Robust chaos and the continuity of attractors. Issue 1 (22nd July 2020)

• Record Type: Journal Article
• Authors: Glendinning, Paul A; Simpson, David J W
• Is Part Of: Transactions of mathematics and its applications. Volume 4:Issue 1(2020)
• Publication Date: 2020-07-22
• Subjects: piecewise-linear -- piecewise-smooth -- border-collision bifurcation -- Hausdorff distance; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1093/imatrm/tnaa002
• Languages: English
• ISSNs: 2398-4945

Abstract

Abstract: As the parameters of a map are varied an attractor may vary continuously in the Hausdorff metric. The purpose of this paper is to explore the continuation of chaotic attractors. We argue that this is not a helpful concept for smooth unimodal maps for which periodic windows fill parameter space densely, but that for many families of piecewise-smooth maps it provides a way to think about changing structures within parameter regions of robust chaos and form a stronger notion of robustness. We obtain conditions for the continuity of an attractor and demonstrate the results with coupled skew tent maps, the Lozi map and the border-collision normal form.