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.

In this thesis we consider a few results related to the Zermelo and Extended Zermelo Model as well as outline some partial results and open problems related thereto. First we will analyze a discrete dynamical system considering under what conditions the convergence of this dynamical system predicts the outcome of the Extended Zermelo Model. In the following chapter we will focus on the Zermelo Model by giving a method for simplifying the derivation of Zermelo ratings for tournaments in terms of specific types of strongly connected components. Following this, the idea of stability of a tournament will be discussed and an upper bound will be obtained on the stability of three-team tournaments. Finally, we will conclude with some partial results related to the topics presented in the previous chapters.

In this thesis we present a numerical and analytical study of modelling extremes in chaotic dynamical systems. We study a range of examples with different dependency structures, and different clustering characteristics. We compare our analysis to the extreme statistics observed for financial returns data, and hence consider the modelling potential of using chaotic systems for understanding financial returns. As part of the study we use the block maxima approach and the peak over threshold method to compute the distribution parameters that arise in the corresponding extreme value distributions. We compare these computations to the theoretical answers, and moreover we obtain error bounds on the rate of convergence of these schemes. In particular we investigate the optimal block size when applying the block maxima method. Since the time series of observations on a dynamical system have dependency we must therefore go beyond the classic approach of studying extremes for independent identically distributed random variables. This is the main purpose of our study. As part of this thesis, we also study clustering in financial returns, and again investigate the potential of using dynamical systems models. Moreover we can also compare numerical quantification of clustering with theoretical approaches. As further work, we measure the dependency structures in our models using a rescaled range analysis. We also make preliminary investigations into record statistics for dynamical systems models, and relate our findings to record statistics in financial data, and to other models (such as random walk models).

Estômatos são poros responsáveis pela troca gasosa entre a folha e o meio externo. A partir da década de 80, experimentos revelaram um complexo padrão espaço temporal na abertura e fechamento dos estômatos. As experiências apontam para uma possível coordenação entre estômatos em algumas áreas da folha chamada de patches. Esse fenômeno é conhecido na literatura como patchy stomatal conductance. Frequentemente a coordenação dinâmica dos estômatos está associada à oscilações temporais na condutãncia estomática (média especial da abertura dos estômatos). Em 1997 Haefner, Buckley e Mott (HBM) publicaram uma análise numérica de um modelo dinâmico para explorar o comportamento complexo dos estômatos. O modelo é baseado em algumas características conhecidas dos estômatos e assume transporte hídrico em uma rede definida por uma geometria simples e bastante restritiva. De acordo com os autores, o modelo reproduz qualitativamente os dados experimentais. Recentemente, Ferraz e Prado mostraram que esse modelo não é capaz de reproduzir os resultados experimentais. Usando ingredientes do modelo sugerido por HBM, Ferraz e Prado sugeriram uma geometria realística de distribuição reservatórios hídricos. Embora essa configuração reproduza os patches, eles permanecem estáticos e nenhuma oscilação é observada. Sem explorar detalhes significativos, Ferraz e Prado afirmaram que a histerese na abertura estomatal poderia explicar vários aspectos dos resultados experimentais. No presente estudo comprovamos, através de uma abordagem computacional baseada em transdutores histeréticos, que a hipótese de histerese na abertura dos estômatos de fato reproduz qualitativamente os dados experimentais. Em nossa abordagem a histerese na abertura dos estômatos é emulada através de operadores chamados de histerons. A robustez da hipótese é testada usando diferentes tipos de histerons. Analisamos a correlação entre os estômatos na rede que simula a superfície da folha. Observamos que a correlação entre estômatos depende da geometria da veia. Uma análise detalhada dos parâmetros envolvidos revela uma dependência entre o período de oscilação na condutância estomática e o déficit de vapor d\'água entre a folha e o meio ambiente. Esta característica subjacente ao modelo pode inspirar novas experiências para testar a hipótese da histerese na abertura dos estômatos. / Stomata are pores on the surface of leaves responsible for controlling the exchange of gas between the plant and the environment. Experiments revealed a complex spatial-temporal pattern in the opening and closing mechanism of stomata. The main feature of the phenomenon is that stomata appear to be synchronized into clusters, known as patches. The dynamical coordination of stomata often involves oscillations in stomatal conductance. In 1997 Haefner, Buckley, and Mott (HBM) published a numerical analysis of a dynamic model to explore the complex behavior of stomata. The model is based on some known features of the stomata, and assumes that water diffuses within the leaves according to a simple geometric arrangement. According to the authors, the model reproduces qualitatively the experimental data. Recently, Ferraz and Prado showed that the computational approach of HBM is not able to reproduce the experimental results. Inspired by this model, Ferraz and Prado introduced a new geometric features that leads to static patches of stomata; however no oscillation was observed and the patches remained static. The authors suggested that hysteresis in stomatal aperture could explain several experimental aspects. We now report a further investigation of the changes suggested by Ferraz and Prado in the original model of HBM. The theoretical approach confirmed that hysteresis in the aperture mechanism of pores reproduces a variety of behaviors of stomatal conductance described in experiments. We explore the hysteresis feature through the formalism of hysteretic transducer. The robustness of the hysteretic assumption is tested by different kinds of hysteresis operators. We analyzed the correlation among stomata in the lattice. We observed that the correlation depends on the geometry of the veins. Finally, the analysis of the model reveals a dependence between the period of oscillation in the stomatal conductance time series and water vapor pressure deficits Δω - an external parameter. Further experiments might explore this underlying feature of the model.

