The present thesis analyses the proposal of CubeHash with spe- cial emphasis on the following papers: "Inside the Hypercube" [1], "Sym- metric States and Their Improved Structure" [7] and "Linearisation Frame- work for Collision Attacks" [6]. The CubeHash algorithm is presented in a concise manner together with a proof that the CubeHash round function R : ({0, 1}32 )32 → ({0, 1}32 )32 is a permutation. The results of [1] and [7] con- cerning the CubeHash symmetric states are reviewed, corrected and substan- tiated by proofs. More precisely, working with a definition of D-symmetric state, based on [7], the thesis proves both that for V = Z4 2 and its linear subspace D, there are 22 |V | |D| D-symmetric states and an internal state x is D-symmetric if and only if the state R(x) is D-symmetric. In response to [1], the thesis presents a step-by-step computation of a lower bound for the num- ber of distinct symmetric states, explains why the improved preimage attack does not work as stated and gives a mathematical background for a search for fixed points in R. The thesis further points out that the linearisation method from [6] fails to consider the equation (A ⊕ α) + β = (A + β) ⊕ α (∗), present during the CubeHash iteration phase. Necessary and sufficient conditions for A being a solution to (∗) are...

In support of society's technological evolution, the study of nonlinear systems in engineering and sciences has become a vital research area. Aiming to contribute in this field, this thesis investigates the behaviour of nonlinear systems using the 'Wiener theories'. As a useful example the Duffing oscillator is investigated in this work. In many real-life applications, nonlinear systems are excited randomly so this work examines systems under white-noise excitation using the Wiener series. Equivalent Linearisation (EL) is a well-known and simple method that approximates a nonlinear system by an equivalent linear system. However, it has deficiencies which this thesis attempts to improve. Initially, the performance of EL for different types of nonlinearities will be assessed and an alternative method to enhance it is suggested. This requires the calculation of the first Wiener kernel of various system defined quantities. The first Wiener kernel, as it will be shown, is the foundation of this research and a central element of the Wiener theory. In this thesis, an analytical proof to explain the interesting behaviour of the first Wiener kernel for a system with nonlinear stiffness is included using an energy transfer approach. Furthermore, the method mentioned above to enhance EL known as the Single-Pole Fit method (SPF) is to be tested for different kinds of systems to prove its robustness and validity. Its direct application to systems with nonlinear stiffness and nonlinear damping is shown as well as its ability to perform for systems with two degrees of freedom where an extension of the SPF method is required to achieve the desired solution. Finally, an investigation to understand and replicate the complex behaviour observed by the first Wiener kernel in the early chapters is carried out. The groundwork for this investigation is done by modelling an isolated nonlinear spring with a series of linear filters and certain nonlinear operations. Subsequently, an attempt is made to relate the principles governing the successful spring model presented to the original nonlinear system. An iterative procedure is used to demonstrate the application of this method, which also enables this new modelling approach to be related to the SPF method.

On s'intéresse dans ce travail à la description des enrichissements des disques de Siegel d'une fraction rationnelle f. Dans un premier temps nous étudions les enrichissements qui sont définis sur un ouvert de la grande orbite d'un disque de Siegel donné. Ce sont nécessairement des applications qui commutent à f là où les compositions ont un sens. Ce sont donc des applications linéaires en coordonnées linéarisantes. Le résultat principal de ce travail est que l'on peut obtenir toutes les applications linéaires en coordonnées linéarisantes définies sur un sous-disque du disque de Siegel de f. Pour démontrer ce résultat nous utilisons la compacité des applications linéarisantes normalisées, le théorème des fonctions implicites dans l'espace des fractions rationnelles de degré fixé et une étude du comportement du rayon d'univalence des applications linéarisantes. Nous identifions également les approches donnant lieu à des enrichissements définis ou à valeurs dans le disque de Siegel tout entier (enrichissements maximaux). Au passage nous généralisons aux limites avec ordre de contact fini par rapport au cercle unité un théorème de JC.Yoccoz sur le comportement du rayon d'univalence pour la famille quadratique lorsque le paramètre converge vers un nombre complexe de module un et d'argument un nombre de Brjuno.Ensuite, nous nous intéressons au cas où f a plusieurs cycles de disques de Siegel. Nous utilisons le théorème de transversalité d'A.Epstein pour décrire les enrichissements de f dans ce cas là. La linéarisabilité de f et la convergence des applications linéarisantes permet de transférer le problème de la description des enrichissements de Siegel de f à un problème de limite géométrique de sous-semigroupes de l'ensemble des nombres complexes non-nuls engendrés par un élément. Nous donnons dans ce travail un modèle topologique de l'adhérence de cet ensemble de sous-semigroupes. Nous déduisons de ces résultats une interprétation en terme de convergence géométrique de dynamiques de polynômes quadratiques et une description des points d'accumulation, pour la topologie de Hausdorff sur les compacts non-vides, des ensembles de Julia lorsque le paramètre tend vers un paramètre de Siegel. / In this work we are interested in giving the description of Siegel discs enrichments of a rational map f. We first study the case of enrichments that are defined on an open subset of the grand orbit of a given Siegel disc. These maps commute with f where it makes sense. Thus they are linear in linearizing coordinates. The main result of this work is that we can obtain all linear maps in linearizing coordinates that are defined in a subdisc of the Siegel disc. For this we use the compactness of the set of normalized linearizing maps, the implicit functions theorem in the space of rational maps with fixed degree and a study on the behaviour on the univalent radius of the linearizing maps. We identify approaches giving enrichments that are defined or take values on the whole Siegel disc (maximal enrichments). We generalize to finite order of contact approaches with respect to the unit circle a theorem of JC.Yoccoz on the behaviour of the univalent radius for the quadratic family when the parameter converges to a complex number of modulus one with argument a Brjuno number.We then focus on the case where f has more than one Siegel disc. We make use of A.Epstein's transversality theorem to describe Siegel enrichments of f in this case. The linearisability of f and the convergence of the linearizing maps reduces the problem of Siegel enrichments description to a geometric limit problem on one generated closed sub-semigroups ofthe set of non zero complex numbers. We give in this work a topological model fot the closure of this set of sub-semigroups.We deduce from these results an interpretation in terms of geometric convergence of quadratic polynomial dynamics and we describe the accumulation points (for the Hausdorff topology on non empty compact subsets) of Julia sets when the parameter converges to a Siegel parameter.

