Made available in DSpace on 2016-04-18T21:39:48Z (GMT). No. of bitstreams: 2 Angelo Schranko de Oliveira1.pdf: 925871 bytes, checksum: 812a592f67dbda8b36f5168fbd5f2598 (MD5) Angelo Schranko de Oliveira2.pdf: 2918106 bytes, checksum: 0969a0bf28b426ce84fe4595d80a73c2 (MD5) Previous issue date: 2009-01-28 / Wolfram Research, Inc. / Cellular automata (CAs) can be defined as discrete dynamical systems over n-dimensional networks of locally connected components, whose evolution occur in a discrete, synchronous and homogeneous fashion. Among their several applications, they have been used as a tool for complex systems modeling governed by fundamental laws of conservation (number-conserving cellular automata) or reversibility (reversible cellular automata). Another fundamental property that can be observed in CAs is regarding to their linearity (linear cellular automata) or nonlinearity. Usually, linear phenomena present low dynamic complexity, however, nonlinear phenoma can present complex behaviours like sensitive dependence on initial conditions and routes to chaos. This work focuses on investigating properties of cellular automata belonging to the intersection of those four classes, namely, reversible, number-conserving, and linear or nonlinear cellular automata. After presenting basic definitions, the notions of number-conserving cellular automata, conservation degree and reversibility are reviewed. Following, a dynamical characterisation parameter which relates the reversibility property of a onedimensional cellular automaton and the pre-images of their basic blocks is introduced, and some proofs of its general properties are given. Empirical observations herein suggest that a cellular automaton is reversible and number-conserving if, and only if, its local transition function is a composition of the local transition functions of the reversible, number-conserving cellular automata with neighbourhood size n=2; such an observation was drawn for neighbourhood sizes n∈{2, 3, 4, 5, 6} and number of states q=2; n∈{2, 3} and q=3; n∈{2, 3} and q=4. A proof for such a conjecture would allow the enumeration between neighbourhood lengths and the quantity of reversible, numberconserving cellular automata in the corresponding space, which can be easily identified by working out the compositions of the local transition functions with n=2. Finally, some relationships between reversible, number-conserving, linear and nonlinear CA rules, their spatio-temporal diagrams and basin of attraction fields are presented. / Autômatos celulares (ACs) podem ser definidos como sistemas dinâmicos sobre redes ndimensionais de componentes localmente conectados, cuja evolução ocorre de forma discreta, síncrona e homogênea. Dentre suas diversas aplicações, têm sido utilizados como ferramenta para modelagem de sistemas complexos regidos por leis fundamentais de conservação (autômatos celulares conservativos) ou reversibilidade (autômatos celulares reversíveis). Outra propriedade fundamental que pode ser observada nos ACs diz respeito à sua linearidade (autômatos celulares lineares) ou nãolinearidade. Fenômenos lineares normalmente apresentam menor complexidade dinâmica, enquanto fenômenos não-lineares podem apresentar propriedades tais como sensibilidade às condições iniciais e rotas para caos. O presente trabalho concentra-se na investigação de propriedades de autômatos celulares unidimensionais pertencentes à interseção dessas quatro classes, isto é, autômatos celulares unidimensionais conservativos, reversíveis, e lineares ou não-lineares. Após definições básicas, são revisitados os conceitos de conservabilidade e reversibilidade. Em seguida, introduz-se um parâmetro de caracterização dinâmica que relaciona a distribuição do número de pré-imagens dos blocos básicos à reversibilidade de autômatos celulares unidimensionais e apresentam-se algumas demonstrações decsuas propriedades gerais. Observações empíricas aqui realizadas sugerem que um autômato celular unidimensional é conservativo e reversível se, e somente se, sua função local de transição de estados é uma composição das funções locais de transição de estado dos autômatos celulares conservativos e reversíveis de vizinhança de comprimento n=2; tal observação foi constatada para vizinhanças de comprimento n∈{2, 3, 4, 5, 6} e quantidade de estados q=2; n∈{2, 3} e q=3; n∈{2, 3} e q=4. Uma demonstração para tal conjectura permitiria estabelecer uma enumeração entre os comprimentos das vizinhanças e a quantidade de autômatos celulares unidimensionais conservativos e reversíveis no espaço correspondente, os quais podem ser facilmente identificados através do cálculo das composições das funções locais de transição de estados com n=2. Por fim, apresentam-se relações entre as classes dos ACs conservativos, reversíveis, lineares e não-lineares, suas dinâmicas espaçotemporais e campos de bacias de atração.

