Piecewise-smooth stochastic systems have attracted a lot of interest in the last decades in engineering science and mathematics. Many investigations have focused only on one-dimensional problems. This thesis deals with simple two-dimensional piecewise-smooth stochastic systems in the absence of detailed balance. We investigate the simplest example of such a system, which is a pure dry friction model subjected to coloured Gaussian noise. The nite correlation time of the noise establishes an additional dimension in the phase space and gives rise to a non-vanishing probability current. Our investigation focuses on stick-slip transitions, which can be related to a critical value of the noise correlation time. Analytical insight is provided by applying the uni ed coloured noise approximation. Afterwards, we extend our previous model by adding viscous friction and a constant force. Then we perform a similar analysis as for the pure dry friction case. With parameter values close to the deterministic stick-slip transition, we observe a non-monotonic behaviour of the probability of sticking by increasing the correlation time of the noise. As the eigenvalue spectrum is not accessible for the systems with coloured noise, we consider the eigenvalue problem of a dry friction model with displacement, velocity and Gaussian white noise. By imposing periodic boundary conditions on the displacement and using a Fourier ansatz, we can derive an eigenvalue equation, which has a similar form in comparison to the known one-dimensional problem for the velocity only. The eigenvalue analysis is done for the case without a constant force and with a constant force separately. Finally, we conclude our ndings and provide an outlook on related open problems.

Piecewise-smooth stochastic systems are widely used in engineering science. However, the theory of these systems is only in its infancy. In this thesis, we take as an example the Brownian motion with dry friction to illustrate dynamical properties of these systems with respect to three interesting topics: (i) weak-noise approximations, (ii) first-passage time (FPT) problems and (iii) functionals of stochastic processes. Firstly, we investigate the validity and accuracy of weak-noise approximations for piecewise-smooth stochastic differential equations (SDEs), taking as an illustrative example the Brownian motion with pure dry friction. For this model, we show that the weak-noise approximation of the path integral correctly reproduces the known propagator of the SDE at lowest order in the noise power, as well as the main features of the exact propagator with higher-order corrections, provided that the singularity of the path integral is treated with some heuristics. We also consider a smooth regularisation of this piecewise-constant SDE and study to what extent this regularisation can rectify some of the problems encountered in the non-smooth case. Secondly, we provide analytic solutions to the FPT problem of the Brownian motion with dry friction. For the pure dry friction case, we find a phase transition phenomenon in the spectrum which relates to the position of the exit point and affects the tail of the FPT distribution. For the model with dry and viscous friction, we evaluate quantitatively the impact of the corresponding stick-slip transition and of the transition to ballistic exit. We also derive analytically the distributions of the maximum velocity till the FPT for the dry friction model. Thirdly, we generalise the so-called backward Fokker-Planck technique and obtain a recursive ordinary differential equation for the moments of functionals in the Laplace space. We then apply the developed results to analyse the local time, the occupation time and the displacement of the dry friction model. Finally, we conclude this thesis and state some related unsolved problems.

Piecewise smooth maps appear as models of various physical, economical and other systems. In such maps bifurcations can occur when a fixed point or periodic orbit crosses or collides with the border between two regions of smooth behaviour as a system parameter is varied. These bifurcations have little analogue in standard bifurcation theory for smooth maps and are often more complex. They are now known as "border collision bifurcations". The classification of border collision bifurcations is only available for one-dimensional maps. For two and higher dimensional piecewise smooth maps the study of border collision bifurcations is far from complete. In this thesis we investigate some of the bifurcation phenomena in two-dimensional continuous piecewise smooth discrete-time systems. There are a lot of studies and observations already done for piecewise smooth maps where the determinant of the Jacobian of the system has modulus less than 1, but relatively few consider models which allow area expansions. We show that the dynamics of systems with determinant greater than 1 is not necessarily trivial. Although instability of the systems often gives less useful numerical results, we show that snap-back repellers can exist in such unstable systems for appropriate parameter values, which makes it possible to predict the existence of chaotic solutions. This chaos is unstable because of the area expansion near the repeller, but it is in fact possible that this chaos can be part of a strange attractor. We use the idea of Markov partitions and a generalization of the affine locally eventually onto property to show that chaotic attractors can exist and are fully two-dimensional regions, rather than the usual fractal attractors with dimension less than two. We also study some of the local and global bifurcations of these attracting sets and attractors.Some observations are made, and we show that these sets are destroyed in boundary crises and some conditions are given.Finally we give an application to a coupled map system.

