Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Um bilhar consiste basicamente de uma partÃcula confinada em uma regiÃo do espaÃo. Trataremos apenas de bilhares em duas dimensÃes na ausÃncia de campos externos e desprezaremos qualquer tipo de forÃas dissipativas, de modo que as colisÃes da partÃcula com as fronteiras do bilhar sÃo elÃsticas. AlÃm disso, as fronteiras sÃo fixas, ou seja, respeitam uma equaÃÃo do tipo $R = R(r, heta)$, onde r e $ heta$ sÃo as coordenadas polares planas. O bilhar à um modelo interessante por vÃrios motivos. Primeiro, à um sistema muito simples (tem poucos graus de liberdade) e de fÃcil visualizaÃÃo. No entanto, possui uma dinÃmica nÃo-trivial com grande riqueza de comportamentos (podendo apresentar comportamento regular, caÃtico ou atà mesmo misto, caso em que coexistem no espaÃo de fase de um Ãnico bilhar regiÃes caÃticas e regulares). Segundo, o tratamento numÃrico desses sistemas nÃo requer integraÃÃo numÃrica de equaÃÃes diferenciais e, portanto, nÃo consume muito tempo de execuÃÃo. AlÃm disso, os bilhares permitem que realizemos investigaÃÃes de carÃter fundamental, por exemplo, podemos estudar como sistemas regulares reagem ao serem levemente perturbados. Especificamente, iremos aplicar uma rugosidade na fronteira do bilhar circular e elÃptico e observar como o espaÃo de fase irà mudar ao sofrer tal perturbaÃÃo. / In this work we are going to study a physical system known as billiard. A billiard is defined to be basically a confined particle in a closed region of the space. We are going to deal with only two-dimensionals billiards in the absence of extern fields and to neglect any kind of dissipative forces, in a way that the colisions of the particle with the boundary are elastics. Beyond that, the boundary are fixed, it means they respect an equation of kind $R(r, heta)$, where $r$ and $ heta$ are the polar coordinates on a plan. A billiard is a very interesting model by several reasons. First, it is a simple system (it has a few degree of freedom) and it is of easy visualization. However, it has a non-trivial dynamics with a big richness of behaviors (from a billiard it could appear regular behavior, chaotic behavior, or even a mixed behavior, where coexist in the phase space of one billiard chaotics and regular regions). Second, the numerical approach of these systems does not require numerical integration of diferential equations and, therefore, does not take too much time of execution. Furthermore, the billiards allow us to perform investigations of fundamental nature, for example, we can study how regular systems react by being slightly disturbed. Especificaly, we perform a rugosity perturbation on the billiard surface and observe how the phase space is going to change.

Neste trabalho é abordado o exemplo proposto por Nicolai Haydn, no qual é dado um exemplo de um deslocamento onde é possível construir in nitas medidas de máxima entropia, além de in nitos estados de equilíbrio. / In this work, we present the example shown by Nicolai Haydn, which is given by subshift where is possible to show in nity measures of maximal entropy, besides in nitely many distinct equilibrium states.

Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T09:38:10Z (GMT). No. of bitstreams: 1 Mereu_AnaCristinadeOliveira_D.pdf: 1463250 bytes, checksum: 9bbe3e5b625f68effb7acc05409359ea (MD5) Previous issue date: 2009 / Resumo: Este trabalho é voltado ao estudo de existência e persistência de órbitas periódicas e órbitas homoclínicas em perturbações de sistemas dinamicos reversíveis. Primeiramente, rompemos a reversibilidade de centros no plano e em dimensões superiores e detectamos condições para a existência de ciclos limites e toros invariantes. A seguir, estudamos a existência de soluções periódicas simétricas de perturbações de uma família de equações diferencias reversíveis. A existência e persistência de órbitas homoclínicas em tais equações também foram discutidas. / Abstract: In this work we study the existence and persistence of some minimal sets in perturbations of reversible systems. First we make non reversible perturbations of centers in R2 and R4 and we detect conditions for the existence of limit cycles and invariant tori. We study the existence of periodic solutions of the perturbations of a family of di_erential equations expressed by x(2n) + a (2n-2)/2 +¿+ a1x(2) + x = 0 ; for n = 2; 3. The existence and persistence of homoclinic orbits in such equations are also discussed. / Doutorado / Geometria e Topologia / Doutor em Matemática

We study certain aspects of the Möbius randomness principle and more specifically the Möbius disjointness conjecture of P. Sarnak. In paper A we establish this conjecture for all orientation preserving circle homeomorphisms and continuous interval maps of zero entropy. In paper B we show, that for all subshifts of finite type with positive topological entropy the Möbius disjointness does not hold. In paper C we study a class of three-interval exchange maps arising from a paper of Bourgain and estimate its Hausdorff dimension. In paper D we consider the Chowla and Sarnak conjectures and the Riemann hypothesis for abstract sequences and study their relationship. / <p>QC 20171016</p>

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.

