Dissertations / Theses on the topic 'Centre manifold theory'

Many chemical reactions can be described as the crossing of an energetic barrier. This process is mediated by an invariant object in phase space. One can construct a normally hyperbolic invariant manifold (NHIM) of the reactive dynamical system which is an invariant sphere that can be considered as the geometric representation of the transition state itself. The NHIM has invariant cylinders (reaction channels) attached to it. This invariant geometric structure survives as long as the invariant sphere is normally hyperbolic. We applied this theory to the hydrogen exchange reaction in three degrees of freedom in order to figure out the reason of the transition state theory (TST) failure. Energies high above the reaction threshold, the dynamics within the transition state becomes partially chaotic. We have found that the invariant sphere first ceases to be normally hyperbolic at fairly low energies. Surprisingly normal hyperbolicity is then restored and the invariant sphere remains normally hyperbolic even at very high energies. This observation shows two different energy values for the breakdown of the TST and the breakdown of the NHIM. This leads to seek another phase space object that is related to the breakdown of the TST. Using theory of the dividing surface including reactive islands (RIs), we can investigate such an object. We found out that the first nonreactive trajectory has been found at the same energy values for both collinear and full systems, and coincides with the first bifurcation of periodic orbit dividing surface (PODS) at the collinear configuration. The bifurcation creates the unstable periodic orbit (UPO). Indeed, the new PODS (UPO) is the reason for the TST failure. The manifolds (stable and centre-stable) of the UPO clarify these expectations by intersecting the dividing surface at the boundary of the reactive island (on the collinear and the three (full) systems, respectively).

This dissertation focuses on the further development of creating accurate numerical models of complex dynamical systems using the holistic discretisation technique [Roberts, Appl. Num. Model., 37:371-396, 2001]. I extend the application from second to fourth order systems and from only one spatial dimension in all previous work to two dimensions (2D). We see that the holistic technique provides useful and accurate numerical discretisations on coarse grids. We explore techniques to model the evolution of spatial patterns governed by pdes such as the Kuramoto-Sivashinsky equation and the real-valued Ginzburg-Landau equation. We aim towards the simulation of fluid flow and convection in three spatial dimensions. I show that significant steps have been taken in this dissertation towards achieving this aim. Holistic discretisation is based upon centre manifold theory [Carr, Applications of centre manifold theory, 1981] so we are assured that the numerical discretisation accurately models the dynamical system and may be constructed systematically. To apply centre manifold theory the domain is divided into elements and using a homotopy in the coupling parameter, subgrid scale fields are constructed consisting of actual solutions of the governing partial differential equation(pde). These subgrid scale fields interact through the introduction of artificial internal boundary conditions. View the centre manifold (macroscale) as the union of all states of the collection of subgrid fields (microscale) over the physical domain. Here we explore how to extend holistic discretisation to the fourth order Kuramoto-Sivashinsky pde. I show that the holistic models give impressive accuracy for reproducing the steady states and time dependent phenomena of the Kuramoto-Sivashinsky equation on coarse grids. The holistic method based on local dynamics compares favourably to the global methods of approximate inertial manifolds. The excellent performance of the holistic models shown here is strong evidence in support of the holistic discretisation technique. For shear dispersion in a 2D channel a one-dimensional numerical approximation is generated directly from the two-dimensional advection-diffusion dynamics. We find that a low order holistic model contains the shear dispersion term of the Taylor model [Taylor, IMA J. Appl. Math., 225:473-477, 1954]. This new approach does not require the assumption of large x scales, formerly absolutely crucial in deriving the Taylor model. I develop holistic discretisation for two spatial dimensions by applying the technique to the real-valued Ginzburg-Landau equation as a representative example of second order pdes. The techniques will apply quite generally to second order reaction-diffusion equations in 2D. This is the first study implementing holistic discretisation in more than one spatial dimension. The previous applications of holistic discretisation have developed algebraic forms of the subgrid field and its evolution. I develop an algorithm for numerical construction of the subgrid field and its evolution for 1D and 2D pdes and explore various alternatives. This new development greatly extends the class of problems that may be discretised by the holistic technique. This is a vital step for the application of the holistic technique to higher spatial dimensions and towards discretising the Navier-Stokes equations.

