Orientadores: Jose Wilson Magalhães Bassani, Antonio Carlos Guimarães de Almeida / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-12T22:48:22Z (GMT). No. of bitstreams: 1 Azevedo_GersonFlorenceCarvalheirade_D.pdf: 2274193 bytes, checksum: 9c8a1fee0426b5335f1a55b9dfcdd33d (MD5) Previous issue date: 2009 / Resumo: Durante o burst epileptiforme e a depressão alastrante (DA), são observados um aumento da [K+]o (concentração extracelular de potássio) e uma diminuição da [Ca2+]o (concentração extracelular de cálcio), evidenciando a participação deste mecanismo não-sináptico nestes padrões oscilatórios anormais. Essas variações nas [K+]o e [Ca2+]o elevam a excitabilidade neuronal. No entanto, não está claro se a alta [K+]o é um fator primário na geração destas atividades neuronais ou se apenas desempenha um papel secundário neste processo. Para melhor compreender a dinâmica não-linear destes padrões, as condições experimentais de alta [K+]o e zero [Ca2+]o foram replicadas em um modelo ampliado de Golomb, referente à região CA1 da formação hipocampal. Importantes mecanismos regulatórios de concentração iônica, como a bomba Na+/K+, a difusão iônica e o sistema de buffer da glia, foram acrescentados ao modelo de Golomb. Dentro destas condições, foi possível simular atividades elétricas neuronais tipicamente apresentadas no burst epileptiforme em sua fase ictal. A DA foi iniciada pela interrupção da atividade da bomba Na+/K+. O bloqueio da bomba Na+/K+ por meio da hipóxia celular é uma manobra experimental para se obter a DA, conhecida também como depressão alastrante hipóxica - DAH. A teoria de bifurcação e o método fast-slow analysis foram utilizados para estudar a interferência do K+ extracelular na excitabilidade celular. Este estudo indicou que o sistema perde a sua estabilidade com o aumento da [K+]o, transitando para um elevado estado de excitabilidade. Este crescimento da [K+]o provoca bifurcações no comportamento dinâmico neuronal, que determinam transições entre diferentes estágios dessas atividades elétricas. No primeiro estágio, o aumento da [K+]o propicia a deflagração do burst epileptiforme e da DA via bifurcações sela-nó e de Hopf supercrítica, respectivamente. Ao longo da atividade neuronal, o nível de excitabilidade é mantido por meio de um crescimento contínuo da [K+]o, que deprime as correntes de K+ em um processo de realimentação positiva. Neste estágio, em relação ao burst epileptiforme, a amplitude e a freqüência dos disparos dos potenciais de ação são alteradas via bifurcação de Hopf supercrítica. No último estágio, com a depressão das correntes de K+, a bomba de Na+/K+ tem uma participação importante no término da atividade neuronal. O burst epileptiforme e a DA são finalizados por meio das bifurcações sela-órbita homoclínica e sela-nó, respectivamente. Portanto, este trabalho sugere que o K+ extracelular pode desempenhar um papel fundamental na dinâmica não-linear do burst epileptiforme e da sua transição para a DA. / Abstract: During the epileptiform burst and the spreading depression (SD), it is observed an increase of [K+]o (extracellular potassium concentration) and a decrease of [Ca2+]o (extracellular calcium concentration), pointing out the participation of this nonsynaptic mechanism in these abnormal oscillatory patterns. These ionic variations raise the neuronal excitability. However, whether the high [K+]o is a primary factor in the beginning of these neuronal activities or just plays a secondary role into this process is unclear. To better understand the nonlinear dynamics of these patterns, the experimental conditions of high [K+]o and zero [Ca2+]o were replicated in an extended Golomb model in which we added important regulatory mechanisms of ion concentration as Na+/K+ pump, ion diffusion and glial buffering. Within these conditions, it was possible to simulate epileptiform burst within the ictal phase. The SD was elicited by the interruption of the Na+/K+ pump activity. The blockage of Na+/K+ pump by cellular hypoxia is an experimental procedure to elicit SD, known as hypoxic spreading depression - HSD. The bifurcation theory and the method of fast-slow analysis were used to study the interference of extracellular K+ in the cellular excitability. This analysis indicates that the system loses its stability at a high [K+]o, transiting to an elevated state of neuronal excitability. This raise of [K+]o provokes bifurcations in the neuronal dynamic behavior, that determine transitions between different stages of these electrical activities. In the initial stage, the increase of [K+]o creates favorable conditions to trigger the epileptiform burst and the SD by saddle-node and supercritical Hopf bifurcations, respectively. During the neuronal activity, the level of excitability is maintained by a continuous growth of [K+]o that depresses K+ currents in a positive feedback way. At this stage, concerning epileptiform burst, the amplitude and frequency of action potentials are changed by supercritical Hopf bifurcation. At the last stage, with the depression of K+ currents, the Na+/K+ pump plays an important role in the end of neuronal activity. The epileptiform burst and SD activities terminate by saddle-homoclinic orbit and saddle-node bifurcations, respectively. Thus, this work suggests that [K+]o may play a fundamental role in the nonlinear dynamics of the epileptiform burst and the transition to SD. / Doutorado / Engenharia Biomedica / Doutor em Engenharia Elétrica

