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.

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.

This thesis tests the ability of the Lorenz's (2005) chaotic model to simulate predictability curve of the ECMWF model calculated from data over the 1986 to 2011 period and demonstrates similarity of the predictability curves for the Lorenz's model with N = 90 variables. This thesis also tests approximations of predictability curves and their differentials, aiming to correct the ECMWF model estimated parameters and thus allow for estimation of the largest Lyapunov exponent, model error and limit value of the predictability curve. The correction is based on comparing the parameters estimated for the Lorenz's and ECMWF and on comparison with the largest Lyapunov exponent (λ=0,35 day-1 ) and limit value of the predictability curve (E∞=8,2) of the Lorenz's model. Parameters are calculated from approximations made by the Quadratic hypothesis with and without model error, as well as by Logarithmic and General hypotheses and by hyperbolic tangent employing corrections with and without model error. Average value of the largest Lyapunov exponent is estimated to be λ=0,37 day-1 for the ECMWF model, limit values of the predictability curves are estimated with lower theoretically derived values and new approach of calculation of model error based on comparison of models is presented.

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Hamiltonian systems typically exhibit a mixed phase space in which regions of regular and chaotic dynamics coexist. The chaotic transport is restricted due to partial barriers, since they only allow for a small flux between different regions of phase space. In systems with a two-dimensional (2D) phase space these partial barriers are well understood. However, in systems with a four-dimensional (4D) phase space their dynamical origin is an open question. Thus, we study these partial barriers and the related chaotic transport in 4D maps. For the chaotic transport, we observe a slow power-law decay of the Poincaré recurrence statistics. This is caused by long-trapped orbits exploring stochastic layers of resonance channels. Moreover, we analyze them and find clear signatures of partial transport barriers. We identify normally hyperbolic invariant manifolds (NHIMs) as the relevant objects determining the flux across these barriers. In addition, NHIMs also form the backbone for the explicit construction of partial barriers. This allows us to determine the flux by measuring the turnstile volume. Moreover, we conjecture the existence of a relevant partial barrier with minimal flux by generalizing a cantorus barrier present in 2D maps. Local properties of the flux are studied and explained in terms of the NHIM. / Hamiltonische Systeme zeigen üblicherweise einen gemischten Phasenraum, in dem Bereiche regulärer und chaotischer Dynamik vorherrschen. Der chaotische Transport wird durch partielle Barrieren behindert, da diese nur einen kleinen Fluss zwischen getrennten Bereichen des Phasenraums zulassen. Für Systeme mit einem zweidimensionalen (2D) Phasenraum sind diese bereits gut verstanden. Hingegen ist deren dynamischer Ursprung in Systemen mit einem vierdimensionalen (4D) Phasenraum noch ungeklärt. In dieser Arbeit betrachten wir deshalb in 4D Abbildungen sowohl chaotischen Transport, als auch partielle Barrieren. Für den chaotischen Transport lässt sich die Verteilung der Poincaré-Rückkehrzeiten durch ein Potenzgesetz beschreiben. Lange Rückkehrzeiten sind dabei auf Trajektorien zurückzuführen, die in den chaotischen Bereichen von Resonanzkanälen verweilen. Für diese stellen wir eindeutige Signaturen von partiellen Barrieren fest. Es zeigt sich, dass normal hyperbolische invariante Mannigfaltigkeiten (NHIM) die maßgeblichen Objekte sind, die den Fluss über partielle Barrieren beschreiben. Anhand dieser lassen sich auch partiellen Barrieren explizit konstruieren, was uns wiederum ermöglicht den Fluss mittels einer Volumenmessung zu bestimmen. Durch die Verallgemeinerung einer Cantorusbarriere, die bereits in 2D Abbildungen auftreten, finden wir eine relevante partielle Barriere mit kleinstem Fluss. Weiterhin betrachten wir die lokale Abhängigkeit des Flusses, welche sich mittels der NHIM beschreiben lässt.

