axisymmetric case). It produces also higher order bifurcations, unstable regions along the familyx ₁, and long period orbits aroundL ₄ andL ₅. The evolution of the various types of orbits is described, as the Jacobi constanth, and the bar amplitude, increase. Of special importance are the infinities of period doubling pitchfork bifurcations. The genealogy of the long and short period orbits is described in detail. There are infinite gaps along the long period orbits producing an infinity of families. All of them bifurcate from the short period family. The rules followed by these families are described. Also an infinity of higher order bridges join the short and long period families. The analogies with the restricted three body problem are stressed.     

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.     

Periodic orbits and bifurcations in the Sitnikov four-body problem

We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the “family parameter” ż₀ varies within a finite interval (while z ₀ tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the z-axis along its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincaré maps, we chart the motion near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of ordered motion surrounded by “sticky” and chaotic orbits as well as orbits which rapidly escape to infinity.     

Families of periodic collision orbits in the general three-body problem

In the general three-body problem, in a rotating frame of reference, a symmetric periodic solution with a binary collision is determined by the abscissa of one body and the energy of the system. For different values of the masses of the three bodies, the symmetric periodic collision orbits form a two-parametric family. In the case of equal masses of the two bodies and small mass of the third body, we found several symmetric periodic collision orbits similar to the corresponding orbits in the restricted three-body problem. Starting with one symmetric periodic collision orbit we obtained two families of such orbits. Also starting with one collision orbit in the Sun-Jupiter-Saturn system we obtained, for a constant value of the mass ratio of two bodies, a family of symmetric periodic collision orbits.     

Numerical investigation of the manifold I qp in the Sun-Jupiter case of the restricted three-body problem

Message and Taylor (1978) have given values of the mean eccentricities and commen-surabilities which correspond to bifurcation orbits of families of symmetric periodic orbits with families of asymmetric periodic orbits in the limit as the mass ratio tends to zero. These bifurcations have been given in a way that they seem to be isolated and unrelated from the whole structure of the periodic orbits of the system.In this paper a numerical investigation of the horizontal stability of the family I and its branches reveals the above bifurcations orbits in the Sun-Jupiter case of the restricted three-body problem and associates these orbits with the whole structure of the system, giving extensive information on them.     

Periodic motions in the spatial Chermnykh restricted three-body problem

Three-dimensional motions in the Chermnykh restricted three-body problem are studied. Specifically, families of three-dimensional periodic orbits are determined through bifurcations of the family of straight-line periodic oscillations of the problem which exists for equal masses of the primaries. These rectilinear oscillations are perpendicular to the plane of the primaries and give rise to an infinite number of families consisting entirely of periodic orbits which belong to the three-dimensional space except their respective one-dimensional bifurcations as well as their planar terminations. Many of the computed branch families are continued in all mass range that they exist.     

Asymmetric Periodic Solutions in the Restricted Problem of Three Bodies

A systematic numerical exploration of the families of asymmetric periodic orbits of the restricted three-body problem when a) the primary bodies are equal and b) for the Earth-Moon mass ratio, is presented. Decades families of asymmetric periodic solutions were found and three of the simplest ones, in the first case, and ten of the second one are illustrated. All of these families consist of periodic orbits which are asymmetric with respect to x-axis while are simple symmetric periodic orbits with respect to y-axis (i.e. the orbit has only one perpendicular intersection at half period with y-axis). Many asymmetric periodic orbits, members of these families, are calculated and plotted. We studied the stability of all the asymmetric periodic orbits we found. These families consist, mainly, of unstable periodic solutions but there exist very small, with respect to x, intervals where these families have stable periodic orbits. We also found, using appropriate Poincaré surface of sections, that a relatively large region of phase space extended around all these stable asymmetric periodic orbits shows chaotic motion.     

Families of Asymmetric Periodic Orbits in the Restricted Three-body Problem

This paper studies the asymmetric solutions of the restricted planar problem of three bodies, two of which are finite, moving in circular orbits around their center of masses, while the third is infinitesimal. We explore, numerically, the families of asymmetric simple-periodic orbits which bifurcate from the basic families of symmetric periodic solutions f, g, h, i, l and m, as well as the asymmetric ones associated with the families c, a and b which emanate from the collinear equilibrium points L ₁, L ₂ and L ₃ correspondingly. The evolution of these asymmetric families covering the entire range of the mass parameter of the problem is presented. We found that some symmetric families have only one bifurcating asymmetric family, others have infinity number of asymmetric families associated with them and others have not branching asymmetric families at all, as the mass parameter varies. The network of the symmetric families and the branching asymmetric families from them when the primaries are equal, when the left primary body is three times bigger than the right one and for the Earth–Moon case, is presented. Minimum and maximum values of the mass parameter of the series of critical symmetric periodic orbits are given. In order to avoid the singularity due to binary collisions between the third body and one of the primaries, we regularize the equations of motion of the problem using the Levi-Civita transformations.     

Non-schubart periodic orbits in the rectilinear three-body problem

In the present paper, in the rectilinear three-body problem, we qualitatively follow the positions of non-Schubart periodic orbits as the mass parameter changes. This is done by constructing their characteristic curves. In order to construct characteristic curves, we assume a set of properties on the shape of areas corresponding to symbol sequences. These properties are assured by our preceding numerical calculations. The main result is that characteristic curves always start at triple collision and end at triple collision. This may give us some insight into the nature of periodic orbits in the N-body problem.     

