Invariant manifolds at infinity of the RTBP and the boundaries of bounded motion
Tóm tắt
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.
Tài liệu tham khảo
Chirikov, B.V., A Universal Instability of Many-Dimensional Oscillator Systems, Phys. Rep., 1979, vol. 52, no. 5, pp. 264–379.
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.
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.
Greene, J. M., A Method for Determining Stochastic Transition, J. Math. Phys., 1979, vol. 620, no. 6, pp. 1183–1201.
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).
Llibre, J. and Simó, C., Oscillatory Solutions in the Planar Restricted Three-Body Problem, Math. Ann., 1980, vol. 248, no. 2, pp. 153–184.
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.
McGehee, R., A Stable Manifold Theorem for Degenerate Fixed Points with Applications to Celestial Mechanics, J. Differential Equations, 1973, vol. 14, pp. 70–88.
Moser, J., Stable and Random Motions in Dynamical Systems, Ann. of Math. Stud., vol. 77, Princeton, N.J.: Princeton Univ. Press, 1973.
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.
Simó, C., Analytical and Numerical Computation of Invariant Manifolds, in Modern Methods in Celestial Mechanics, D. Benest, C. Froeschlé (Eds.), Gif-sur-Yvette: Ed. Frontières, 1990, pp. 285–330. (Also available at www.maia.ub.es/dsq/2004, no. 2.)
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.
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.
Szebehely, V. G., Theory of Orbits, New York: Acad. Press, 1967.
Treschev, D., Multidimensional Symplectic Separatrix Maps, J. Nonlinear Sci., 2002, vol. 12, no. 1, pp. 27–58.
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.