L'augmentation des performances des futurs systèmes dynamiques nécessite la prise en compte des phénomènes physiques non linéaires. Cette thèse apporte un éclairage et des contributions sur deux sujets complémentaires liés aux phénomènes dynamiques non linéaires. Le mémoire de thèse est divisé en deux parties.La première partie porte sur les non-linéarités des amplificateurs de puissance dans le cadre d'applications destinées aux télécommunications ou à la diffusion audio-visuelle. Plusieurs méthodes de modélisation et de linéarisation des amplificateurs de puissance ont été conçues et discutées. Un banc de test a été développé afin d'évaluer les méthodes sur des amplificateurs réels. La robustesse de ces techniques à un mauvais alignement temporel des signaux ainsi que leur capacité à faire face à des artefacts spectraux ont été évaluées. Par ailleurs, nous avons effectué une étude théorique sur l'existence et la prise en compte de solutions multiples dans l'approche adaptative par apprentissage indirect. La deuxième partie traite des systèmes dynamiques non linéaires qui présentent des solutions chaotiques. Ces systèmes sont bien connus, mais les techniques d'identification de ces solutions manquent de fiabilité ou nécessitent une puissance de calcul importante. Dans cette thèse, plusieurs méthodes utilisant également le calcul parallèle sont présentées. Les systèmes à commande différentielle fractionnaire sont brièvement discutés. Il est aussi montré, qu'il existe des systèmes liés à des fonctions de transfert non linéaires avec quantification pour lesquels les méthodes d'analyse classiques échouent / In order to achieve better performance of modern communication devices, that have to be operated on its physical limits, the nonlinear phenomena need to be taken into the account. This thesis brings insight into two different subjects related with nonlinear dynamical phenomena. The thesis itself is divided into two parts : the first part is focused on the domain of nonlinear power amplifiers from the system point of view. Several methods for modelization and linearization of power amplifiers have been designed and discussed. A test-bench has been assembled in order to evaluate the proposed methods on real power amplifiers. Then the robustness to time misalignment in the system and the ability to deal with spectral artifacts in the system of presented methods have been evaluated. Also a theoretical study has been conducted on the existence and management of multiple solutions in the frame of adaptive indirect learning approach. The second part deals with nonlinear dynamical systems that are exhibiting chaotic solutions. Such systems are well known, but techniques for identifying reliable such solutions are either missing or are computational intense. In this thesis several methods using also parallel computing are presented. Systems with fractional differential order are briefly discussed. It is as well shown, that there exists systems related with quantified nonlinear transfer functions for which the standard analyzing methods fails

The master's thesis deals mainly with continuous nonlinear dynamical systems that exhibit chaotic behavior. The main goal is to create algorithms for chaos detection and their subsequent testing on known models. Most of the thesis is devoted to the estimation of the Lyapunov exponents, further it deals with the estimation of the fractal dimension of an attractor and summarizes the 0--1 test. The thesis includes three algorithms created in MATLAB -- an algorithm for estimating the largest Lyapunov exponent and two algorithms for estimating the entire Lyapunov spectra. These algorithms are then tested on five continuous dynamical systems. Especially the error of estimation, speed of these algorithms and properties of Lyapunov exponents in different areas of system behavior are investigated.

Diplomová práce pojednává o chaotických dynamických systémech se zvláštním zaměřením na jejich synchronizaci. Proces synchronizace je aplikován použitím dvou různých metod, a to - metodou úplné synchronizace na dva Lorenzovy systémy a metodou negativní zpětné vazby na dva Rösslerovy systémy. Dále je prozkoumána možná aplikace synchronizace chaotických systémů v oblasti soukromé komunikace, která je doplněná algoritmy v prostředí MATLAB.

In this thesis the Lyapunov exponents of random dynamical systems are presented and investigated. The main results are: 1. In the space of all unbounded linear cocycles satisfying a certain integrability condition, we construct an open set of linear cocycles have simple Lyapunov spectrum and no exponential separation. Thus, unlike the bounded case, the exponential separation property is nongeneric in the space of unbounded cocycles. 2. The multiplicative ergodic theorem is established for random difference equations as well as random differential equations with random delay. 3. We provide a computational method for computing an invariant measure for infinite iterated functions systems as well as the Lyapunov exponents of products of random matrices. / In den vorliegenden Arbeit werden Lyapunov-Exponented für zufällige dynamische Systeme untersucht. Die Hauptresultate sind: 1. Im Raum aller unbeschränkten linearen Kozyklen, die eine gewisse Integrabilitätsbedingung erfüllen, konstruieren wir eine offene Menge linearer Kyzyklen, die einfaches Lyapunov-Spektrum besitzen und nicht exponentiell separiert sind. Im Gegensatz zum beschränkten Fall ist die Eingenschaft der exponentiellen Separiertheit nicht generisch in Raum der unbeschränkten Kozyklen. 2. Sowohl für zufällige Differenzengleichungen, als auch für zufällige Differentialgleichungen, mit zufälligem Delay wird ein multiplikatives Ergodentheorem bewiesen. 3.Eine algorithmisch implementierbare Methode wird entwickelt zur Berechnung von invarianten Maßen für unendliche iterierte Funktionensysteme und zur Berechnung von Lyapunov-Exponenten für Produkte von zufälligen Matrizen.

