Many physical phenomena can be described by nonlinear models. The last few decades have seen an enormous growth of the applicability of nonlinear models and of the development of related nonlinear concepts. This has been driven by modern computer power as well as by the discovery of new mathematical techniques, which include two contrasting themes: (i) the theory of dynamical systems, most popularly associated with the study of chaos, and (ii) the theory of integrable systems associated, among other things, with the study of solitons. In this dissertation, we study two nonlinear models. One is the 1-dimensional vibrating string satisfying wtt − wxx = 0 with van der Pol boundary conditions. We formulate the problem into an equivalent first order hyperbolic system, and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary conditions. Thus, the problem is reduced to the discrete iteration problem of the type un+1 = F (un). Periodic solutions are investigated, an invariant interval for the Abel equation is studied, and numerical simulations and visualizations with different coefficients are illustrated. The other model is the Korteweg-de Vries-Burgers (KdVB) equation. In this dissertation, we proposed two new approaches: One is what we currently call First Integral Method, which is based on the ring theory of commutative algebra. Applying the Hilbert-Nullstellensatz, we reduce the KdVB equation to a first-order integrable ordinary differential equation. The other approach is called the Coordinate Transformation Method, which involves a series of variable transformations. Some new results on the traveling wave solution are established by using these two methods, which not only are more general than the existing ones in the previous literature, but also indicate that some corresponding solutions presented in the literature contain errors. We clarify the errors and instead give a refined result.

Periodic and chaotic dynamical systems follow deterministic equations such as Newton's laws of motion. To distinguish the difference between two systems, the initial conditions have an important role. Chaotic behaviors or dynamics are characterized by sensitivity to initial conditions. Mathematically, a chaotic system is defined as a system very sensitive to initial conditions. A small difference in initial conditions causes unpredictability in the final outcome. If error is measured from the initial state, the relative error grows exponentially. Prediction becomes impossible and finally, chaotic systems can come to become stochastic system. To make chaotic motion, the number of variables in the system should be above three and there should be non-linear terms coupling several of the variables in the equation of motion. Phase space is defined as the space spanned by the coordinate and velocity vectors. In our case, mixing zone is phase space. With the above characteristics - the initial condition sensitivity of a chaotic system, our plan is to find most efficient chaotic stirrer. In this thesis, we present four methods to measure mixing state based on the chaotic dynamics theory. The Lyapunov exponent is a measure of the sensitivity to initial conditions and can be used to calculate chaotic strength. We can decide the chaotic state with one real number and measure efficiency of the chaotic mixer and find the optimum frequency. The Poincare section method provides a means for viewing the phase space diagram so that the motion is observed periodically. To do this, the trajectory is sectioned at regular intervals. With the Poincare section method, we can find 'islands' considered as bad mixed zones so that the mixing state can be measured qualitatively. With the chaotic dynamics theory, the initial length of the interface can grow exponentially in a chaotic system. We will show the above characteristics of the chaotic system to prove as fact that our model is an efficient chaotic mixer. The final goal for making chaotic stirrer is how to implement efficient dispersed particles. The box counting method is focused on measurement of the particles dispersing state. We use snap shots of the mixing process and with these snap shots, we devise a plan to measure particles' dispersing rate using the box-counting method.

All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local stretching/contraction and the smearing due to noise. My goal is to determine the `finest attainable' partition for a given hyperbolic dynamical system and a given weak additive white noise. That is achieved by computing the local eigenfunctions of the Fokker-Planck evolution operator in linearized neighborhoods of the periodic orbits of the corresponding deterministic system, and using overlaps of their widths as the criterion for an optimal partition. The Fokker-Planck evolution is then represented by a finite transition graph, whose spectral determinant yields time averages of dynamical observables. The method applies in principle to both continuous- and discrete-time dynamical systems. Numerical tests of such optimal partitions on unimodal maps support my hypothesis.

The structure of wavefunctions of quantum systems strongly depends on the underlying classical dynamics. In this text a selection of articles on eigenfunctions in systems with fully chaotic dynamics and systems with a mixed phase space is summarized. Of particular interest are statistical properties like amplitude distribution and spatial autocorrelation function and the implication of eigenfunction structures on transport properties. For systems with a mixed phase space the separation into regular and chaotic states does not always hold away from the semiclassical limit, such that chaotic states may completely penetrate into the region of the regular island. The consequences of this flooding are discussed and universal aspects highlighted.

<p>Since the late 1960’s, the hydromechanical term turbulence has been a part of the business administration vocabulary, but until the late 1980’s and early 1990’s, a relatively small amount of research was dedicated to this field. These studies and more contemporary ones conclude that where the business environment is paradoxical and of fast- changing and chaotic nature, successful corporate strategies are shaped by strategic flexibility founded in high innovation rates, networks and alliances, and organisational elasticity and adaptiveness. From this perspective, the purpose of this study was to track and examine the strategy formation processes of a company operating in a turbulent context, and to contribute to an understanding of how these turbulent conditions can be managed. The study was conducted with a hermeneutic, systems- oriented, longitudinal case-study method and with a contextcontent- process perspective in which the process was the key factor. To a large extent, our conclusions coincide with those of other researchers. Forming multidimensional networks and alliances coloured by voluntary initiatives and full attention seem to be an extremely important contribution to survival in turbulent contexts. Nevertheless, it is equally important to break up and build new alliances as the initial objectives of the arrangement have expired or been reached. Furthermore, in contrast to other researchers’ observations, we conclude that high innovation rates do not necessarily lead to a greater potential to be successful in a turbulent context. The issue is instead to present a product offering flexible in itself developed and marketed by a flexible organisation. Innovation rates are decided by self-initiated and unofficial activity on part of the r&d teams and other coworkers, and management’s task is to facilitate for this corporate creativity to develop.</p>

