In the reopening of fluid occluded airways, the pressure gradient due to the propagation of an air bubble causes extensive epithelial cell damage. The mechanism of cell necrosis and biotransport may be further understood by characterizing the flow fields near the tip of a semi-infinite bubble propagating through a fluid-filled bifurcation. A symmetric microfluidic pulmonary bifurcation model was fabricated for optical diagnostics with an instantaneous μ-PIV/ shadowgraphy microscopy system. Data handling and processing techniques were developed to calculate interfacial characteristics of multiphase flow from the microscopy system and accuracy was quantified through varying the apparatus set up. Differences in the interfacial geometric characteristics were quantified for changes in static and dynamic surface tension in comparisons of water, SDS, and Infasurf that may reflect changes in the mechanical stress that stimulate, and potentially damage, epithelial cells that line the airways. From these results, the asymmetrical tendencies of opening a symmetric pulmonary bifurcation model were quantified. It was found that pulmonary surfactant stabilized symmetric bifurcations that opened asymmetrically without the aid of surfactant. / acase@tulane.edu

Le fonctionnement d'un neurone, cellule fondamentale du système nerveux, intéresse de nombreuses disciplines scientifiques. Il existe ainsi des modèles mathématiques qui décrivent leur comportement par des systèmes d'EDO. Plusieurs de ces modèles peuvent ensuite être couplés afin de pouvoir étudier le comportement de réseaux, systèmes complexes au sein desquels émergent des propriétés. Ce travail présente, dans un premier temps, les principaux mécanismes régissant ce fonctionnement afin d'en comprendre la modélisation. Plusieurs modèles sont alors présentés, jusqu'à celui de Hindmarsh-Rose (1984), qui présente une dynamique lente-rapide. C'est sur l'étude numérique mais également théorique de la dynamique asymptotique et transitoire de ce dernier modèle, que se concentre la seconde partie de cette thèse. Dans une troisième partie, des réseaux d'interactions sont construits en couplant les systèmes dynamiques précédemment étudiés. L'étude du phénomène de synchronisation complète au sein de ces réseaux montre l'existence de propriétés émergentes pouvant être caractérisées par des lois de puissance. Enfin, un algorithme de détection de la synchronisation de bursts est proposé.

Un point d'aplatissement d'une courbe (réelle $C^\infty$ ou complexe holomorphe) de l'espace projectif (réel ou complexe) de dimension $n$ est un point de la courbe pour lequel l'hyperplan osculateur à un contact plus élevé qu'en un point ordinaire. Pour $n=2$, les points d'aplatissement sont communément appelés les points d'inflexions.\\ Dans la première partie de la thèse étudie les familles de courbes par rapport aux points d'aplatissement.\\ On introduit une notion de forme normale par rapport aux aplatissements des fibres d'une application $f:(\KM^n,0) \to (\KM^{n-1},0)$, avec $\KM=\RM$ ou $\KM=\CM$,\\ Ensuite, on commence la classification des germes d'applications (réels $C^\infty$ ou complexes holomorphes) $f:(\KM^n,0) \to (\KM^{n-1},0)$, avec $\KM=\RM$ ou $\KM=\CM$, par rapport aux points d'aplatissement des fibres de l'application. On introduit la notion de déformation verselle par rapport aux aplatissements, et on calcule ces déformations pour les fonctions de Morse de deux variables. Enfin, on définit ``les invariants fondamentaux de topologie projective'' d'un germe $f:(\KM^n,0) \to (\KM^{n-1},0)$ et on calcule ces invariants pour les éléments de la classification.\\ Dans une deuxième partie, on tente d'inclure la théorie des aplatissements des courbes en développant la théorie de propagation des fronts d'onde. par le biais d'un théorème de déformations verselles pour les applications legendriennes. On généralise des résultats de Kazarian sur les courbes spatiales au cas variétés de dimension quelconque. Notamment, on démontre un théorème sur la bifurcation des courbes paraboliques de certaines familles de surfaces dans l'espace projectif.

