Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2018-03-28T11:58:56Z No. of bitstreams: 2 Dissertação - Jeferson Arley Poveda Contreras - 2018.pdf: 763599 bytes, checksum: 6800571168e0aa9de85d151e4c912725 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-03-29T11:29:24Z (GMT) No. of bitstreams: 2 Dissertação - Jeferson Arley Poveda Contreras - 2018.pdf: 763599 bytes, checksum: 6800571168e0aa9de85d151e4c912725 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-03-29T11:29:24Z (GMT). No. of bitstreams: 2 Dissertação - Jeferson Arley Poveda Contreras - 2018.pdf: 763599 bytes, checksum: 6800571168e0aa9de85d151e4c912725 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-03-07 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / The goal of this work is study limit cycles of piecewise smooth vector fields. First, we present the basic theory, passing through the areas of analysis, qualitative theory of differential equations and algebra. We also present basic concepts of Filippov fields, which are indispensable for the study of piecewise smooth fields. In chapter one, was the main topic, a general method for finding limit cycles will be described; in the second chapter limit cycles are found in a piecewise smooth vector field with non-degenerate center being perturbed by a piecewise polynomial vector field. In the fourth chapter, we study limit cycles in piecewise smooth Hamiltonian fields. / O objetivo deste trabalho é estudar ciclos limite de campos de vetores suaves por parte. Primeiro apresentaremos a teoria básica, passando pelas áreas de análise, teoria qualitativa das equações diferenciais e álgebra. Apresentamos também conceitos básicos de campos de Filippov, os quais são imprescindíveis para o estudo dos campos suaves por partes. No capítulo dos, como tópico principal, será descrito um método geral para encontrar ciclos limite; no segundo três são encontrados ciclos limites em um campo de vetores suave por partes com um centro não degenerado sendo perturbado por um polinômio. No quarto capitulo estudaremos os ciclos limites de campos de vetores Hamiltonianos por parte.

Submitted by Liliane Ferreira (ljuvencia30@gmail.com) on 2018-06-14T11:12:47Z No. of bitstreams: 2 Tese - João Lopes Cardoso Filho - 2018.pdf: 1729607 bytes, checksum: 8279e98ec23b68bab062f8c812957bf4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-06-15T10:25:16Z (GMT) No. of bitstreams: 2 Tese - João Lopes Cardoso Filho - 2018.pdf: 1729607 bytes, checksum: 8279e98ec23b68bab062f8c812957bf4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-06-15T10:25:16Z (GMT). No. of bitstreams: 2 Tese - João Lopes Cardoso Filho - 2018.pdf: 1729607 bytes, checksum: 8279e98ec23b68bab062f8c812957bf4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-05-18 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / In this work we exhibit canonical forms for 2D codimension one piecewise smooth vector Fields (PSVF). All possible orientations and codimension one scenarios were covered. Also the intrinsic objects that characterize each one of the canonical forms were presented. Also we present topological distinct canonical forms for a larger class for symmetric PSVF where the set of fixed points is contained in the variety os discontinuity. Finally we analyze the simultaneous occurrence of sliding and crossing limit cycle in the case where the piecewise linear vector fields presents a continuum of periodic orbits. / Neste trabalho exibiremos inicialmente as formas canônicas para campos vetoriais suaves por partes (PSVF) no plano. Todas os possíveis cenários de codimensão um são abordados. Também apresentamos formas canônicas topologicamente distintas para uma classe de PSVF com simetria onde o conjunto de pontos fixos está contido na variedade de descontinuidade. Finalmente, analisaremos a ocorrência simultânea de ciclos limite costurantes e deslizantes no caso linear por partes que apresentam um contínuo de órbitas periódicas.

