Singular-Perturbation Homoclinic Hyperchaos

  • O. E. Rossler
Part of the International Centre for Mechanical Sciences book series (CISM, volume 319)


The relaxation-oscillation or singular-perturbation principle, respectively, lends itself to the generation of chaotic flows out of two 2-D flows connected via two 1-D thresholds (“reinjection principle”). In the same vein, two 3-D flows can be overlaid to generate hyperchaos using two 2-D thresholds. A special case in spiral-flow-based singular-perturbation chaos is “homoclinic” reinjection, a codimension-one situation that is covered by Shil’nikov’s theorem after time inversion. An analogous reinjection in screw-flow-based hyperchaos is of codimension two. An infinitely often folded (in the limit noninvertible) hyperhorseshoe exists near the homoclinic trajectory. The latter can be part of a strictly self-similar basin boundary.


Finite Type Basin Boundary Chaotic Flow Homoclinic Trajectory Extended Neighborhood 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Rossler, O.E. (1976). Chaotic behavior in simple reaction systems. Z. Naturforsch. 31 a, 259–264.MathSciNetGoogle Scholar
  2. 2.
    Rossler, O.E. (1979) Chaos. In: Structural Stability in Physics, ( W. Göttinger and H. Eikemeier, eds.), pp. 290–309. New York: Springer-Verlag.CrossRefGoogle Scholar
  3. 3.
    Shil’nikov, L.P. (1965). A case of the existence of a countable number of periodic motions. Soviet Math. Dokl. 6, 163–166.Google Scholar
  4. 4.
    Rossler, O.E. (1979). An equation for hyperchaos. Phys. Lett. 71 A, 155–157.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Smale, S. (1967). Differentiable dynamical systems. Bull. Amer. Math. Soc. 73, 747–817.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Shilnikov, L.P. (1970). A contribution to the problem of the structure of an extended neighborhood of a structurally stable equilibrium of saddle-focus type. Math. USSR Sbornik 10, 91–102.CrossRefGoogle Scholar
  7. 7.
    Rossler, O.E., C. Kahlert, J. Parisi, J. Peinke and B. Röhricht (1986). Hyperchaos and Julia sets. Z. Naturforsch. 41a, 819–322.MathSciNetGoogle Scholar
  8. 8.
    Rossler, O.E., J.L. Hudson, M. Klein and C. Mira (1990). Self-similar basin boundary in a continuous system. In: Nonlinear Dynamics in Engineering Systems, ( W. Schiehlen, ed.), pp. 265–273. New York: Springer-Verlag.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 1991

Authors and Affiliations

  • O. E. Rossler
    • 1
  1. 1.University of TubingenTubingenGermany

Personalised recommendations