Abstract This study fills an urgent need for qualitative analyses of relationships resulting in human change. It is a result of sixteen years of independent study by the author. It combines postgraduate study of nonlinear methodology, applied research of children’s pretend play, experience in educational psychology and Gestalt-counselling, as well as the practical training of graduate students at the Karelian State Pedagogical University (Petrozavodsk, Russia), and the Kajaani Department of Teacher Education (Kajaani, Finland). In this thesis, an attempt is made to reveal the fundamental reality of relationships between human beings. Using theories of helping relationships and data from developmental psychology, a qualitative nonlinear dynamical model of human relationships is elaborated. The scientific findings of Kurt Lewin and the Gestalt-therapy theory are widely used. To illustrate the explanatory potential of the proposed relationship model and the possibility of qualitative analyses, children’s pretend play is analyzed. In the first chapter, the basic connectedness between humans is studied. The author is focused on theories of relationships and their application to the organizing of relationships’ flow. The second chapter is devoted to detailed analyses of dynamic features of these theories and Kurt Lewin’s conception of tension system. The ontological philosophy of relationships is briefly reviewed. This helps to formulate the main problem of the research – how is a nonlinear phenomenological model of human relationships possible? In the third chapter, a new nonlinear dynamic model of human relationships is elaborated. Several conceptions from Lewin’s dynamic psychology and Gestalt-therapy are further developed in the model. A number of examples are analyzed. Video-data on children’s pretend play is analyzed in the fourth chapter. In the subsequent discussions some advantages and shortcomings of the suggested dynamic nonlinear model are examined. / Tiivistelmä Tämä tutkimus pyrkii vastaamaan kysymykseen miten inhimilliset suhteet voivat johtaa laadullisiin muutoksiin. Työssä paneudutaan ihmisten välisten suhteiden psykologisiin perusteisiin. Siinä kehitellään ihmisten välisten suhteiden ei-lineaarinen dynaaminen malli käyttäen kehityspsykologian ja auttamissuhteiden teorioita. Analyysi pohjautuu Kurt Lewinin ja hahmoterapian teoreettisiin oivalluksiin. Kehitellyn mallin selitysvoiman ja laadullisen analyysin mahdollisuuksien osoittamiseksi mallia sovelletaan lasten juonellisen roolileikin erittelyyn. Ensimmäisessä luvussa pohditaan esimerkkien avulla ihmisten välisten kontaktien perusluonnetta. Erityisesti keskitytään suhteiden teorioihin ja niiden sovelluksiin suhteiden jatkumon rakentamiseksi. Toinen luku paneutuu näiden teorioiden kuvaamien suhteiden dynaamisten piirteiden yksityiskohtaiseen tarkasteluun ja Kurt Lewinin ”tension system” käsitteeseen. Siinä esitellään myöskin lyhyesti suhteiden yksilökehityksen filosofiaa. Tältä pohjalta muotoillaan tutkimuksen pääongelma: Kuinka inhimillisten suhteiden ei-lineaarinen fenomenologinen malli on mahdollinen? Kolmannessa luvussa kehitellään uusi ei-lineaarinen inhimillisten suhteiden malli. Mallissa on kehitelty ja annettu uusi tulkinta useille Lewinin dynaamisen psykologian ja hahmoterapian käsitteille. Kehittelyä on tuettu käytännön esimerkein. Neljännessä luvussa on analysoitu lasten juonellisen roolileikin videotallenteita mallia käyttäen. Pohdinta tuo esille joitakin uuden mallin etuja ja jatkokehittelyn tarpeita.

