Le fonctionnement d'un neurone, unité fondamentale du système nerveux, intéresse de nombreuses disciplines scientifiques. Il existe ainsi des modèles mathématiques qui décrivent leur comportement par des systèmes d'EDO ou d'EDP. Plusieurs de ces modèles peuvent ensuite être couplés afin de pouvoir étudier le comportement de réseaux, systèmes complexes au sein desquels émergent des propriétés. Ce travail présente, dans un premier temps, les principaux mécanismes régissant ce fonctionnement pour en comprendre la modélisation. Plusieurs modèles sont alors présentés, jusqu'à celui de FitzHugh-Nagumo (FHN), qui présente une dynamique très intéressante.C'est sur l'étude théorique mais également numérique de la dynamique asymptotique et transitoire du modèle de FHN en EDO, que se concentre la seconde partie de cette thèse. A partir de cette étude, des réseaux d'interactions d'EDO sont construits en couplant les systèmes dynamiques précédemment étudiés. L'étude du phénomène de synchronisation identique au sein de ces réseaux montre l'existence de propriétés émergentes pouvant être caractérisées par exemple par des lois de puissance. Dans une troisième partie, on se concentre sur l'étude du système de FHN dans sa version EDP. Comme la partie précédente, des réseaux d'interactions d'EDP sont étudiés. On entreprend dans cette partie une étude théorique et numérique. Dans la partie théorique, on montre l'existence de l'attracteur global dans l'espace L2(Ω)nd et on donne des conditions suffisantes de synchronisation. Dans la partie numérique, on illustre le phénomène de synchronisation ainsi que l'émergence de lois générales telles que les lois puissances ou encore la formation de patterns, et on étudie l'effet de l'ajout de la dimension spatiale sur la synchronisation. / The neuron, a fundamental unit in the nervous system, is a point of interest in many scientific disciplines. Thus, there are some mathematical models that describe their behavior by ODE or PDE systems. Many of these models can then be coupled in order to study the behavior of networks, complex systems in which the properties emerge. Firstly, this work presents the main mechanisms governing the neuron behaviour in order to understand the different models. Several models are then presented, including the FitzHugh-Nagumo one, which has a interesting dynamic. The theoretical and numerical study of the asymptotic and transitory dynamics of the aforementioned model is then proposed in the second part of this thesis. From this study, the interaction networks of ODE are built by coupling previously dynamic systems. The study of identical synchronization phenomenon in these networks shows the existence of emergent properties that can be characterized by power laws. In the third part, we focus on the study of the PDE system of FHN. As the previous part, the interaction networks of PDE are studied. We have in this section a theoretical and numerical study. In the theoretical part, we show the existence of the global attractor on the space L2(Ω)nd and give the sufficient conditions for identical synchronization. In the numerical part, we illustrate the synchronization phenomenon, also the general laws of emergence such as the power laws or the patterns formation. The diffusion effect on the synchronization is studied.

Potential theory is the name given to the broad field of analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, Green’s functions, potentials and capacity. In this text, our main goal will be to gain a deeper understanding towards complex dynamics, the study of dynamical systems defined by the iteration of analytic functions, using the tools and techniques of potential theory. We will restrict ourselves to holomorphic polynomials in C. At first, we will discuss briefly about harmonic and subharmonic functions. In course, potential theory will repay its debt to complex analysis in the form of some beautiful applications regarding the Julia sets (defined in Chapter 8) of a certain family of polynomials, or a single one. We will be able to provide an explicit formula for computing the capacity of a Julia set, which in some sense, gives us a finer measurement of the set. In turn, this provides us with a sharp estimate for the diameter of the Julia set. Further if we pick any point w from the Julia set, then the inverse images q−n(w) span the whole Julia set. In fact, the point-mass measures with support at the discrete set consisting of roots of the polynomial, (qn-w) will eventually converge to the equilibrium measure of the Julia set, in the weak*-sense. This provides us with a very effective insight into the analytic structure of the set. Hausdorff dimension is one of the most effective notions of fractal dimension in use. With the help of potential theory and some ergodic theory, we can show that for a certain holomorphic family of polynomials varying over a simply connected domain D, one can gain nice control over how the Hausdorff dimensions of the respective Julia sets change with the parameter λ in D.

