[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.

Für einen R^d-wertigen stochastischen Prozess X auf R bezeichne X_t den Segmentprozess X_t:={X(t+u): u = 0. Es wird folgende affine stochastische Differentialgleichung mit unendlichem Gedächtnis betrachtet: dX(t)=L(X_t)dt + dW(t) für t >= 0, X_0=F, (A) wobei L:B -> R^d ein lineares stetiges Funktional, W einen Wiener-Prozess mit Werten in R^d sowie B einen semi-normierten linearen Unterraum von {f:(-00, 0] -> R^d} bezeichnen. Die Anfangsbedingung F ist eine B-wertige Zufallsvariable. Die Lösung X der Gleichung (A) lässt sich mittels einer Formel der Variation der Konstanten darstellen. Für die Existenz einer stationären Lösung werden hinreichende und notwendige Bedingungen vorgestellt. Für eine spezielle Klasse von Funktionalen L kann Gleichung (A) auf ein System gewöhnlicher stochastischer Gleichungen ohne Gedächtnis reduziert werden. Diese Reduktion wird im Detail untersucht, insbesondere gewinnt man hierdurch ein einfaches äquivalentes Kriterium für die Existenz stationärer Lösungen von Gleichungen mit Funktionalen L dieser Klasse. Durch Einbettung der Gleichung (A) in den Bidualraum B** gelingt die Bestimmung der Lyapunov-Exponenten der Lösung. Hierzu wird ein neuer Zusammenhang der Lösung der sogenannten adjungierten Gleichung von (A) und einer Spektralzerlegung des Raumes B benutzt. Die Untersuchung der stetigen Abhängigkeit der Lösung von dem Funktional L und der Anfangsbedingung F ermöglicht die Behandlung anwendungsorientierter Aspekte. In Verbindung mit den Ergebnissen über reduzierbare Gleichungen wird ein Verfahren zur Approximation der Lösung von Gleichung (A) durch Ornstein-Uhlenbeck-Prozesse vorgestellt. Eine allgemeine Klasse von Ito-Differentialgleichungen mit nichtlinearen vergangenheitsabhängigen Drift- und Dispersionskoeffizienten wird eingeführt, in der die Gleichung (A) als eine spezielle affine Gleichung verstanden werden kann. Für diese allgemeinen Gleichungen wird ein Existenz- und Eindeutigkeitssatz nachgewiesen. / For an R^d-valued stochastic process X denote the segment process by X_t:={X(t+u): u = 0. We consider the following affine stochastic differential equation with infinite delay: dX(t)=L(X_t)dt + dW(t) for t >= 0, X_0= F, (A) where L:B -> R^d denotes a linear continuous functional, W denotes a Wiener process with values in R^d and B is a semi-normed linear subspace of {f: (-00, 0] -> R^d}. The initial condition F is a B-valued random variable. The solution X of equation (A) can be represented by a variation of constants formula. We provide sufficient and necessary conditions for the existence of a stationary solution. For a special class of functionals L the equation (A) can be reduced to a system of ordinary stochastic differential equations without memory. This reduction is studied in detail. In particular, we deduce a simple equivalent condition for the existence of stationary solutions of equations with functionals L in this class. The embedding of equation (A) into the bidualspace B** enables us to calculate the Lyapunov exponents of the solution. For this purpose we exploit a new connection between the solution of the so-called adjoint equation of (A) and a spectral decompositon of the space B. By considering the continuous dependence of the solution on the functional L and the initial condition F we obtain results useful in applications. In conjunction with results on reducible equations we establish an approximation scheme for the solution of equation (A) by Ornstein-Uhlenbeck processes. Moreover, we introduce a general class of Ito differential equations with non-linear drift and dispersion hereditary coefficients. We deduce a result on the existence of unique solutions for this general class of equations. Equation (A) can be regarded as a special affine equation in this class.

[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).

