Hamiltonian–Hopf Bifurcation and KAM Tori for the Restricted Four-Body Problem

• Xuemei Li ^[1] ; Qiyao Mai ^[1]
  1. [1] Hunan Normal University

     Hunan Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 4, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper we investigate the planar circular restricted (3 + 1)-body problem. A massless particle moves under the Newtonian gravitational influence of three primaries.

    These primaries revolve in circular orbits around their common center of mass within the same plane, forming a collinear configuration. We establish the existence and nonlinear stability of equilibrium points under the condition that two of the primaries have equal masses, denoted by μ. Through analytical and numerical methods, we identify a critical mass parameterμ1 ≈ 0.0743 where aHamiltonian–Hopf bifurcation occurs at non-collinear equilibrium points. Furthermore, we prove the existence of Hill-type periodic orbits and KAM tori surrounding these periodic orbits in this circular restricted four-body problem.

• Referencias bibliográficas
  • 1. Albouy, A., Kaloshin, V.: Finiteness of central configurations of five bodies in the plane. Ann. Math. 175(2), 535–588 (2012)
  • 2. Alrebdi, H.I., Dubeibe, F.L., Zotos, E.E.: Equilibrium points and their stability in a new generalized planar version of the collinear...
  • 3. Arenstorf, R.F.: Central configurations of four bodies with one inferior mass. Celest. Mech. 28(1–2), 9–15 (1982)
  • 4. Baltagiannis, A.N., Papadakis, K.E.: Families of periodic orbits in the restricted four-body problem. Astrophys. Space Sci. 336(2), 357–367...
  • 5. Bardin, B.S., Volkov, E.V.: Stability study of a relative equilibrium in the planar circular restricted fourbody problem. In: IOP Conference...
  • 6. Bardin, B.S., Volkov, E.V.: The Lyapunov stability of central configurations of the planar circular restricted four-body problem. Cosm....
  • 7. Barros, J.F., Leandro, E.S.: The set of degenerate central configurations in the planar restricted fourbody problem. SIAM J. Math. Anal....
  • 8. Barros, J.F., Leandro, E.S.: Bifurcations and enumeration of classes of relative equilibria in the planar restricted four-body problem....
  • 9. Bottke, W.F., Marschall, R., Nesvorný, D., Vokrouhlický, D.: Origin and evolution of Jupiter’s trojan asteroids. Space Sci. Rev. 219(8),...
  • 10. Budzko, D.A.: Linear stability analysis of equilibrium solutions of restricted planar four-body problem. In: Computer Algebra Systems...
  • 11. Burgoyne, N., Cushman, R.: Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues. Celest. Mech. 8(4), 435–443...
  • 12. Cushman, R., Kelley, A., Koçak, H.: Versal normal form at the Lagrange equilibrium L4. J. Differential Equ. 64(3), 340–374 (1986)
  • 13. Doedel, E.J., Romanov, V.A., Paffenroth, R.C., et al.: Elemental periodic orbits associated with the libration points in the circular...
  • 14. Emery, J.P., Binzel, R.P., Britt, D.T., Brown, M.E., Howett, C.J., Martin, A.C., Wong, I.: Surface compositions of trojan asteroids. Space...
  • 15. Euler, L.: De motu rectilineo trium corporum se mutuo attrahentium. Novi Comment. Acad/. Sci. Petropol. 11, 144–151 (1767)
  • 16. Grebenikov, E.A., Gadomski, L., Prokopenya, A.N.: Studying the stability of equilibrium solutions in a planar circular restricted four-body...
  • 17. Hampton, M., Moeckel, R.: Finiteness of relative equilibria of the four-body problem. Invent. Math. 163(2), 289–312 (2006)
  • 18. Han, Y., Li, Y., Yi, Y.: Invariant tori in Hamiltonian systems with high order proper degeneracy. Ann. Henri Poincaré 10(8), 1419–1436...
  • 19. Henrard, J.: Lyapunov’s center theorem for resonant equilibrium. J. Differential Equ. 14(3), 431–441 (1973)
  • 20. Idrisi, M.J., Suraj, M.S., Ershkov, S., Abouelmagd, E.I., Beza, D.: A generalized framework for the collinear restricted four-body problem...
  • 21. Lagrange, J.L.: Mécanique Analytique. Mallet-Bachelier (1853)