Spiral waves occur in several natural phenomena, including reaction fronts in two-dimension excitable media. In this thesis we attempt to characterize the motion of the spiral tip of a rigidly rotating wave and a linearly travelling wave in the context of a lattice perturbation. This system can be reduced to its center manifold, which allows us to describe the system as ordinary differential equations. This in turn means dynamical systems methods are appropriate to describe the motion of the tip. It is in such a context that we work on spiral waves. We study perturbed rotating waves and travelling waves using standard techniques from dynamical systems theory.

In this thesis we consider the traditional Van der Pol Oscillator with a forcing dependent on a delay in feedback. The delay is taken to be a nonlinear function of both position and velocity which gives rise to many different types of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes place at certain parameter values using methods of centre manifold reduction of DDEs and normal form theory. We present numerical simulations that have been accurately predicted by the phase portraits in the Zero-Hopf bifurcation to confirm our numerical results and provide a physical understanding of the oscillator with the delay in feedback.

Dimensionality reduction methods allow for the study of high-dimensional systems by producing low-dimensional descriptions that preserve the relevant structure and features of interest. For dynamical systems, attractors are particularly important examples of such features, as they govern the long-term dynamics of the system, and are typically low-dimensional even if the state space is high- or infinite-dimensional. Methods for reduction need to be able to determine a suitable reduced state space in which to describe the attractor, and to produce a reduced description of the corresponding dynamics. In the presence of a parameter space, a system can possess a family of attractors. Parameters are important quantities that represent aspects of the physical system not directly modelled in the dynamics, and may take different values in different instances of the system. Therefore, including the parameter dependence in the reduced system is desirable, in order to capture the model's full range of behaviour. Existing methods typically involve algebraically manipulating the original differential equation, either by applying a projection, or by making local approximations around a fixed-point. In this work, we take more of a geometric approach, both for the reduction process and for determining the dynamics in the reduced space. For the reduction, we make use of an existing secant-based projection method, which has properties that make it well-suited to the reduction of attractors. We also regard the system to be a manifold and vector field, consider the attractor's normal and tangent spaces, and the derivatives of the vector field, in order to determine the desired properties of the reduced system. We introduce a secant culling procedure that allows for the number of secants to be greatly reduced in the case that the generating set explores a low-dimensional space. This reduces the computational cost of the secant-based method without sacrificing the detail captured in the data set. This makes it feasible to use secant-based methods with larger examples. We investigate a geometric formulation of the problem of dimensionality reduction of attractors, and identify and resolve the complications that arise. The benefit of this approach is that it is compatible with a wider range of examples than conventional approaches, particularly those with angular state variables. In turn this allows for application to non-autonomous systems with periodic time-dependence. We also adapt secant-based projection for use in this more general setting, which provides a concrete method of reduction. We then extend the geometric approach to include a parameter space, resulting in a family of vector fields and a corresponding family of attractors. Both the secant-based projection and the reproduction of dynamics are extended to produce a reduced model that correctly responds to the parameter dependence. The method is compatible with multiple parameters within a given region of parameter space. This is illustrated by a variety of examples.

This thesis analyses numerical methods used in finding solutions of diferential equations. Numerical methods are viewed as discrete dynamical systems that give useful information on continuous dynamical systems defined by systems of (ordinary) diferential equations. We analyse non-standard finite difference schemes that have no spurious fixed-points compared to the dynamical system under consideration, the linear stability/instability property of the fixed-points being the same for both the discrete and continuous systems. We obtain a sharper condition for the elementary stability of the schemes. For more complex dynamical systems which are dissipative, we design schemes that replicate this property. Furthermore, we investigate the impact of the above analysis on the numerical solution of partial differential equations. We specifically focus on reaction-diffusion equations that arise in many fields of engineering and applied sciences. Often their solutions enjoy the follow- ing essential properties: Stability/instability of the fixed points for the space independent equation, the conservation of energy for the stationary equation, and boundedness and positivity. We design new non-standard finite diference schemes which replicate these properties. Our construction make use of three strategies: the renormalization of the denominator of the discrete derivative, non-local approximation of the nonlinear terms and simple functional relation between step sizes. Numerical results that support the theory are provided. Copyright / Thesis (PhD)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

