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Subject: search.filters.subject.Bifurcation theory

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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.
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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.
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    Globally resonant homoclinic tangencies : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand
    (Massey University, 2022) Muni, Sishu Shankar
    The attractors of a dynamical system govern its typical long-term behaviour. The presence of many attractors is significant as it means the behaviour is heavily dependent on the initial conditions. To understand how large numbers of attractors can coexist, in this thesis we study the occurrence of infinitely many stable single-round periodic solutions associated with homoclinic connections in two-dimensional maps. We show this phenomenon has a relatively high codimension requiring a homoclinic tangency and 'global resonance', as has been described previously in the area-preserving setting. However, unlike in that setting, local resonant terms also play an important role. To determine how the phenomenon may manifest in bifurcation diagrams, we also study perturbations of a globally resonant homoclinic tangency. We find there exist sequences of saddle-node and period-doubling bifurcations. Interestingly, in different directions of parameter space, the bifurcation values scale differently resulting in a complicated shape for the stability region for each periodic solution. In degenerate directions the bifurcation values scale substantially slower as illustrated in an abstract piecewise-smooth C¹ map.
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    Dynamical effects of degree correlations in networks of type I model neurons : a dissertation presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Auckland, New Zealand
    (Massey University, 2020) Bläsche, Christian
    The complex behaviour of human brains arises from the complex interconnection of the well-known building blocks -- neurons. With novel imaging techniques it is possible to monitor firing patterns and link them to brain function or dysfunction. How the network structure affects neuronal activity is, however, poorly understood. In this thesis we study the effects of degree correlations in recurrent neuronal networks on self-sustained activity patterns. Firstly, we focus on correlations between the in- and out-degrees of individual neurons. By using Theta Neurons and Ott/Antonsen theory, we can derive a set of coupled differential equations for the expected dynamics of neurons with equal in-degree. A Gaussian copula is used to introduce correlations between a neuron’s in- and out-degree, and numerical bifurcation analysis is used determine the effects of these correlations on the network's dynamics. We find that positive correlations increase the mean firing rate, while negative correlations have the opposite effect. Secondly, we turn to degree correlations between neurons -- often referred to as degree assortativity -- which describes the increased or decreased probability of connecting two neurons based on their in-or out-degrees, relative to what would be expected by chance. We present an alternative derivation of coarse-grained degree mean field equations utilising Theta Neurons and the Ott/Antonsen ansatz as well, but incorporate actual adjacency matrices. Families of degree connectivity matrices are parametrised by assortativity coefficients and subsequently reduced by singular value decomposition. Thus, we efficiently perform numerical bifurcation analysis on a set of coarse-grained equations. To our best knowledge, this is the first time a study examines the four possible types of degree assortativity separately, showing that two have no effect on the networks' dynamics, while the other two can have a significant effect.
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    Path-following methods for boundary value problems and their applications to combustion equations : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University
    (Massey University, 1993) Scotter, Miguel David
    This thesis is primarily concerned with the numerical techniques involved in bifurcation analysis, in particular with the software package AUTO developed by Eusebius Doede! which performs this analysis on dynamical systems. The techniques of AUTO are investigated and applied to a steady state heat equation. The chosen equation can be solved by analytical methods for some boundary conditions. Initially AUTO was successfully applied to such problems, which have analytical solutions confirming its reliability. The software was then used to solve dynamical system problems which do not have known analytical solutions. These problems necessitated a modification to AUTO for non-autonomous systems. The modified version of AUTO was shown to be successful in finding solutions to these problems.