Based on the generating matrix to the Tribonacci sequence, the Tribonacci cat map is a discrete chaotic dynamical system, similar to Arnold's discrete cat map, but on three dimensional space. In this thesis, this new mapping is introduced and the properties of its matrix are presented. The main results of the investigation prove how the size of the domain of the map affects its period and explore the orbit lengths of non-trivial points. Different upper bounds to the map are studied and proved, and a conjecture based on numerical calculations is proposed. The Tribonacci cat map is used for applications such as 3D image encryption and colour encryption. In the latter case, the results provided by the mapping are compared to those from a generalised form of the map.

Avec l'avènement de machine parallèles multi-coeurs, de nombreux algorithmes doivent être modifiés ou conçus pour s'adapter à ces architectures. Ces algorithmes consistent pour la plupart à diviser le problème original en plusieurs petits sous-problèmes et à les distribuer sur les différentes unités de calcul disponibles. La résolution de ces petits sous-problèmes peut être exécutée en parallèle, des communications entre les unités de calcul étant indispensables pour assurer la convergence de ces méthodes.Ma thèse propose de nouveaux algorithmes parallèles pour résoudre de grands systèmes linéaires.Les algorithmes proposés sont ici basés sur la méthode du gradient. Deux points fondamentaux de la méthode du gradient sont la direction de descente de la solution approchée et la valeur du pas de descente, qui détermine la modification à effectuer à chaque itération. Nous proposons dans cette thèse de calculer la direction et le pas indépendamment et localement sur chaque unité de calcul, ce qui nécessite moins de synchronisation entre les processeurs, et par suite rend chaque itération simple et plus rapide, et rend son extension dans un contexte asynchrone possible.Avec les paramètres d'échelle appropriés pour le pas des longueurs, la convergence peut être démontrée pour les deux versions synchrone et asynchrone des algorithmes. De nombreux tests numériques illustrent l’efficacité de ces méthodes.L'autre partie de ma thèse propose d'utiliser une méthode d'extrapolation pour accélérer les méthodes itératives classiques avec retard. Bien que les séquences de vecteur générées par des méthodes itératives asynchrones générales classiques ne peut être accélérée, nous sommes en mesure de démontrer que, une fois le modèle de calcul et de communication fixés au cours de l’exécution, la séquence de vecteurs générés peut être accéléré. De nombreux tests numériques illustrent l’efficacité de ces accélérations dans le cas des méthodes avec retard. / With the increase of architectures composed of multi-cores, many algorithms need to revisited and be modified to exploit the power of these new architectures. These algorithms divide the original problem into “small pieces” and distribute these pieces to different processors at disposal, thus communications among them are indispensible to assure the convergence. My thesis mainly focus on solving large sparse systems of linear equations in parallel with new methods. These methods are based on the gradient methods. Two key parameters of the gradient methods are descent direction and step-length of descent for each iteration. Our methods compute the directions locally, which requires less synchronization and computation, leading to faster iterations and make easy asynchronization possible. Convergence can be proved in both synchronized or asynchronized cases. Numerical tests demonstrate the efficiency of these methods. The other part of my thesis deal with the acceleration of the vector sequences generated by classical iterative algorithms. Though general chaotic sequences may not be accelerated, it is possible to prove that with any fixed retard pattern, then the generated sequence can be accelerated. Different numerical tests demonstrate its efficiency.