Orientador: Tiago de Carvalho / Resumo: Nesta tese abordaremos aspectos qualitativos e dinâmicos de problemas envolvendo campos de vetores suaves por partes, também conhecidos como campos descontínuos. Primeiramente, apresentamos aplicações da teoria de campos de vetores descontínuos em modelos de tratamento intermitente de Câncer e Vírus da Imunodeficiência Humana onde exibimos a existência de singularidades típicas e órbitas periódicas. Ainda no contexto de aplicações, revisitamos um modelo predador-presa descontínuo de modo a concluir que o mesmo tem um comportamento caótico através da existência de uma órbita de Shilnikov. Posteriormente, respondemos questões sobre existência de conjuntos minimais e caóticos para campos de vetores descontínuos na esfera bidimensional. Em seguida, partimos ao estudo de bifurcação de ciclos limites em campos de vetores descontínuos tri e bidimensionais. No primeiro caso, perturbamos um campo descontínuo tangente a uma folheação por toros de modo a gerar uma quantidade finita ou infinita de ciclos limites. No segundo caso, estudamos uma família de campos descontínuos apresentando uma dobra-dobra invisível de costura, sua ciclicidade e a relação entre os coeficientes de Lyapunov desta família e sua regularização. Além disso, estudamos campos vetoriais suaves por partes Hamiltonianos contendo uma dobra-dobra invisível de costura donde apresentamos uma fórmula explícita para o cálculo dos cinco primeiros coeficientes de Lyapunov, além de explorar os diagramas de bifurcação gerados pe... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: In this work we discuss qualitative and dynamic features of problems involving piecewise smooth vector fields, also known as discontinuous vector fields. Firstly, we present applications of discontinuous vector field theory in Human Immunodeficiency Virus and Cancer intermittent treatment models where we exhibit typical singularities and periodic orbits. Moreover, we revisit a discontinuous predator-prey model in order to conclude that it has a chaotic behavior through the existence of a Shilnikov orbit. Next, we answer questions about the existence of minimal and chaotic sets in the bidimensional sphere for discontinuous vector fields. Subsequently, we investigate the creation of limit cycles in three and two-dimensional discontinuous vector fields. In the first case, we perturb a discontinuous vector field tangent to a foliation composed by topological nested tori to generate a finite or infinite number of limit cycles. In the second case, we analyze a family of discontinuous vector fields containing a crossing invisible fold-fold, their cyclicity and the relation between the Lyapunov coefficients of this family and their regularization. Also, we study general piecewise Hamiltonian vector fields presenting a crossing invisible fold-fold where we give an explicit formula for the computation of the five first Lyapunov coefficients in addition to the investigation of the bifurcation diagrams. / Doutor

Title: Approximation, numerical realization and qualitative analysis of contact problems with friction Author: Tomáš Ligurský Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Jaroslav Haslinger, DrSc., Department of Numerical Mathe- matics Abstract: This thesis deals with theoretical analysis and numerical realization of dis- cretized contact problems with Coulomb friction. First, discretized 3D static contact prob- lems with isotropic and orthotropic Coulomb friction and solution-dependent coefficients of friction are analyzed by means of the fixed-point approach. Existence of at least one solution is established for coefficients of friction represented by positive, bounded and con- tinuous functions. If these functions are in addition Lipschitz continuous and upper bounds of their values together with their Lipschitz moduli are sufficiently small, uniqueness of the solution is guaranteed. Second, properties of solutions parametrized by the coefficient of friction or the load vector are studied in the case of discrete 2D static contact problems with isotropic Coulomb friction and coefficient independent of the solution. Conditions under which there exists a local Lipschitz continuous branch of solutions around a given reference point are established due to two variants of the...

