TEIXEIRA, Rivania Maria do Nascimento. Leis de escala em mapeamentos discretos. 2016. 85 f. Tese (Doutorado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2016. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2016-07-18T18:20:56Z No. of bitstreams: 1 2016_tese_rmnteixeira.pdf: 5826571 bytes, checksum: 6cde4ae78436e469a71b8b9608331776 (MD5) / Approved for entry into archive by Edvander Pires (edvanderpires@gmail.com) on 2016-07-18T18:22:35Z (GMT) No. of bitstreams: 1 2016_tese_rmnteixeira.pdf: 5826571 bytes, checksum: 6cde4ae78436e469a71b8b9608331776 (MD5) / Made available in DSpace on 2016-07-18T18:22:35Z (GMT). No. of bitstreams: 1 2016_tese_rmnteixeira.pdf: 5826571 bytes, checksum: 6cde4ae78436e469a71b8b9608331776 (MD5) Previous issue date: 2016 / In this work we are going to investigate the scale formalism in discret mappings. In 1D mappings, we explore the asymptotic decays to the steady state with focus in three types of bifurcation: transcriptical, pitchfork and period-doubling. We identify this behavior through a well defined generalized homogeneous function with critical exponents. Next to the bifurcation point, the decay to the fix point occurs by an exponential function, which is given by a power law that is independent of the non-linearity mapping. The numerical results obtained agree with the analytical results. We also apply the scale formalism in conservatives and dissipatives bidimensional mappings. In the conservative case, our goal was analyze the behavior of the chaotics orbits next to the phase transition from the integrable to the non-integrable. Next to that transition, we describe the dynamical system using a generalized homogeneous function for which we found a power law that describe the behavior of the criticality. Through a phenomenological discussion, we found critical exponents in agree with the analytical description. In the dissipative case, our main goal was to investigate the influence of a dissipative term in the dynamics, causing a phase transition - suppression of unlimited difusion of the action variable. Following a phenomenological approach with an analytical description, we were able to determine the critical exponents using a generalized homogeneous function. / Neste trabalho investigamos algumas aplicações do formalismo de escala em mapeamentos discretos. Exploramos os decaimentos assintóticos ao estado estacionário com foco em três tipos de bifurcações em mapeamentos unidimensionais: bifurcação transcrítica, bifurcação supercrítica de forquilha e bifurcação de duplicação de período. Caracterizamos este comportamento através de uma função homogênea generalizada com expoentes críticos bem definidos. Próximo ao ponto de bifurcação o decaimento ao ponto fixo ocorre através de uma função exponencial cujo o tempo de relaxação é caracterizado por uma lei de potência que independe da não linearidade do mapa. Os resultados obtidos numericamente harmonizam com os resultados analíticos. Aplicamos também o formalismo de escala em mapeamentos bidimensionais conservativos e dissipativos. No caso conservativo, nosso objetivo foi analisar o comportamento de órbitas caóticas próximas à transição de fase de integrável para não integrável. Próximo à esta transição, descrevemos o sistema dinâmico utilizando uma função homogênea generalizada para a qual encontramos um lei de escala que descreve o comportamento da ação quadrática média próximo à transição. Através de uma discussão fenomenológica, encontramos expoentes críticos que corroboram com a descrição analítica. No caso dissipativo, nosso principal objetivo foi investigar a influência na dinâmica ao ser introduzido um termo dissipativo, causando a supressão da difusão ilimitada da variável ação quadrática média. Seguimos uma descrição fenomenológica acompanhada de uma descrição analítica e assim, determinamos os expoentes críticos usando uma função homogênea generalizada.

