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No description available.

This thesis contains a number of new results on the topological and geometric properties of certain invariant sets in the dynamics of entire functions, inspired by recent work of Rippon and Stallard. First, we explore the intricate structure of the spider's web fast escaping sets associated with certain transcendental entire functions. Our results are expressed in terms of the components of the complement of the set (the 'holes' in the web). We describe the topology of such components and give a characterisation of their possible orbits under iteration. We show that there are uncountably many components having each of a number of orbit types, and we prove that components with bounded orbits are quasiconformally homeomorphic to components of the filled Julia set of a polynomial. We prove that there are singleton periodic components and that these are dense in the Julia set. Next, we investigate the connectedness properties of the set of points K( f) where the iterates of an entire function f are bounded. We describe a class of transcendental entire functions for which K( f) is to- tally disconnected if and only if each component of K (f) containing a critical point is aperiodic. Moreover we show that, for such functions, if K(f) is disconnected then it has uncountably many components. We give examples of functions for which K(f) is totally disconnected, and we use quasiconformal surgery to construct a function for which K(f) has a component with empty interior that is not a singleton. Finally we show that, if the Julia set of a transcendental entire function is locally connected, then it must take the form of a spider's web. In the opposite direction, we prove that a spider's web Julia set is always locally connected at a dense subset of buried points. We also show that the set of buried points (the residual Julia set) can be a spider's web.

Le séchage est une opération incontournable pour la valorisation énergétique des boues résiduaires. La flexibilité pour ajuster la teneur en matière sèche finale de la boue est un critère important pour le choix d'une technologie. Cet objectif est difficile à atteindre pour les sécheurs à palettes. La modélisation du processus est alors essentielle. Malheureusement, le comportement rhéologique des boues est complexe et la mécanique des fluides numérique est hors de portée. La notion de Distribution des Temps de Séjour est employée ici pour caractériser l'écoulement. Un protocole fiable et reproductible a été établi et mis en œuvre sur un pilote de laboratoire. Des injections Dirac d'oxyde de titane et de sels métalliques, avec la spectrométrie de fluorescence X comme méthode de détection, ont été employées pour caractériser les DTS du solide anhydre et de la boue humide. Pré-Mélanger la boue pâteuse, pour disperser le traceur par exemple, modifie la structure du matériau. Ceci a été mis en évidence par des mesures de distribution en taille des particules et par des caractérisations rhéologiques. Cependant, des expériences de séchage en batch ont montré que ce pré-Mélange n'a aucune influence sur la cinétique et sur la phase plastique. Nous avons montré que le solide anhydre et le solide humide s'écoulent de la même manière. Une seconde méthode, basée sur une détection par conductimétrie, a alors été développée. Plus facile à mettre en œuvre et moins onéreuse, cette méthode s'avère tout aussi fiable que la première. L'influence de la durée de stockage de la boue, avant séchage, a été évaluée. Le temps de séjour de la boue dans le sécheur double quand la durée de stockage passe de 24h à 48h. Finalement, un modèle d'écoulement, basé sur la théorie de chaînes de Markov, a été développé. L'écoulement du solide anhydre est décrit par une chaîne de n cellules parfaitement mélangées, n correspondant au nombre de palettes. Les probabilités de transition entre les cellules sont régies par deux paramètres : le ratio de recyclage interne, R, et la masse de solides retenus, MS. R est déterminé par la relation de Van der Laan et MS est identifié par ajustement du modèle aux données expérimentales. Le modèle décrit de manière satisfaisante les DTS. La masse de solides retenus