The vital signs monitoring data of an infant receiving intensive care are a rich source of information about its health condition. One major concern about the state of health of such patients is the onset of neonatal sepsis, a life-threatening bloodstream infection. As early signs are subtle and current diagnosis procedures involve slow laboratory testing, sepsis detection based on the monitored physiological dynamics is a clinically significant task. This challenging problem can be thoroughly modelled as real-time inference within a machine learning framework. In this thesis, we develop probabilistic dynamical models centred around the goal of providing useful predictions about the onset of neonatal sepsis. This research is characterised by the careful incorporation of domain knowledge for the purpose of extracting the infant’s true physiology from the monitoring data. We make two main contributions. The first one is the formulation of sepsis detection as learning and inference in an Auto-Regressive Hidden Markov Model (AR-HMM). The model investigates the extent to which physiological events observed in the patient’s monitoring traces could be used for the early detection of neonatal sepsis. In addition, the proposed approach involves exact marginalisation over missing data at inference time. When applying the ARHMM on a real-world dataset, we found that it can produce effective predictions about the onset of sepsis. Second, both sepsis and clinical event detection are formulated as learning and inference in a Hierarchical Switching Linear Dynamical System (HSLDS). The HSLDS models dynamical systems where complex interactions between modes of operation can be represented as a twolevel hidden discrete hierarchical structure. For neonatal condition monitoring, the lower layer models clinical events and is controlled by upper layer variables with semantics sepsis/nonsepsis. The model parameterisation and estimation procedures are adapted to the specifics of physiological monitoring data. We demonstrate that the performance of the HSLDS for the detection of sepsis is not statistically different from the AR-HMM, despite the fact that the latter model is given “ground truth” annotations of the patient’s physiology.

Les processus auto-stabilisants sont définis comme des solutions d'équations différentielles stochastiques dont le terme de dérive contient à la fois le gradient d'un potentiel ainsi qu'un terme non-linéaire au sens de McKean qui attire le processus vers sa propre loi de distribution. On dispose de nombreux résultats lorsque l'environnement est convexe. L'objet de ce travail est de les étendre autant que possible au cas général notamment lorsque le paysage contient plusieurs puits. Des différences fondamentales sont constatées.Le premier chapitre prouve l'existence d'une solution forte. Le second s'intéresse aux lois de probabilités d'une telle solution. En particulier, l'existence et la non-unicité des mesures stationnaires sont mises en évidence sous des hypothèses faibles. Les chapitres trois et quatre sont affectés au comportement de ces mesures lorsque le coefficient de diffusion tend vers 0.Le chapitre cinq met en relation le processus auto-stabilisant avec des systèmes particulaires via une « propagation du chaos ». Il est ainsi possible de transposer certains résultats du système de particules sur le processus non-markovien et réciproquement. Le chapitre six est dédié au dénombrement exact des mesures stationnaires.Le chapitre sept est employé pour l'étude du comportement en temps long. D'une part, un résultat de convergence dans un cas simple est fourni. D'autre part, un principe de grandes déviations est mis en évidence par l'utilisation des résultats de Freidlin et Wentzell / Self-stabilizing processes are defined as the solutions of stochastic differential equations which drift term contains the gradient of a potential and a term nonlinear in the sense of McKean which attracts the process to its own law distribution. There are many results if the landscape is convex. The purpose of this work is to extend these in the general case especially when the landscape contains contains several wells. Essential differences are found.The first chapter proves the strong existence of a solution. The second one deals with the probability measure of the solution. Particularly, the existence and the non-uniqueness of the stationary measures are highlighted under weak assumptions. Chapter three and four are assigned to the asymptotic analysis in the small noise limit of these measures.Chapter five connects the self-stabilizing process and some particle systems via a « propagation of chaos ». It is thus possible to translate some results from the particle systems to the non-markovian process and reciprocally.Chapter seven is used to study the long time behavior. In one hand, a convergence's result is provided in a simple case. In the other hand, a large deviations principle is highlighted by using the results of Freidlin and Wentzell