Submitted by Aelson Maciera (aelsoncm@terra.com.br) on 2017-06-05T20:04:17Z No. of bitstreams: 1 TeseEJH.pdf: 21600349 bytes, checksum: 9be6866d2a15e1ea7aa6325d99e7fa4c (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-06-06T13:12:02Z (GMT) No. of bitstreams: 1 TeseEJH.pdf: 21600349 bytes, checksum: 9be6866d2a15e1ea7aa6325d99e7fa4c (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-06-06T13:12:09Z (GMT) No. of bitstreams: 1 TeseEJH.pdf: 21600349 bytes, checksum: 9be6866d2a15e1ea7aa6325d99e7fa4c (MD5) / Made available in DSpace on 2017-06-06T13:20:52Z (GMT). No. of bitstreams: 1 TeseEJH.pdf: 21600349 bytes, checksum: 9be6866d2a15e1ea7aa6325d99e7fa4c (MD5) Previous issue date: 2016-12-05 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / We study the existence and multiplicity of nontrivial solutions for two classes of elliptic problems. The first problem covers a general class of operators with variables exponents where the nonlinearitv has subcritical growth. The second problem is a nonlocal elliptic problem where the nonlinearitv has critical growth. ... continua / Neste trabalho estudamos a existência e multiplicidade de soluções não triviais para duas classes de problemas elípticos. O primeiro problema elíptico que estudamos abrange uma classe geral de operadores com expoentes variáveis onde a não linearidade possui crescimento subcrítico. O segundo problema trata de uma equação não local envolvendo uma ampla classe de operadores onde a não linearidade possui crescimento sublinear/superlinear, mais um termo com crescimento crítico. ... continua

Estudos sobre o comportamento dinâmico de estruturas não lineares são até os dias de hoje motivo de extensas pesquisas em todo o mundo. Desde o início do desenvolvimento da teoria das oscilações não lineares buscou-se compreender os mecanismos básicos, como perturbações que provocassem respostas complexas nas estruturas flexíveis. Este trabalho apresenta um estudo teórico e experimental do comportamento dinâmico de uma semi-asa de um avião acoplada a uma turbina com a hélice desbalanceada, esse sistema é representado através de uma viga engastada excitada por uma fonte não ideal localizada na extremidade oposta ao engaste. Entende-se como sistema não ideal aquele que considera que a excitação é influenciada pela própria resposta do sistema. Para sistemas dinâmicos não ideais, deve-se adicionar uma equação que descreva como a fonte não ideal interage com o sistema. Considera-se na equação do sistema apenas o primeiro modo de vibrar. Os resultados de simulação numérica apresentados são obtidos utilizando o software Matlab® 8.0 e o parâmetro de controle a ser analisado é o torque do motor. Os resultados que mostram o comportamento dinâmico do sistema são o histórico no tempo, plano de fase, FFT e para identificar o comportamento caótico calculam-se os expoentes de Lyapunov. O gráfico que mostra a presença do efeito Sommerfeld (salto) no sistema também é apresentado. Na parte experimental, apresenta-se todo o procedimento experimental, assim como os resultados: Histórico no tempo, plano de fase reconstruído, FFT, expoentes de Lyapunov e as análises que ilustram a presença do efeito Sommerfeld no experimento. / Studies about the dynamic behaviour of nonlinear structures have been to this date subject of extensive research all around the world. Since the beginning of the development of the nonlinear oscillation theory one has tried to understand the basic mechanisms, like disruptions that would cause complex answers on flexible structures. This paper presents a theoretical and practical study of the dynamic behaviour of a semi-wing of an airplane installed on a turbine with unbalanced propellers; this system is represented through a cantilever beam excited by a non-ideal source located at the end opposite to the coupling. As a non-ideal system we mean the one that considers that the excitement is influenced by the system\'s response itself. For non-ideal dynamic systems, one must add an equation that describes how the non-ideal source interacts with the system. Only the first vibrating mode is considered in the system\'s equation. The numeric simulation results shown are obtained by using the Matlab® 8.0 software and the control parameter to be analyzed is the motor torque. The results that show the dynamic behaviour of the system are time history, phase plan, FFT and to identify the chaotic behaviour the Lyapunov\'s indexes are calculated. The graphic that shows the presence of the Sommerfeld effect (jump) in the system is also presented. In the experimental part, all the practical procedure is presented, as well as experimental results, like, for example: Time history, phase plan reconstruction, FFT, Lyapunov exponents and the analyses that illustrate the presence of the Sommerfeld effect on the experiment.

