Symmetry-breaking for a restricted -body problem in the Maxwell-ring configuration
We investigate the motion of a massless body interacting with the Maxwell relative equilibrium, which consists of bodies of equal mass at the vertices of a regular polygon that rotates around a central mass. The massless body has three equilibrium -orbits from which families of Lyapunov orbits emerge. Numerical continuation of these families using a boundary value formulation is used to construct the bifurcation diagram for the case , also including some secondary and tertiary bifurcating families. We observe symmetry-breaking bifurcations in this system, as well as certain period-doubling bifurcations.
In his 1859 essay Ma () Maxwell proposed a model to study the rings of Saturn. His model consists of bodies of equal mass at the vertices of a regular polygon that rotates around a massive body at the center. Maxwell used Fourier analysis and dispersion relations in the determination of the stability of the ring. The Maxwell equilibrium has been studied in several papers since then. In particular, Moeckel proved in Moekel92 () that the equilibrium is stable if and the body at the center massive enough. See also GaIz13 (); VaKo07 (); Ro00 () and references therein.
In this paper we consider the motion of a satellite under the gravitational effect of the Maxwell equilibrium. Several papers have been devoted to study the stability and bifurcation of periodic solutions for the restricted -body problem in the Maxwell configuration. For example, a study of the existence and linear stability of equilibrium positions can be found in BaEl04 (), an analysis of the bifurcation of planar and vertical families of periodic solutions in GaIz10 (), and a numerical exploration in Ka08 ().
Given that a change of stability occurs at when , we present a numerical exploration of the motion of the satellite for this stable system, taking . We follow the planar and vertical Lyapunov families that were proved to exist in GaIz10 (), and we compute new, secondary families that bifurcate from the Lyapunov families. These numerical results allow us to construct a bifurcation diagram in Figure 2 that shows primary, secondary, and tertiary bifurcating families. The results also allow us to present the isotropy lattice in Figure 3 which shows the symmetries of these families.
The numerical computations in this paper are done using continuation methods and boundary value techniques for determining the periodic orbits that emanate from the equilibrium orbits. Python scripts that make the AUTO software perform the calculations reported in this paper will be made freely available. Similar techniques have been applied to the restricted -body problem; see for example DoVa (), where a detailed bifurcation diagram with various families of periodic orbits can be found.
This paper is organized as follows. In Section 1 we recall some key results from the literature concerning the equilibria and the Lyapunov orbits of the problem. In Section 2 we present a bifurcation diagram and an isotropy lattice for the restricted -body problem in the Maxwell configuration, with and . In Section 3 we describe the isotropy groups of the Lyapunov families. In Section 4 we address secondary bifurcations, and in Section 5 describe some tertiary families. We also present evidence of the existence of invariant tori foliated by periodic orbits. Finally, in Section 6, we consider the breaking of symmetries of planar interplanetary periodic orbits.
1 The restricted -body problem
The Maxwell relative equilibrium consists of a body of mass at , and bodies of mass located at , , where . These positions and masses correspond to a relative equilibrium solution of Newton’s equations, , when the masses are renormalized by BaEl04 (); GaIz10 (), where ,
The equation of a satellite in rotating coordinates, , is
The first term of the potential in Equation (2) corresponds to the centrifugal force, the second term is the interaction with the mass , and the third term models the interaction with the primaries of mass ; see BaEl04 () and GaIz10 ().
The equilibria of equation (1) are critical points of , defined in Equation (2). Moreover, due to the particular form of the Maxwell configuration, the potential is -invariant. In this respect the existence of three -orbits of equilibria for any value of is proved in BaEl04 (); GaIz10 (). For small , two additional -orbits of equilibria appear close to the origin GaIz10 (); see Figure 1.
