Chaos and the continuum limit in the gravitational N-body problem. II. Nonintegrable potentials

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 65, Issue 6, 2002, Pages

  Sideris, Ioannis V ^a   Kandrup, Henry E ^a  

^a University of Florida ^*  (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CHAOTIC ENSEMBLES; GRAVITATIONAL N-BODY PROBLEMS; MACROSCOPIC CHAOS; PHASE MIXING;

COMPUTER SIMULATION; CONVERGENCE OF NUMERICAL METHODS; ENERGY CONSERVATION; LYAPUNOV METHODS; POISSON EQUATION; WHITE NOISE;

CHAOS THEORY;

EID: 41349084318     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.65.066203     Document Type: Article

References (31)

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13. C. L. Bohn, H. E. Kandrup, and R. A. Kishek, in Proceedings of Snowmass, Snowmass, Co, edited by C. M. Sazama, eConf C01/06/30, M302 (2001).
14. The paper by D. Merritt and M. Valluri, Astrophys. J. 471, 82 (1996) entailed an extension to three dimensions of ideas introduced in
15. H. E. Kandrup and M. E. Mahon, Phys. Rev. E 49, 3735 (1994) in the context of a simple two-dimensional model, namely, the sixth order truncation of the Toda lattice potential.
16. Compare with G. Bertin, Dynamics of Galaxies (Cambridge University Press, Cambridge, 2000)
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21. C. Siopis, Ph.D. thesis, University of Florida, 1998.
22. See A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (Springer, Berlin, 1992).
23. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).
24. A. Griner, W. Strittmatter, and J. Honerkamp, J. Stat. Phys. 51, 95 (1989).
25. Following, G. Contopoulos, Astron. J. 76, 147 (1971), the term “sticky” is used here to designate segments of chaotic orbits which, because of topological obstructions such as cantori, are trapped temporarily near regular phase space regions. Such sticky orbit segments typically have finite time Lyapunov exponents that are considerably smaller than those associated with unsticky, wildly chaotic segments; and they tend to be more regular in visual appearance, characterized by considerably lower complexities 6. Visual examples of stickiness are given in Fig. 1 in 10.
26. These ensembles sampled Merritt and Fridman’s 17 shell 8, for which, in the continuum limit, (Formula presented).
27. Modulo the case of a constant density sphere, where every unperturbed orbit has the same unperturbed frequencies, so that linear phase mixing is impossible.
28. Compare with Fig. 9 in C. Siopis, B. L. Eckstein, and H. E. Kandrup, Ann. N.Y. Acad. Sci. 867, 41 (1998).
29. The ensembles sampled the lowest energy shell considered by Merritt and Fridman 17, for which (Formula presented).
30. C. L. Bohn and J. R. Delayen, Phys. Rev. E 50, 1516 (1994).
31. One might, naively, suppose that the relative number of regular and chaotic orbits in each ensemble would provide a more satisfactory probe. However, as discussed in Ref. 18, stickiness is so pronounced in the Dehnen potential that integrations for times as short as (Formula presented)–(Formula presented) are not always sufficient to make sharp distinctions between regularity and chaos. For the lower energy orbits, (Formula presented) was computed by integrating for a time (Formula presented); for higher energy, (Formula presented).