[pt] Os expoentes de Lyapunov são uma ferramenta bastante utilizada quando busca-se entender o comportamento de sistemas dinâmicos, em particular de cociclos lineares. De fato, concentramo-nos no expoente maximal, pois este determina o comportamento geral do sistema, de modo que sua positividade pode ser um indicativo de que estamos lidando com um sistema caótico. Nesse sentido estudamos um teorema provado por Michael Herman, que fornece uma cota inferior para o expoente de Lyapunov maximal de uma classe de cociclos lineares definidos por rotações no círculo. A prova deste resultado utiliza um processo de complexificação do cociclo e um argumento de subharmonicidade. Surpreendentemente, essa cota inferior é na verdade uma identidade, o que foi provado posteriormente por Avila e Bochi. Como será mostrado nesta dissertação, o argumento para obter a identidade depende crucialmente da harmonicidade, e não da mera subharmonicidade de certas funções associadas às iterações do cociclo. / [en] Lyapunov exponents are a widely used tool when trying to understand the behavior of dynamical systems in general, and in particular that of linear cocycles. We focus on the maximal exponent, as it determines the general behavior of the system, in that its positivity can be an indication that we are dealing with a chaotic system. In this sense, we study a theorem obtained by Michael Herman, providing a lower bound on the maximal Lyapunov exponent of a class of linear cocycles defined by circle rotations. The proof of this result employs the complexification of the cocycle and an argument based on subharmonicity. Surprisingly, this lower bound is in fact an identity, which was proven later by Avila and Bochi. As it will be shown in this dissertation, the argument for obtaining this identity depends crucially on the harmonicity, as opposed to the mere subharmonicity of certain functions associated with the iterates of the cocycle.

Diese Dissertation untersucht Aspekte des Zusammenspiels von ergodischem Langzeitver- halten und der Glättungseigenschaft dynamischer Systeme, die von stochastischen Differen- tialgleichungen (SDEs) mit Sprüngen erzeugt sind. Im Speziellen werden SDEs getrieben von Lévy-Prozessen und der Marcusschen kanonischen Gleichung untersucht. Ein vari- ationeller Ansatz für den Malliavin-Kalkül liefert eine partielle Integration, sodass eine Variation im Raum in eine Variation im Wahrscheinlichkeitsmaß überführt werden kann. Damit lässt sich die starke Feller-Eigenschaft und die Existenz glatter Dichten der zuge- hörigen Markov-Halbgruppe aus einer nichtstandard Elliptizitätsbedingung an eine Kom- bination aus Gaußscher und Sprung-Kovarianz ableiten. Resultate für Sprungdiffusionen auf Untermannigfaltigkeiten werden aus dem umgebenden Euklidischen Raum hergeleitet. Diese Resultate werden dann auf zufällige dynamische Systeme angewandt, die von lin- earen stochastischen Differentialgleichungen erzeugt sind. Ruelles Integrierbarkeitsbedin- gung entspricht einer Integrierbarkeitsbedingung an das Lévy-Maß und gewährleistet die Gültigkeit von Oseledets multiplikativem Ergodentheorem. Damit folgt die Existenz eines Lyapunov-Spektrums. Schließlich wird der top Lyapunov-Exponent über eine Formel der Art von Furstenberg–Khasminsikii als ein ergodisches Mittel der infinitesimalen Wachs- tumsrate über die Einheitssphäre dargestellt. / The present thesis investigates certain aspects of the interplay between the ergodic long time behavior and the smoothing property of dynamical systems generated by stochastic differential equations (SDEs) with jumps, in particular SDEs driven by Lévy processes and the Marcus’ canonical equation. A variational approach to the Malliavin calculus generates an integration-by-parts formula that allows to transfer spatial variation to variation in the probability measure. The strong Feller property of the associated Markov semigroup and the existence of smooth transition densities are deduced from a non-standard ellipticity condition on a combination of the Gaussian and a jump covariance. Similar results on submanifolds are inferred from the ambient Euclidean space. These results are then applied to random dynamical systems generated by linear stochas- tic differential equations. Ruelle’s integrability condition translates into an integrability condition for the Lévy measure and ensures the validity of the multiplicative ergodic theo- rem (MET) of Oseledets. Hence the exponential growth rate is governed by the Lyapunov spectrum. Finally the top Lyapunov exponent is represented by a formula of Furstenberg– Khasminskii–type as an ergodic average of the infinitesimal growth rate over the unit sphere.

Dizertační práce se zabývá elektronicky nastavitelnými oscilátory, studiem nelineárních vlastností spojených s použitými aktivními prvky a posouzením možnosti vzniku chaotického signálu v harmonických oscilátorech. Jednotlivé příklady vzniku podivných atraktorů jsou detailně diskutovány. V doktorské práci je dále prezentováno modelování reálných fyzikálních a biologických systémů vykazujících chaotické chování pomocí analogových elektronických obvodů a moderních aktivních prvků (OTA, MO-OTA, CCII ±, DVCC ±, atd.), včetně experimentálního ověření navržených struktur. Další část práce se zabývá možnostmi v oblasti analogově – digitální syntézy nelineárních dynamických systémů, studiem změny matematických modelů a odpovídajícím řešením. Na závěr je uvedena analýza vlivu a dopadu parazitních vlastností aktivních prvků z hlediska kvalitativních změn v globálním dynamickém chování jednotlivých systémů s možností zániku chaosu v důsledku parazitních vlastností použitých aktivních prvků.