Orientador: Edson Denis Leonel
Resumo: Este estudo objetiva provar que sistemas dinâmicos de dimensão N, de codimensão um e satisfazendo as condições do teorema da bifurcação de Hopf, podem ser expressos em uma forma analítica simplificada que preserva a topologia do espaço de fases da configuração original, na vizinhança do ponto de equilíbrio. A esta forma simplificada é atribuído o nome de forma normal. Para tanto, foi utilizado a teoria da variedade central, necessária para reduzir a dimensão de sistemas à sua variedade bidimensional, e o teorema das formas normais, utilizando-se como método para determinar a forma simplificada da variedade central associada aos sistemas dinâmicos, atendendo as condições do teorema da bifurcação de Hopf. A partir da análise dos resultados aqui encontrados foi possível construir a prova matemática de que sistemas de dimensão N, atendendo as condições do teorema de Hopf, podem ser reescritos em uma expressão analítica geral e simplificada. Enfim, através deste estudo foi possível resumir todos os resultados aqui obtidos em um teorema geral que, além de reduzir a custosa tarefa de obtenção de formas normais, abrange sistemas N-dimensionais com ocorrência da bifurcação de Hopf.
Abstract: In this work we prove the following: consider a N-dimensional system that is reduced to its center manifold. If it is proved the system satisfies the conditions of Hopf bifurcation theorem, then the original system of differential equations is rewritten in a simpler analytical expression that preserves the phase space topology. This last is also known as the normal form. The center manifold is used to derive a reduced order expression, and the normal form theory is applied to simplify the form of the dynamics on the center manifold. The key results here allow constructing a general mathematical proof for the normal form of N-dimensional systems reduced to its center manifold. In the class of dynamical systems under Hopf bifurcations, the present work reduces the work done to obtain normal forms.
Mestre

In this thesis, both theoretical and application oriented results are obtained for differential equations with discontinuities of different types: impulsive differential equations, differential equations with piecewise constant argument of generalized type and differential equations with discontinuous right-hand sides. Several qualitative problems such as stability, Hopf bifurcation, center manifold reduction, permanence and persistence are addressed for these equations and also for Lotka-Volterra predator-prey models with variable time of impulses, ratio-dependent predator-prey systems and logistic equation with piecewise constant argument of generalized type. For the first time, by means of Lyapunov functions coupled with the Razumikhin method, sufficient conditions are established for stability of the trivial solution of differential equations with piecewise constant argument of generalized type. Appropriate examples are worked out to illustrate the applicability of the method. Moreover, stability analysis is performed for the logistic equation, which is one of the most widely used population dynamics models. The behaviour of solutions for a 2-dimensional system of differential equations with discontinuous right-hand side, also called a Filippov system, is studied. Discontinuity sets intersect at a vertex, and are of the quasilinear nature. Through the B&
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equivalence of that system to an impulsive differential equation, Hopf bifurcation is investigated. Finally, the obtained results are extended to a 3-dimensional discontinuous system of Filippov type. After the existence of a center manifold is proved for the 3-dimensional system, a theorem on the bifurcation of periodic solutions is provided in the critical case. Illustrative examples and numerical simulations are presented to verify the theoretical results.

Este trabalho apresenta um estudo qualitativo das equações diferenciais nãolineares que descrevem o sincronismo de fase nos PLLs de 3ª ordem que compõem redes OWMS de topologia mista, Estrela Simples e Cadeia Simples. O objetivo é determinar, através da Teoria de Bifurcações, os valores ou relações entre os parâmetros constitutivos da rede que permitam a existência e a estabilidade do estado síncrono, quando são aplicadas, no oscilador mestre, duas funções de excitação muito comuns na prática: o degrau e a rampa de fase. Na determinação da estabilidade dos pontos de equilíbrio, sob o ponto de vista de Lyapunov, a existência de pontos de equilíbrio não-hiperbólicos não permite uma aproximação linear e, nesses casos, é aplicado o Teorema da Variedade Central. Essa técnica de simplificação de sistemas dinâmicos permite fazer uma aproximação homeomórfica em torno desses pontos, preservando a orientação no espaço de fases e possibilitando determinar localmente suas estabilidades.
This work presents a qualitative study of the non-linear differential equations that describe the synchronous state in 3rd order PLLs that compose One-way masterslave time distribution networks with Single Star and Single Chain topologies. Using bifurcation theory, the dynamical behavior of third-order phase-locked loops employed to extract the syncronous state in each node is analyzed depending on constitutive node parameters when two usual inputs, the step and the ramp phase pertubations, are supposed to appear in the master node. When parameter combinations result in non hyperbolic synchronous states, from Lyapunov point of view, the linear approximation does not provide any information about the local behavior of the system. In this case, the center manifold theorem permits the construction of an equivalent vector field representing the asymptotic behavior of the original system in the neighborhood of these points. Thus, the local stability can be determined.