Dans cette thèse, nous étudions un modèle de décroissance (decay) d'un système quantique à plusieurs niveaux appelé le modèle de Friedrichs. Dans un premier travail, nous considérons un couplage d'un kaon avec un environnement décrit par un continuum d'énergie. On montre que les oscillations du kaon entre les états K_1, K_2, leur decay et la violation CP sont bien décrits par ce type de modèle. Ensuite, nous appliquons à ce modèle le formalisme de l'opérateur de temps qui décrit la résonance, c'est-à-dire la probabilité de survie des états instables. Enfin, nous considérons un gaz de Lorentz comme un ensemble de boules de billard avec des collisions élastiques contre des obstacles et un système de sphères dures en dimension 2. Nous étudions la simulation numérique de la dynamique du système et calculons l'augmentation de l'entropie de non-équilibre au cours du temps sous l'effet des collisions et sa relation avec les exposants de Lyapounov positifs.

Les travaux présentés dans ce mémoire portent sur l'optimisation de la structure et de la commande de systèmes d'émulation de puissance, appelés Charges Actives. Afin de présenter de très bonnes performances dynamiques ainsi qu'une très grande robustesse, ces Charges Actives utilisent des modulateurs et régulateurs de courant (MRC) et de tension (MRT). Ces procédés font partie de la classe des régulateurs auto-oscillant et sont donc par nature fortement non-linéaires. Aussi, pour que leur application à la Charge Active soit optimale, le premier point abordé dans ce mémoire traite de la modélisation de ces régulateurs et de l'identification des différents problèmes éventuels, inhérents à leurs non-linéarités. Il est alors apparut que des phénomènes de synchronisation et d'instabilité de la fréquence de commutation peuvent apparaître si certaines conditions ne sont pas respectées. Le second point abordé est la généralisation de ces procédés de modulation à des systèmes quelconques, basée sur une méthode de synthèse en mode de glissement. De cette étude, une nouvelle structure de modulation et de régulation de tension est proposée, permettant de répondre plus efficacement aux problématiques posées par la Charge Active. Les résultats expérimentaux obtenus sur un prototype de Charge Active montrent les très grandes performances de ce nouveau procédé, contribuant ainsi à l'amélioration de la qualité et de la précision des anciennes et nouvelles générations de Charges Actives.

Thèse doctorat : Mathématiques : La Rochelle : 2003. / Bibliogr. p. 113-119.

Reproduction de : Thèse de doctorat : Lasers, Molécules, Rayonnement atmosphérique : Lille 1 : 2004. / N° d'ordre (Lille 1) : 3554. Titre provenant de la page de titre du document numérisé. Bibliogr. p. 185-187.

We numerically investigate electron transport properties in an electron waveguide which can be constructed in 2DEG of the heterostructure of GaAs and AlGaAs. We apply R-matrix theory to solve a Schrödinger equation and construct a S-matrix, and we then calculate conductance of an electron waveguide. We study single impurity scattering in a waveguide. A [delta]-function model as a single impurity is very attractive, but it has been known that [delta]-function potential does not give a convergent result in two or higher space dimensions. However, we find that it can be used as a single impurity in a waveguide with the truncation of the number of modes. We also compute conductance for a finite size impurity by using R-matrix theory. We propose an appropriate criteria for determining the cut-off mode for a [delta]-function impurity that reproduces the conductance of a waveguide when a finite impurity presents. We find quantum scattering echoes in a ripple waveguide. A ripple waveguide (or cavity) is widely used for quantum chaos studies because it is easy to control a particle's dynamics. Moreover we can obtain an exact expression of Hamiltonian matrix with for the waveguide using a simple coordinate transformation. Having an exact Hamiltonian matrix reduces computation time significantly. It saves a lot of computational needs. We identify three families of resonance which correspond to three different classical phase space structures. Quasi bound states of one of those resonances reside on a hetero-clinic tangle formed by unstable manifolds and stable manifolds in the phase space of a corresponding classical system. Resonances due to these states appear in the conductance in a nearly periodic manner as a function of energy. Period from energy frequency gives a good agreement with a prediction of the classical theory. We also demonstrate wavepacket dynamics in a ripple waveguide. We find quantum echoes in the transmitted probability of a wavepacket. The period of echoes also agrees with the classical predictions. We also compute the electron transmission probability through a multi-ripple electron waveguide. We find an effect analogous to the Dicke effect in the multi-ripple electron waveguide. We show that one of the S-matrix poles, that of the super-radiant resonance state, withdraws further from the real axis as each ripple is added. The lifetime of the super-radiant state, for N quantum dots, decreases as [1/N] . This behavior of the lifetime of the super-radiant state is a signature of the Dicke effect. / text