Systeme mit Zeitverzögerung sind dadurch charakterisiert, dass deren zukünftige Entwicklung durch den Zustand zum aktuellen Zeitpunkt nicht eindeutig festgelegt ist. Die Historie des Zustands muss in einem Zeitraum bekannt sein, dessen Länge Totzeit genannt wird und die Gedächtnislänge festlegt. In dieser Arbeit werden fundamentale Effekte untersucht, die sich ergeben, wenn die Totzeit zeitlich variiert wird. Im ersten Teil werden zwei Klassen periodischer Totzeitvariationen eingeführt. Da diese von den dynamischen Eigenschaften einer eindimensionalen iterierten Abbildung abgeleitet werden, die über die Totzeit definiert wird, werden die Klassen entsprechend der zugehörigen Dynamik konservativ oder dissipativ genannt. Systeme mit konservativer Totzeit können in Systeme mit konstanter Totzeit transformiert werden und besitzen gleiche charakteristische Eigenschaften. Dagegen weisen Systeme mit dissipativer Totzeit fundamentale Unterschiede z.B. in der Tangentialraumdynamik auf. Im zweiten Teil werden diese Ergebnisse auf Systeme angewendet, deren Totzeit im Vergleich zur internen Relaxationszeit des Systems groß ist. Es zeigt sich, dass ein durch dissipative Totzeitvariationen induzierter Mechanismus, genannt resonanter Dopplereffekt, unter anderem zu neuen Arten chaotischer Dynamik führt. Diese sind im Vergleich zur bekannten chaotischen Dynamik in Systemen mit konstanter Totzeit sehr niedrig-dimensional. Als Spezialfall wird das so genannte laminare Chaos betrachtet, dessen Zeitreihen durch nahezu konstante Phasen periodischer Dauer gekennzeichnet sind, deren Amplitude chaotisch variiert. Im dritten Teil dieser Arbeit wird auf der Basis experimenteller Daten und durch die Analyse einer nichtlinearen retardierten Langevin-Gleichung gezeigt, dass laminares Chaos robust gegenüber Störungen wie zum Beispiel Rauschen ist und experimentell realisiert werden kann. Es werden Methoden zur Zeitreihenanalyse entwickelt, um laminares Chaos in experimentellen Daten ohne Kenntnis des erzeugenden Systems zu detektieren. Mit diesen Methoden ist selbst dann eine Detektion möglich, wenn das Rauschen so stark ist, dass laminares Chaos mit bloßem Auge nur schwer erkennbar ist.:1. Introduction 2. Dissipative and conservative delays in systems with time-varying delay 3. Laminar Chaos and the resonant Doppler effect 4. Laminar Chaos: a robust phenomenon 5. Summary and concluding remarks A. Appendix / In systems with time-delay, the evolution of a system is not uniquely determined by the state at the current time. The history of the state must be known for a time period of finite duration, where the duration is called delay and determines the memory length of the system. In this work, fundamental effects arising from a temporal variation of the time-delay are investigated. In the first part, two classes of periodically time-varying delays are introduced. They are related to a specific dynamics of a one-dimensional iterated map that is defined by the time-varying delay. Referring to the related map dynamics the classes are called conservative or dissipative. Systems with conservative delay can be transformed into systems with constant delay, and thus have the same characteristic properties as constant delay systems. In contrast, there are fundamental differences, for instance, in the tangent space dynamics, between systems with dissipative delay and systems with constant delay. In the second part, these results are applied to systems with a delay that is considered large compared to the internal relaxation time of the system. It is shown that a mechanism induced by dissipative delays leads to new kinds of regular and chaotic dynamics. The dynamics caused by the so-called resonant Doppler effect is fundamentally different from the behavior known from systems with constant delay. For instance, the chaotic attractors in systems with dissipative delay are very low-dimensional compared to typical ones arising in systems with constant delay. An example of this new kind of low-dimensional dynamics is given by the so-called Laminar Chaos. It is characterized by nearly constant laminar phases of periodic duration, where the amplitude varies chaotically. In the third part of this work, it is shown that Laminar Chaos is a robust phenomenon, which survives perturbations such as noise and can be observed experimentally. Therefore experimental data is provided and a nonlinear delayed Langevin equation is analyzed. Using the robust features that characterize Laminar Chaos, methods for time series analysis are developed, which enable us to detect Laminar Chaos without the knowledge of the specific system that has generated the time series. By these methods Laminar Chaos can be detected even for comparably large noise strengths, where the characteristic properties are nearly invisible to the eye.:1. Introduction 2. Dissipative and conservative delays in systems with time-varying delay 3. Laminar Chaos and the resonant Doppler effect 4. Laminar Chaos: a robust phenomenon 5. Summary and concluding remarks A. Appendix