Orientador: Tiago de Carvalho / Resumo: O objetivo desta dissertação é abordar aspectos qualitativos de sistemas de equações diferenciais ordinárias e sistemas contínuos suaves por partes aplicados à dinâmica do Vírus da Imunodeficiência Humana (HIV). Neste trabalho, apresentamos um modelo matemático que descreve a dinâmica do HIV no corpo humano e o analisamos através da matriz da próxima geração e teoria de estabilidade, com a finalidade de prever se a doença fica ou não controlada. Posteriormente, estudamos um sistema de equações diferenciais ordinárias usado para modelar a dinâmica do vírus para diferentes tipos de tratamentos. Tal modelo foi explorado qualitativamente de duas maneiras: por um sistema contínuo (pelo método de Korobeinikov) e por um descontínuo (pelas convenções de Filippov). Analisamos o comportamento dinâmico de terapias antirretrovirais, visando a diminuição das concentrações virais no sangue, de acordo com a análise da estabilidade realizada. / Abstract: The goal of this dissertation is to study qualitative aspects about systems of ordinary differential equations and piecewise smooth systems applied to the dynamic of Human Immunodeficiency Virus (HIV). In this work, we present a mathematical model that describes the dynamic of HIV in the human body and we analyze this model by next-generation matrix and stability theory in order to predict if the disease becomes stable, and thus stop virus transmission. In addition, we studied another system of ordinary differential equations that were proposed to model the HIV dynamics assuming different therapies. We have explored qualitatively the model by two distinct approaches: a continuous system (by Korobeinikov method) and a discontinuous system (by Filippov theory). Due to the stability analysis, it was possible to understand the dynamics of anti-retroviral therapies, which are responsible for decreasing the concentration of detectable HIV in blood. / Mestre

abstract: Diabetes is a disease characterized by reduced insulin action and secretion, leading to elevated blood glucose. In the 1990s, studies showed that intravenous injection of fatty acids led to a sharp negative response in insulin action that subsided hours after the injection. The molecule associated with diminished insulin signalling response was a byproduct of fatty acids, diacylglycerol. This dissertation is focused on the formulation of a model built around the known mechanisms of glucose and fatty acid storage and metabolism within myocytes, as well as downstream effects of diacylglycerol on insulin action. Data from euglycemic-hyperinsulinemic clamp with fatty acid infusion studies are used to validate the qualitative behavior of the model and estimate parameters. The model closely matches clinical data and suggests a new metric to determine quantitative measurements of insulin action downregulation. Analysis and numerical simulation of the long term, piecewise smooth system of ordinary differential equations demonstrates a discontinuous bifurcation implicating nutrient excess as a driver of muscular insulin resistance. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics for the Life and Social Sciences 2017

Submitted by JÚLIO HEBER SILVA (julioheber@yahoo.com.br) on 2017-06-28T17:43:40Z No. of bitstreams: 2 Dissertação - Marcos Tsujii - 2015.pdf: 919903 bytes, checksum: e6e6bb36b7e9700b446e1807f1854651 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Cláudia Bueno (claudiamoura18@gmail.com) on 2017-07-07T19:52:00Z (GMT) No. of bitstreams: 2 Dissertação - Marcos Tsujii - 2015.pdf: 919903 bytes, checksum: e6e6bb36b7e9700b446e1807f1854651 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-07-07T19:52:00Z (GMT). No. of bitstreams: 2 Dissertação - Marcos Tsujii - 2015.pdf: 919903 bytes, checksum: e6e6bb36b7e9700b446e1807f1854651 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2015-03-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we will study the dynamics in smooth vector elds, in vector elds near the boundary and in piecewise-smooth vector elds and each of their most popular types of bifurcations up to now. / Neste trabalho, estudaremos a dinâmica em campos de vetores suaves, em campos de vetores em variedades com bordo e em campos de vetores suaves por partes e cada um dos seus respectivos tipos de bifurcações mais conhecidos.