Neste trabalho nós estudamos os expoentes de Lyapunov de aplicações f : Td → Td homotópicas a uma aplicação Anosov linear e a continuidade absoluta de folheações. Nós mostramos para algumas classes de homotopia de aplicações que a soma dos expoentes de Lyapunov está limitado pela soma dos expoentes de Lyapunov da aplicação Anosov linear. Além disso, admitindo uma propriedade conhecida como densidade uniformemente limitada (UBD) nas folheações, mostramos uma igualdade entre a soma dos expoentes de Lyapunov de f e do Anosov linear. Também construímos um conjunto C1 aberto de difeomorfismos parcialmente hiperbólicos do toro T4, preservando volume, com folheação central bidimensional não compacta e não absolutamente contínua. Ainda construímos um exemplo parcialmente hiperbólico com folhas centrais bidimensionais, não compactas onde a desintegração do volume ao longo da folheação central não é nem Lebesgue nem atômica. / In this work we study the Lyapunov exponents of maps f : Td → Td homotopic to a linear Anosov map. We proof for some homotopic classes of maps which the sum of Lyapunov exponents is bounded by the sum of the Lyapunov exponents of the linear Anosov map. Moreover, by assuming a property known as uniformly bounded density (UBD) in the foliations, we show an equality between the sum of the Lyapunov exponents of f and the linear Anosov. We also construct an C1 open set of volume preserving partially hyperbolic diffeomorphisms with non compact two dimensional center foliation and non absolutely continuous. We still build an example of partially hyperbolic diffeomorphism with non compact bidimensional center leaves where the disintegration of volume along the center foliation is neither Lebesgue nor atomic.

Pour un n-uplet de nombres réels, vu comme un point de l'espace projectif, on définit pour chaqueindice d entre 0 et n-1 deux exposants d'approximation diophantienne (un ordinaire et un uniforme)qui mesurent l'approximabilité de celui-ci par des sous-espaces rationnels de dimension d dansl'espace projectif. Il se trouve que ces 2n exposants ne sont pas indépendants les uns des autres.Cette thèse s'inscrit dans l'étude du spectre de tout ou partie de ces exposants, qui a fait l'objet denombreux travaux récents. On utilise notamment les outils récents de la géométrie paramétriquedes nombres pour étudier le spectre des exposants uniforme, et on traite un cas pondéré endimension 2. / Given a n-tuple of real numbers, seen as a point in the projective space, one can define for eachindex d between 0 and n-1 two exponents of diophantine approximation (an ordinary and auniform) which measure the approximability of this n-tuple by rational subspaces of dimension d inthe projective space. These 2n exponents are not independant. This thesis is part of the study fromthe spectrum of all or part of these exponents, which have been much studied recently. We userecent tools coming from the parametric geometry of numbers to study the spectrum of the uniformexponents, and deal with a twisted case in dimension two.

The de Broglie-Bohm (BB) approach to quantum mechanics gives trajectories similar to classical trajectories except that they are also determined by a quantum potential. The quantum potential is a "fictitious potential" in the sense that it is part of the quantum kinetic energy. We use quantum trajectories to treat quantum chaos in a manner similar to classical chaos. For the kicked rotor, which is a bounded system, we use the Benettin et al. method to calculate both classical and quantum Lyapunov exponents as a function of control parameter K and find chaos in both cases. Within the chaotic sea we find in both cases nonchaotic stability regions for K equal to multiples of π. For even multiples of π the stability regions are associated with classical accelerator mode islands and for odd multiples of π they are associated with new oscillator modes. We examine the structure of these regions. Momentum diffusion of the quantum kicked rotor is studied with both BB and standard quantum mechanics (SQM). A general analytical expression is given for the momentum diffusion at quantum resonance of both BB and SQM. We obtain agreement between the two approaches in numerical experiments. For the case of nonresonance the quantum potential is not zero and must be included as part of the quantum kinetic energy for agreement. The numerical data for momentum diffusion of classical kicked rotor is well fit by a power law DNβ in the number of kicks N. In the anomalous momentum diffusion regions due to accelerator modes the exponent β(K) is slightly less than quadratic, except for a slight dip, in agreement with an upper bound (K2/2)N2. The corresponding coefficient D(K) in these regions has three distinct sections, most likely due to accelerator modes with period greater than one. We also show that the local Lyapunov exponent of the classical kicked rotor has a plateau for a duration that depends on the initial separation and then decreases asymptotically as O(t-1lnt), where t is the time. This behavior is consistent with an upper bound that is determined analytically.