Human movement is inherently complex, requiring the control and coordination of many neurophysiological and biomechanical degrees-of-freedom, and the extent to which individuals exhibit variation in their movement patterns is captured by the construct of motor variability (MV). MV is being used increasingly to describe movement quality and function among clinical populations and elderly individuals. However, current evidence presents conflicting views on whether increased MV offers benefits or is a hindrance to performance. To better understand the utility of MV for performance prediction, we focused on current research needs in the military domain. Dismounted soldiers, in particular, are expected to perform at a high level in complex environments and under demanding physical conditions. Hence, it is critical to understand what strategies allow soldiers to better adapt to fatigue and diverse environmental factors, and to develop predictive tools for estimating changes in soldier performance. Different aspects of performance such as motor learning, experience, and adaptability to fatigue were investigated when soldiers performed various gait tasks, and gait variability (GV) was quantified using four different types of measures (spatiotemporal, joint kinematics, detrended fluctuation analysis, and Lyapunov exponents). During a novel obstacle course task, we found that frontal plane coordination variability of the hip-knee and knee-ankle joint couples exhibited strong association with rate of learning the novel task, explaining 62% of the variance, and higher joint kinematic variability during the swing phase of baseline gait was associated with faster learning rate. In a load carriage task, GV measures were more sensitive than average gait measures in discriminating between experience and load condition: experienced cadets exhibited reduced GV (in spatiotemporal measures and joint kinematics) and lower long-term local dynamic stability at the ankle, compared to the novice group. In the final study investigating multiple measures of obstacle performance, and variables predictive of changes in performance following intense whole-body fatigue, joint kinematic variability of baseline gait explained 28-59% of the variance in individual performances changes. In summary, these results support the feasibility of anticipating and augmenting task performance based on individual motor variability. This work also provides guidelines for future research and the development of training programs specifically for improving military training, performance prediction, and performance enhancement. / Doctor of Philosophy / All people move with some level of inherent variability, even when doing the same activity, and the extent to which individuals exhibit variation in their movement patterns is captured by the construct of motor variability (MV). MV is being increasingly used to describe movement quality and function among clinical populations and elderly individuals. However, it is still unclear whether increased MV offers benefits or is a hindrance to performance. To better understand the utility of MV for performance prediction, we focused on current research needs in the military domain. Dismounted soldiers, in particular, are expected to perform at a high level in complex environments and under demanding physical conditions. Hence, it is critical to understand what strategies allow soldiers to better adapt to fatigue and diverse environmental factors, and to develop tools that might predict changes in soldier performance. Different aspects of performance were investigated, including learning a new activity, experience, and adaptability to fatigue, and gait variability was quantified through different approaches. When examining how individual learn a novel obstacle course task, we found that certain aspects of gait variability had strong associations with learning rate. In a load carriage task, variability measures were determined to be more sensitive to difference in experience level and load condition compared to typical average measures of gait. Specifically, variability increased with load, and the experienced group was less variable overall and more stable in the long term. Lastly, a subset of gait variability measures were associated with individual differences in fatigue-related changes in performance during an obstacle course. In summary, the results presented here support that it may be possible to both anticipate and enhance task performance based on individual variability. This work also provides guidelines for future research and the development of training programs specifically for improving military training, performance prediction, and performance enhancement.

