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Zeitschrift für Naturforschung A

A Journal of Physical Sciences

Editor-in-Chief: Holthaus, Martin

Editorial Board: Fetecau, Corina / Kiefer, Claus

IMPACT FACTOR 2016: 1.432

CiteScore 2017: 1.30

SCImago Journal Rank (SJR) 2017: 0.403
Source Normalized Impact per Paper (SNIP) 2017: 0.632

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Volume 31, Issue 12


Different Types of Chaos in Two Simple Differential Equations

Otto E. Rössler
  • Institute for Physical and Theoretical Chemistry, Division of Theoretical Chemistry, University of Tübingen
  • Other articles by this author:
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Published Online: 2014-06-02 | DOI: https://doi.org/10.1515/zna-1976-1231


Different types of chaotic flow are possible in the 3-dimensional state spaces of two simple non-linear differential equations. The first equation consists of a 2-variable, double-focus subsystem complemented by a linearly coupled third variable. It produces at least three types of chaos: Lorenzian chaos, "sandwich" chaos, and "horseshoe" chaos. Two figure 8-shaped chaotic regimes of the latter type are possible simultaneously, running through each other like 2 links of a chain. In the second equation, a transition between two different types of horseshoe chaos (spiral chaos and screw chaos) is possible. While sandwich chaos allows for a genuine strange attractor, the same has not yet been demonstrated for horseshoe chaos. Unlike the situation in the analogous 1-dimensional case, an emergent period-3 solution is not necessarily stable in the horseshoe. Since chaos is a "super-oscillation" (emergent with the third dimension), the existence of "super-chaos" is postulated for the nect level.

A six-minute, super-8 sound film, demonstrating the different behavioral modes and their bifurcations in the 2 equations, has been prepared. Chaos sounds as musical as a snore.

About the article

Received: 1976-11-10

Published Online: 2014-06-02

Published in Print: 1976-12-01

Citation Information: Zeitschrift für Naturforschung A, Volume 31, Issue 12, Pages 1664–1670, ISSN (Online) 1865-7109, ISSN (Print) 0932-0784, DOI: https://doi.org/10.1515/zna-1976-1231.

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© 1946 – 2014: Verlag der Zeitschrift für Naturforschung. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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