In GaIz10 (), taking advantage of the symmetries of the equations, the authors analyze the bifurcation of periodic solutions from the -orbits of equilibria. Here we state these results for the equilibria , and of the three -orbits:
[Ize & García-Azpeitia GaIz10 ().] The libration point , has one global bifurcation family of planar periodic solutions that will be denoted by , and one global family of vertical solutions, denoted by . Similarly, the libration point has one global family of planar periodic solutions, denoted , and one family of vertical solutions, . For , the equilibrium has two global bifurcations of planar solutions, one of which has longer period, denoted , and another planar family of shorter period, . There is also a bifurcating family of vertical periodic solutions that will be denoted by . Moreover, as a consequence of the symmetries, the shape of all vertical solutions close to the equilibrium resembles a spatial figure eight.
The global property guarantees that the family is a continuum that either goes to infinity in Sobolev norm or period, ends in a collision, or ends at a bifurcation point. Indeed, each one of these possibilities appears in the numerical continuation of the families, as illustrated in the bifurcation diagram in Figure 2.
2 Breaking of symmetries
In this section we discuss the breaking of symmetries of Equation (1) for the case and , as observed in the numerically computed Lyapunov families that emerge from the libration points and the secondary families that bifurcate from them.
The -periodic solutions of Equation (1) are zeros of the map
defined in a set of -periodic collisionless functions ; see GaIz10 (). Since the potential is -invariant and the equations are autonomous, the map is equivariant under the action of given by
where . In addition the equations are symmetric with respect to reflection of about the plane, while reversing time, and with respect to reflection of about the plane. In this regard we define the reflections and by
where and . Therefore the map is equivariant under the full symmetry group
We will use the property that the group orbit of a function ,
is isomorphic to , where is the isotropy group defined as
For example, the libration equilibria , for , have group orbits and isotropy group
We therefore present the breaking of symmetries only for the libration points , .
The bifurcation diagram for the case is given in Figure 2. The blue lines represent the vertical Lyapunov families and the planar Lyapunov families , . Planar families are positioned in the plane of the bodies, as are the black lines that represent planar interplanetary orbits. The red lines are the result of a secondary symmetry-breaking, with two solutions per equilibrium, as for the Halo orbits and the Axial orbits , . The green lines correspond to a tertiary symmetry-breaking, and as such they have a trivial isotropy group and four symmetry-related branches per libration point.
: Some of the families in the bifurcation diagram that end at a tetrahedron, in fact terminate as a heteroclinic orbit. For the restricted three-body problem similar heteroclinic orbits are given in CaDo12 (); LoMa00 (), and the existence of heteroclinic connections is proved in LiMa (). In future work we will present many other heteroclinic connections that we have located by continuation of orbits in stable/unstable manifolds. Furthermore, in future work we will present evidence of families of planar orbits that interconnect the planar families and , as in Henrard (). All results are accompanied by scripts that allow their reproduction.
The breaking of symmetries in the bifurcation diagram gives rise to the lattice of isotropy groups of bifurcating orbits (isotropy lattice) in Figure 3.
3 Lyapunov families
The first bifurcation occurs when the -symmetry is broken, giving rise to the Lyapunov families (the blue lines in Figure 2) from the equilibria , . The number of symmetry-related Lyapunov branches is equal to the order of , which is , since each isotropy group is isomorphic to .
Hereafter denotes that is a subgroup of . The isotropy group of the planar Lyapunov orbits that emerge from the libration equilibria is
These periodic solutions have the property that is even, is odd, and . In particular, we observe that the planar orbits are invariant under the transformation that takes to .
For the vertical Lyapunov orbits the isotropy group is
Here a solution is fixed by if it satisfies
which is equivalent to assuming that is a -periodic even function, is a -periodic odd function, and . Therefore these solutions follow the planar -periodic curve twice; one time with the spatial coordinate and a second time with . This fact was proved in GaIz10 ().
