×

Invariant manifolds at infinity of the RTBP and the boundaries of bounded motion. (English) Zbl 1342.37033

Summary: Invariant manifolds of a periodic orbit at infinity in the planar circular RTBP are studied. To this end we consider the intersection of the manifolds with the passage through the barycentric pericenter. The intersections of the stable and unstable manifolds have a common even part, which can be seen as a displaced version of the two-body problem, and an odd part which gives rise to a splitting. The theoretical formulas obtained for a Jacobi constant \(C\) large enough are compared to direct numerical computations showing improved agreement when \(C\) increases. A return map to the pericenter passage is derived, and using an approximation by standard-like maps, one can make a prediction of the location of the boundaries of bounded motion. This result is compared to numerical estimates, again improving for increasing \(C\). Several anomalous phenomena are described.

MSC:

37D05 Dynamical systems with hyperbolic orbits and sets
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
70F07 Three-body problems
37M05 Simulation of dynamical systems
37N05 Dynamical systems in classical and celestial mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chirikov, B.V., A Universal Instability of Many-Dimensional Oscillator Systems, Phys. Rep., 1979, vol. 52, no. 5, pp. 264-379. · doi:10.1016/0370-1573(79)90023-1
[2] Fox, A.M. and Meiss, J.D., Critical Invariant Circles in Asymmetric and Multiharmonic Generalized Standard Maps, Commun. Nonlinear Sci. Numer. Simul., 2014, vol. 19, no. 4, pp. 1004-1026. · Zbl 1457.37058 · doi:10.1016/j.cnsns.2013.07.028
[3] Galante, J. and Kaloshin, V., Destruction of Invariant Curves in the Restricted Circular Planar Three-Body Problem by Using Comparison of Action, Duke Math. J., 2011, vol. 159, no. 2, pp. 275-327. · Zbl 1269.70016 · doi:10.1215/00127094-1415878
[4] Greene, J. M., A Method for Determining Stochastic Transition, J. Math. Phys., 1979, vol. 620, no. 6, pp. 1183-1201. · doi:10.1063/1.524170
[5] Guardia, M., Martín, P., and Seara, T.M., Oscillatory Motions for the Restricted Planar Circular Three-Body Problem, Preprint, available at http://arxiv.org/abs/1207.6531 (2014).
[6] Llibre, J. and Simó, C., Oscillatory Solutions in the Planar Restricted Three-Body Problem, Math. Ann., 1980, vol. 248, no. 2, pp. 153-184. · Zbl 0505.70010 · doi:10.1007/BF01421955
[7] Martínez, R. and Pinyol, C., Parabolic Orbits in the Elliptic Restricted Three Body Problem, J. Differential Equations, 1994, vol. 111, no. 2, pp. 299-339. · Zbl 0804.70009 · doi:10.1006/jdeq.1994.1084
[8] McGehee, R., A Stable Manifold Theorem for Degenerate Fixed Points with Applications to Celestial Mechanics, J. Differential Equations, 1973, vol. 14, pp. 70-88. · Zbl 0264.70007 · doi:10.1016/0022-0396(73)90077-6
[9] Moser, J., Stable and Random Motions in Dynamical Systems, Ann. of Math. Stud., vol. 77, Princeton, N.J.: Princeton Univ. Press, 1973. · Zbl 0271.70009
[10] Sánchez, J., Net, M., and Simó, C., Computation of Invariant Tori by Newton-Krylov Methods in Large-Scale Dissipative Systems, Phys. D, 2010, vol. 239, nos. 3-4, pp. 123-133. · Zbl 1183.37137 · doi:10.1016/j.physd.2009.10.012
[11] Simó, C.; Benest, D. (ed.); Froeschlé, C. (ed.), Analytical and Numerical Computation of Invariant Manifolds, 285-330 (1990), Gif-sur-Yvette
[12] Simó, C. and Treschev, D., Stability Islands in the Vicinity of Separatrices Of Near-Integrable Symplectic Maps, Discrete Contin. Dyn. Syst. Ser. B, 2008, vol. 10, nos. 2-3, pp. 681-698. · Zbl 1153.37406
[13] Sitnikov, K.A., The Existence of Oscillatory Motions in the Three-Body Problems, Soviet Phys. Dokl., 1960, vol. 5, pp. 647-650; see also: Dokl. Akad. Nauk SSSR, 1960, vol. 133, no. 2, pp. 303-306. · Zbl 0108.18603
[14] Szebehely, V. G., Theory of Orbits, New York: Acad. Press, 1967. · Zbl 1372.70004
[15] Treschev, D., Multidimensional Symplectic Separatrix Maps, J. Nonlinear Sci., 2002, vol. 12, no. 1, pp. 27-58. · Zbl 1022.37041 · doi:10.1007/s00332-001-0460-2
[16] Zaslavskii, G. M. and Chirikov, B. V., Stochastic Instability of Non-Linear Oscillations, Soviet Phys. Uspekhi, 1972, vol. 14, pp. 549-568; see also: Uspekhi Fiz. Nauk, 1971, vol. 105, pp. 3-39. · Zbl 1156.34335 · doi:10.1070/PU1972v014n05ABEH004669
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.