[pt] Uma medida de probabilidade com suporte compacto em um grupo de matrizes determina uma sequência de matrizes aleatórias i.i.d. Considere o processo multiplicativo correspondente e suas médias geométricas. O teorema de Furstenberg-Kesten, análogo da lei dos grandes números neste cenário, garante que as médias geométricas desse processo multiplicativo convergem quase certamente para uma constante, chamada de expoente de Lyapunov maximal da medida dada. Este conceito pode ser reformulado no contexto mais geral da teoria ergódica usando cociclos lineares aleatórios sobre o shift de Bernoulli. Uma questão natural diz respeito às propriedades de regularidade do expoente de Lyapunov como uma função dos seus dados. Sob uma condição de irredutibilidade e em um cenário específico (que foi posteriormente generalizado por vários autores) Le Page estabeleceu a continuidade de Holder do expoente de Lyapunov. Recentemente, Baraviera e Duarte obtiveram uma prova direta e elegante deste tipo de resultado. Seu argumento usa a fórmula de Furstenberg e as propriedades de regularidade da medida estacionária. Seguindo sua abordagem, neste trabalho obtemos um novo resultado mostrando que, sob a mesma hipótese de irredutibilidade, o expoente de Lyapunov depende Hölder continuamente da medida, relativamente à métrica de Wasserstein, generalizando assim o resultado de Baraviera e Duarte. / [en] A compactly supported probability measure on a group of matrices determines a sequence of i.i.d. random matrices. Consider the corresponding multiplicative process and its geometric averages. Furstenberg-Kesten s theorem, the analogue of the law of large numbers in this setting, ensures that the geometric averages of this multiplicative process converge almost surely to a constant, called the maximal Lyapunov exponent of the given measure. This concept can be reformulated in the more general context of ergodic theory using random linear cocycles over the Bernoulli shift. A natural question concerns the regularity properties of the Lyapunov exponent as a function of the data. Under an irreducibility condition and in a specific setting (which was later generalized by various authors) Le Page established the Holder continuity of the Lyapunov exponent. Recently, Baraviera and Duarte obtained a direct and elegant proof of this type of result. Their argument uses Furstenberg s formula and the regularity properties of the stationary measure. Following their approach, in this work we obtain a new result showing that under the same irreducibility hypothesis, the Lyapunov exponent depends Holder continuously on the measure, relative to the Wasserstein metric, thus generalizing the result of Baraviera and Duarte.

[pt] Os resultados fundamentais da teoria ergódica – o teorema de Birkhoff e o teorema de Kingman – se referem a convergência em quase todo ponto de um processo ergódico aditivo e subaditivo, respectivamente. É bem conhecido que dado um sistema unicamente ergódico e um observável contínuo, as médias de Birkhoff correspondentes convergem em todo ponto e uniformemente. Desta forma, é natural também se perguntar o que acontece com o teorema de Kingman quando o sistema é unicamente ergódico. O primeiro objetivo desta dissertação é responder a essa pergunta utilizando o trabalho de A. Furman. Mais ainda, apresentamos algumas extensões e aplicações desse resultado para cociclos lineares, que foram obtidas por S. Jitomirskaya e R. Mavi. Nosso segundo objetivo é provar um novo resultado sobre taxas de convergências de médias de Birkhoff, para um certo tipo de processo unicamente ergódico: uma translação diofantina no toro com um observável Holder contínuo. / [en] The fundamental results in ergodic theory – the Birkhoff theorem and the Kingman theorem – refer to the almost everywhere convergence of additive and respectively subadditive ergodic processes. It is well known that given a uniquely ergodic system and a continuous observable, the corresponding Birkhoff averages converge everywhere and uniformly. It is therefore natural to ask what happens with Kingman s theorem when the system is uniquely ergodic. The first objective of this dissertation is to answer this question following the work of A. Furman. Moreover, we present some extensions and applications of this result for linear cocycles, which were obtained by S. Jitomirskaya and R. Mavi. Our second objective is to prove a new result regarding the rate of convergence of the Birkhoff averages for a certain type of uniquely ergodic process: a Diophantine torus translation with Holder continuous observable.

[pt] O teorema de Benedicks e Carleson afirma que para a família quadrática existe um conjunto de parâmetros, com medida positiva, para os quais o expoente de Lyapunov é positivo no ponto crítico. Nesta dissertação apresentamos uma demonstração rigorosa e detalhada desse célebre resultado. Uma parte importante da demonstração é o estudo do comportamento quase periódico de um conjunto de órbitas. Além disso, um argumento de grandes desvios é utilizado para mostrar que os parâmetros que não satisfazem a propriedade desejada formam um conjunto pequeno. Tais técnicas apresentam um interesse intrínseco, já que têm se mostrado muito proveitosas para o estudo de outros problemas em sistemas dinâmicos. Combinando o teorema de Benedicks e Carleson ao teorema de Singer, conclui-se que para um conjunto de parâmetros com medida positiva, a função quadrática correspondente não admite atratores periódicos, indicando um comportamento caótico. Neste trabalho, também são estudados critérios para a positividade do expoente de Lyapunov de cociclos quase periódicos de Schrodinger, como o teorema de Herman. O estudo de cociclos de Schrodinger representa um importante tópico na área de física matemática. Mais ainda, algumas das generalizações de tais critérios utilizam as técnicas de Benedicks-Carleson. / [en] The Benedicks and Carleson theorem states that for the quadratic family there exists a set of parameters, with positive measure, for which the Lyapunov exponent is positive at the critical point. In this dissertation we present a rigorous and detailed proof of this famous result. An important part of the proof is the study of the quasi periodic behavior of a set of orbits. In addition, a large deviation argument is used to show that parameters which do not satisfy the desired property form a small set. Such techniques have an intrinsic interest, as they have proven fruitful in the study of other problems in dynamical systems. Combining Benedicks-Carlesons theorem with Singers theorem, we conclude that for a set of parameters with positive measure, the corresponding quadratic function does not admit periodic attractors, indicating its chaotic behavior. In this work we also study criteria for the positivity of the Lyapunov exponent of quasi-periodic Schrodinger cocycles, such as Hermans theorem. The study of the Schrodinger cocycles represents an important topic in mathematical physics. Moreover, some of the generalizations of such criteria use the techniques of Benedicks-Carleson.