Cette thèse contient des contributions originales à la théorie du Contrôle Optimal en temps discret et en horizon infini du point de vue de Pontryagin. Il y a 5 chapitres dans cette thèse. Dans le chapitre 1, nous rappelons des résultats préliminaires sur les espaces de suites à valeur dans et des résultats de Calcul Différentiel. Dans le chapitre 2, nous étudions le problème de Contrôle Optimal, en temps discret et en horizon infini avec la contrainte asymptotique et avec le système autonome. En utilisant la structure d'espace affine de Banach de l'ensemble des suites convergentes vers 0, et la structure d'espace vectoriel de Banach de l'ensemble des suites bornées, nous traduisons ce problème en un problème d'optimisation statique dam des espaces de Banach. Après avoir établi des résultats originaux sur les opérateurs de Nemytskii sur les espaces de suites et après avoir adapté à notre problème un théorème d'existence de multiplicateurs, nous établissons un nouveau principe de Pontryagin faible pour notre problème. Dans le chapitre 3, nous établissons un principe de Pontryagin fort pour les problèmes considérés au chapitre 2 en utilisant un résultat de Ioffe-Tihomirov. Le chapitre 4 est consacré aux problèmes de Contrôle Optimal, en temps discret et en horizon infini, généraux avec plusieurs critères différents. La méthode utilisée est celle de la réduction à l'horizon fini, initiée par J. Blot et H. Chebbi en 2000. Les problèmes considérés sont gouvernés par des équations aux différences ou des inéquations aux différences. Un nouveau principe de Pontryagin faible est établi en utilisant un résultat récent de J. Blot sur les multiplicateurs à la Fritz John. Le chapitre 5 est consacré aux problèmes multicritères de Contrôle Optimal en temps discret et en horizon infini. De nouveaux principes de Pontryagin faibles et forts sont établis, là-aussi en utilisant des résultats récents d'optimisation, sous des hypothèses plus faibles que celles des résultats existants. / This thesis contains original contributions to the optimal control theory in the discrete-time framework and in infinite horizon following the viewpoint of Pontryagin. There are 5 chapters in this thesis. In Chapter 1, we recall preliminary results on sequence spaces and on differential calculus in normed linear space. In Chapter 2, we study a single-objective optimal control problem in discrete-time framework and in infinite horizon with an asymptotic constraint and with autonomous system. We use an approach of functional analytic for this problem after translating it into the form of an optimization problem in Banach (sequence) spaces. Then a weak Pontyagin principle is established for this problem by using a classical multiplier rule in Banach spaces. In Chapter 3, we establish a strong Pontryagin principle for the problems considered in Chapter 2 using a result of Ioffe and Tihomirov. Chapter 4 is devoted to the problems of Optimal Control, in discrete time framework and in infinite horizon, which are more general with several different criteria. The used method is the reduction to finite-horizon initiated by J. Blot and H. Chebbi in 2000. The considered problems are governed by difference equations or difference inequations. A new weak Pontryagin principle is established using a recent result of J. Blot on the Fritz John multipliers. Chapter 5 deals with the multicriteria optimal control problems in discrete time framework and infinite horizon. New weak and strong Pontryagin principles are established, again using recent optimization results, under lighter assumptions than existing ones.

Diplomová práce pojednává o teorii chaotických dynamických systémů, speciálně se pak zabývá Rösslerovým systémem. Kromě standardních výpočtů spojených s bifurkační analýzou se práce zaměřuje na problém stabilizace, konkrétně na stabilizaci rovnovážných bodů. Ke stabilizaci je využita základní metoda zpětnovazebního řízení s časovým zpožděním. Významnou část práce tvoří zavedení a implementace obecné metody pro hledání vhodné volby parametrů vedoucí k úspěšné stabiliaci. Dalším diskutovaným tématem je možnost synchronizace dvou Rösslerových systémů pomocí různých synchronizačních schémat.

This master's thesis deals with qualitative analysis of nonlinear differential equations of second order. For autonomous equations some basic notions of Hamiltonian systems (mainly construction of phase portrait) are presented. For non-autonomous equations the method of lower and upper functions for periodic boundary value problem is used. These notions are then applied to a model of mechanical oscillator, a question of existence of solutions to autonomous and non-autonomous nonlinear differential equations is studied.

This master‘s thesis deals with analysis of electronic dynamical systems exhibiting chaotic solution. In introduction, some basic concepts for better understanding of dynamical systems are explained. After introduction, current knowledge from the world of circuits exhibiting chaotic solutions are discussed. The best-known chaotic systems are analyzed numerically in Matlab software. Numerical analysis and experimental verification were demonstrated at C class transistor amplifier, which confirmed the chaotic behavior and generation of a strange attractor.

