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    The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps
    (Elsevier Ltd, 2024-07) Ghosh I; McLachlan RI; Simpson DJW
    We study two-dimensional, two-piece, piecewise-linear maps having two saddle fixed points. Such maps reduce to a four-parameter family and are well known to have a chaotic attractor throughout open regions of parameter space. The purpose of this paper is to determine where and how this attractor undergoes bifurcations. We explore the bifurcation structure numerically by using Eckstein's greatest common divisor algorithm to estimate from sample orbits the number of connected components in the attractor. Where the map is orientation-preserving the numerical results agree with formal results obtained previously through renormalisation. Where the map is orientation-reversing or non-invertible the same renormalisation scheme appears to generate the bifurcation boundaries, but here we need to account for the possibility of some stable low-period solutions. Also the attractor can be destroyed in novel heteroclinic bifurcations (boundary crises) that do not correspond to simple algebraic constraints on the parameters. Overall the results reveal a broadly similar component-doubling bifurcation structure in the orientation-reversing and non-invertible settings, but with some additional complexities.
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    An equation-free approach for heterogeneous networks : this dissertation is submitted for the degree of Doctor of Philosophy, School of Mathematical and Computational Sciences, Massey University
    (Massey University, 2023) Zafar, Sidra
    Oscillators exist in many biological, chemical and physical systems. Often when oscillators with different periods of oscillations are coupled, they synchronize and oscillate with the same period. Examples include groups of synchronously flashing fireflies or chirping crickets. There are two questions of interest in this work. (1) Under what conditions will a system of coupled oscillators synchronize? and (2) Can a large system of synchronized oscillators be represented by a smaller number of variables? We study these questions for Kuramoto like models which are coupled in different ways. Examples include spatially extended and all-all coupling. We study conditions under which synchronization occurs in small and large networks by varying the coupling strength, calculating stabilities of synchronized solutions and creating bifurcation diagrams of the steady state solution as a function of coupling strength in one and two-dimensions. We use an equation free approach to approximate the coarse scale behavior of a large, coupled network, for which the equations are known, by a low dimensional description of variables, for which no governing equations are available in closed form. Our results show that a small number of variables can reproduce the behavior of the stable solutions in the full systems. However, the equation-free approach did not work as well for the unstable solutions. Possible reasons for this are explored in the thesis.