The electroencephalogram (EEG) is a commonly used tool for studying the emergent electrical rhythms of the brain. It has wide utility in psychology, as well as bringing a useful diagnostic aid for neurological conditions such as epilepsy. It is of growing importance to better understand the emergence of these electrical rhythms and, in the case of diagnosis of neurological conditions, to find mechanistic differences between healthy individuals and those with a disease. Mathematical models are an important tool that offer the potential to reveal these otherwise hidden mechanisms. In particular Neural Mass Models (NMMs), which describe the macroscopic activity of large populations of neurons, are increasingly used to uncover large-scale mechanisms of brain rhythms in both health and disease. The dynamics of these models is dependent upon the choice of parameters, and therefore it is crucial to be able to understand how dynamics change when parameters are varied. Despite they are considered low-dimensional in comparison to micro-scale neural network models, with regards to understanding the relationship between parameters and dynamics NMMs are still prohibitively high dimensional for classical approaches such as numerical continuation. We need alternative methods to characterise the dynamics of NMMs in high dimensional parameter spaces. The primary aim of this thesis is to develop a method to explore and analyse the high dimensional parameter space of these mathematical models. We develop an approach based on statistics and machine learning methods called decision tree mapping (DTM). This method is used to analyse the parameter space of a mathematical model by studying all the parameters simultaneously. With this approach, the parameter space can efficiently be mapped in high dimension. We have used measures linked with this method to determine which parameters play a key role in the output of the model. This approach recursively splits the parameter space into smaller subspaces with an increasing homogeneity of dynamics. The concepts of decision tree learning, random forest, measures of importance, statistical tests and visual tools are introduced to explore and analyse the parameter space. We introduce formally the theoretical background and the methods with examples. The DTM approach is used in three distinct studies to: • Identify the role of parameters on the dynamic model. For example, which parameters have a role in the emergence of seizure dynamics? • Constrain the parameter space, such that regions of the parameter space which give implausible dynamic are removed. • Compare the parameter sets to fit different groups. How does the thalamocortical connectivity of people with and without epilepsy differ? We demonstrate that classical studies have not taken into account the complexity of the parameter space. DTM can easily be extended to other fields using mathematical models. We advocate the use of this method in the future to constrain high dimensional parameter spaces in order to enable more efficient, person-specific model calibration.

Dynamics with choice is a generalization of discrete-time dynamics where instead of the same evolution operator at every time step there is a choice of operators to transform the current state of the system. Many real life processes studied in chemical physics, engineering, biology and medicine, from autocatalytic reaction systems to switched systems to cellular biochemical processes to malaria transmission in urban environments, exhibit the properties described by dynamics with choice. We study the long-term behavior in dynamics with choice. We prove very general results on the existence and properties of global compact attractors in dynamics with choice. In addition, we study the dynamics with restricted choice when the allowed sequences of operators correspond to subshifts of the full shift. One of practical consequences of our results is that when the parameters of a discrete-time system are not known exactly and/or are subject to change due to internal instability, or a strategy, or Nature's intervention, the long term behavior of the system may not be correctly described by a system with "averaged" values for the parameters. There may be a Gestalt effect.

