Browsing by Author "Ghosh, Indranil"

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    Robust chaos in piecewise-linear maps : a thesis submitted in partial fulfillment for the award of the degree of Doctor of Philosophy, School of Mathematical and Computational Sciences, Massey University, Palmerston North, New Zealand
    (Massey University, 2024-05-31) Ghosh, Indranil
    Piecewise-linear maps describe the dynamical behaviour of a wide variety of physical systems that switch between different modes of evolution, such as optimal control systems, mechanical systems with contact events, and social and economics systems involving decisions or constraints. This thesis focuses on a canonical form for two-dimensional continuous piecewise-linear maps, known as the border-collision normal form. Recent work showed that where the normal form is orientation-preserving it can exhibit chaotic dynamics that is robust in the sense that it occurs throughout an open region of four-dimensional parameter space. In this thesis we first use renormalisation to partition this region by the number of connected components of the chaotic attractor, revealing previously undescribed bifurcation structure in a succinct way. Next, we prove that in part of this region the attractor satisfies Devaney's definition of chaos, strengthening existing results. Here we also show that the one-dimensional stable manifold of a fixed point densely fills a two-dimensional area of phase space, and identify a heteroclinic bifurcation, not described previously, at which the attractor undergoes a crisis and may be destroyed. We then generalise the results to the orientation-reversing and non-invertible parameter regimes of the normal form by developing new ways of constructing trapping regions and invariant expanding cones that establish the existence of chaotic attractors. Bifurcations of the attractor are explored numerically by using Eckstein's greatest common divisor algorithm and comparing the results to those generated through renormalisation. Finally we extend the study to higher dimensional maps by constructing a novel trapping region for the $N$-dimensional border-collision normal form.