Geometry and topology of escape. I. Epistrophes

Chaos

Volumn 13, Issue 3, 2003, Pages 880-891

  Mitchell, K A ^a   Handley, J P ^a   Tighe, B ^a   Delos, J B ^a   Knudson, S K ^a  

^a The College of William and Mary   (United States)

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EID: 0141678055     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1598311     Document Type: Article

References (66)

1. C. Jaffé, S. D. Ross, M. W. Lo, J. Marsden, D. Farrelly, and T. Uzer, Phys. Rev. Lett. 89, 011101 (2002).
2. C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, Phys. Rev. Lett. 69, 506 (1992); C. M. Marcus, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, Chaos 3, 643 (1993); H. Ishio and J. Burgdörfer, Phys. Rev. B 51, 2013 (1995); X. Yang, H. Ishio, and J. Burgdörfer, ibid. 52, 8219 (1995); C. D. Schwieters, J. A. Alford, and J. B. Delos, ibid. 54, 10652 (1996).
3. C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, Phys. Rev. Lett. 69, 506 (1992); C. M. Marcus, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, Chaos 3, 643 (1993); H. Ishio and J. Burgdörfer, Phys. Rev. B 51, 2013 (1995); X. Yang, H. Ishio, and J. Burgdörfer, ibid. 52, 8219 (1995); C. D. Schwieters, J. A. Alford, and J. B. Delos, ibid. 54, 10652 (1996).
4. C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, Phys. Rev. Lett. 69, 506 (1992); C. M. Marcus, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, Chaos 3, 643 (1993); H. Ishio and J. Burgdörfer, Phys. Rev. B 51, 2013 (1995); X. Yang, H. Ishio, and J. Burgdörfer, ibid. 52, 8219 (1995); C. D. Schwieters, J. A. Alford, and J. B. Delos, ibid. 54, 10652 (1996).
5. C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, Phys. Rev. Lett. 69, 506 (1992); C. M. Marcus, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, Chaos 3, 643 (1993); H. Ishio and J. Burgdörfer, Phys. Rev. B 51, 2013 (1995); X. Yang, H. Ishio, and J. Burgdörfer, ibid. 52, 8219 (1995); C. D. Schwieters, J. A. Alford, and J. B. Delos, ibid. 54, 10652 (1996).
6. C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, Phys. Rev. Lett. 69, 506 (1992); C. M. Marcus, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, Chaos 3, 643 (1993); H. Ishio and J. Burgdörfer, Phys. Rev. B 51, 2013 (1995); X. Yang, H. Ishio, and J. Burgdörfer, ibid. 52, 8219 (1995); C. D. Schwieters, J. A. Alford, and J. B. Delos, ibid. 54, 10652 (1996).
7. J. U. Noeckel and A. D. Stone, Nature (London) 385, 45 (1997).
8. M. J. Davis and S. K. Gray, J. Chem. Phys. 84, 5389 (1986); M. J. Davis, ibid. 83, 1016 (1985).
9. M. J. Davis and S. K. Gray, J. Chem. Phys. 84, 5389 (1986); M. J. Davis, ibid. 83, 1016 (1985).
10. A. Tiyapan and C. Jaffé, J. Chem. Phys. 99, 2765 (1993).
11. A. Tiyapan and C. Jaffé, J. Chem. Phys. 101, 10393 (1994).
12. A. Tiyapan and C. Jaffé, J. Chem. Phys. 103, 5499 (1995).
13. P. M. Koch and D. R. Mariani, Phys. Rev. Lett. 46, 1275 (1981); see also A. Mühlpfordt, U. Even, E. Rabani, and R. D. Levine, Phys. Rev. A 51, 3922 (1995).
14. P. M. Koch and D. R. Mariani, Phys. Rev. Lett. 46, 1275 (1981); see also A. Mühlpfordt, U. Even, E. Rabani, and R. D. Levine, Phys. Rev. A 51, 3922 (1995).
15. C. Jaffé, D. Farrelly, and T. Uzer, Phys. Rev. A 60, 3833 (1999).
16. C. Jaffé, D. Farrelly, and T. Uzer, Phys. Rev. Lett. 84, 610 (2000).
17. C. Jaffe and T. Uzer, J. Phys. Chem. A 105, 2783 (2001).
18. S. Wiggins, L. Wiesenfeld, C. Jaffé, and T. Uzer, Phys. Rev. Lett. 86, 5478 (2001).
19. G. M. Lankhuijzen and L. D. Noordam, Phys. Rev. Lett. 76, 1784 (1996).
20. F. Robicheaux and J. Shaw, Phys. Rev. Lett. 77, 4154 (1996); Phys. Rev. A 56, 278 (1997).
21. F. Robicheaux and J. Shaw, Phys. Rev. Lett. 77, 4154 (1996); Phys. Rev. A 56, 278 (1997).
22. S. Wiggins, Chaotic Transport in Dynamical Systems (Springer, New York, 1992).
23. E. A. Jackson, Perspectives of Nonlinear Dynamics, Vol. II (Cambridge University Press, Cambridge, 1990).
24. M. Courtney, H. Jiao, N. Spellmeyer, D. Kleppner, J. Gao, and J. B. Delos, Phys. Rev. Lett. 74, 1538 (1995).