No description available.

The sigma-delta modulator is a popular signal amplitude quantization error (or noise) shaper used in oversampling analogue-to-digital and digital-to-analogue converter systems. The shaping of the noise frequency spectrum is performed by feeding back the quantization errors through a time delay element filter and feedback loop in the circuit, and by the addition of a possible stochastic dither signal at the quantizer. The aim in audio systems is to limit audible noise and distortions in the reconverted analogue signal. The formulation of the sigma-delta modulator as a discrete dynamical system provides a useful framework for the mathematical analysis of such a complex nonlinear system, as well as a unifying basis from which to consider other systems, from pseudorandom number generators to stochastic resonance processes, that yield equivalent formulations. The study of chaos and other complementary aspects of internal dynamical behaviour in previous research has left important issues unresolved. Advancement of this study is naturally facilitated by the dynamical systems approach. In this thesis, the general order feedback/feedforward sigma-delta modulator with multi-bit quantizer (no overload) and general input, is modelled and studied mathematically as a dynamical system. This study employs pertinent topological methods and relationships, which follow centrally from the symmetry of the circle map interpretation of the error state space dynamcis. The main approach taken is to reduce the nonlinear system into local or special case linear systems. Systems of sufficient structure are shown to often possess structured random, or random-like behaviour. An adaptation of Devaney's definition of chaos is applied to the model, and an extensive investigation of the conditions under which the associated chaos conditions hold or do not hold is carried out. This seeks, in part, to address the unresolved research issues. Chaos is shown to hold if all zeros of the noise transfer function lie outside the unit circle of radius two, provided the input is either periodic or persistently random (mod delta). When the filter satisfies a certain continuity condition, the conditions for chaos are extended, and more clear cut classifications emerge. Other specific chaos classifications are established. A study of the statistical properties of the error in dithered quantizers and sigma-delta modulators is pursued using the same state space model. A general treatment of the steady state error probability distribution is introduced, and results for predicting uniform steady state errors under various conditions are found. The uniformity results are applied to RPDF dithered systems to give conditions for a steady state error variance of delta squared over six. Numerical simulations support predictions of the analysis for the first-order case with constant input. An analysis of conditions on the model to obtain bounded internal stability or instability is conducted. The overall investigation of this thesis provides a theoretical approach upon which to orient future work, and initial steps of inquiry that can be advanced more extensively in the future.

Observed in many physical systems, coarsening is an orderly decrease in the number of localized structures, such as particles, drops, shear bands, solitons, or point defects. Coarsening is a type of pattern formation in which the characteristic length scale between features grows while the total number of features decreases. These phenomena have been studied in many problems and several mathematical techniques for modeling these phenomena have been developed. This dissertation examines the aggregation of drops in the thin film equation, where drops may coarsen through two general mechanisms: collision and collapse. A series of simplifications to model this process is developed. Slender-body asymptotics is applied to the Navier-Stokes equations for fluid motion in order to derive the Reynolds lubrication equation. The lubrication equation is in turn simplified to a coarsening dynamical system (CDS) model for interacting drops through solvability conditions for a perturbation about a drop-type steady state. Lastly, the dynamical system is averaged into an ensemble model to describe the dynamics of the distribution of drop sizes. The ensemble model takes the form of an integro-differential equation for the distribution function, much like the model of Ostwald ripening proposed by Lifshitz and Slyozov. A convenient choice of scaling yields an intermediate asymptotic self-similar solution. This solution is compared to numerical simulations of the ensemble model and histograms of drop masses from the CDS model. The early-time dynamics before similarity are explored by varying the initial distribution of drop sizes. Interesting far-from-similarity ``stairstep'' behavior is observed in the coarsening rate when the initial distribution has a very small variance. A well-chosen initial condition with a fractal-like structure is shown to replicate the stairstep behavior. At very long times, the mean drop size grows large, requiring the inclusion of gravity in the model. The CDS model parameters are modified as a result of the dependence of drop shapes on both size and gravity. The new dynamical system predicts the coarsening rate slowing from a power law to an inverse logarithmic rate. The energy liberated by each coarsening event is shown to approach a gravity-dependent constant as the mean drop mass increases. This suggests a reason for the coarsening slow-down. / Dissertation