The first part of this Thesis deals with the analysis of piecewise-smooth mechanical systems and the definition of special stability criteria in presence of non-conservative follower forces. To illustrate the peculiar stability properties of this kind of dynamical system, a reference 2 d.o.f. structure has been considered, composed of a rigid bar, with one and constrained to slide, without friction, along a curved profile, whereas the other and is subject to a follower force. In particular, the curved constraint is assumed to be composed of two circular profiles, with different and opposite curvatures, defining two separated subsystems. Due to this jump in the curvature, located at the junction point between the curved profiles, the entire mechanical structure can be modelled by discontinuous equations of motion, the differential equations valid in each subsystem can be combined, leading to the definition of a piecewise-smooth dynamical system. When a follower force acts on the structure, an unexpected and counterintuitive behaviour may occur: although the two subsystems are stable when analysed separately, the composed structure is unstable and exhibits flutter-like exponentially-growing oscillations. This special form of instability, previously known only from a mathematical point of view, has been analysed in depth from an engineering perspective, thus finding a mechanical interpretation based on the concept of non-conservative follower load. Moreover, the goal of this work is also the definition of some stability criteria that may help the design of these mechanical piecewise-smooth systems, since classical theorems cannot be used for the investigation of equilibrium configurations located at the discontinuity. In the literature, this unusual behaviour has been explained, from a mathematical perspective, through the existence of a discontinuous invariant cone in the phase space. For this reason, starting from the mechanical system described above, the existence of invariant cones in 2 d.o.f. mechanical systems is investigated through Poincaré maps. A complete theoretical analysis on piecewise-smooth dynamical systems is presented and special mathematical properties have been discovered, valid for generic 2~d.o.f. piecewise-smooth mechanical systems, which are useful for the characterisation of the stability of the equilibrium configurations. Numerical tools are implemented for the analysis of a 2~d.o.f. piecewise-smooth mechanical system, valid for piecewise-linear cases and extendible to the nonlinear ones. A numerical code has been developed, with the aim of predicting the stability of a piecewise-linear dynamical system a priori, varying the mechanical parameters. Moreover, “design maps” are produced for a given subset of the parameters space, so that a system with a desired stable or unstable behaviour can easily be designed. The aforementioned results can find applications in soft actuation or energy harvesting. In particular, in systems devoted to exploiting the flutter-like instability, the range of design parameters can be extended by using piecewise-smooth instead of smooth structures, since unstable flutter-like behaviour is possible also when each subsystem is actually stable. The second part of this Thesis deals with the numerical analysis of an elastic cloak for transient flexural waves in Kirchhoff-Love plates and the design of special metamaterials for this goal. In the literature, relevant applications of transformation elastodynamics have revealed that flexural waves in thin elastic plates can be diverted and channelled, with the aim of shielding a given region of the ambient space. However, the theoretical transformations which define the elastic properties of this “invisibility cloak” lead to the presence of a strong compressive prestress, which may be unfeasible for real applications. Moreover, this theoretical cloak must present, at the same time, high bending stiffness and a null twisting rigidity. In this Thesis, an orthotropic meta-structural plate is proposed as an approximated elastic cloak and the presence of the prestress has been neglected in order to be closer to a realistic design. With the aim of estimating the performance of this approximated cloak, a Finite Element code is implemented, based on a sub-parametric technique. The tool allows the investigation of the sensitivity of specific stiffness parameters that may be difficult to match in a real cloak design. Moreover, the Finite Element code is extended to investigate a meta-plate interacting with a Winkler foundation, to analyse how the substrate modulus transforms in the cloak region. This second topic of the Thesis may find applications in the realization of approximated invisibility cloaks, which can be employed to reduce the destructive effects of earthquakes on civil structures or to shield mechanical components from unwanted vibrations.

