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Chaos and wild chaos in Lorenz-type systems. (English) Zbl 1318.37009
AlSharawi, Ziyad (ed.) et al., Theory and applications of difference equations and discrete dynamical systems. ICDEA, Muscat, Oman, May 26–30, 2013. Berlin: Springer (ISBN 978-3-662-44139-8/hbk; 978-3-662-44140-4/ebook). Springer Proceedings in Mathematics & Statistics 102, 75-98 (2014).
Summary: 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 R. Bamón, J. Kiwi, and J. Rivera [“Wild Lorenz like attractors”, Preprint, arXiv:math/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.
For the entire collection see [Zbl 1297.39001].

MSC:
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
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