Symbolic dynamics in the planar three-body problem. (English) Zbl 1229.70033
Summary: A chaotic invariant set is constructed for the planar three-body problem. The orbits in the invariant set exhibit many close approaches to triple collision and also excursions near infinity. The existence proof is based on finding appropriate “windows” in the phase space which are stretched across one another by flow-defined Poincaré maps.
References:
| [1] | Euler, L., De motu rectilineo trium corporum se mutuo attahentium, Novi Comm. Acad. Sci. Imp. Petrop., 1767, vol. 11, pp. 144–151. |
| [2] | Lagrange, J.L., Ouvres, vol.6. Paris: Gauthier-Villars, 1873. |
| [3] | Sundman, K.F., Nouvelles recherches sur le problèm des trois corps, Acta Soc. Sci. Fenn, 1909, vol. 35, pp. 1–27. · JFM 40.1017.07 |
| [4] | Siegel, C.L., Der Dreierstos, Ann. Math., 1941, vol. 42, pp. 127–168. · Zbl 0024.36404 · doi:10.2307/1968991 |
| [5] | Siegel, C. and Moser, J., Lectures on Celestial Mechanics, New York: Springer-Verlag, 1971. · Zbl 0312.70017 |
| [6] | McGehee, R., Triple Collision in the Collinear Three Body Problem, Inv. Math., 1974, vol. 27, pp. 191–227. · Zbl 0297.70011 · doi:10.1007/BF01390175 |
| [7] | McGehee, R., Singularities in Classical Celestial Mechanics, in Proc. of Int. Cong. Math., Helsinki, 1978, pp. 827–834. |
| [8] | Devaney, R., Triple Collision in the Planar Isosceles 3-Body Problem, Inv. Math., 1980, vol. 60, pp. 249–267. · Zbl 0438.58014 · doi:10.1007/BF01390017 |
| [9] | Devaney, R., Singularities in Classical Mechanical Systems, in Katok, A., Ed., Ergodic Theory and Dynamical Systems I, Proceedings-Special Year, Maryland 1979–1980, Boston: Birkhauser, 1981, pp. 211–333. |
| [10] | Irigoyen, J.M., La variété de collision triple dans le cas isoscèle du problème des trois corps, C. R. Acad. Sci. Paris, 1980, vol. 290, pp. B489–B492. · Zbl 0447.70014 |
| [11] | Lacomba, E. and Losco, L., Triple Collision in the Isosceles Three-Body Problem, Bull. Amer. Math. Soc., 1980, vol. 3, pp. 710–714. · Zbl 0446.58007 · doi:10.1090/S0273-0979-1980-14802-8 |
| [12] | Moeckel, R., Orbits of the Three-Body Problem which Pass Infinitely Close to Triple Collision, Amer. Jour. Math, 1981, vol. 103, pp. 1323–1341. · Zbl 0475.70008 · doi:10.2307/2374233 |
| [13] | Simó, C., Analysis of Triple Collision in the Isosceles Problem, Marcel Dekker, 1981, pp. 203–224. |
| [14] | Moeckel, R., Chaotic Dynamics Near Triple Collision, Arch. Rat. Mech., 1989, vol. 107, pp. 37–69. · Zbl 0697.70021 · doi:10.1007/BF00251426 |
| [15] | Sitnikov, K., Existence of Oscillating Motions for the Three Body Problem, Dokl. Akad. Nauk USSR, 1960, vol. 133, pp. 303–306. |
| [16] | Alexeev, V.M., Quasirandom Dynamical Systems. I. Quasirandom Diffeomorphisms, Math. USSR-Sb., 1968, vol. 5, pp. 73–128. · Zbl 0198.56903 · doi:10.1070/SM1968v005n01ABEH002587 |
| [17] | Alexeev, V.M., Quasirandom Dynamical Systems. II. One-Dimensional Nonlinear Oscillations in a Field with Periodic Perturbation, Math. USSR-Sb., 1968, vol. 6, pp. 505–560. · Zbl 0198.57001 · doi:10.1070/SM1968v006n04ABEH001074 |
| [18] | V. M. Alexeev. Quasirandom Dynamical Systems. III. Quasirandom Oscillations of One-Dimensional Oscillators, Math. USSR-Sb., 1969, vol. 7, pp. 1–43. · Zbl 0198.57002 · doi:10.1070/SM1969v007n01ABEH001076 |