Orientadores: Romis Ribeiro de Faissol Attux, Diogo Coutinho Soariano, Filipe Ieda Fazanaro / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-27T04:17:12Z (GMT). No. of bitstreams: 1 Santos_OdairVieirados_D.pdf: 3365036 bytes, checksum: 967476ee0acb26bf5197de95e344c6c9 (MD5) Previous issue date: 2015 / Resumo: O problema de análise do sincronismo entre sistemas dinâmicos se reveste de enorme importância prática, uma vez que ocorre em uma miríade de contextos naturais e artificiais. Para avaliar a ocorrência desse fenômeno, é usual lançar mão dos expoentes de Lyapunov condicionados, que refletem, em sua definição, o acoplamento entre sistemas. Nesta tese, é proposto um novo método para cálculo desses expoentes, que pode ser extremamente útil quando há descontinuidades ou quando o cálculo da matriz jacobiana é proibitivo. Esse método tem por base o método das dinâmicas clonadas para estimação do espectro convencional de Lyapunov. A nova proposta é validada em diversos cenários, partindo de sistemas clássicos e chegando aos modelos neuronais de Hindmarsh-Rose e Hodgkin-Huxley. O estudo do sincronismo uni- e bidirecional desses modelos, aliás, constitui também uma contribuição original do trabalho / Abstract: The problem of analyzing synchronism between dynamical systems is of enormous practical significance, as this phenomenon arises in several natural and artificial domains. To characterize the occurrence of synchronism, it is usual to employ the conditioned Lyapunov exponents, which reflect, in their definition, the coupling between systems. In this thesis, a new method for calculating these exponents is proposed, which can be extremely useful in the presence of discontinuities or when to calculate the Jacobian matrix is prohibitive. This method is based on the cloned dynamics approach for estimating the conventional Lyapunov spectrum. The new proposal is validated in a number of scenarios, ranging from classical systems to the Hindmarsh-Rose and Hodgkin-Huxley neuron models. The study of the uniand bidirectional synchronism of these models are apropos also an original contribution of this work / Doutorado / Automação / Doutor em Engenharia Elétrica

L'objet de cette thèse est l'introduction, l'étude et l'utilisation des cônes complexes multidimensionnels. Dans un premier temps, nous étudions la grassmannienne des espaces de Banach. Nous définissons une notion de bonne décomposition pour les espaces de dimension p et nous démontronsl'équivalence entre la distance de Hausdorff sur la grassmannienne et la distance fournie par une norme sur l'algèbre extérieure.Dans un deuxième temps, nous définissons les cônes complexes p-dimensionnels ainsi qu'une jauge sur les sous-espaces de dimension p de ces cônes. Nous montrons alors un principe de contraction pour cette jauge. Cela nous permet de prouver, pour un opérateur contractant un tel cône, l'existence d'un trou spectral séparant les p valeurs propres dominantes du reste du spectre. Nous utilisons cette théorie pourdémontrer un théorème de régularité analytique pour les exposants de Lyapunov d'un produit aléatoire d'opérateurs contractant un même cône.Nous donnons également une comparaison entre la distance de Hausdorff entre espaces vectoriels et notre jauge.Enfin, nous introduisons une notion de cône dual pour les cônes p-dimensionnels. Dans ce cadre, nous prouvons que les propriétéstopologiques d'un cône se traduisent en propriétés topologiques sur son dual, et réciproquement. Nous complétons le théorème de régularitéprécédent en démontrant l'existence et la régularité d'une décomposition de l'espace en "espace lent" et "espace rapide". / The subject of this thesis is the introduction, the study and the applications of multidimensional complex cones. First, we study the grassmannian of Banach space. We define a notion of right decomposition for p-dimensional spaces and we prove the equivalence between theHausdorff distance on the grassmannian and the distance given by a norm on the exterior algebra.Then, we define p-dimensional complex cones and a gauge on the subspaces of dimension p of these cones. We show a contraction principle for thisgauge. This allows us to prove, for an operator contracting such a cone, the existence of a spectral gap which isolate the p leading eigenvaluesfrom the rest of the spectrum. We use this theory to prove a theorem of analytic regularity for Lyapunov exponents of a random product ofoperators contracting a cone. We also give a comparison between the Hausdorff distance for vector spaces and our gauge.Finally, we introduce a notion of dual cone for p-dimensional cones. In this setting, we prove that the topological properties of a cone translateinto topological properties for its dual and conversely. We complete the previous regularity theorem by proving the existence and the regularity ofa dominated splitting of the space into a "fast space" and a "slow space".