Klassische Methoden für die direkte Berechnung von Hopf Punkten und andere Singularitaten basieren auf der Auswertung und Faktorisierung der Jakobimatrix. Dieses stellt ein Hindernis dar, wenn die Dimensionen des zugrundeliegenden Problems gross genug ist, was oft bei Partiellen Diferentialgleichungen der Fall ist. Die betrachteten Systeme haben die allgemeine Darstellung f ( x(t), α) für t grösser als 0, wobei x die Zustandsvariable, α ein beliebiger Parameter ist und f glatt in Bezug auf x und α ist. In der vorliegenden Arbeit wird ein Matrixfreies Schema entwicklet und untersucht, dass ausschliesslich aus Produkten aus Jakobimatrizen und Vektoren besteht, zusammen mit der Auswertung anderer Ableitungsvektoren erster und zweiter Ordnung. Hiermit wird der Grenzwert des Parameters α, der zuständig ist für das Verlieren der Stabilität des Systems, am Hopfpunkt bestimmt. In dieser Arbeit wird ein Gleichungssystem zur iterativen Berechnung des Hopfpunktes aufgestellt. Das System wird mit einer skalaren Testfunktion φ, die aus einer Projektion des kritischen Eigenraums bestimmt ist, ergänzt. Da das System f aus einer räumlichen Diskretisierung eines Systems Partieller Differentialgleichungen entstanden ist, wird auch in dieser Arbeit die Berechung des Fehlers, der bei der Diskretisierung unvermeidbar ist, dargestellt und untersucht. Zur Bestimmung der Hopf-Bedingungen wird ein einzelner Parameter gesteuert. Dieser Parameter wird unabhängig oder zusammen mit dem Zustandsvektor in einem gedämpften Iterationsschritt neu berechnet. Der entworfene Algorithmus wird für das FitzHugh-Nagumo Model erprobt. In der vorliegenden Arbeit wird gezeigt, wie für einen kritischen Strom, das Membranpotential eine fortschreitende Welle darstellt.
Classical methods for the direct computation of Hopf bifurcation points and other singularities rely on the evaluation and factorization of Jacobian matrices. In view of large scale problems arising from PDE discretization systems of the form f( x (t), α ), for t bigger than 0, where x are the state variables, α are certain parameters and f is smooth with respect to x and α, a matrix-free scheme is developed based exclusively on Jacobian-vector products and other first and second derivative vectors to obtain the critical parameter α causing the loss of stability at the Hopf point. In the present work, a system of equations is defined to locate Hopf points, iteratively, extending the system equations with a scalar test function φ, based on a projection of the eigenspaces. Since the system f arises from a spatial discretization of an original set of PDEs, an error correction considering the different discretization procedures is presented. To satisfy the Hopf conditions a single parameter is adjusted independently or simultaneously with the state vector in a deflated iteration step, reaching herewith both: locating the critical parameter and accelerating the convergence rate of the system. As a practical experiment, the algorithm is presented for the Hopf point of a brain cell represented by the FitzHugh-Nagumo model. It will be shown how for a critical current, the membrane potential will present a travelling wave typical of an oscillatory behaviour.

Neste trabalho, são estudados os problemas de sincronismo de fase nas redes mestre-escravo de via única (OWMS), nas topologias Estrela Simples, Cadeia Simples e mista, através da Teoria Qualitativa de Equações Diferenciais, com ênfase no Teorema da Variedade Central. Através da Teoria das Bifurcações, analisa-se o comportamento dinâmico das malhas de sincronismo de fase (PLL) de segunda ordem que compõem cada rede, frente às variações nos seus parâmetros constitutivos. São utilizadas duas funções de excitação muito comuns na prática: o degrau e a rampa de fase, aplicadas pelo nó mestre. Em cada caso, discute-se a existência e a estabilidade do estado síncrono. A existência de pontos de equilíbrio não-hiperbólicos, não permite uma aproximação linear, e nesses casos é aplicado o Teorema da Variedade Central. Através dessa rigorosa técnica de simplificação de sistemas dinâmicos é possível fazer uma aproximação homeomórfica em torno desses pontos, preservando a orientação no espaço de fases. Desse modo, é possível determinar, localmente, suas estabilidades.
This work presents stability analysis of the syncronous state for three types of one-way master-slave time distribution network topologies: single star, single chain and both of them, mixed. Using bifurcation theory, the dynamical behavior of second-order phase-locked loops employed to extract the syncronous state in each node is analyzed in function of the constitutive parameters. Two usual inputs, the step and the ramp phase pertubations, are supposed to appear in the master node and, in each case, the existence and stability of the syncronous state are studied. For parameter combinations resulting in non hyperbolic synchronous states, the linear approximation does not provide any information, even about the local behaviour of the system. In this case, the center manifold theorem permits the construction of an equivalent vector field representing the asymptotic behaviour of the original system in the neighborhood of these points. Thus, the local stability can be determined.

There are basically two types of variables in population modelling, global and local variables. The former describes the behavior of the entire population while the latter describes the behavior of individuals within this population. The description of the population using local variables is more detailed, but it is also computationally costly. In many cases to study the dynamics of this population, it is sufficient to focus only on global variables. In applied sciences, to achieve this, the method of aggregation of variables is used. One of methods used to mathematically justify variables aggregation is the centre manifold theory. In this dissertation we provide detailed proofs of basic results of the centre manifold theory and discuss some examples of applications in population modelling.
Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2009.