25. G. Raithel and H. Walther, Phys. Rev. A 49, 1646 (1994); G. Raithel, M. Fauth, and H. Walther, ibid. 44, 1898 (1991).
26. G. Raithel and H. Walther, Phys. Rev. A 49, 1646 (1994); G. Raithel, M. Fauth, and H. Walther, ibid. 44, 1898 (1991).
27. A. Holle, J. Main, G. Wiebusch, H. Rottke, and K. H. Welge, Phys. Rev. Lett. 61, 161 (1988).
28. A. Kips, W. Vassen, W. Hogervorst, and P. A. Dando, Phys. Rev. A 58, 3043 (1998).
29. U. Eichmann, K. Richter, D. Wintgen, and W. Sandner, Phys. Rev. Lett. 61, 2438 (1988).
30. N. Spellmeyer, D. Kleppner, M. R. Haggerty, V. Kondratovich, J. B. Delos, and J. Gao, Phys. Rev. Lett. 79, 1650 (1997).
31. M. L. Du and J. B. Delos, Phys. Rev. A 38, 1896 (1988).
32. J. Gao and J. B. Delos, Phys. Rev. A 46, 1455 (1992).
33. J. Main, G. Wiebusch, K. Welge, J. Shaw, and J. B. Delos, Phys. Rev. A 49, 847 (1994).
34. We also have in mind applications to the semi-classical propagation of quantum wave functions, for which it is natural to consider an initial curve (Lagrangian manifold) in phase space (Ref. 56). Lines of initial conditions are also common in studies of classical chaotic scattering (as in Refs. 34-39).
35. R. S. MacKay, J. D. Meiss, and I. C. Percival, Physica D 13, 55 (1984).
36. V. Rom-Kedar, Physica D 43, 229 (1990).
37. V. Rom-Kedar, Nonlinearity 7, 441 (1994),
38. D. W. Noid, S. K. Gray, and S. A. Rice, J. Chem. Phys. 84, 2649 (1986).
39. J.-M. Petit and M. Henon, Icarus 66, 536 (1986).
40. B. Eckhardt and C. Jung, J. Phys. A 19, L829 (1986).
41. B. Eckhardt, J. Phys. A 20, 5971 (1987).
42. C. Jung and H. J. Scholz, J. Phys. A 20, 3607 (1987).
43. B. Rückerl and C. Jung, J. Phys. A 27, 55 (1994).
44. B. Rückerl and C. Jung, J. Phys. A 27, 6741 (1994).
45. C. Lipp and C. Jung, J. Phys. A 28, 6887 (1995).
46. C. Jung, C. Lipp, and T. H. Seligman, Ann. Phys. (Leipzig) 275, 151 (1999).
47. T. Bütikofer, C. Jung, and T. H. Seligman, Phys. Lett. A 265, 76 (2000).
48. P. Gaspard and S. Rice, J. Chem. Phys. 90, 2225 (1989).
49. P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge, 1998).
50. J. Palis, Topology 8, 385 (1969).
51. We thank R. MacKay for bringing this to our attention.
52. R. W. Easton, Trans. Am. Math. Soc. 294, 719 (1986).
53. An end point could also lie on u, in which case it would be on the boundary of the complex and no epistrophe would converge upon it. See Fig. 1.
54. n are points in the phase plane. 47 Merriam-Webster's Collegiate Dictionary (Merriam-Webster, Springfield, MA, 1999).
55. American College Dictionary (Random House, New York, 1947).
56. 0 coincides with the stable manifold over an interval of nonzero length, a nongeneric situation.
57. 0 is viewed as an open set, even though it is not an open subset of the phase plane.
58. J. G. Hocking and G. S. Young, Topology (Addison-Wesley, Reading, MA, 1961), pp. 97-100.
59. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, Chichester, 1990); H.-O. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science (Springer, New York, 1992).
60. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, Chichester, 1990); H.-O. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science (Springer, New York, 1992).
61. K. R. Meyer, Trans. Am. Math. Soc. 149, 95 (1970).
62. K. R. Meyer and G. R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem (Springer, New York, 1992), pp. 224-225.
63. J. Moser, Commun. Pure Appl. Math. 9, 673 (1956).
64. V. P. Maslov and M. V. Fedoriuk, Semi-classical Approximation in Quantum Mechanics (Reidel, Boston, 1981); J. B. Delos, Adv. Chem. Phys. 65, 161 (1966); M. V. Berry, Proc. R. Soc. London, Ser. A 413, 183 (1987).
65. V. P. Maslov and M. V. Fedoriuk, Semi-classical Approximation in Quantum Mechanics (Reidel, Boston, 1981); J. B. Delos, Adv. Chem. Phys. 65, 161 (1966); M. V. Berry, Proc. R. Soc. London, Ser. A 413, 183 (1987).
66. V. P. Maslov and M. V. Fedoriuk, Semi-classical Approximation in Quantum Mechanics (Reidel, Boston, 1981); J. B. Delos, Adv. Chem. Phys. 65, 161 (1966); M. V. Berry, Proc. R. Soc. London, Ser. A 413, 183 (1987).