Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics

Chaos

Volumn 10, Issue 2, 2000, Pages 427-461

  Koon, Wang Sang ^a   Martin W, Lo ^b   Marsden, Jerrold E ^a   Ross, Shane D ^a  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034195465     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.166509     Document Type: Article

References (42)

1. P. Holmes, "Poincaré, celestial mechanics, dynamical systems theory and chaos," Phys. Rep. 193, 137-163 (1990).
2. C. Simo, "Dynamical systems methods for space missions on a vicinity of collinear libration points," Hamiltonian Systems with Three or More Degrees of Freedom, edited by C. Simó (Kleuwer Academic, Dordrecht, 1999), pp. 223-241.
3. E. Belbruno and B. Marsden, "Resonance hopping in comets," Astron. J. 113, 1433-1444 (1997).
4. M. Lo and S. Ross, "SURFing the solar system: Invariant manifolds and the dynamics of the solar system," JPL IOM 312/97, 1997, pp. 2-4.
5. J. Llibre, R. Martinez, and C. Simó, "Transversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem," J. Diff. Eqns. 58, 104-156 (1985).
6. M. Lo et al., Genesis Mission Design, Paper No. AIAA 98-4468, 1998.
7. C. Conley, "On some new long periodic solutions of the plane restricted three body problem," Commun. Pure Appl. Math. 16, 449-467 (1963).
8. C. Conley, "Low energy transit orbits in the restricted three-body problem," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 16, 732-746 (1968).
9. R. P. McGehee, "Some homoclinic orbits for the restricted three-body problem," Ph.D. thesis, University of Wisconsin, 1969.
10. J. Moser, Stable and Random Motions in Dynamical Systems with Special Emphasis on Celestial Mechanics (Princeton University Press, Princeton, 1973).
11. R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed. (Addison-Wesley, New York, 1978).
12. K. R. Meyer and R. Hall, Hamiltonian Mechanics and the n-Body Problem (Springer-Verlag, Applied Mathematical Sciences, Berlin, 1992).
13. J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics, 2nd ed. (Springer-Verlag, Berlin, 1994), Vol. 17.
14. V. Szebehely, Theory of Orbits (Academic, New York, 1967).
15. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathmatics Science (Springer-Verlag, Berlin, 1990), Vol. 2.
16. G. Gómez, A. Jorba, J. Masdemont, and C. Simó, Study Refinement of Semi-Analytical Halo Orbit Theory, Final Report, ESOC Contract No.: 8625/89/D/MD(SC), Barcelona, April, 1991.
17. S. Wiggins, "Global dynamics, phase space transport, orbits homoclinic to resonances, and applications," Fields Institute Monographs, American Mathematical Society, 1993.
18. R. L. Devaney, "Singularities in classical mechanical systems," Ergodic Theory and Dynamical Systems, edited by A. Katok (Birkhäuer, Basel, 1981), pp. 211-333.
19. M. Lo and S. Ross, "Low energy interplanetary transfers using invariant manifolds of L1, L2 and halo orbits," AAS/AIAA Space Flight Mechanics Meeting, Monterey, California, 9-11 February 1998.
20. D. F. Lawden, Optimal Trajectories for Space Navigation (Butterworth, London, 1963).
21. W. S. Koon and J. E. Marsden, "Optimal control for holonomic and nonholonomic mechanical systems with symmetry and Lagrangian reduction," SIAM J. Control Optim. 35, 901-929 (1997).
22. A. M. Bloch and J. E. Marsden, "Controlling homoclinic orbits," Theor. Comput. Fluid Dyn. 1, 179-190 (1989).
23. A. M. Bloch, N. Leonard, and J. E. Marsden, "Stabilization of mechanical systems using controlled Lagrangians," Proc. CDC 36, 2356-2361 (1997).
24. J. M. Wendlandt and J. E. Marsden, "Mechanical integrators derived from a discrete variational principle," Physica D 106, 223-246 (1997).
25. C. Kane, J. E. Marsden, and M. Ortiz, "Symplectic energy momentum integrators," J. Math. Phys. 40, 3353-3371 (1999).
26. C. Kane, J. E. Marsden, M. Ortiz, and M. West, "Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems, Int. J. Num. Math. Engin, (to appear).
27. J. E. Marsden and J. Scheurle, "Pattern evocation and geometric phases in mechanical systems with symmetry," Dyn. Stab. Syst. 10, 315-338 (1995).
28. J. E. Marsden, J. Scheurle, and J. Wendlandt, "Visualization of orbits and pattern evocation for the double spherical pendulum," ICIAM 95: Mathematical Research, edited by K. Kirchgässner, O. Mahrenholtz, and R. Mennicken (Academic-Verlag, Berlin, 1996), Vol. 87, pp. 213-232.
29. S. Dermott et al., "A circumsolar ring of asteroidal dust in resonant lock with the Earth," Nature (London) 369, 719-723 (1994).
30. E. Belbruno and J. Miller, Sun-perturbed earth-to-moon transfers with ballistic capture," J. Guid. Control Dyn. 16, 770-775 (1993).
31. A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, and G. Sánchez de Alvarez, "Stabilization of rigid body dynamics by internal and external torques," Automatica 28, 745-756 (1992).
32. P. J. Enright and B. A. Conway, "Discrete approximations to optimal trajectories using direct transcription and nonlinear programming," J. Guid. Control Dyn. 15, 994-1002 (1992).
33. A. L. Herman and B. A. Conway, "Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules," J. Guid. Control Dyn. 19, 592-599 (1996).
34. L. Hiday-Johnston and K. Howell, "Transfers between libration-point orbits in the elliptic restricted problem," Celest. Mech. Dyn. Astron. 58, 317-337 (1994).
35. C. Howell, B. Barden, and M. Lo, "Application of dynamical systems theory to trajectory design for a libration point mission," J. Astronaut. Sci. 45, 161-178 (1997).
36. J. E. Marsden, O. M. O'Reilly, F. J. Wicklin, and B. W. Zombro, "Symmetry, stability, geometric phases, and mechanical integrators," Nonlinear Sci. Today 1, 4-11, 14-21 (1991).
37. C. K. McCord, K. R. Meyer, and Q. Wang, The Integral Manifolds of the Three Body Problem, Mem. Am. Math. Soc. (1998), Vol. 628.
38. R. P. McGehee, "Triple collision in the collinear three-body problem," Invent. Math. 27, 191-227 (1974).
39. R. W. Easton and R. McGehee, "Homoclinic phenomena for orbits doubly asymptotic to an invariant three-sphere," Indiana Univ. Math. J. 28, 211-240 (1979).
40. J. E. Marsden, Lectures on Mechanics, London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, 1992), Vol. 174.
41. R. P. McGehee, "Parabolic orbits in the three-body problem. Dynamical systems," Proceedings Symposium, of the University Bahia, Salvador, 1971 (Academic, New York, 1973), pp. 249-251.
42. M. Henon, "Chaotic scattering modelled by an inclined billiard," Physica D 33, 132-156 (1988).