| [19] | Moser, J., Stable and Random Motions in Dynamical Systems, vol. 77 of Annals of Math. Studies, Princeton: Princeton Univ. Press, 1973. |
| [20] | McGehee, R., A Stable Manifold Theorem for Degenerate Fixed Points with Applications to Celestial Mechanics, JDE, 1973, vol. 14, pp. 70–88. · Zbl 0264.70007 · doi:10.1016/0022-0396(73)90077-6 |
| [21] | Llibre, J. and Simó, C., Some Homoclinic Phenomena in the Three-Body Problem, JDE, 1980, vol. 37, pp. 444–465. · Zbl 0445.70005 · doi:10.1016/0022-0396(80)90109-6 |
| [22] | Xia, Z., Melnikov Method and Transversal Homoclinic Points in the Restricted Three-Body Problem, JDE, 1992, vol. 96, pp. 170–184. · Zbl 0741.70007 · doi:10.1016/0022-0396(92)90149-H |
| [23] | Easton, R. and McGehee, R., Homoclinic Phenomena for Orbits Doubly Asymptotic to an Invariant Three-Sphere, Ind. Jour. Math., 1979, vol. 28, pp. 211–240. · Zbl 0435.58010 · doi:10.1512/iumj.1979.28.28015 |
| [24] | Easton, R., Parabolic Orbits for the Planar Three Body Problem, JDE, 1984, vol. 52, pp. 116–134. · Zbl 0487.70014 · doi:10.1016/0022-0396(84)90138-4 |
| [25] | Robinson, C., Homoclinic Orbits and Oscillation for the Planar Three-Body Problem, JDE, 1984, vol. 52, pp. 356–377. · Zbl 0495.70025 · doi:10.1016/0022-0396(84)90168-2 |
| [26] | Robinson, C., Stable Manifolds in Hamiltonian Systems, in Meyer, K. and Saari, D., Eds., Proceedings of the AMS Summer Research Conference on Hamiltonian Systems, vol. 81 of Contemporary Mathematics, Providence: AMS, 1987, pp. 77–98. |
| [27] | Easton, R., Capture Orbits and Melnikov Integrals in the Planar 3-Body Problem, Celestial Mech. Dynam. Astronom., 1991, vol. 50, pp. 283–297. · doi:10.1007/BF00048768 |
| [28] | Quillen, J., An Application of the Melnikov Method to the Planar Problem, PhD thesis, Univ. of Colorodo, 1986. |
| [29] | Xia, Z., Arnold Diffusion and Oscillatory Solutions in the Planar Three-Body Problem, JDE, 1994, vol. 110, pp. 289–321. · Zbl 0804.70011 · doi:10.1006/jdeq.1994.1069 |
| [30] | Moeckel, R., Heteroclinic Phenomena in the Isosceles Three-Body Problem, SIAM Jour. Math. Anal., 1984, vol. 15, pp. 857–876. · Zbl 0593.70009 · doi:10.1137/0515065 |
| [31] | Moser, J., Regularization of Kepler’s Problem and the Averaging Method on a Manifold, Comm. Pure Appl. Math, 1970, vol. 23, pp. 609–636. · Zbl 0193.53803 · doi:10.1002/cpa.3160230406 |
| [32] | Devaney, R., Structural Stability of Homothetic Solutions of the Collinear N-Body Problem, Cel. Mech., 1979, vol. 19, pp. 391–404. · Zbl 0411.70006 · doi:10.1007/BF01231016 |
| [33] | Robinson, C. and Saari, D., N-Body Spatial Parabolic Orbits Asymptotic to Collinear Central Configurations, JDE, 1983, vol. 48, pp. 434–459. · Zbl 0493.70020 · doi:10.1016/0022-0396(83)90103-1 |
| [34] | Moeckel, R., Orbits Near Triple Collision in the Three-Body Problem, Ind. Jour. of Math., 1983, vol. 32, pp. 221–239. · Zbl 0508.70008 · doi:10.1512/iumj.1983.32.32020 |
| [35] | Moeckel, R., Spiralling Invariant Manifolds, Jour. Diff. Eq., 1984, vol. 66, pp. 189–207. · Zbl 0634.58016 · doi:10.1016/0022-0396(87)90031-3 |
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