Estômatos são poros responsáveis pela troca gasosa entre a folha e o meio externo. A partir da década de 80, experimentos revelaram um complexo padrão espaço temporal na abertura e fechamento dos estômatos. As experiências apontam para uma possível coordenação entre estômatos em algumas áreas da folha chamada de patches. Esse fenômeno é conhecido na literatura como patchy stomatal conductance. Frequentemente a coordenação dinâmica dos estômatos está associada à oscilações temporais na condutãncia estomática (média especial da abertura dos estômatos). Em 1997 Haefner, Buckley e Mott (HBM) publicaram uma análise numérica de um modelo dinâmico para explorar o comportamento complexo dos estômatos. O modelo é baseado em algumas características conhecidas dos estômatos e assume transporte hídrico em uma rede definida por uma geometria simples e bastante restritiva. De acordo com os autores, o modelo reproduz qualitativamente os dados experimentais. Recentemente, Ferraz e Prado mostraram que esse modelo não é capaz de reproduzir os resultados experimentais. Usando ingredientes do modelo sugerido por HBM, Ferraz e Prado sugeriram uma geometria realística de distribuição reservatórios hídricos. Embora essa configuração reproduza os patches, eles permanecem estáticos e nenhuma oscilação é observada. Sem explorar detalhes significativos, Ferraz e Prado afirmaram que a histerese na abertura estomatal poderia explicar vários aspectos dos resultados experimentais. No presente estudo comprovamos, através de uma abordagem computacional baseada em transdutores histeréticos, que a hipótese de histerese na abertura dos estômatos de fato reproduz qualitativamente os dados experimentais. Em nossa abordagem a histerese na abertura dos estômatos é emulada através de operadores chamados de histerons. A robustez da hipótese é testada usando diferentes tipos de histerons. Analisamos a correlação entre os estômatos na rede que simula a superfície da folha. Observamos que a correlação entre estômatos depende da geometria da veia. Uma análise detalhada dos parâmetros envolvidos revela uma dependência entre o período de oscilação na condutância estomática e o déficit de vapor d\'água entre a folha e o meio ambiente. Esta característica subjacente ao modelo pode inspirar novas experiências para testar a hipótese da histerese na abertura dos estômatos. / Stomata are pores on the surface of leaves responsible for controlling the exchange of gas between the plant and the environment. Experiments revealed a complex spatial-temporal pattern in the opening and closing mechanism of stomata. The main feature of the phenomenon is that stomata appear to be synchronized into clusters, known as patches. The dynamical coordination of stomata often involves oscillations in stomatal conductance. In 1997 Haefner, Buckley, and Mott (HBM) published a numerical analysis of a dynamic model to explore the complex behavior of stomata. The model is based on some known features of the stomata, and assumes that water diffuses within the leaves according to a simple geometric arrangement. According to the authors, the model reproduces qualitatively the experimental data. Recently, Ferraz and Prado showed that the computational approach of HBM is not able to reproduce the experimental results. Inspired by this model, Ferraz and Prado introduced a new geometric features that leads to static patches of stomata; however no oscillation was observed and the patches remained static. The authors suggested that hysteresis in stomatal aperture could explain several experimental aspects. We now report a further investigation of the changes suggested by Ferraz and Prado in the original model of HBM. The theoretical approach confirmed that hysteresis in the aperture mechanism of pores reproduces a variety of behaviors of stomatal conductance described in experiments. We explore the hysteresis feature through the formalism of hysteretic transducer. The robustness of the hysteretic assumption is tested by different kinds of hysteresis operators. We analyzed the correlation among stomata in the lattice. We observed that the correlation depends on the geometry of the veins. Finally, the analysis of the model reveals a dependence between the period of oscillation in the stomatal conductance time series and water vapor pressure deficits Δω - an external parameter. Further experiments might explore this underlying feature of the model.