In this thesis I explore three new topics in Dynamical Systems. In Chapters 2 and 3 I investigate the dynamics of a family of asynchronous linear systems. These systems are of interest as models for asynchronous processes in economics and computer science and as novel ways to solve linear equations. I find a tight sandwich of bounds relating the Lyapunov exponents of these asynchronous systems to the eigenvalue of their synchronous counterparts. Using ideas from the theory of IFSs I show how the random behavior of these systems can be quickly sampled and go some way to characterizing the associated probability measures. In Chapter 4 I consider another family of random linear dynamical system but this time over the Max-plus semi-ring. These models provide a linear way to model essentially non-linear queueing systems. I show how the topology of the queue network impacts on the dynamics, in particular I relate an eigenvalue of the adjacency matrix to the throughput of the queue. In Chapter 5 I consider non-smooth systems which can be used to model a wide variety of physical systems in engineering as well as systems in control and computer science. I introduce the Moving Average Transformation which allows us to systematically 'smooth' these systems enabling us to apply standard techniques that rely on some smoothness, for example computing Lyapunov exponents from time series data.

The main purpose of this thesis to examine the controllability and observability of the linear discrete fractional systems. First we introduce the problem and continue with the review of some basic definitions and concepts of fractional calculus which are widely used to develop the theory of this subject. In Chapter 3, we give the unique solution of the fractional difference equation involving the Riemann-Liouville operator of real order between zero and one. Additionally we study the sequential fractional difference equations and describe the way to obtain the state-space repre- sentation of the sequential fractional difference equations. In Chapter 4, we study the controllability and observability of time-invariant linear nabla fractional systems.We investigate the time-variant case in Chapter 5 and we define the state transition matrix in fractional calculus. In the last chapter, the results are summarized and directions for future work are stated.

In this dissertation we develop aspects of ergodic theory for C*-dynamical systems for which the C*-algebras are allowed to be noncommutative. We define four ergodic properties, with analogues in classic ergodic theory, and study C*-dynamical systems possessing these properties. Our analysis will show that, as in the classical case, only certain combinations of these properties are permissable on C*-dynamical systems. In the second half of this work, we construct concrete noncommutative C*-dynamical systems having various permissable combinations of the ergodic properties. This shows that, as in classical ergodic theory, these ergodic properties continue to be meaningful in the noncommutative case, and can be useful to classify and analyse C*-dynamical systems. / Dissertation (MSc)--University of Pretoria, 2013. / gm2014 / Mathematics and Applied Mathematics / unrestricted

In this work, we studied three selected problems in FRW spacetime. In the first part, we analysed the motion of a test particle in the homogeneous and isotropic universe. We presented a framework in which one can derive the uniformly accelerated trajectory and geodesic motion if a scale factor for a given spacetime is provided as a function of coordinate time. By applying the confomal time transformation, we were able to convert second order differential equations of motion in FRW spacetime to first order differential equations. From this, we managed to obtain a formalism to derive the uniformly accelerated trajectory of a test particle in spatially curved FRW spacetime. The second part of this work is devoted to dynamical cosmology. In particular, we analyse the cases of barotropic fluids and non-minimally coupled scalar field in spatially curved FRW spacetime. First, we set up the dynamical systems for an unspecified EoS of a barotropic fluid case and an unspecified positive potential for a non-minimal coupled scalar field case. For both of these systems, we determined well-defined dynamical variables valid for all curvatures. In the framework of these general setups we discovered several characteristic features of the systems, such as invariant subsets, symmetries, critical points and their...

This study was directed toward the development of a bistable wall attachment Flip-Flop device which was capable of directionally controlling particle flow. The particles were transported by a fluid stream which under the influence of wall attachment. The dominant criteria in the development of the device was the achievement of the highest recovery of particles at the active output, without destroying the wall attachment of the fluid stream The experiment was conducted in several distinct stages; each of which was concerned with at least one aspect of wa1l attachment or particle flow. Results derived from one test were used to develop the criteria for the next experimental arrangement. Two experimental models were constructed: one of plywood with only one attachment wall, and one of plexiglas which had two attachment walls and was bistable. The plywood model was used in testing wall attachment and particle recovery as a function of the attachment wall angle. From these tests it was concluded that the optimum wall angle was 18 degrees from the center line of the device. Observations of particle action in the plywood model led to the incorporation of additional features in the plexiglas model. They were: an extended nozzle, the elimination of the separation bubble, and the development of smooth transitions at the corners. The plexiglas model was used to investigate optimum splitter location, the effect of jet velocity on recovery efficiency, the effect of vents on the performance of the device, and the performance of the device using a water jet. In the last stages of testing, moving parts and additional output features were used in conjunction with the bistable device to improve the collection efficiency. Some observations resulting from the data gathered in the various tests include: 1. The higher the jet velocity, the greater the wall attachment. 2. The higher the density and viscosity of the fluid stream the greater the recovery of particles at the active output. 3. Particles with large inertial forces were controlled less by the attached jet stream. The addition 0f vents in the device may produce greater particle recovery. 5. The use of moving parts and variations in the output leg design can produce 100 percent particle recovery. This study indicated that it was possible to control the directional flow of particles with the bistable wall attachment device which was developed. However, the pure fluid bistable device could not achieve 100 percent recovery of particles. The addition of moving parts or variations in the output leg design can produce 100 percent recovery of the particles. The use of a bistable device could provide simplicity, reliability and adaptability in transporting materials for industrial processes.