Mathematical models provide unique insights to real-world problems. Within the context of infectious diseases, models are used to explore the dynamics of infections and control mechanisms. Human papillomavirus (HPV) globally infects about 630 million people, many of these infections develop into cancers and genital warts. Vaccines are available to protect against the most prevalent and devastating strains of HPV. The introduction of this vaccine as part of a national immunization program in Canada is a complex decision for policy-makers in which mathematical models can play a key role. We use the current recommendations provided by the World Health Organization to explore the integral role mathematical models have in the decision to incorporate the HPV vaccine within a national immunization program. We then provide a review of the literature discussing the role of mathematical models in the decision to include a vaccine in a national immunization program within the context of the HPV vaccine. Next, we evaluate the current standing of mathematical models used within the context of HPV immunization, to highlight the types of models used, underlying assumptions and general recommendations made about these immunization programs. Then, we create and analyze a model to explore the possibility of bettering the current HPV vaccine strategy in Canada. We focus on the effects of the grade of vaccination and the number of doses required to eradicate the targeted strains of HPV.

Orientador: Bianca Morelli Rodolfo Calsavara / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-26T15:10:31Z (GMT). No. of bitstreams: 1 SalesFilho_Nazime_M.pdf: 36937949 bytes, checksum: 4ceff2992bbc8648a89104715aac602e (MD5) Previous issue date: 2015 / Resumo: Neste trabalho será analisado um modelo matemático que descreve a propagação da dengue. Tal modelo é dado por um sistema de equações diferenciais ordinárias não lineares sujeitas a condições iniciais, que descreve duas populações: a de mosquitos e a humana. A população de mosquitos é dividida em duas subpopulações: fase aquática, incluindo os ovos, larvas e pupas, e fase alada, que é subdividida em mosquitos suscetíveis e infectados. A população humana é dividida em subpopulações de suscetíveis, infectados e removidos. No modelo citado é assumido que a população de mosquito e a população humana atingiram homogeneidade espacial, isto é, não há movimentação destas populações influenciando na disseminação da doença. O principal interesse neste trabalho é analisar qualitativamente o comportamento das populações em torno dos pontos de equilíbrio do sistema. Para este fim, além do uso de ferramentas analíticas também foram realizadas simulações numéricas utilizando o software Maple. Dessa forma foi possível obter informações sobre a disseminação da dengue, sob algumas hipóteses, mesmo sem obtermos solução explícita do sistema / Abstract: In this work it will be analyzed a mathematical model describing propagation of dengue disease. This model is given by a system of nonlinear ordinary differential equations, subjected to initial conditions, involving two populations: one of mosquitos and another of humans. The mosquitos population is divided in two subpopulations: the aquatic phase, including eggs, larvae and pupae, and the winged phase, that is divided in susceptible and infected mosquitos. The human population is divided in subpopulations of susceptible, infected and removed. In the cited model it is assumed that the mosquito and human populations achieved spatial homogeneity, i.e., there is no movement of these populations affecting the disease dissemination. The main interest of this work is to analyze qualitatively the populations behavior around the equilibrium points of the system. To this end, in addition to the use of analytical tools, numerical simulations were performed by using Maple software. In this way, it was possible to obtain information about dengue dissemination, under some hypotheses, even without obtaining explicit solution for the system / Mestrado / Matematica Aplicada e Computacional / Mestre em Matemática Aplicada e Computacional

In 2019, Adams, Ehlting, and Edwards developed a four-variable system of ordinary differential equations modelling phenylalanine metabolism in plants according to Michaelis-Menten kinetics. Analysis of the model suggested that when a series of reactions known as the Shikimate Ester Loop (SEL) is included, phenylalanine flux into primary metabolic pathways is prioritized over flux into secondary metabolic pathways when the availability of shikimate, a phenylalanine precursor, is low. Adams et al. called this mechanism of metabolic regulation the Precursor Shutoff Valve (PSV). Here, we attempt to simplify Adams and colleagues’ model by reducing the system to three variables and replacing the Michaelis-Menten terms with piecewise-defined approximations we call ramp functions. We examine equilibria and stability in this simplified model, and show that PSV-type regulation is still present in the version with the SEL. Then, we define a class of systems structurally similar to the simplified Adams model called biochemical ramp systems. We study the properties of the Jacobian matrices of these systems and then explore equilibria and stability in systems of n ≥ 2 variables. Finally, we make several suggestions regarding future work on biochemical ramp systems. / Graduate