Families of periodic orbits in the planar three-body problem

A periodic orbit of the restricted circular three-body problem, selected arbitrarily, is used to generate a family of periodic motions in the general three-body problem in a rotating frame of reference, by varying the massm ₃ of the third body. This family is continued numerically up to a maximum value of the mass of the originally small body, which corresponds to a mass ratiom ₁:m ₂:m ₃?5:5:3. From that point on the family continues for decreasing massesm ₃ until this mass becomes again equal to zero. It turns out that this final orbit of the family is a periodic orbit of the elliptic restricted three body problem. These results indicate clearly that families of periodic motions of the three-body problem exist for fixed values of the three masses, since this continuation can be applied to all members of a family of periodic orbits of the restricted three-body problem. It is also indicated that the periodic orbits of the circular restricted problem can be linked with the periodic orbits of the elliptic three-body problem through periodic orbits of the general three-body problem.     

Three-dimensional periodic oscillations generating from plane periodic ones around the collinear Lagrangian points

The three families of three-dimensional periodic oscillations which include the infinitesimal periodic oscillations about the Lagrangian equilibrium pointsL ₁,L ₂ andL ₃ are computed for the value =0.00095 (Sun-Jupiter case) of the mass parameter. From the first two vertically critical (|a _v |=1) members of the familiesa, b andc, six families of periodic orbits in three dimensions are found to bifurcate. These families are presented here together with their stability characteristics. The orbits of the nine families computed are of all types of symmetryA, B andC. Finally, examples of bifurcations between families of three-dimensional periodic solutions of different type of symmetry are given.     

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.     

Characteristics of resonant periodic orbits

We study the families of periodic orbits in a time-independent two-dimensional potential field symmetric with respect to both axes. By numerical calculations we find characteristic curves of several families of periodic orbits when the ratio of the unperturbed frequencies isA ^1/2/B ^1/2=2/1. There are two groups of characteristic curves: (a) The basic characteristic and the characteristics which bifurcate from it. (b) The characteristics which start from the boundary line and the axisx=0.     

Periodic Motions and their Stability in a N = ν + 1-body Regular Polygonal Configuration

We present a systematic investigation of the parametric evolution of both retrograde and direct families of periodic motions as well as their stability in the inner region of the peripheral primaries of the planar N-body regular polygonal configuration (ring model). In particular, we study the change of the bifurcation points as well as the change of the size and dynamical structure of the rings of stability for different values of the parameters ν = N?1 (number of peripheral primaries) and β (mass ratio). We find some types of bifurcations of families of periodic motions, namely period doubling pitchfork bifurcations, as well as bifurcations of symmetric and non-symmetric periodic orbits of the same period. For a given value of N ? 1, the intervals Δx and ΔC of the rings of stability (where the periodic orbits are stable) of both retrograde and direct families increase with β increasing, while for a given value of β, the interval ΔC decreases with increasing N ? 1. In general, it seems that the dynamical properties of the system depend on the ratio (N ? 1)/β. The size of each ring of stability tends to zero as the ratio (N ? 1)/β → ∞, that is, if N ? 1→∞ or β → 0, the size of each ring of stability tends to zero (Δx → 0 and ΔC → 0) and, in general, the retrograde and direct families tend to disappear. This study gives us interesting information about the evolution of these two families and the changes of the bifurcation patterns since, for example, in some cases the stability index A oscillates between ?1 ≤ Α ≤ + 1. Each time the family becomes critically stable a new dynamical structure appears. The ratios of the Jacobian constant C between the successive critical points, C _i /C _i+1, tend to 1. All the above depend on the parameters N ? 1, β and show changes in the topology of the phase space and in the dynamical properties of the system.     

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.     

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.     

Families of asymmetric periodic solutions of the restricted problem of three bodies for the Sun-Jupiter mass ratio and their relationship with the symmetric families

Message derived a method to detect bifurcations of a family of asymmetric periodic solutions from a family of symmetric periodic solutions in the restricted problem of three bodies for the limiting case when the second body has zero mass. This is used to examine several small integer commensurabilities. A total of 21 exterior and 21 interior small integer commensurabilities are examined and bifurcations (two in number) are found to exist only for exterior commensurabilities (q+1):1,q=1, 2,, 7. On investigating other commensurabilities of this form for values ofq up to 50 two bifurcations are still found to exist for each. The eccentricities of the two bifurcation orbits are given for eachq up to 20. For a Sun-Jupiter mass ratio the complete family of asymmetric periodic solutions associated withq=1, 2,..., 5, and the initial segments of the asymmetric family withq=6, 7,..., 12, have been numerically determined. The family associated withq=5 contains some unstable orbits but all orbits in the other four complete families are stable. The five complete families each begin and end on the same symmetric family. The network of asymmetric and symmetric families close to the commensurabilities (q+1):1,q=1, 2,..., 5 is discussed.     

Bifurcations of triple-periodic orbits

We consider families of periodic orbits in potentials symmetric with respect to thex-axis. The characteristics of triple-periodic orbits (i.e. orbits intersecting thex-axis three times) that bifurcate from the central characteristic do not have their maximum or minimum energy (or perturbation) at the point of intersection. We explain theoretically that this happens only for triple-periodic orbits and not for any other type of resonant periodic orbits and verify this fact by numerical calculations.     

Periodic orbits in the restricted four-body problem with two equal masses

The restricted (equilateral) four-body problem consists of three bodies of masses m ₁, m ₂ and m ₃ (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh’s critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them.