The Ph.D. Thesis ¿Operators on weighted spaces of holomorphic functions¿ presented here treats different areas of functional analysis such as spaces of holomorphic functions, infinite dimensional holomorphy and dynamics of operators. After a first chapter that introduces the notation, definitions and the basic results we will use throughout the thesis, the text is divided into two parts. A first one, consisting of Chapters 1 and 2, focused on a study of weighted (LB)-spaces of entire functions on Banach spaces, and a second one, corresponding to Chapters 3 and 4, where we consider differentiation and integration operators acting on different classes of weighted spaces of entire functions to study its dynamical behaviour. In what follows, we give a brief description of the different chapters: In Chapter 1, given a decreasing sequence of continuous radial weights on a Banach space X, we consider the weighted inductive limits of spaces of entire functions VH(X) and VH0(X). Weighted spaces of holomorphic functions appear naturally in the study of growth conditions of holomorphic functions and have been investigated by many authors since the work of Williams in 1967, Rubel and Shields in 1970 and Shields and Williams in 1971. We determine conditions on the family of weights to ensure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study Hörmander algebras of entire functions defined on a Banach space and we give a description of them in terms of sequence spaces. We also focus on algebra homomorphisms between these spaces and obtain a Banach-Stone type theorem for a particular decreasing family of weights. Finally, we study the spectra of these weighted algebras, endowing them with an analytic structure, and we prove that each function f ¿ VH(X) extends naturally to an analytic function defined on the spectrum. Given an algebra homomorphism, we also investigate how the mapping induced between the spectra acts on the corresponding analytic structures and we show how in this setting composition operators have a different behavior from that for holomorphic functions of bounded type. This research is related to recent work by Carando, García, Maestre and Sevilla-Peris. The results included in this chapter are published by Beltrán in [14]. Chapter 2 is devoted to study the predual of VH(X) in order to linearize this space of entire functions. We apply Mujica¿s completeness theorem for (LB)-spaces to find a predual and to prove that VH(X) is regular and complete. We also study conditions to ensure that the equality VH0(X) = VH(X) holds. At this point, we will see some differences between the finite and the infinite dimensional cases. Finally, we give conditions which ensure that a function f defined in a subset A of X, with values in another Banach space E, and admitting certain weak extensions in a space of holomorphic functions can be holomorphically extended in the corresponding space of vector-valued functions. Most of the results obtained have been published by the author in [13]. The rest of the thesis is devoted to study the dynamical behaviour of the following three operators on weighted spaces of entire functions: the differentiation operator Df(z) = f (z), the integration operator Jf(z) = z 0 f(¿)d¿ and the Hardy operator Hf(z) = 1 z z 0 f(¿)d¿, z ¿ C. In Chapter 3 we focus on the dynamics of these operators on a wide class of weighted Banach spaces of entire functions defined by means of integrals and supremum norms: the weighted spaces of entire functions Bp,q(v), 1 ¿ p ¿ ¿, and 1 ¿ q ¿ ¿. For q = ¿ they are known as generalized weighted Bergman spaces of entire functions, denoted by Hv(C) and H0 v (C) if, in addition, p = ¿. We analyze when they are hypercyclic, chaotic, power bounded, mean ergodic or uniformly mean ergodic; thus complementing also work by Bonet and Ricker about mean ergodic multiplication operators. Moreover, for weights satisfying some conditions, we estimate the norm of the operators and study their spectrum. Special emphasis is made on exponential weights. The content of this chapter is published in [17] and [15]. For differential operators ¿(D) : Bp,q(v) ¿ Bp,q(v), whenever D : Bp,q(v) ¿ Bp,q(v) is continuous and ¿ is an entire function, we study hypercyclicity and chaos. The chapter ends with an example provided by A. Peris of a hypercyclic and uniformly mean ergodic operator. To our knowledge, this is the first example of an operator with these two properties. We thank him for giving us permission to include it in our thesis. The last chapter is devoted to the study of the dynamics of the differentiation and the integration operators on weighted inductive and projective limits of spaces of entire functions. We give sufficient conditions so that D and J are continuous on these spaces and we characterize when the differentiation operator is hypercyclic, topologically mixing or chaotic on projective limits. Finally, the dynamics of these operators is investigated in the Hörmander algebras Ap(C) and A0 p(C). The results concerning this topic are included by Bonet, Fernández and the author in [16]. / Beltrán Meneu, MJ. (2014). Operators on wighted spaces of holomorphic functions [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/36578 / TESIS / Premios Extraordinarios de tesis doctorales

