Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-21T18:17:18.509Z Has data issue: false hasContentIssue false
  • English
  • Français

Conditional Entropy: A Tool to Explore the Phase Space

Published online by Cambridge University Press:  12 April 2016

P. Cincotta  and
C. Simó
P. Cincotta
Affiliation:
Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseo del Bosque, 1900 La Plata, Argentina
C. Simó
Affiliation:
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spaine-mail:carles@maia.ub.es

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we show that the Conditional Entropy of nearby orbits may be a useful tool to explore the phase space associated to a given Hamiltonian. The arc length parameter along the orbits, instead of the time, is used as a random variable to compute the entropy. In the first part of this work we summarise the main analytical results to support this tool while, in the second part, we present numerical evidence that this technique is able to localise (stable) periodic and quasiperiodic orbits, ‘aperiodic’ orbits (chaotic motion) and unstable periodic orbits (the ‘source’ of chaotic motion). Besides, we show that this technique provides a measure of chaos which is similar to that given by the largest Lyapunov Characteristic Number. It is important to remark that this method is very simple to compute and does not require long time integrations, just realistic physical times.

Keywords

ChaosLyapunov Characteristic NumberEntropy
Type
Analytical and Numerical Tools
Information
International Astronomical Union Colloquium , Volume 172: Impact of Modern Dynamics in Astronomy , 1999 , pp. 195 - 209
Copyright
Copyright © Kluwer 1999

References

Arnold, V. & Avez, A.: 1989, Ergodic Problems of Classical Mechanics, (New York: Addison-Wesley), 2nd. ed.Google Scholar
Cincotta, P. & Simó, C.: 1998, preprintGoogle Scholar
Contopoulos, G. & Voglis, N.: 1996, Cel. Mech. & Dynam. Astron., 64, 1 Google Scholar
Fraser, A. & Swinney, H.: 1986, Phys. Rev. A, 33, 1134 Google Scholar
Hénon, M. & Heiles, C.: 1964, AJ, 69, 73 Google Scholar
Katz, A.: 1967, Principles of Statistical Mechanics, The Information Theory Approach, (San Francisco: W. H. Freeman & Co.)Google Scholar
Laskar, J.: 1990, Icarus, 88, 266 Google Scholar
Laskar, J.: 1993, Physica D, 67, 257 Google Scholar
Merritt, D. & Valluri, M.: 1996, ApJ, 471, 82 Google Scholar
Núñez, J., Cincotta, P. & Wachlin, F.: 1996, Cel. Mech. & Dynam. Astron., 64, 43 Google Scholar
Papaphilippou, Y. & Laskar, J.: 1998, A&A, 329, 451 Google Scholar
Udry, S. & Pfenniger, D.: 1988, A&A, 198, 135 Google Scholar
Voglis, N., Contopoulos, G. & Efthymiopoulos, C.: 1999, Cel. Mech. & Dynam. Astron., in press.Google Scholar
Wachlin, F. & Ferraz-Mello, S.: 1998, MNRAS, 298, 22 CrossRefGoogle Scholar