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Chaos and Wild Chaos in Lorenz-Type Systems

  • Hinke M OsingaEmail author
  • Bernd Krauskopf
  • Stefanie Hittmeyer
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 102)

Abstract

This contribution provides a geometric perspective on the type of chaotic dynamics that one finds in the original Lorenz system and in a higher-dimensional Lorenz-type system. The latter provides an example of a system that features robustness of homoclinic tangencies; one also speaks of ‘wild chaos’ in contrast to the ‘classical chaos’ where homoclinic tangencies accumulate on one another, but do not occur robustly in open intervals in parameter space. Specifically, we discuss the manifestation of chaotic dynamics in the three-dimensional phase space of the Lorenz system, and illustrate the geometry behind the process that results in its description by a one-dimensional noninvertible map. For the higher-dimensional Lorenz-type system, the corresponding reduction process leads to a two-dimensional noninvertible map introduced in 2006 by Bamón, Kiwi, and Rivera-Letelier [arXiv 0508045] as a system displaying wild chaos. We present the geometric ingredients—in the form of different types of tangency bifurcations—that one encounters on the route to wild chaos.

Keywords

Phase Portrait Chaotic Attractor Lorenz System Homoclinic Bifurcation Lorenz Attractor 
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.

Notes

Acknowledgments

The work on the Lorenz system presented here has been performed in collaboration with Eusebius Doedel. We acknowlegde his contribution to the computation of the Lorenz attractor as shown in Fig. 1 and of the Lorenz manifold on the sphere in Figs. 2 and 3; moreover, the leaves of the stable foliation in Fig. 5 were computed with AUTO demo files that he developed recently. HMO and BK thank the organisers of ICDEA 2013 for their support, financial and otherwise.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Hinke M Osinga
    • 1
    Email author
  • Bernd Krauskopf
    • 1
  • Stefanie Hittmeyer
    • 1
  1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand

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