Dynamical systems possess an interesting and complex behaviour that have attracted a number of researchers across different fields, such as Biology, Economics and most importantly in Engineering. The complex and unpredictability of nonlinear customary behaviour or the chaotic behaviour, makes it strange to analyse them. This thesis presents the analysis of the system of nonlinear differential equations of the so--called Lu--Chen--Cheng system. The system has similar dynamical behaviour with the famous Lorenz system. The nature of equilibrium points and stability of the system is presented in the thesis. Examples of chaotic dynamical systems are presented in the theory. The thesis shows the dynamical structure of the Lu--Chen--Cheng system depending on the particular values of the system parameters and routes to chaos. This is done by both the qualitative and numerical techniques. The bifurcation diagrams of the Lu--Chen--Cheng system that indicate limit cycles and chaos as one parameter is varied are shown with the help of the largest Lyapunov exponent, which also confirms chaos in the system. It is found out that most of the system's equilibria are unstable especially for positive values of the parameters $a, b$. It is observed that the system is highly sensitive to initial conditions. This study is very important because, it supports the previous findings on chaotic behaviours of different dynamical systems.

Rank one subshifts are dynamical systems generated by a regular combinatorial process based on sequences of positive integers called the cut and spacer parameters. Despite the simple process that generates them, rank one subshifts comprise a generic set and are the source of many counterexamples. As a result, measure theoretic rank one subshifts, called rank one transformations, have been extensively studied and investigations into rank one subshifts been the basis of much recent work. We will answer several open problems about rank one subshifts. We completely classify the maximal equicontinuous factor for rank one subshifts, so that this factor can be computed from the parameters. We use these methods to classify when large classes of rank one subshifts have mixing properties. Also, we completely classify the situation when a rank one subshift can be a factor of another rank one subshift.

The aim of this dissertation is to investigate ergodic properties, in particular unique ergodicity, in a noncommutative setting, that is in C*-dynamical systems. Fairly recently Abadie and Dykema introduced a broader notion of unique ergodicity, namely relative unique ergodicity. Our main focus shall be to present their result for arbitrary abelian groups containing a F lner sequence, and thus generalizing the Z-action dealt with by Abadie and Dykema, and also to present examples of C*-dynamical systems that exhibit variations of these (uniquely) ergodic notions. Abadie and Dykema gives some characterizations of relative unique ergodicity, and among them they state that a C*-dynamical system that is relatively uniquely ergodic has a conditional expectation onto the xed point space under the automorphism in question, which is given by the limit of some ergodic averages. This is possible due to a result by Tomiyama which states that any norm one projection of a C*-algebra onto a C*-subalgebra is a conditional expectation. Hence the rst chapter is devoted to the proof of Tomiyama's result, after which some examples of C*-dynamical systems are considered. In the last chapter we deal with unique and relative unique ergodicity in C*-dynamical systems, and look at examples that illustrate these notions. Speci cally, we present two examples of C*-dynamical systems that are uniquely ergodic, one with an R2-action and the other with a Z-action, an example of a C*-dynamical system that is relatively uniquely ergodic but not uniquely ergodic, and lastly an example of a C*-dynamical system that is ergodic, but not uniquely ergodic. / Dissertation (MSc)--University of Pretoria, 2013. / gm2014 / Mathematics and Applied Mathematics / unrestricted

We study a two-dimensional Fermi-Pasta-Ulam lattice in the long-amplitude, small-wavelength limit. The one-dimensional lattice has been thoroughly studied in this limit, where it has been established that the dynamics of the lattice is well-approximated by the Korteweg–De Vries (KdV) equation for timescales of the order ε^−3. Further it has been shown that solitary wave solutions of the FPU lattice in the one dimensional case are well approximated by solitary wave solutions of the KdV equation. A two-dimensional analogue of the KdV equation, the Kadomtsev–Petviashvili (KP-II) equation, is known to be a good approximation of certain two-dimensional FPU lattices for similar timescales, although no proof exists. In this thesis we present a rigorous justification that the KP-II equation is the long-amplitude, small-wavelength limit of a two-dimensional FPU model we introduce, analogous to the one-dimensional FPU system with quadratic nonlinearity. We also prove that the cubic KP-II equation is the limit of a model analogous to a one-dimensional FPU system with cubic nonlinearity. Further we study whether stability of line solitons in the KP-II equation extends to stability of one-dimensional FPU solitary waves in the two-dimensional lattices. / Thesis / Doctor of Philosophy (PhD)

This work deals with a class of nonlinear dynamical systems exhibiting both continuous and discrete dynamics, which is called as hybrid dynamical system. We provide a broader framework of generalized hybrid dynamical systems allowing us to handle issues on modeling, stability and interconnections. Various sufficient stability conditions are proposed by extensions of direct Lyapunov method. We also explicitly show Lyapunov formulations of the nonlinear small-gain theorems for interconnected input-to-state stable hybrid dynamical systems. Applications on modeling and stability of hybrid dynamical systems are given by effective strategies of vaccination programs to control a spread of disease in epidemic systems. / Entwicklung eines Frameworks für hybride dynamische Systeme zur Decomkosition oder Komposition solcher Systeme. Untersuchung der Stabilität von gekoppelten hybriden Systemen.