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Theory and Applications of Difference Equations and Discrete Dynamical Systems pp 75–98Cite as

Chaos and Wild Chaos in Lorenz-Type Systems

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((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.

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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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Authors and Affiliations

  1. Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand

    Hinke M Osinga, Bernd Krauskopf & Stefanie Hittmeyer

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  1. Hinke M Osinga
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  2. Bernd Krauskopf
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Correspondence to Hinke M Osinga .

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Editors and Affiliations

  1. Sultan Qaboos University Dept. of Math. and Statistics, Muscat, Oman

    Ziyad AlSharawi

  2. University of Arizona Dept. of Mathematics, Tucson, Arizona, USA

    Jim M. Cushing

  3. Trinity University Dept. of Mathematics, San Antonio, Texas, USA

    Saber Elaydi

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Osinga, H.M., Krauskopf, B., Hittmeyer, S. (2014). Chaos and Wild Chaos in Lorenz-Type Systems. In: AlSharawi, Z., Cushing, J., Elaydi, S. (eds) Theory and Applications of Difference Equations and Discrete Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 102. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44140-4_4

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