Ces dernières années, l'intérêt scientifique porté aux différents aspects des systèmes autonomes est en pleine croissance. Des voitures autonomes jusqu'à l'Internet des objets, il est clair que la capacité de systèmes à prendre des décision de manière autonome devient cruciale. De plus, ces systèmes opéreront avec des ressources limitées. Dans cette thèse, ces systèmes sont étudiés sous l'aspect de la théorie de l'information, dans l'espoir qu'une compréhension fondamentale de leurs limites et de leurs utilisations pourrait aider leur conception par les futures ingénieurs.Dans ce travail, divers problèmes de décision binaire distribuée et collaborative sont considérés. Deux participants doivent "déclarer" la mesure de probabilité de deux variables aléatoires, distribuées conjointement par un processus sans mémoire et désignées par $vct{X}^n=(X_1,dots,X_n)$ et $vct{Y}^n=(Y_1,dots,Y_n)$. Cette décision et prise entre deux mesures de probabilité possibles sur un alphabet fini, désignés $P_{XY}$ et $P_{bar{X}bar{Y}}$. Les prélèvements marginaux des variables aléatoires, $vct{X}^n$ et $vct{Y}^n$ sont supposés à être disponibles aux différents sites .Il est permis aux participants d'échanger des quantités limitées d'information sur un canal parfait avec un contraint de débit maximal. Durant cette thèse, la nature de cette communication varie. La communication unidirectionnelle est considérée d'abord, suivie par la considération de communication bidirectionnelle, qui permet des échanges interactifs entre les participants. / In recents years, interest has been growing in research of different autonomous systems. From the self-dring car to the Internet of Things (IoT), it is clear that the ability of automated systems to make autonomous decisions in a timely manner is crucial in the 21st century. These systems will often operate under stricts constains over their resources. In this thesis, an information-theoric approach is taken to this problem, in hope that a fundamental understanding of the limitations and perspectives of such systems can help future engineers in designing them.Throughout this thesis, collaborative distributed binary decision problems are considered. Two statisticians are required to declare the correct probability measure of two jointly distributed memoryless process, denoted by $vct{X}^n=(X_1,dots,X_n)$ and $vct{Y}^n=(Y_1,dots,Y_n)$, out of two possible probability measures on finite alphabets, namely $P_{XY}$ and $P_{bar{X}bar{Y}}$. The marginal samples given by $vct{X}^n$ and $vct{Y}^n$ are assumed to be available at different locations.The statisticians are allowed to exchange limited amounts of data over a perfect channel with a maximum-rate constraint. Throughout the thesis, the nature of communication varies. First, only unidirectional communication is allowed. Using its own observations, the receiver of this communication is required to first identify the legitimacy of its sender by declaring the joint distribution of the process, and then depending on such authentication it generates an adequate reconstruction of the observations satisfying an average per-letter distortion. Bidirectional communication is subsequently considered, in a scenario that allows interactive communication between the participants.

In this thesis we study two seperate yet related three parameter-families of continuously differentiable maps from the unit circle to unit circle which have a single critical point. For one of the families we show that there is a set of positive measure of parameters such that there is a set of positive measure for which all points in the latter set, the derivative experiences exponential growth. We do so by applying a similar methodology to what Michael Benedicks and Lennart Carleson used to study the quadratic family. For the other family we attempt to show a similar but weaker result using a similar method, but do not manage to do so. We expound on what difficulties the latter family provides and what features Benedicks and Carleson used for the quadratic family that we do not have available. / I den här uppsatsen studerar vi två olika men relaterede treparameterfamiljer av kontinuerligt differentierbara avbildningar från enhetscirkeln till enhetscirkeln som har exakt en kritisk punkt. For den ena familjen visar vi att det finns en mängd av positivt mått av parametrar sådana att det finns en mängd av positivt mått så att för varje punkt i den senarenämnde mängden erfar derivatan exponentiell tillväxt. Vi uppnår detta genom att använda en metod som liknar den som Michael Benedicks och Lennart Carleson använde för att studera den kvadratiska familjen. För den andra familjen försöker vi visa ett liknande men svagare resultat genom att använda en liknande metodik men misslyckas. Vi diskuterar och förklarar vilka svårigheter den senare familjen ger och vilka egenskaper som Benedicks och Carleson använder sig av hos den kvadratiska familjen som vår familj saknar