Preprint

Chaos and wild chaos in Lorenz-type systems

Hinke M. Osinga, Bernd Krauskopf, and Stefanie Hittmeyer

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 robust homoclinic tangencies; one also speaks of `wild chaos´ in contrast to the `classical chaos´ where homoclinic tangencies can only occur densely, and not 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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