Self-Similar Basin Boundary in a Continuous System

  • O. E. Rossler
  • J. L. Hudson
  • M. Klein
  • C. Mira
Conference paper
Part of the International Union of Theoretical and Applied Mechanics book series (IUTAM)


A four-variable ordinary differential equation with hyperchaos is investigated further numerically. A self-similar Sierpinski-type fractal is found in a plane of initial conditions if the prospective fate of each point (whether it escapes through the one or the other escape hole in the exploded hyperchaotic attractor) is used for a coloring criterion. A basin boundary of the same qualitative shape therefore exists in either this equation or a closely related one. All hyper-chaotic systems are eligible for an analogous investigation — both numerically and, if possible, experimentally.


Chaotic Attractor Stable Manifold Basin Boundary Slow Manifold Hyperchaotic System 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Smale (1967). Bull. Amer. Math. Soc. 73, 747.CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    B. Mandelbrot, The Fractal Geometry of Nature. Freeman, San Francisco 1983.Google Scholar
  3. 3.
    O.E. Rössler and C. Mira (1981), Higher-order chaos in a constrained differential equation with an explicit cross section, extended abstract in: Tagungsbericht 40/1981, pp. 9–10. Mathematisches Forschungsinstitut Oberwolfach, 7620 Oberwolfach-Walke, West Germany.Google Scholar
  4. 4.
    O.E. Rössler (1983). Z. Naturforsch. 38 a, 788.MATHADSMathSciNetGoogle Scholar
  5. 5.
    C. Mira (1978), Complex dynamics generated by a third-order differential equation (in French), in: Proc. “Equadiff 78” ( R. Conti, G. Sestini and G. Villari, Eds.), pp. 25–36. Florence, Italy.Google Scholar
  6. 6.
    O.E. Rössler (1976). Z. Naturforsch. 31 a, 259.ADSGoogle Scholar
  7. 7.
    M. Hénon (1976). Commun. Math. Phys. 50, 69.CrossRefMATHADSGoogle Scholar
  8. 8.
    C. Mira (1979). C. R. Acad. Sc. Paris 288 A, 591.Google Scholar
  9. 9.
    I. Gumowski and C. Mira (1975). C. R. Acad. Sc. Paris 280 A, 905.MATHMathSciNetGoogle Scholar
  10. 10.
    O.E. Rössler, C. Kahlert, J. Parisi, J. Peinke and B. Röhricht (1986). Z. Naturforsch. 41 a, 819.Google Scholar
  11. 11.
    O.E. Rössler, J. Hudson, M. Klein and R. Wais (1988), Self-similar basin boundary in an invertible system (folded-towel map), in: Dynamic Patterns in Complex Systems ( J.A.S. Kelso, A.J. Mandell and M.F. Shlesinger, Eds.), pp. 209–218. World Scientific, Singapore.Google Scholar
  12. 12.
    O.E. Rössler and J.L. Hudson (1989), Self-similarity in hyperchaotic data, in: Chaotic Dynamics in Brain Function ( E. Basar, Ed.), pp. 113–121. Springer-Verlag, Berlin.Google Scholar
  13. 13.
    O.E. Rössler (1979). Phys. Lett. 71 A, 155.Google Scholar
  14. 14.
    O.E. Rössler, J.L. Hudson and M. Klein (1989). J. Phys. Chem. 93, 2858.Google Scholar
  15. 15.
    O.E. Rössler and C. Kahlert (1987). Z. Naturforsch. 42 a, 324.Google Scholar
  16. 16.
    P.M. Battelino, C. Grebogi, E. Ott, J.A. Yorke and E.D. Yorke (1988). Physica 32 D, 296.CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    O.E. Rössler (1985), Example of an axiom-A ODE, in: Chaos, Fractals and Dynamics ( P. Fischer and W.R. Smith, Eds.), pp. 105–114. M. Dekker, New York.Google Scholar
  18. 18.
    F. Moon, Personal communication 1989.Google Scholar
  19. 19.
    G. Baier, K. Wegmann and J.L. Hudson (1989). Phys. Lett. A (submitted).Google Scholar

Copyright information

© Springer-Verlags Berlin Heidelberg 1990

Authors and Affiliations

  • O. E. Rossler
    • 1
  • J. L. Hudson
    • 2
  • M. Klein
    • 1
  • C. Mira
    • 3
  1. 1.Institute for Physical and Theoretical ChemistryU. of TubingenTubingenGermany
  2. 2.Department of Chemical EngineeringUniversity of VirginiaCharlottesvilleUSA
  3. 3.Systèmes Nonlinéaires et ApplicationsINSA of ToulouseToulouse CedexFrance

Personalised recommendations