Tato práce je přehledem zpožděných diferenciálních rovnic v dynamických systémech. Počínaje obecným přehledem zpožděných diferenciálních rovnic představujeme koncept zpožděných diferenciálů a použití jeho modelů, od biologie a populační dynamiky po fyziku a inženýrství. Poskytneme také přehled Dynamické systémy a diferenciální rovnice zpoždění v dynamických systémech. Oblastí pro modelování s rovnicemi zpožďovacích diferenciálů je Epidemiologie. Důraz je kladen na vývoj epidemiologického modelu Susceptible-Infected-Removed (SIR) bez časového zpoždění. Analyzujeme naše dva modely v rovnováze a lokální stabilitě pomocí předpokládaných dat COVID -19. Výsledky by byly porovnány mezi modelem bez zpoždění a modelem se zpožděním.

Die vorliegende Arbeit befasst sich mit einer Vermutung von Sarnak aus dem Jahre 2010 über die Orthogonalität von durch deterministische dynamische Systeme induzierte Folgen zur Möbiusschen μ-Funktion. Ihre Hauptresultate sind zum einen der Ergodensatz mit Möbiusgewichten, welcher eine maßtheoretische (schwächere) Version von Sarnaks Vermutung darstellt, und zum anderen die bereits gesicherte Gültigkeit der genannten Vermutung in Spezialfällen, wobei hier exemplarisch unter anderem der Thue–Morse Shift und Schiefprodukterweiterungen von rationalen Rotationen auf dem Kreis gewählt worden sind. Zum Zwecke der Motivation zeigen wir, dass eine gewisse Wachstumsabschätzung für die Mertensfunktion äquivalent ist zum Primzahlsatz und skizzieren ein Resultat, welches die Äquivalenz einer weiteren solchen Abschätzung zur Riemannschen Vermutung liefert, um auf diese Weise die Bedeutung der Möbiusfunktion für die Zahlentheorie herauszustellen. Da sie für das Verständnis von Sarnaks Vermutung unerlässlich ist, geben wir eine Einführung in die Theorie der Entropie dynamischer Systeme auf Grundlage der Definitionen von Adler–Konheim–McAndrew, Bowen–Dinaburg und Kolmogorov–Sinai. Ferner berechnen wir die topologische Entropie des Thue–Morse Shifts und von Schiefprodukterweiterungen von Rotatione auf dem Kreis. Wir studieren die ergodische Zerlegung T-invarianter Maße auf kompakten metrischen Räumen mit stetiger Transformation T, welche wir für den Beweis des Ergodensatzes mit Möbiusgewichten benötigen. Sodann beweisen wir den genannten gewichteten Ergodensatz. Wir geben eine hinreichende Bedingung an für das Erfülltsein von Sarnaks Vermutung in einem gegebenen dynamischen System, welche im anschließenden Kapitel Anwendung findet. So wird nachgewiesen, dass Sarnaks Vermutung im Falle des Thue–Morse Shifts und von Schiefprodukterweiterungen von rationalen Rotationen auf dem Kreis erfüllt ist. Abschließend wird gezeigt, dass Sarnaks Vermutung sich als Konsequenz aus einer Vermutung von Chowla ergibt. / The thesis in hand deals with a conjecture of Sarnak from 2010 about the orthogonality of sequences induced by deterministic dynamical systems to the Möbius μ-function. Its main results are the ergodic theorem with Möbius weights, which is a measure theoretic (weaker) version of Sarnak’s conjecture, and the already assured validity of Sarnak’s conjecture in special cases, where we have exemplarily chosen the Thue–Morse shift and skew product extensions of rational rotations on the significance of the Möbius function for number theory. Since it is essential for the understanding of Sarnak’s conjecture we give an introduction to the theory of entropy of dynamical systems based on the definitions of Adler–Konheim–McAndrew, Bowen–Dinaburg and Kolmogorov–Sinai. Furthermore, we calculate the topological entropy of the Thue–Morse shift and of skew product extensions of rotations on the circle. We study the ergodic decomposition for T-invariant measures on compact metric spaces with continuous transformations T, which we will need for the proof of the ergodic theorem with Möbius weights. Thereafter, we prove the namely weighted ergodic theorem. We give a sufficient condition for Sarnak’s conjecture to hold for a given dynamical system, which we make use of in the following chapter. Thereupon, it is varified that Sarnak’s conjecture holds for the Thue–Morse shift and for skew product extensions of rational rotations on the circle. Lastly, it is shown that Sarnak’s conjecture from one of Chowla.