identifiée est toujours plus faible que la quantité mesurée expérimentalement. Une partie de la boue, collée aux parois du sécheur et au rotor, agit comme un volume mort. / Drying is an unavoidable operation prior to sludge valorization in incineration, pyrolysis or gasification. The flexibility to adapt the solid content of the dried sludge to the demand is a major requirement of any drying system. This objective is difficult to reach for paddle dryers. Modeling the process is thus essential. Unfortunately, sludge rheological behavior is complex and computational fluid dynamics is out of reach for the time being. The concept of Residence Time Distribution (RTD) is used here to investigate sludge flow pattern in a paddle dryer. A reliable and reproducible protocol was established and implemented on a lab-Scale continuous dryer. Pulse injections of titanium oxide and of salt metals, with X-Ray fluorescence spectroscopy as detection method, were used to characterize the RTD of anhydrous solid and wet sludge, respectively. Premixing the pasty sludge, for tracer powder dispersion for instance, changes the structure of the material. This was highlighted through the measurements of particle size distributions and characterization of rheological properties. However, drying experiments performed in batch emphasized that premixing does not have any influence on the kinetic and the sticky phase. The RTD curves of the anhydrous solid are superimposed on those of the moist sludge. Consequently, a simpler protocol, based on pulse injection of chloride sodium and offline conductivity measurements, was established. Easier to implement in industry and cheaper, this method proves to be as reliable as the first one. The influence of storage duration prior to drying was assessed. The mean residence time doubles when the storage duration changes from 24h to 48h. Finally, a model based on the theory of Markov chains has been developed to represent the RTD. The flow of anhydrous solids is described by a chain of n perfectly mixed cells, n corresponding to the number of paddles. The transition probabilities between the cells are governed by two parameters: the ratio of internal recirculation, R, and the solids hold-Up, MS. R is determined from the Van der Laan's relation and MS is identified by fitting the model to the experimental RTD. The model describes the flow pattern with a good accuracy. The computed hold-Up is lower than the experimental one. Part of the sludge is stuck to the walls of the dryer, acting as dead volumes in the process.

Dynamische Systeme mit Gedächtnis spielen in verschiedensten Anwendungen und Forschungsgebieten eine wesentliche Rolle. Gedächtnis bedeutet dabei, dass das zukünftige Systemverhalten nicht nur durch den aktuellen Zustand festgelegt wird, sondern im Allgemeinen auch durch vergangenen Zustände. Ein prominenter Vertreter für dieses Verhalten ist die Hysterese. Aufgrund der unterschiedlichen Mechanismen, welche zum Auftreten von Hysterese führen können, haben sich eine Vielzahl an Modellen etabliert, um diese zu beschreiben und zu modellieren. Zwei häufig verwendete Modelle sind dabei das Random Field Ising-Model und das Preisach-Model. Beide Modelle unterscheiden sich grundlegend in der Art, wie es zu Hysterese kommt. Während beim Random Field Ising-Model Hysterese aufgrund der Wechselwirkung benachbarter Spins auftritt, benutzt das Preisach-Model hingegen eine Vielzahl an elementaren bistabilen Relais, um komplexes hysteretisches Verhalten abzubilden. Trotz dieser Unterschiedlichkeit zeigen beide Modelle ähnliche Eigenschaften wie return point memory und wipe-out. Wir wollen in dieser Arbeit das dynamische Verhalten eines einfachen harmonischen Oszillators untersuchen, welcher mithilfe eines Feedback-Loops an ein hysteretisches Spinsystem gekoppelt wird. Es soll das Verhalten dieses Hybrid-Systems, das sowohl aus kontinuierlichen als auch aus diskreten Variablen besteht, für verschieden große Spinsysteme untersucht werden. Wir konzentrieren uns dabei auf drei vereinfachte Spinkonfigurationen. Dies ermöglicht uns, unter Verwendung der Preisach-Theorie, den Limes eines unendlich großen Spinsystems analytisch zu beschreiben. Wir zeigen, dass sich das Verhalten von dynamischen