Le problème de la maîtrise du transport de l'énergie électrique a donné naissance au projet FACTS (Flexible Alternative Current Transmission System ! Pour améliorer la flexibilité des réseaux de transport existants. Le ST A TCOM (ST ATic COMD*~ n s ator) est un dispositif FACTS dédié à la compensation d'énergie réactive transitant sur le réseau . L'évolution des composants d'électronique de puissance a apporté des solutions technologiques pour la réalisation des structures onduleurs du STATCOM. En tenant compte des diverses structures présentées, un modèle dynnamique est construit en utilisant la théorie du modèle moyen généralisé. Il est ensuite validé avec le modèle topologique, lequel décrit le comportement fin de l'onduleur. Afin d' avoir un contrôle robuste du courant réactif du dispositif, une loi de commande non linéaire est élaborée à partir de la théorie de la linéarisation par bouclage. La linéarisation est obtenue au détriment des comportements dynamiques du courant actif et de la tension continue de l' onduleur. Une optimisation de la loi de conunande est proposée afin de maîtriser le comportement dynamique de toutes les variables du dispositif. Cette loi de commal1de est validée sur le modèle topologique après avoir abordé le problème des filtres de mesure. La mise en place de la linéarisation par bouclage nécessite une connaissance des valeurs des composants de la structure. Une estimation ainsi qu'une correction de l'erreur commise sur ces grandeurs sont alors proposées. Le modèle du ST A TCOM avec ses lois de commande est ensuite inséré dans un logiciel destiné à l' étude du comportement dynamique de réseaux (EUROSTAG). A cet effet, le modèle mis au point prend en compte le comportement dynamique de la structure du ST A TCOM, et apporte une richesse supplémentaire pour l 'étude dynamique des réseaux. L'intérêt de la loi de commande optimisée est nus en évidence comparativement aux solutions classiques.

Three mathematical models, two of primary alkaline battery cathode discharge, and one of primary alkaline battery discharge, are developed, presented, solved and investigated in this thesis. The primary aim of this work is to improve our understanding of the complex, interrelated and nonlinear processes that occur within primary alkaline batteries during discharge. We use perturbation techniques and Laplace transforms to analyse and simplify an existing model of primary alkaline battery cathode under galvanostatic discharge. The process highlights key phenomena, and removes those phenomena that have very little effect on discharge from the model. We find that electrolyte variation within Electrolytic Manganese Dioxide (EMD) particles is negligible, but proton diffusion within EMD crystals is important. The simplification process results in a significant reduction in the number of model equations, and greatly decreases the computational overhead of the numerical simulation software. In addition, the model results based on this simplified framework compare well with available experimental data. The second model of the primary alkaline battery cathode discharge simulates step potential electrochemical spectroscopy discharges, and is used to improve our understanding of the multi-reaction nature of the reduction of EMD. We find that a single-reaction framework is able to simulate multi-reaction behaviour through the use of a nonlinear ion-ion interaction term. The third model simulates the full primary alkaline battery system, and accounts for the precipitation of zinc oxide within the separator (and other regions), and subsequent internal short circuit through this phase. It was found that an internal short circuit is created at the beginning of discharge, and this self-discharge may be exacerbated by discharging the cell intermittently. We find that using a thicker separator paper is a very effective way of minimising self-discharge behaviour. The equations describing the three models are solved numerically in MATLABR, using three pieces of numerical simulation software. They provide a flexible and powerful set of primary alkaline battery discharge prediction tools, that leverage the simplified model framework, allowing them to be easily run on a desktop PC.