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This thesis concerns the analysis of dynamical systems suitable to be modelled by piecewise-smooth differential equations. In such systems the continuous-in-time dynamics is interrupted by discrete-in-time jumps in the state or governing equations of motion. Not only can this framework be used to describe existing systems with strong nonlinear behaviour such as impacts and friction, but the non-smooth properties can be exploited to design new mechanical devices. As suggested in this work it opens up the possibility of, for example, fast limit switches and energy transfer mechanisms. Particularly, the dynamics at the onset of low-velocity impacts in systems with recurrent dynamics, so called grazing bifurcations in impact-oscillators, are investigated. As previous work has shown, low-velocity impacts is a strong source of instability to the dynamics, and efforts to control the behaviour is of importance. This problem is approached in two ways in this work. One is to investigate the influence of parameter variations on the dynamic behaviour of the system. The other is to implement low-cost control strategies to regulate the dynamics at the grazing bifurcation. The control inputs are of impulsive nature, and utilizes the natural dynamics of the system to the greatest extent. The scientific contributions of this work is collected in five appended papers. The first paper consists of an experimental verification of a map that captures the correction to the smooth dynamics induced by an impact, known in the literature as the discontinuity map. It is shown that the lowest order expansion of the map accurately captures the transient growth rate of impact velocities. The second paper presents a constructive proof of a control algorithm for a rather large class of impact oscillators. The proof is constructive in the sense that it gives control parameters which stabilizes the dynamics at the onset of low-velocity impacts. In the third paper a piecewise-smooth quarter-car model is derived, and the control strategy is implemented to reduce impact velocities in the suspension system. In the fourth and fifth papers the grazing bifurcation of an impact oscillator with dry friction type damping is investigated. It turns out that the bifurcation is triggered by the disappearance of an interval of stable stick solutions. A condition on the parameters of the system is derived which differentiates between stable and unstable types of bifurcation scenarios. Additionally, a low-cost control strategy is proposed, similar to the one previously mentioned, to regulate the bifurcation scenario. / QC 20100811

L'objectif de cette thèse est la compréhension et la modélisation du cortex visuel des mammifères avec des modèles à taux de décharge, appelés les champs neuronaux, où les fines spécificités du calcul neuronal sont négligées. Cette thèse est divisée en trois parties, deux parties théoriques et une plus appliquée. Dans la première partie, nous examinons les états stationnaires des équations des champs neuronaux en utilisant des outils topologiques et la théorie des bifurcations. Nous sommes particulièrement intéressés par le nombre de ces états étant donné un stimulus parce que tous ces états sont des représentations corticales du stimulus. Toutefois, selon les paramètres, les équations du champ de neurones peuvent avoir des solutions stationnaires multiples qui sont autant de représentations corticales du stimulus. Si plus d'une solution stable existe, nous avons montré comment distinguer une de ces activités corticales comme étant la représentation '' principale'' du stimulus et les autres comme des illusions neuronales. L'étude aboutit à un schéma numérique pour calculer les différentes solutions stationnaires du réseau de champ de neurones en utilisant plusieurs paramètres de continuation : ce schéma est utile pour sa capacité à détecter les bifurcations Saddle-Node. Dans la deuxième partie de cette thèse, nous étudions les effets des retards de propagation de l'information, les principaux résultats théoriques étant la preuve d'un théorème de variété centrale. Cependant, le résultat le plus utile est une formule analytique pour les courbes de bifurcation de Hopf dans le plan (vitesse de propagation - délai synaptique). Ces courbes indiquent les paramètres qui produisent des oscillations spontanées des neurones. L'étude du réseau sans cette formule analytique est très laborieuse. Elle a été utilisée pour révéler la structure très complexe du diagramme de bifurcation dans les réseaux de neurones avec retards de propagation. Enfin, dans la dernière partie de cette thèse, nous étudions trois modèles de cortex visuel auxquels nous appliquons les outils développés dans les parties précédentes. Le premier modèle est le Ring Model of orientation tuning pour lequel nous avons découvert l'existence d'un seuil de perception et expliqué comment il peut être observé expérimentalement. Le second modèle étudié est celui Blumenfeld et al., très proche dans sa formulation du modèle précédent, et se fonde sur des données expérimentales (fournis par G. Masson et le laboratoire de F.Chavane à l'INT, Marseille, FRANCE). Nous avons montré comment les symétries imparfaites de la connectivité affectent les réponses du réseau. En particulier, nous avons montré comment le réseau parvient à produire une réponse en accord avec le stimulus malgré ses préférences internes. Enfin, le dernier modèle que nous avons étudié est un modèle de l'aire visuelle V1 que nous avons développé, dans la lignée du travail de Bressloff et al. Nous avons appliqué à ce modèle, les outils mathématiques et informatiques développés dans les parties précédentes. Ce nouveau modèle ne dispose pas d'une connectivité dépendant de l'orientation préférée des neurones. En particulier, nous avons montré que si la vitesse de propagation de l'information était trente fois plus lente, une instabilité pourrait se développer entraînant des illusions périodiques en temps.