No description available.

We propose a prototype model of market dynamics in which all functional relationships are linear. We take into account three borders, defined by linear functions, which are intrinsic to the economic reasoning: non-negativity of prices; downward rigidity of capacity (depreciation) and a capacity constraint for the production decision. Given the linear specification, the borders are the only source for the emerging of cyclical and more complex dynamics. In particular, we discuss centre bifurcations, border collision bifurcations and degenerate flip bifurcations - dynamic phenomena the occurrence of which are intimately related to the existence of borders. / Series: Department of Economics Working Paper Series

We explore the effects of Brexit on trade patterns and on the spatial distribution of industry between the United Kingdom and the European Union and within the EU. Our study adopts a new economic geography (NEG) perspective developing a linear model with three regions, the UK and two separated regions composing the EU. The 3-region framework and linear demands allow for different trade patterns. Two possible ante-Brexit situations are possible, depending on the interplay between local market size, local competition and trade costs: industrial agglomeration or dispersion. Considering a soft and a hard Brexit scenario, the ante-Brexit situation is altered substantially, depending on which scenario prevails. UK firms could move to the larger EU market, even in the peripheral region, reacting to the higher trade barriers, relocation representing a substitute for trade. Alternatively, some EU firms could move in the more isolated UK market finding shelter from the competition inside the EU. We also consider the post-Brexit scenario of deeper EU integration, leading to a weakening of trade links between the EU and the UK. Our analysis also reveals a highly complex bifurcation sequence leading to many instances of multistability, intricate basins of attraction and cyclical and chaotic dynamics. / Series: Department of Economics Working Paper Series

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2016-06-02T21:00:54Z No. of bitstreams: 2 Dissertação - Jeidy Johana Jimenez Ruiz - 2016.pdf: 946402 bytes, checksum: 0a36384eddfdcc5620d74725a24dd86a (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-06-03T11:43:02Z (GMT) No. of bitstreams: 2 Dissertação - Jeidy Johana Jimenez Ruiz - 2016.pdf: 946402 bytes, checksum: 0a36384eddfdcc5620d74725a24dd86a (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) / Made available in DSpace on 2016-06-03T11:43:02Z (GMT). No. of bitstreams: 2 Dissertação - Jeidy Johana Jimenez Ruiz - 2016.pdf: 946402 bytes, checksum: 0a36384eddfdcc5620d74725a24dd86a (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) Previous issue date: 2016-03-04 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / The study under existence and uniqueness of limit cycles of equations systems differential is a very active research topic in the qualitative theory of dynamical systems. In this theme we study this topic in discontinuous dynamic systems. Let’s make this in Liénard differentials equation systems, allowing a line of discontinuity. Furthermore, we present the known method of Averaging firstly in your classic version, that is, for class fields at least C2, we study also to generalized version, to piecewise- smooth dynamical systems. As a result, we use this tool to determine the number of limit cycles that can bifurcate of a planar center, inside the equation Liénard differentials equation class. / O estudo sobre existência e unicidade de ciclos limites de sistemas de equações diferenciais é um tópico de grande interesse na teoria qualitativa de sistemas dinâmicos. Nesta dissertação, estudamos este tópico em sistemas dinâmicos descontínuos. Vamos fazer esta análise em sistemas de equações diferenciais de Liénard, permitindo uma linha de descontinuidade. Além disso, vamos apresentar o conhecido método Averaging de primeira ordem, em primeiro lugar na sua versão clássica, isto é, para campos de classe pelo menos C2, depois apresentaremos também a versão generalizada, para sistemas diferenciais definidos por partes. Como resultado, fazemos uso desta ferramenta para determinar o número de ciclos limites que podem bifurcar de um centro planar, dentro da classe de equações diferenciais de Liénard.