Optimal periodic orbits of chaotic systems occur at low period

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

Volumn 54, Issue 1, 1996, Pages 328-337

  Hunt, Brian R ^a,b   Ott, Edward ^a,b  

^a University of Maryland ^*  (United States)

^b UNIVERSITY OF MARYLAND   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4544258791     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.54.328     Document Type: Article

References (16)

1. E. Ott, C. Grebogi and James A. Yorke, Phys. Rev. Lett.64, 1196 (1990).
2. J. C. Alexander, I. Kan, J. A. Yorke and Z. You, Int. J. Bifurc. Chaos2, 795 (1992);
3. E. Ott, J. C. Sommerer, J. C. Alexander, I. Kan and J. A. Yorke, Phys. Rev. Lett.71, 4134 (1993);
4. J. C. Sommerer and E. Ott, Nature365, 136 (1993).
5. E. Ott, J. C. Sommerer, J. C. Alexander, I. Kan and J. A. Yorke, Physica D76, 384 (1994).
6. Another related situation occurs in the ``bubbling transition” 5,6 first noted by Ashwin 5. This transition is marked by the onset of temporally intermittent bursting in the presence of small noise (see Refs. 3,5 and the paper of N. Platt, S. M. Hammel and J. F. Heagy, Phys. Rev. Lett.72, 3498 (1994)).
7. P. Ashwin, J. Buescu and I. Stewart, Phys. Lett.193A, 126 (1994).
8. S. C. Venkataramani et al., Phys. Rev. E (to be published); Y.-C. Lai et al., Phys. Rev. Lett. (to be published).
9. J. A. Yorke (private communication)has suggested that the bifurcation is typically mediated by a periodic orbit, and this conjecture is supported by this paper.
10. A different but related result has been obtained by B. R. Hunt and J. A. Yorke, who argue that a certain type of crisis bifurcation of a chaotic attractor is typically mediated by a low period unstable periodic orbit [Trans. Am. Math. Soc.325, 141 (1991)].
11. This line of argument is supported by the work of K. Sigmund, who proved that for hyperbolic systems, every invariant measure can be approximated arbitrarily well by the δ-function measure on a periodic orbit [Am. J. Math.94, 31 (1972)].
12. L.-S. Young, Erg. Th. Dyn. Syst.2, 109 (1982).
13. J. L. Kaplan and J. A. Yorke, in Functional Differential Equations and Approximation of Fixed Points, edited by H.-O. Peitgen and H.-O. Walter, Lecture Notes in Mathematics Vol. 730 (Springer, Berlin, 1979), p. 204.
14. O. Biham and W. Wenzel, Phys. Rev. Lett.63, 819 (1989);
15. see also P. Grassberger, H. Kantz and U. Moenig, J. Phys. A22, 5217 (1989).
16. Some of the results in this paper were reported in abbreviated form in B. R. Hunt and E. Ott, Phys. Rev. Lett.76, 2254 (1996).