Abstract
We study the distribution of regular and irregular periodic orbits on a Poincaré surface of section of a simple Hamiltonian system of 2 degrees of freedom. We explain the appearance of many lines of periodic orbits that form Farey trees. There are also lines that are very close to the asymptotic curves of the unstable periodic orbits. Some regular orbits, sometimes stable, are found inside the homoclinic tangle. We explain this phenomenon, which shows that the homoclinic tangle does not cover the whole area around an unstable orbit, but has gaps. Inside the lobes only irregular orbits appear, and some of them are stable. We conjecture that the opposite is also true, i.e. all irregular orbits are inside lobes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Artuso, R., Aurell, E. and Cvitanovic, P.: 1990, Nonlinearity 3, 325.
Barbanis, B. and Contopoulos, G.: 1995, Astron. Astrophys. 294, 33.
Contopoulos, G.: 1970, Astron. J. 75, 96.
Contopoulos, G. and Polymilis, C.: 1996, Celest. Mech. Dyn. Astron. 63, 189.
Cvitanovic, P.: 1988, Phys. Rev. Lett. 61, 2729.
Cvitanovic, P.: 1992, in Kim, J. H. and Stringer, J. ‘Applied Chaos’, Wiley, New York, 413.
Eckhardt, B. and Grossman, S.: 1994, Phys. Rev. E. 50, 4571.
Guckenheimer, J. and Holmes, P.: 1983, ‘Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields’, Springer Verlag, New York.
Niven, I. and Zuckerman, H.: 1960, An Introduction to the Theory of Numbers, Wiley, New York.
Szebehely, V.: 1967, Theory of Orbits, Academic Press, N. York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Contopoulos, G., Grousouzakou, E. & Polymilis, C. Distribution of periodic orbits and the homoclinic tangle. Celestial Mech Dyn Astr 64, 363–381 (1996). https://doi.org/10.1007/BF00054553
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00054553