Работа содержит исследования трех моделей живых систем с дискретным временем. В первой главе рассматривается одномерная модель нейронной активности, задаваемая кусочно-гладким отображением. Показывается, что в случае одномерного отображения наличие случайного возмущения приводит к появлению всплесков (спайкингу). Исследуются два механизма генерации спайков, вызванных добавлением случайного возмущения в один из параметров. Иллюстрируется, что сосуществование двух аттракторов является не единственной причиной возникновения спайкинга. Для прогнозирования уровня интенсивности шума, необходимого для генерации спайков, применяется метод доверительных областей, который основан на функции стохастической чувствительности. Также находятся основные характеристики межспайковых интервалов в зависимости от интенсивности шума. Вторая глава работы посвящена применению метода функции стохастической чувствительности к аттракторам кусочно-гладкого одномерного отображения, описывающего динамику численности популяции. Первым этапом исследования является параметрический анализ возможных режимов детерминированной модели: определение зон существования устойчивых равновесий и хаотических аттракторов. Для определения параметрических границ хаотического аттрактора применяется теория критических точек. В случае, когда на систему оказывает влияние случайное воздействие, на основе техники функции стохастической чувствительности дается описание разброса случайных состояний вокруг равновесия и хаотического аттрактора. Проводится сравнительный анализ влияния параметрического и аддитивного шума на аттракторы системы. С помощью техники доверительных интервалов изучаются вероятностные механизмы вымирания популяции под действием шума. Анализируются изменения параметрических границ существования популяции под действием случайного возмущения. В третьей главе проводится анализ возможных динамических режимов детерминированной и стохастической модели Лотки-Вольтерры. В зависимости от двух параметров системы строится карта режимов. Изучаются параметрические зоны существования устойчивых равновесий, циклов, замкнутых инвариантных кривых, а также хаотических аттракторов. Описываются бифуркации удвоения периода, Неймарка--Саккера и кризиса. Демонстрируется сложная форма бассейнов притяжения. Помимо детерминированной системы подробно изучается стохастическая, описывающая влияние внешнего случайного воздействия. В случае хаоса дан алгоритм нахождения критических линий, описывающих границу хаотического аттрактора. Опираясь на найденную чувствительность аттракторов, строятся доверительные полосы и эллипсы, позволяющие описать разброс случайных состояний вокруг детерминированного аттрактора. / The work contains study of three models of biological systems with discrete time. In the first chapter a one-dimensional model of neural activity defined by a piecewise-smooth map is considered. It is shown that in the case of a one-dimensional model, the presence of a random disturbance leads to a spike generation. Two mechanisms of spike generation caused by the presence of a random disturbance in one of the parameters are investigated. It is illustrated that the coexistence of two attractors is not the only reason of spiking. To predict the level of noise intensity needed to generate spikes, the confidence-domain method is used, which is based on the stochastic sensitivity function. The main characteristics of interspike intervals depending on the intensity of the noise are also described. The second chapter is devoted to the application of the method of the stochastic sensitivity function to attractors of a piecewise-smooth one-dimensional map, which describes the population dynamics. The first stage of the study is a parametric analysis of the possible regimes of the deterministic model: determining the zones of existence of stable equilibria and chaotic attractors. The theory of critical points is used to determine the parametric boundaries of a chaotic attractor. In the case where the system is affected by a random noise, based on the stochastic sensitivity function, a description of the spread of random states around equilibrium and a chaotic attractor is given. A comparative analysis of the influence of parametric and additive noise on the attractors is carried out. Using the technique of confidence intervals, the probabilistic mechanisms of extinction of a population under the influence of noise are studied. Changes in the parametric boundaries of the existence of population under the influence of random disturbance are analyzed. In the third chapter the possible dynamic modes of the Lotka-Volterra model in determi\-nistic and stochastic cases are analyzed. Depending on the two parameters of the system, bifurcation diagram is constructed. Parametric zones of the existence of stable equilibria, cycles, closed invariant curves, and also chaotic attractors are studied. The bifurcations of the period doubling, Neimark--Sacker and the crisis are described. The complex shape of the basins of attraction is demonstrated. In addition to the deterministic system, the stochastic system is studied in detail, which describes the influence of external random disturbance. In the case of chaos, an algorithm for finding critical lines describing the boundary of a chaotic attractor is given. Based on the stochastic sensitivity function, confidence bands and ellipses are constructed to describe the spread of random states around a deterministic attractor.