Homogenization and other multiscale modelling techniques empower scientist and engineers to build efficient macroscale mathematical models for simulating materials with complicated microstructure. But the modelling methodology rarely systematically derives the boundary conditions for macroscale model. This thesis aims to systematically derive boundary conditions for macroscale models without heuristic arguments. I start by building a smooth macroscale model for a one-dimensional discrete diffusion system with rapidly varying microscale diffusivity, finite scale separation, and Dirichlet boundary conditions. I apply both centre manifold theory and homogenization theory to build the macroscale model. Both theories find same macroscale model. I then apply modern dynamical system theory to derive macroscopic boundary conditions for this class of diffusion problems. The results suggest a specific Robin boundary condition is a good choice for the macroscale model. I extend my methodology to a linear two-strand diffusion problem. My method finds macroscale boundary conditions for the microscale two-strand problem with different classes of microscale boundary conditions such as specified flux and mixed microscale boundary conditions. The two-strand problem has a more complicated eigen structure than the single strand problems but my method performs well. I also show this method is suitable for continuous problems, such as a class of continuous heterogeneous wave partial differential equations. Furthermore, I apply this technique to wave equations with periodic elasticities and densities and with arbitrary periodicity and number of strands. The algebra in these problem is tedious so I extensively implement computer algebra to find the corresponding macroscale models and boundary conditions. Finally, I consider nonlinearity by analysing the macroscale modelling and the derivation of macroscale boundary conditions for a nonlinear heat exchanger. The proposed technique provides a systematic tool for deriving macroscale boundary conditions for multiscale models. In comparison with heuristically proposed boundary conditions, my derived boundary conditions improve the accuracy of multiscale models in physical science and engineering.
Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2015.

The thesis is based on using dynamical systems theories and techniques to study the qualitative dynamics of herpes simplex virus type 2 (HSV-2), a sexually-transmitted disease of major public health significance. A deterministic model for the interaction of the virus with the immune system in the body of an infected individual (in vivo) is designed first of all. It is shown, using Lyapunov function and LaSalle's Invariance Principle, that the virus-free equilibrium of the model is globally-asymptotically stable whenever a certain biological threshold, known as the reproduction number, is less than unity. Furthermore, the model has at least one virus-present equilibrium when the threshold quantity exceeds unity. Using persistence theory, it is shown that the virus will always be present in vivo whenever the reproduction threshold exceeds unity. The analyses (theoretical and numerical) of this model show that a future HSV-2 vaccine that enhances cell-mediated immune response will be effective in curtailling HSV-2 burden in vivo. A new single-group model for the spread of HSV-2 in a homogenously-mixed sexually-active population is also designed. The disease-free equilibrium of the model is globally-asymptotically stable when its associated reproduction number is less than unity. The model has a unique endemic equilibrium, which is shown to be globally-stable for a special case, when the reproduction number exceeds unity. The model is extended to incorporate an imperfect vaccine with some therapeutic benefits. Using centre manifold theory, it is shown that the resulting vaccination model undergoes a vaccine-induced backward bifurcation (the epidemiological importance of the phenomenon of backward bifurcation is that the classical requirement of having the reproduction threshold less than unity is, although necessary, no longer sufficient for disease elimination. In such a case, disease elimination depends upon the initial sizes of the sub-populations of the model). Furthermore, it is shown that the use of such an imperfect vaccine could lead to a positive or detrimental population-level impact (depending on the sign of a certain threshold quantity). The model is extended to incorporate the effect of variability in HSV-2 susceptibility due to gender differences. The resulting two-group (sex-structured) model is shown to have essentially the same qualitative dynamics as the single-group model. Furthermore, it is shown that adding periodicity to the corresponding autonomous two-group model does not alter the dynamics of the autonomous two-group model (with respect to the elimination of the disease). The model is used to evaluate the impact of various anti-HSV control strategies. Finally, the two-group model is further extended to address the effect of risk structure (i.e., risk of acquiring or transmitting HSV-2). Unlike the two-group model described above, it is shown that the risk-structured model undergoes backward bifurcation under certain conditions (the backward bifurcation property can be removed if the susceptible population is not stratified according to the risk of acquiring infection). Thus, one of the main findings of this thesis is that risk structure can induce the phenomenon of backward bifurcation in the transmission dynamics of HSV-2 in a population.

Imagine fluid moving through a long pipe or channel, and we inject dye or solute into this pipe. What happens to the dye concentration after a long time? Initially, the dye just moves along downstream with the fluid. However, it is also slowly diffusing down the pipe and towards the edges as well. It turns out that after a long time, the combined effect of transport via the fluid and this slow diffusion results in what is effectively a much more rapid diffusion process, lengthwise down the stream. If 0