Fractal dimensions of chaotic saddles of dynamical systems

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

Volumn 54, Issue 5, 1996, Pages 4819-4823

  Hunt, Brian R ^a   Ott, Edward ^b   Yorke, James A ^a,c  

^a UNIVERSITY OF MARYLAND   (United States)

^b UNIVERSITY OF MARYLAND   (United States)

^c Bldg 142   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001600430     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/physreve.54.4819     Document Type: Article

References (25)

1. (a) J. L. Kaplan and J. A. Yorke, in Functional Differential Equations and Approximations of Fixed Points, edited by H.-O. Peitgen and H.-O. Walter, Lecture Notes in Mathematics Vol. 730 (Springer, Berlin, 1979), p. 204. For further discussion and development see
2. (b) J. D. Farmer, E. Ott, and J. A. Yorke, Physica D 7, 153 (1983);
3. (c) J. C. Alexander and J. A. Yorke, Ergod. Theory Dyn. Syst. 4, 1 (1984);
4. (d) P. Frederickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, J. Diff. Eq. 49, 185 (1983).
5. The Kaplan-Yorke formula for the dimension is known to fail in special cases. However, these cases are claimed to be atypical in that arbitrary small changes of the map restore agreement (e.g., see [1(c)]).
6. L.-S. Young, Ergod. Theor. Dyn. Syst. 2, 109 (1982).
7. F. Ledrappier, Commun. Math. Phys. 81, 229 (1981).
8. F. Ledrappier and L.-S. Young, Ann. Math. 122, 509 (1985); 122, 540 (1985). A brief discussion of the partial dimension formalism is given by J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
9. F. Ledrappier and L.-S. Young, Ann. Math. 122, 509 (1985); 122, 540 (1985). A brief discussion of the partial dimension formalism is given by J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
10. F. Ledrappier and L.-S. Young, Ann. Math. 122, 509 (1985); 122, 540 (1985). A brief discussion of the partial dimension formalism is given by J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
11. F. Ledrappier and L.-S. Young, Commun. Math. Phys. 117, 529 (1988).
12. H. Kantz and P. Grassberger, Physica D 17, 75 (1985).
13. G. H. Hsu, E. Ott, and C. Grebogi, Phys. Lett. A 127, 199 (1988).
14. N/2.
15. F. Mitschke, Phys. Rev. A 41, 1169 (1990).
16. For work making use of the theory of chaotic scattering to treat tracer convection by a two-dimensional Lagrangian chaotic fluid flow see the papers by C. Jung and E. Ziemniak, J. Phys. A 25, 3929 (1992); C. Jung, T. Tél, and E. Ziemniak, CHAOS 3, 555 (1993); E. Ziemniak, C. Jung, and T. Tél, Physica D 76, 123 (1994); A. Péntek, Z. Toroczkai, T. Tél, C. Grebogi, and J. A. Yorke, Phys. Rev. E 51, 4076 (1995).
17. For work making use of the theory of chaotic scattering to treat tracer convection by a two-dimensional Lagrangian chaotic fluid flow see the papers by C. Jung and E. Ziemniak, J. Phys. A 25, 3929 (1992); C. Jung, T. Tél, and E. Ziemniak, CHAOS 3, 555 (1993); E. Ziemniak, C. Jung, and T. Tél, Physica D 76, 123 (1994); A. Péntek, Z. Toroczkai, T. Tél, C. Grebogi, and J. A. Yorke, Phys. Rev. E 51, 4076 (1995).
18. For work making use of the theory of chaotic scattering to treat tracer convection by a two-dimensional Lagrangian chaotic fluid flow see the papers by C. Jung and E. Ziemniak, J. Phys. A 25, 3929 (1992); C. Jung, T. Tél, and E. Ziemniak, CHAOS 3, 555 (1993); E. Ziemniak, C. Jung, and T. Tél, Physica D 76, 123 (1994); A. Péntek, Z. Toroczkai, T. Tél, C. Grebogi, and J. A. Yorke, Phys. Rev. E 51, 4076 (1995).
19. For work making use of the theory of chaotic scattering to treat tracer convection by a two-dimensional Lagrangian chaotic fluid flow see the papers by C. Jung and E. Ziemniak, J. Phys. A 25, 3929 (1992); C. Jung, T. Tél, and E. Ziemniak, CHAOS 3, 555 (1993); E. Ziemniak, C. Jung, and T. Tél, Physica D 76, 123 (1994); A. Péntek, Z. Toroczkai, T. Tél, C. Grebogi, and J. A. Yorke, Phys. Rev. E 51, 4076 (1995).
20. The arguments in this section generalize those of Refs. [8,9]. See also E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993), pp. 176-179.
21. This argument (as in the Kaplan-Yorke argument) might superficially appear to yield the box-counting dimension rather than the information dimension. We note, however, that the information dimension may be viewed as the box-counting dimension of the minimal set covering most of the measure (e.g., see [1b]). Furthermore, the expansions and contractions specified by the Lyapunov exponents are only valid for typical orbits (i.e., for those orbits on the attractor that have most of the measure).
22. Equation (17) can also be obtained from the partial dimension treatment of Ref. [5] by maximizing D over the possible values of the partial dimensions.
23. E. Ott, T. Sauer, and J. A. Yorke, Phys. Rev. A 39, 4212 (1989) (see in particular the Appendix).
24. R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, and M. A. Rubio, Phys. Rev, Lett. 60, 979 (1988).
25. L. M. Pecora and T. L. Carroll, CHAOS 6, 432 (1996).