Unstable dimension variability and complexity in chaotic systems

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

Volumn 59, Issue 4, 1999, Pages R3807-R3810

  Lai, Ying Cheng ^a  

^a UNIVERSITY OF KANSAS   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001460304     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.59.R3807     Document Type: Article

References (34)

1. R. Badii and A. Politi, Complexity: Hierarchical Structures and Scaling in Physics (Cambridge University Press, Cambridge, England, 1997).
2. J. P. Crutchfield and K. Young, Phys. Rev. Lett. 63, 105 (1989).
3. P. Grassberger, in Fifth Mexican School on Statistical Mechanics, edited by F. Ramos-Gómez (World Scientific, Singapore, 1991).
4. R. Badii, in Chaotic Dynamics: Theory and Practice, edited by T. Bountis (Plenum, New York, 1992).
5. L. Poon and C. Grebogi, Phys. Rev. Lett. 75, 4023 (1995).
6. Y.-C. Lai and C. Grebogi, Phys. Rev. E 54, 4667 (1996).
7. P. Bergé and M. Dubois, Phys. Lett. A 93, 365 (1983).
8. P. E. Rapp, J. Neurosci. 14, 4731 (1994).
9. F. T. Arecchi, Phys. Rev. Lett. 65, 2531 (1990).
10. C. S. Daw, Phys. Rev. Lett. 75, 2308 (1995).
11. F. Romeiras, C. Grebogi, E. Ott, and W. P. Dayawansa, Physica D 58, 165 (1992).
12. It has been a widely accepted notion that in deterministic chaotic systems, unstable periodic orbits constitute the skeleton of the chaotic sets such as chaotic attractors or nonattracting chaotic saddles. See, for example, D. Auerbach, P. Cvitanović, J.-P. Eckmann, G. H. Gunaratne, and I. Procaccia, Phys. Rev. Lett. 58, 2387 (1987).
13. R. Abraham and S. Smale, Proc. Symp. Pure Math. 14, 5 (1970).
14. S. P. Dawson, C. Grebogi, T. Sauer, and J. A. Yorke, Phys. Rev. Lett. 73, 1927 (1994).
15. T. Sauer, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 79, 59 (1997).
16. E. J. Kostelich, I. Kan, C. Grebogi, E. Ott, and J. A. Yorke, Physica D 109, 81 (1997).
17. K. T. Alligood, T. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer-Verlag, New York, 1997).
18. See, for example, J. C. Alexander, J. A. Yorke, Z. You, and I. Kan, Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 795 (1992)
19. P. Ashwin, J. Buescu, and I. N. Stewart, Phys. Lett. A 193, 126 (1994)
20. Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995).
21. See, for example, J. F. Heagy, T. L. Carroll, and L. M. Pecora, Phys. Rev. E 50, 1874 (1994).
22. J. F. Heagy, T. L. Carroll, and L. M. Pecora, Phys. Rev. Lett. 73, 3528 (1994)
23. Phys. Rev. Lett.L. M. Pecora and T. L. Carroll, 80, 2109 (1998).
24. M. Hénon, Commun. Math. Phys. 50, 69 (1976).
25. O. Biham and W. Wenzel, Phys. Rev. Lett. 63, 819 (1989).
26. Y.-C. Lai and R. L. Winslow, Physica D 74, 353 (1994).
27. The basin of the attractor in M is riddled. It is in fact a fat fractal [Y.-C. Lai and C. Grebogi, Phys. Rev. E 53, 1371 (1996)] that exhibits a hierarchy of structures.
28. This is analogous to the concept of contrast that quantifies visibility of fringes in optical interferometry.
29. C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. A 37, 1711 (1988)
30. Y.-C. Lai, Y. Nagai, and C. Grebogi, Phys. Rev. Lett. 79, 652 (1997).
31. K. Ikeda, Opt. Commun. 30, 257 (1979)
32. S. M. Hammel, C. K. R. T. Jones, and J. Moloney, J. Opt. Soc. Am. B 2, 552 (1985).
33. P. Schmelcher and F. K. Diakonos, Phys. Rev. Lett. 78, 4733 (1997)
34. P. SchmelcherF. K. DiakonosPhys. Rev. E 57, 2739 (1998).