This thesis is intended to provide fundamental information for the construction and analysis of rotordynamic theoretical models, and their comparison the experimental systems. Finite Element Method (FEM) is used to construct models using Timoshenko beam elements with viscous and hysteretic internal damping. Eigenvalues and eigenvectors of state space equations are used to perform stability analysis, produce critical speed maps, and visualize mode shapes. Frequency domain analysis of theoretical models is used to provide Bode diagrams and in experimental data full spectrum cascade plots. Experimental and theoretical model analyses are used to optimize the control algorithm for an Active Magnetic Bearing on an overhung rotor.

This master's thesis deals with solving fractional-order ordinary differential equations by the Adomian decomposition method. A part of the work is therefore devoted to the theory of equations containing differential operators of non-integer order, especially the Caputo operator. The next part is devoted to the Adomian decomposition method itself, its properties and implementation in the case of Chen system. The work also deals with bifurcation analysis of this system, both for integer and non-integer case. One of the objectives is to clarify the discrepancy in the literature concerning the fractional-order Chen system, where experiments based on the use of the Adomian decomposition method give different results for certain input parameters compared with numerical methods. The clarification of this discrepancy is based on recent theoretical knowledge in the field of fractional-order differential equations and their systems. The conclusions are supported by numerical experiments, own code implementing the Adomian decomposition method on the Chen system was used.

Title: Bifurcation of Ordinary Differential Equations from Points of Fučík Spectrum Author: Vendula Exnerová Department: Department of Mathematical Analysis Supervisor: doc. RNDr. Jana Stará, CSc., Department of Mathematical Analysis MFF UK, Prague Abstract: The main subject of the thesis is the Fučík spectrum of a system of two differential equations of the second order with mixed boundary conditions. In the first part of the thesis there are described Fučík spectra of problems of a differential equation with Dirichlet, mixed and Neumann boundary conditions. The other part deals with systems of two differential equations. It attends to basic properties of systems and their nontrivial solutions, to a possibility of a reduction of number of parameters and to a dependance of a problem with mixed boundary condition on one with Dirichlet boundary conditions. The thesis takes up the results of E. Massa and B. Ruff about the Dirichlet problem and improves some of their proofs. In the end the Fučík spectrum of a problem with mixed boundary conditions is described as the union of countably many continuously differentiable surfaces and there is proven that this spectrum is closed.

The solving of stiff systems is still a contemporary sophisticated problem. The basic problem is the absence of precise definition of stiff systems. A question is also how to detect the stiffness in a given system of differential equations. Implicit numerical methods are commonly used for solving stiff systems. The stability domains of these methods are relatively large but the order of them is low.   The thesis deals with numerical solution of ordinary differential equations, especially numerical calculations using Taylor series methods. The source of stiffness is analyzed and the possibility how to reduce stiffness             in systems of ordinary differential equations (ODEs) is introduced. The possibility of detection stiff systems using explicit Taylor series terms is analyzed. The stability domains of explicit and implicit Taylor series are presented. The solutions of stiff systems using implicit Taylor series method are presented in many examples. The multiple arithmetic must be used in many cases. The new suitable parallel algorithm based on implicit Taylor series method with recurrent calculation of Taylor series terms and Newton iteration method (ITMRN) is proposed.

This thesis presents a numerical approach based on generalized fractional-order Chebyshev wavelets for solving fractional-order optimal control problems. The exact value of the Riemann– Liouville fractional integral operator of the generalized fractional-order Chebyshev wavelets is computed by applying the regularized beta function. We apply the given wavelets, the exact formula, and the collocation method to transform the studied problem into a new optimization problem. The convergence analysis of the proposed method is provided. The present method is extended for solving fractional-order, distributed-order, and variable-order optimal control problems. Illustrative examples are considered to show the advantage of this method in comparison with the existing methods in the literature.