Partial transport barriers in the chaotic sea of Hamiltonian systems restrict classical chaotic transport, as they only allow for a small flux between phase-space regions. In two-dimensional (2D) symplectic maps, the most restrictive partial barriers are based on a cantorus, the remnants of a broken one-dimensional (1D) torus forming a Cantor set. Quantum mechanically for 2D symplectic maps one has a universal transition from impeded to unimpeded transport. The scaling parameter is the ratio of flux to the Planck cell of size h, so quantum transport is suppressed if h is much bigger than the flux while mimicking classical transport if it is much smaller. Whether a transition exists in higher-dimensional systems and how it scales is still an open question and will be answered in this talk. In a four-dimensional (4D) symplectic map, the cantorus is generalized to a normally hyperbolic invariant manifold (NHIM) with the structure of a cantorus. Using the general flux formula, we consider higher-order periodic NHIMs to approximate the global flux across a partial barrier. One naively expects that the scaling parameter of the universal transition is the same, but now with a Planck cell h squared. We show that due to classical diffusive transport along resonance channels, the quantized system exhibits dynamical localization and the localization length modifies the scaling parameter. / Partielle Transportbarrieren in der chaotischen See von Hamiltonischen Systemen schränken den klassischen chaotischen Transport ein, indem sie nur einen kleinen Fluss zwischen Phasenraumregionen zulassen. In zweidimensionalen (2D) symplektischen Abbildungen basieren die restriktivsten partiellen Barrieren auf einem Cantorus, die Cantor-Menge der Überreste eines zerstörten ein-dimensionalen (1D) Torus. In quantisierten 2D symplektischen Abbildungen findet man einen universellen Übergang von eingeschränktem zu uneingeschränktem Transport. Der Skalierungsparameter ist das Verhältnis vom Fluss zur Planck-Zelle der Größe h, so dass der quantenmechanische Transport unterdrückt ist, wenn h sehr viel größer ist als der Fluss, während klassischer Transport nachgeahmt wird, wenn er sehr viel kleiner ist. Ob jedoch auch ein universeller Übergang in höherdimensionalen Systemen existiert und wie er skaliert, ist bislang ungeklärt und wird in dieser Arbeit untersucht. In einer vierdimensionalen (4D) symplektischen Abbildung ist die Verallgemeinerung des Cantorus eine normal hyperbolische invariante Mannigfaltigkeit (NHIM) mit der Struktur eines Cantorus. Wir betrachten periodische NHIMs höherer Ordnung um den globalen Fluss durch eine partielle Barriere mit der allgemeinen Flussformel zu approximieren. Naiverweise erwartet man, dass der Skalierungsparameter des universellen Übergangs gleich ist, jedoch mit der neuen Größe der Planck-Zelle h quadriert. Wir zeigen, dass aufgrund von klassischen, diffusiven Transport entlang von Resonanzkanälen das quantisierte System dynamische Lokalisierung aufweist und die Lokalisierungslänge Einfluss auf den Skalierungsparameter hat.

Astrophysical observations of the cosmic microwave background point to inconsistencies in the standard model of cosmology, and a primordial accelerated expansion of the universe known as inflation has been suggested as a solution. Unfortunately, observational evidence of inflation is lacking, and there exists hundreds of models that populate the inflationary landscape. In this thesis, we explore three of these and see what constraints are set on them in order to account for observations. We find that two of the models have regimes of trans-planckian nature, while the third leads to a non-invertible equation. / Astronomiska observationer av den kosmiska bakgrundsstrålningen tyder på bristfälligheter i standardkosmologin, vilket har lett till förslaget om en accelererande expansion i de tidigaste skeden av universum känd som inflation. I avsaknaden av observationella bevis finns det numera hundratals inflationsmodeller, och i detta arbete kommer vi att rikta fokuset mot tre av dessa avvilka två visar sig överstiga transplanckianska värden medan den sista leder till en ekvation som inte är inverterbar.