Abstract
The paper deals with the study of a satellite attracted by n primary bodies, which form a relative equilibrium. We use orthogonal degree to prove global bifurcation of planar and spatial periodic solutions from the equilibria of the satellite. In particular, we analyze the restricted three body problem and the problem of a satellite attracted by the Maxwell’s ring relative equilibrium.
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Arribas M., Elipe A.: Bifurcations and equilibria in the extended n-body ring problem. Mech. Res. Comm. 31(1), 1–8 (2004)
Bang D., Elmabsout B.: Representations of complex functions, means on the regular n-gon and applications to gravitational potential. J. Phys. 36(45), 11435–11450 (2003)
Bang D., Elmabsout B.: Restricted n + 1-body problem: existence and stability of relative equilibria. Celest. Mech. Dyn. Astron. 89(4), 305–318 (2004)
Bardin B.: On motions near the Lagrange equilibrium point L 4 in the case of Routh’s critical mass ratio. Celest. Mech. Dyn. Astron. 82, 163–177 (2002)
Barrio R., Blesa F., Serrano S.: Qualitative analysis of the (N + 1)-body ring problem. Chaos, Solitons and Fractals 36, 1067–1088 (2008)
Doedel E.J., Romanov V.A., Paffenroth R.C., Keller H.B., Dichmann D.J., Galán-Vioque J., Vanderbauwhede A.: Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem. Bifurcation and Chaos 17(8), 2625–2677 (2007)
Efthymiopoulos C.: Formal integrals and Nekhoroshev stability in a mapping model for the Trojan asteroids. Celest. Mech. Dyn. Astron. 92, 29–52 (2005)
Érdi B., Forgács-Dajka E., Nagy I., Rajnai R.: A parametric study of stability and resonances around L 4 in the elliptic restricted three-body problem. Celest. Mech. Dyn. Astron. 104, 145–158 (2009)
García-Azpeitia, C.: Aplicación del grado ortogonal a la bifurcación en sistemas hamiltonianos. UNAM. PhD thesis (2010)
García-Azpeitia, C., Ize, J.: Global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for the n-body problem. Submitted, 2011 (2011)
Gómez G., Simó C., Llibre J., Martinez R.: Dynamics and Mission Design near Libration Points, vol. 1–4. World Scientific Monograph Series in Mathematics. World Scientific, Singapore (2000)
Ize J.: Topological bifurcation. Prog. Nonlinear Differ. Equ. their Appl. 15, 341–463 (1995)
Ize, J., Vignoli, A.: Equivariant degree theory. De Gruyter Series in Nonlinear Analysis and Applications 8, Walter de Gruyter (2003)
Kalvouridis T.J.: Particle motions in Maxwell’s ring dynamical systems. Celest. Mech. Dyn. Astron. 102(1–3), 191–206 (2008)
Kasdin N., Gurfil P., Kolemen E.: Canonical modelling of relative spacecraft motion via epicyclic orbital elements. Celest. Mech. Dyn. Astron. 92(4), 337–370 (2005)
Llibre J., Stoica C.: Comet and Hill-type periodic orbits in restricted (N + 1)-body problems. J. Differ. Equ. 250, 1747–1766 (2011)
Maciejewski A., Rybicki S.: Global bifurcations of periodic solutions of the restricted three body problem. Celest. Mech. Dyn. Astron. 88, 293–324 (2004)
Marchal C.: The Three-Body Problem, Studies in Astronautics vol. 4. Elsevier, Amsterdam (1990)
Mavraganis A.G., Kalvouridis T.J.: A proper choice of variables for the study of satellite’s close approaches in a ring assembly of N massive bodies. Planet. Space Sci. 55, 401–406 (2007)
Meyer K.: Periodic Solutions of the N-Body Problem (Lecture Notes in Mathematics vol. 1719). Springer, Berlin (1999)
Meyer K., Hall G.R.: An Introduction to Hamiltonian Dynamical Systems. Springer, Berlin (1991)
Pinotsis A.D.: Evolution and stability of the theoretically predicted families of periodic orbits in the N-body ring problem. Astron. Astrophys. 432, 713–729 (2005)
Pinotsis A.D.: Instabilities and bifurcations of the families of collision periodic orbits in the restricted three-body problem. Planet. Space Sci. 55(4), 401–406 (2009)
Roberts, G.E.: Linear stability in the 1 + n-gon relative equilibrium. In: Delgado, J. (ed.) Hamiltonian Systems and Celestial Mechanics. HAMSYS-98. Proceedings of the 3rd International Symposium, World Sci. Monogr. Ser. Math. vol. 6, pp. 303–330. World Scientific, 2000 (2000)
Sandor Z., Érdi B., Efthymiopoulos B., Efthymiopoulos C.: The phase space structure around L 4 in the restricted three-body problem. Celest. Mech. Dyn. Astron. 78, 113–123 (2000)
Sicardy B.: Stability of the triangular Lagrange points beyong Gaschau’s value. Celest. Mech. Dyn. Astron. 107, 145–155 (2010)
Siegel C., Moser J.: Lectures on Celestial Mechanics. Springer, Berlin (1971)
Vanderbei R.J., Kolemen E.: Linear stability of ring systems. The Astron. J. 133, 656–664 (2007)
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García-Azpeitia, C., Ize, J. Global bifurcation of planar and spatial periodic solutions in the restricted n-body problem. Celest Mech Dyn Astr 110, 217–237 (2011). https://doi.org/10.1007/s10569-011-9354-2
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DOI: https://doi.org/10.1007/s10569-011-9354-2