Cílem této práce je se seznámení s možnostmi programu Matlab z hlediska detailní analýzy deterministických dynamických systémů. Jedná se především o analýzu časové posloupnosti a o nalezení Lyapunových exponentů. Dalším cílem je navrhnout algoritmus umožňující specifikovat chování systému na základě znalosti příslušných diferenciálních rovnic. To znamená, nalezení chaotických systémů.

Dans cette thèse, nous étudions les sujets suivants :- la stabilité ergodique pour les systèmes conservatifs ;- la généricité de l'existence d'exposants positifs pour certains produits tordus avec fibres de dimension deux ;- rigidité des mesures $u$-Gibbs pour certains systèmes partiellement hyperboliques ;- la transitivité robuste.Nous donnons une preuve de la stabilité ergodique pour certains systèmes partiellement hyperboliques sans utiliser l'accessibilité. Ces systèmes ont été introduits par Pierre Berger et Pablo Carrasco, et ils ont les propriétés suivantes : ils possèdent une direction centrale bidimensionnelle ; ils sont non-uniformément hyperboliques avec un exposant positif et un exposant négatif le long de la direction centrale pour presque tout point, et la décomposition d'Oseledets n'est pas dominée.Dans un autre travail, nous donnons des critères de stabilité ergodique pour des systèmes ayant une décomposition dominée. En particulier, nous explorons la notion d'hyperbolicité par chaîne introduite par Sylvain Crovisier et Enrique Pujals. À l'aide de cette notion, nous donnons des critères explicites de stabilité ergodique et nous donnons quelques applications.Dans un travail commun avec Mauricio Poletti, nous prouvons que le produit aléatoire de difféomorphismes de surface conservatifs possède génériquement une région avec des exposants positifs. Nos résultats s'appliquent également aux produits tordus plus généraux.Nous étudions également les perturbations dissipatives de l'exemple de Berger-Carrasco. Nous classifions toutes les mesures $u$-Gibbs qui peuvent apparaître dans un voisinage de l'exemple. Dans ce voisinage, nous prouvons que toute mesure $u$-Gibbs est soit l'unique mesure SRB du système, soit la désintégration dans le feuilletage central est atomique. Dans un travail commun avec Pablo Carrasco, nous prouvons que cet exemple est robustement transitif (en fait robustement topologiquement mélangeant). / In this thesis we study the following topics:-stable ergodicity for conservative systems;-genericity of the existence of positive exponents for some skew products with two dimensional fibers;-rigidity of $u$-Gibbs measure for certain partially hyperbolic systems;-robust transitivity.We give a proof of stable ergodicity for a certain partially hyperbolic system without using accessibility. This system was introduced by Pierre Berger and Pablo Carrasco, and it has the following properties: it has a two dimensional center direction; it is non-uniformly hyperbolic having both a positive and a negative exponent along the center for almost every point, and the Oseledets decomposition is not dominated.In a different work, we find criteria of stable ergodicity for systems with a dominated splitting. In particular, we explore the notion of chain-hyperbolicity introduced by Sylvain Crovisier and Enrique Pujals. With this notion we give explicit criteria of stable ergodicity, and we give some applications.In a joint work with Mauricio Poletti, we prove that the random product of conservative surface diffeomorphisms generically has a region with positive exponents. Our results also hold for more general skew products.We also study dissipative perturbations of the Berger-Carrasco example. We classify all the $u$-Gibbs measures that may appear inside a neighborhood of the example. In this neighborhood, we prove that any $u$-Gibbs measure is either the unique SRB measure of the system or it has atomic disintegration along the center foliation. In a joint work with Pablo Carrasco, we prove that this example is robustly transitive (indeed robustly topologically mixing).