Advertisement

Bi-Fractal Basin Boundaries in Invertible Systems

  • O. E. Rössler
  • C. Kahlert
  • J. L. Hudson
Part of the NATO ASI Series book series (NSSA, volume 138)

Abstract

A Lauwerier-type limiting case of a folded-towel diffeomorphism is considered. Its existence confirms the conjecture made previously that differentiable dynamical systems may possess bifractal (self-similar, in two directions) basin boundaries. A procedure how to search for such boundaries in realistic systems is indicated.

Keywords

Chaotic Attractor Basin Boundary Lyapunov Characteristic Exponent Invertible System Unstable Periodic Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S.W. McDonald, C. Grebogi, E. Ott, and J.A. Yorke, Physica 17D: 125 (1985).Google Scholar
  2. [2]
    C. Grebogi, personal communication, August (1986).Google Scholar
  3. [3]
    O.E. Rössler, C. Kahlert, J. Parisi, J. Peinke, and B. Röhricht, Z. Naturforsch. 41a: 819 (1986).Google Scholar
  4. [4]
    C. Kahlert, and O.E. Rössler, Analogues to a Julia boundary away from analyticity, Z. Naturforsch. 42a: 324 (1987).Google Scholar
  5. [5]
    M.F. Barnsley, and A.N. Harrington, Physica 15D: 421 (1985).Google Scholar
  6. [6]
    O.E. Rössler, and C. Mira, Higher-order chaos in a constrained differential equation with an explicit cross section, Topologie: Spezialtagung: Dynamical Systems, September 13–19, 1981, Mathematisches Forschungsinstitut Oberwolfach Tagungsberichte (1981).Google Scholar
  7. [7]
    O.E. Rössler, Z. Naturforsch. 38a: 788 (1986).Google Scholar
  8. [8]
    O.E. Rössler, Phys. Lett. 71A: 155 (1979).CrossRefGoogle Scholar
  9. [9]
    S. Swale, Bull. Amer. Math. Soc. 73: 747 (1967).CrossRefGoogle Scholar
  10. [10]
    H.A. Lauwerier, Physica 21D: 146 (1986).Google Scholar
  11. [11]
    O.E. Rössler, Chaos, in “Structural Stability in Physics” W. Göttinger, and H. Eikermeier, eds., pp 290–309, Springer-Verlag, Berlin (1979).CrossRefGoogle Scholar
  12. [12]
    O.E. Rössler, “Chaos, The World of Nonperiodic Oscillations,” Unpublished manuscript (1981).Google Scholar
  13. [13]
    K. Stefanski, preprint, Nicholas Copernicus University (1983).Google Scholar
  14. [14]
    H. Froehling, J.P. Crutchfield, D. Farmer, N.H. Packard, and R.Shaw, Physica 3D: 605 (1981).Google Scholar
  15. [15]
    J.L. Hudson, H. Killory, and O.E. Rössler, Hyperchaos in an explicit reaction system, preprint, September (1986).Google Scholar
  16. [16]
    R. Shaw, “The Dripping Faucet as a Model Chaotic System”, Aerial Press, Santa Cruz, Calif. (1985).Google Scholar
  17. [17]
    J. Peinke, B. Röhricht, A. Mühlbach, J. Parisi, C. Nöldeke, R.P. Huebener, and O.E. Rössler, Z. Naturforsch. 40a: 562 (1985).Google Scholar
  18. [18]
    O.E. Rössler, J.L. Hudson, and J.A. Yorke, Z. Naturforsch. 41a: 979 (1986).Google Scholar
  19. [19]
    J. Peinke, J. Parisi, B. Röhricht, and O.E. Rössler, Instability of the Mandelbrot set, Z. Naturforsch. 42a: 263 (1987).Google Scholar
  20. [20]
    B. Röhricht, J. Parisi, J. Peinke, and O.E. Rössler, Z. Phys. B. - Condensed Matter 65: 259 (1986).CrossRefGoogle Scholar
  21. [21]
    O.E. Rössler, Z. Naturforsch. 31a: 1168 (1976).Google Scholar
  22. [22]
    B. Röhricht et al., Self-similar fractal in the parameter space of a Rashevsky-Turing morphogenetic system, in preparation.Google Scholar
  23. [23]
    B. Mandelbrot, “The Fractal Geometry of Nature”, Freeman, San Francisco (1982).Google Scholar

Copyright information

© Springer Science+Business Media New York 1987

Authors and Affiliations

  • O. E. Rössler
    • 1
  • C. Kahlert
    • 1
  • J. L. Hudson
    • 2
  1. 1.Institute for Physical and Theoretical ChemistryUniversity of TubingenTubingenWest Germany
  2. 2.Department of Chemical EngineeringUniversity of VirginiaCharlottesvilleUSA

Personalised recommendations