Systemen gekoppelt an ein endliches Spinsystem im Allgemeinen von Systemen gekoppelt an ein unendliches Spinsystem unterscheidet. Im Zuge dessen werden wir eine Methode vorstellen, um Lyapunov Spektren für dynamische Systeme mit preisachartiger Hysterese und glatter Dichte zu bestimmen. Wir zeigen weiterhin, dass bestimmte relevante Größen wie fraktale Dimension und Magnetisierung im Allgemeinen kein selbstmittelndes Verhalten aufweisen. Diese Resultate können erhebliche Auswirkungen auf die Vergleichbarkeit und Interpretation von Theorie und Experiment bei dynamischen Systemen mit Hysterese haben. / Dynamical systems with memory play a huge role in technical applications as well as in different research fields. In general memory means, the systems' behavior is not only determined by its last state, but also by the history of previous states. One prominent example of such behavior is the hysteresis. Caused by the many reasons for hysteretic behavior, multiple models for hysteresis have been developed over the past hundred years. Two commonly used models are the Random Field Ising Model and the Preisach model. Both models differ in the way, how the memory is build into the system. Whereas, the Random Field Ising Model shows hysteresis because of the interaction between nearby spins, the complex hysteresis of the Preisach model is build by a superposition of elementary bi-stable relays. Besides these differences, both models show similar hysteric behavior like return point memory and wipe-out. In this work, we want to investigate the dynamical behavior of a simple harmonic oscillator coupled to Ising spins in a closed loop way, showing hysteresis. The system consists of discrete and continuous degrees of freedom, and therefore it has a hybrid character. Concentrating on three simplified spin interactions, on one hand we investigate the dynamical properties of the system for a varying finite number of spins and on the other hand we use the Preisach model to calculate the limit of an infinite number of spins. We find, that dynamical systems coupled to a finite and infinite number of spins, respectively, in general behave differently. Thereby, we develop a method to determine the whole Lyapunov spectrum for systems with Preisach like hysteresis and a smooth density. Furthermore, we show that some dynamical properties like the fractal dimension and the magnetization in general do not show self-averaging. These findings could have a huge impact on the comparability and interpretation of theoretical and experimental results in the context of dynamical systems with hysteresis.

Η παρούσα εργασία αφορά στη μελέτη Χαμιλτώνιων συστημάτων Ν μη γραμμικών ταλαντωτών, όπως είναι αυτό των Fermi Pasta και Ulam (FPU), με στόχο την βαθύτερη κατανόηση της δυναμικής των σχεδόν-περιοδικών τροχιών και του ρόλου των αντίστοιχων τόρων στο χώρο φάσεων, καθώς αυξάνουμε την ενέργεια Ε και τον αριθμό βαθμών ελευθερίας Ν του συστήματος. Το βασικό μας αποτέλεσμα είναι ότι υπάρχουν τόροι χαμηλής διάστασης, που προκύπτουν από τη συνέχεια των αντίστοιχων του γραμμικού συστήματος, οι οποίοι ευθύνονται για τις FPU επαναλήψεις και εμποδίζουν την ισοκατανομή της ενέργειας μεταξύ όλων των κανονικών τρόπων ταλάντωσης. Αναλύοντας ευστάθεια αυτών των τόρων, μπορέσαμε να δώσουμε μια πληρέστερη ερμηνεία στο Παράδοξο των FPU, συνδέοντας και συμπληρώνοντας έτσι δύο από τις επικρατέστερες ερμηνείες του εν λόγω φαινομένου. / The present work concerns the study of Hamiltonian systems of N nonlinear coupled oscillators, as it is the one by Fermi Pasta and Ulam (FPU), in order to understand the dynamics of quasi-periodic orbits and the role of their corresponding tori in phase space, as we increase the energy E and the number N of the degrees of freedom. Our fundamental result is that there exist tori of low dimension, that come from the continuation of the corresponding tori of the linear system, which are responsible for the FPU recurrences and prevent the system from equipartition of the energy among all normal modes. By investigating the stability of these tori, we achieved to provide a more complete explanation for the FPU paradox, connecting and supplementing in this way two of the most dominant approaches for this paradox.