  • 22. Leandro, E.S.: On the central configurations of the planar restricted four-body problem. J. Differential Equ. 226(1), 323–351 (2006)
  • 23. Lerman, L.M., Markova, A.P.: On stability at the Hamiltonian Hopf bifurcation. Regul. Chaotic Dyn. 14(1), 148–162 (2009)
  • 24. Llibre, J., Pa¸sca, D., Valls, C.: The circular restricted 4-body problem with three equal primaries in the collinear central configuration...
  • 25. Markeev, A.P.: Libration Points in Celestial Mechanics and Astrodynamics. Nauka, Moscow (1978)
  • 26. Markeyev, A.P.: The problem of the stability of the equilibrium position of a Hamiltonian system at 3:1 resonance. J. Appl. Math. Mech....
  • 27. Meyer, K.R.: Periodic solutions of the N-body problem. J. Differential Equ. 39(1), 2–38 (1981)
  • 28. Meyer, K.R., Palacián, J.F., Yanguas, P.: Geometric averaging of Hamiltonian systems: periodic solutions, stability, and KAM tori. SIAM...
  • 29. Meyer, K.R., Schmidt, D.S.: Periodic orbits near L4 for mass ratios near the critical mass ratio of Routh. Celest. Mech. 4(1), 99–109...
  • 30. Meyer, K.R., Schmidt, D.S.: The stability of the Lagrange triangular point and a theorem of Arnold. J. Differential Equ. 62(2), 222–236...
  • 31. Meyer, K.R., Schmidt, D.S.: Invariant manifolds in the Hamiltonian–Hopf bifurcation. Qual. Theory Dyn. Syst. 19(1), (2020)
  • 32. Michalodimitrakis, M.: The circular restricted four-body problem. Astrophys. Space Sci. 75(2), 289– 305 (1981)
  • 33. Oberti, P.: Lagrangian satellites of tethys and dione. II—theory of motion. Astron. Astrophys. 228(1), 275–283 (1990)
  • 34. Palacios, M., Arribas, M., Abad, A., Elipe, A.: Symmetric periodic orbits in theMoulton-Copenhagen problem. Celest. Mech. Dyn. Astron....
  • 35. Palmore, J.I.: Collinear relative equilibria of the planar n-body problem. Celest. Mech. 28(1–2), 17–24 (1982)
  • 36. Papadakis, K.E.: Families of asymmetric periodic solutions in the restricted four-body problem. Astrophys. Space Sci. 361(12), 377 (2016)
  • 37. Papadakis, K.E.: Families of three-dimensional periodic solutions in the circular restricted four-body problem. Astrophys. Space Sci....
  • 38. Pedersen, P.: Librationspunkte im restringierten Vierkörperproblem. Publ.MindreMedd. Kobenhavns Observatorium 137, 1–80 (1944)
  • 39. Pedersen, P.: Stabilitätsuntersuchungen im restringierten vierkörperproblem. Publ. Mindre Medd. Kobenhavns Observatorium 159, 1–38 (1952)
  • 40. Robutel, P., Rambaux, N., Castillo-Rogez, J.: Analytical description of physical librations of saturnian coorbital satellites janus and...
  • 41. Robutel, P., Rambaux, N., El Moutamid, M.: Influence of the coorbital resonance on the rotation of the Trojan satellites of Saturn. Celest....
  • 42. Roels, J.: Families of periodic solutions near a Hamiltonian equilibrium when the ratio of two eigenvalues is 3. J. Differential Equ....
  • 43. Schmidt, D.S.: Periodic solutions near a resonant equilibrium of a Hamiltonian system. Celest. Mech. 9(1), 81–103 (1974)
  • 44. Simó, C.: Relative equilibrium solutions in the four body problem. Celest. Mech. 18(2), 165–184 (1978)
  • 45. Strömgren, E.: Connaissance actuelle des orbites dans le probleme des trois corps. Bull. Astr. Observatoire de Paris 9(1), 87–130 (1933)
  • 46. Szebehely, V., Ingram, D.S., Cochran, J.E.:Mean motions and characteristic exponents at the libration points. Astron. J. 75, 92 (1970)
  • 47. Vander Meer, J.C.: Bifurcation at nonsemisimple 1:-1 resonance. Z. Angew. Math. Phys. (ZAMP) 37(3), 425–437 (1986)
  • 48. Vander Meer, J.C.: The Hamiltonian Hopf bifurcation. In: The Hamiltonian Hopf Bifurcation, pp. 66–83. Springer (2006)
  • 49. Wiggins, S.: Nonlinear Differential Equations and Dynamical Systems. Universitex. Springer-Verlag, Berlin (1990)
  • 50. Zepeda Ramírez, J.A., Alvarez-Ramírez, M.: Equilibrium points and their linear stability in the planar equilateral restricted four-body...