4 Secondary families
The bifurcations from the Lyapunov families coincide with the second breaking of symmetries. The families that emanate from such bifurcation points (the red lines in Figure 2) have three kinds of isotropy groups, each one isomorphic to . The number of symmetry-related branches is equal to . Therefore, there are two such branches per equilibrium.
There is a symmetry-breaking from the planar families to solutions with isotropy group
These solutions have vertical component . In this case the -symmetry is broken, so these planar orbits are asymmetric with respect to the transformation that takes to . Such solutions are observed for the third bifurcating family along and the fourth family that bifurcates from ; see Figure 4.
The Halo families and bifurcate from the Lyapunov families and , respectively. Each of these has isotropy group
i.e., their solutions have the property that is even and is odd. Thus these spatial orbits are invariant under the transformation that takes to .
The Axial families , and similarly families , bifurcate from , for . Here the symmetry-breaking is from the group to the group
This means that the Axial solutions satisfy or, setting , that
Then is even, and and are odd. Therefore these spatial orbits are invariant under the transformation that takes to .
5 Tertiary families
Tertiary symmetry-broken families (the green lines in Figure 2) correspond to families that bifurcate from solutions with isotropy groups (6), (7), and (8). Since these groups are isomorphic to , the tertiary symmetry-broken solutions have the trivial isotropy group. Therefore, the number of symmetry-related branches of the tertiary families is , i.e. four per equilibrium. In particular, the symmetry-breaking bifurcations along the families and give rise to families that generate invariant surfaces.
This is the case of the surface generated by the family in Figure 5 that reconnects to after a complete loop around the central body, but to an orbit that is symmetric to the original one. Following this surface, i.e., its orbits, for a second loop around the central body, we obtain a double surface that interconnects the pairs of Axial bifurcation orbits that emanate from each of the libration points symmetry-related to . The surface generated by the orbits of the family contains an additional bifurcation orbits that connect to Halo families . Consequently there is a continuous path in the bifurcation diagram between any of the symmetry-related libration points and (see the bifurcation diagram Figure 2).
The surface generated by the orbits of the family in Figure 5 is similar to that of that contains a set of bifurcation orbits along the Axial families . These bifurcation orbits are distinct from the bifurcation orbits along that are interconnected via the family . The family has extra bifurcation orbits that connect to Halo-like families that we do not describe here.
Several families bifurcate from the family with trivial isotropy group, one of which is illustrated in the bottom right panel of Figure 6. This family is similar to and in that it connects one Axial family, with isotropy group , to one Halo-like family, with isotropy group . In future work we will report other families bifurcating from that connect two Axial-like families with isotropy group .
6 Interplanetary orbits
Several families connect to planar families that enclose more than one body and that do not correspond to a planar Lyapunov family. Such families are referred to as interplanetary in Ka08 (), and indicated by black lines in the bifurcation diagram of Figure 2.
The family has the symmetry group (5) and connects to a planar family in Figure 7 via a reverse period-doubling bifurcation, i.e., the vertical family arises from the planar family via a period-doubling bifurcation. Indeed, the family bifurcates from the interplanetary family with isotropy group
These planar solutions are -periodic and their orbits are invariant under the transformation that takes to . This is consistent with the symmetry-breaking phenomenon, since the isotropy group (5) is contained in the isotropy group of the interplanetary orbits (9).
The Lyapunov families and also connect to planar interplanetary families, and , respectively, via a period-doubling bifurcation. Actually, and correspond to the same family, i.e., , and moreover, both and bifurcate from exactly the same orbit along , via a degenerate period-doubling bifurcation with two Floquet multipliers at .
Another family that ends in an interplanetary family is ; see Figure 7. Here the isotropy group of the planar family is
These solutions have the property that is even, is odd, and , so that the orbits are planar and symmetric with respect to the transformation that takes to . Therefore the Axial family , with group , corresponds to a symmetry-breaking from the interplanetary family . Acknowledgements. We thank Ramiro Chavez Tovar for his assistance in producing the bifurcation diagram.
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