Massey Documents by Type

Permanent URI for this communityhttps://mro.massey.ac.nz/handle/10179/294

Browse

Has files: YesSubject: search.filters.subject.490105 Dynamical systems in applications

Search Results

Now showing 1 - 4 of 4
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    A dynamical systems model for optimizing rotational grazing : a thesis presented in partial fulfilment of the requirements for the degree of Ph. D. in Mathematics at Massey University
    (Massey University, 1993) Woodward, Simon
    This thesis considers modelling agricultural grazing using dynamical systems. It is in five chapters, some of which have been or will be published in international refereed journals. The first chapter considers grazing a two-paddock system at low pasture mass in order to maximise herbage conservation and/or herbage intake. For the latter objective, there is an optimal swap-over time which depends on the initial herbage masses and the stocking densities. In general, optimal swap-over gives only small improvements in herbage intake compared to continuous grazing or rotational grazing in which animals spend equal time in each paddock. The second chapter applies this to comparing continuous, rotational, and optimal grazing strategies over a range of stocking rates. As stocking rate increases optimal rotational grazing can increase herbage intake. The third chapter deals with grazing a multi-paddock system in order to maximise intake. Animals are shifted at regular time intervals. Stocking rate and average initial herbage have the greatest effect on herbage growth, conservation, and intake. Grazing strategy effects are less significant. However, traditional strategies of rotational grazing perform poorly in some cases, and in these cases a "greedy" grazing strategy can give imporoved production. The difficulties of finding optimal strategies are discussed. The fourth chapter examines modelling senescence in grazed grass pasture using a differential-delay equation where senescence rates are explicitly dependent on leaf age. A simple differential-delay model is formulated and appraised by comparison to data from a published grazing experiment. This simple model describes subtle features of pasture dynamics. The fifth chapter uses this delay model to make a simple comparison between rotational and continuous grazing. The average rate of senescence is higher under rotational grazing and this is exacerbated by delay effects. For this reason, production is likely to be lower under rotational grazing.
  • Thumbnail Image
    Item
    Computational Studies in the Few-Body Problem : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics at Massey University, Albany, New Zealand
    (Massey University, 2019) Chopovda, Valerie
    In this research we investigate the gravitational four-body problem which describes the motion in the system of four stars moving under their mutual gravitational forces. It involves studies of the dynamics of few-body systems and finding periodic orbits, and subsequently an analysis of their stability. One of the challenges of the problem is the necessity of using regularisation algorithms in order to avoid singularities when there is a possibility of collision or close encounter between stars. One of the featured solutions of the problem is the collinear Schubart orbit discovered for systems of three [59] and four [70] bodies. This orbit has been shown to form families of collinear and planar orbits for three bodies by Henon [32]. Sweatman discovered Schubart orbits in the collinear symmetric four-body problem [70, 71]. In this work we generate the families of Schubart orbits starting from the planar orbit obtained by Sweatman [73]. Utilising the symmetries of the four-body Schubart orbits, we solve the Caledonian symmetric four-body problem (CS4BP). Initially we consider the case of equal masses. This is later extended to the case of pairwise symmetric masses. The family is parametrised by two parameters: the mass of the outer bodies and the distance between the two closest non-symmetric bodies. The collinear orbits are collisional, but there are no collisions when the orbits evolve to planar motion. The planar family of pairwise symmetric masses is bounded by the line of the collinear Schubart orbits. Within its boundaries, there are four regions separated by two special types of orbits present in the family: the equal-mass orbits and the double choreography orbits. Two of the regions are symmetrical to the other two. We perform a linear stability analysis of the discovered solutions both in and out of the plane. We also distinguish the influence of symmetrical and non-symmetrical perturbations on the orbits. An algorithm for the orbit search and orbital stability analysis is presented.