This thesis is concerned with the modeling and numerical study of nonsmooth dynamical systems (NSDS). The first part of the thesis deals with the modeling of some DC-DC power converters using the complementarity formalism. This mathematical theoretical framework allows us to ensure existence and uniqueness of solutions in a "natural" and synthetic way. Specifically, it works pretty well in power electronic converters because it incorporates generalized discontinuous conduction modes (GDCM), characterized by a reduction of the dimension of the effective dynamics. For systems with a single diode, analytical state-space conditions for the presence of a GDCM are stated and simulation results, showing a variety of behaviours, such as persistent or re-entering GDCM, are also presented. Furthermore, the analysis and simulation of a parallel resonant converter (PRC), which has four diodes, illustrate the convenience of the complementarity formalism to simulate electrical systems with a large number of ideal diodes. We also present the simulation of a boost converter with a sliding mode control, even though a general control theory for complementarity systems is not still developed.In the second part of the thesis we focus on the bifurcation analysis in NSDS, and in particular, we have studied different mechanical systems which involve impacts and dry-friction. It is known that nonsmooth or discontinuous dynamical systems can exhibit the bifurcations also exhibited by smooth systems. In addition to these, there are also some novel transitions so-called discontinuity-induced bifurcations (DIBs) which are unique to these systems. We have investigated the complex behaviour occurring in an impacting mechanical system. DIBs such as corner impact bifurcations and transitions from complete to uncomplete chattering motions have been analysed in detail. Another type of DIBs recently classified are the so-called sliding bifurcations. Such bifurcations are a characteristic feature of so-called Filippov systems. We present detailed examples of all the different sliding bifurcation scenarios in a dry friction oscillator using a measured friction characteristic firstly introduced by Popp. Furthermore, a codimension-two degenerate switching-sliding bifurcation is displayed. In this case of degenerate switching-sliding bifurcation two curves of codimension-one sliding bifurcations, crossing-sliding and adding-sliding, branch out from the codimension-two point. Also, a cusp smooth codimension-two bifurcation is shown and coexistence of periodic orbits in the region between both fold codimension-one curves is studied.We have also investigated the dynamic behaviour of the two-block Burridge model for earthquake simulations. Previous numerical studies investigated by Ruina verified that, with a friction force of Coulomb type, the system presents only periodic behaviour. We show that chaotic regions can be observed in a symmetric configuration even if a Coulomb friction is considered with the relaxation of one of the assumptions assumed in the seismological literature. Furthermore, we have studied the behaviour of the system with asymmetric configuration. Different periodic solutions and regions of chaos can be observed varying the asymmetry of the system. With respect to the bifurcation point of view, we have analysed several smooth bifurcations (smooth and DIBs) observed in this system.Chapter 6 of this thesis presents the SICONOS software platform dedicated to simulation of NSDS. We give an overview of the SICONOS software and the way NSDS are modeled and simulated within the platform. Routines for analysis (stability, bifurcations, invariant manifolds,.) of NSDS implemented in the platform are explained in detail. To conclude this part, several representative samples are shown in order to illustrate the SICONOS platform abilities.Conclusion and some open problems are presented in the last chapter.

Poorly damped electromechanical modes detection in a power system and corresponding stability margins prediction are very important in power system planning and operation, and can provide significant help to power system operators with preventing stability problems. <br /><br /> Stochastic subspace identification is proposed in this thesis as a technique to extract the critical mode(s) from the measured ambient noise without requiring artificial disturbances (e. g. a line outage), allowing these critical modes to be used as an on-line index, which is referred here to as System Identification Stability Indices (SISI) to predict the closest oscillatory instability. The SISI is not only independent of system models and truly representative of the actual system, but also computationally efficient. In addition, readily available signals in a power system and several identification methods are categorized, and merits and pitfalls of each one are addressed in this work. <br /><br /> The damping torque of linearized models of power systems is studied in this thesis as another possible on-line security index. This index is estimated by means of proper system identification techniques applied to both power system transient response and ambient noise. The damping torque index is shown to address some of drawbacks of the SISI. <br /><br /> This thesis also demonstrates the connection between the second order statistical properties, including confidence intervals, of the estimated electromechanical modes and the variance of model parameters. These analyses show that Monte-Carlo type of experiments or simulations can be avoided, hence resulting in a significant reduction in the number of samples. <br /><br /> In these types of studies, the models available in simulation packages are extremely important due to their unquestionable impact on modal analysis results. Hence, in this thesis, the validity of generator subtransient model and a typical STATCOM transient stability (TS) model are also investigated by means of system identification, illustrating that under certain conditions the STATCOM TS model can yield results that are too optimistic, which can lead to errors in power system planning and operation. <br /><br /> In addition to several small test systems used throughout this thesis, the feasibility of the proposed indices are tested on a realistic system with 14,000 buses, demonstrating their usefulness in practice.

Poorly damped electromechanical modes detection in a power system and corresponding stability margins prediction are very important in power system planning and operation, and can provide significant help to power system operators with preventing stability problems. <br /><br /> Stochastic subspace identification is proposed in this thesis as a technique to extract the critical mode(s) from the measured ambient noise without requiring artificial disturbances (e. g. a line outage), allowing these critical modes to be used as an on-line index, which is referred here to as System Identification Stability Indices (SISI) to predict the closest oscillatory instability. The SISI is not only independent of system models and truly representative of the actual system, but also computationally efficient. In addition, readily available signals in a power system and several identification methods are categorized, and merits and pitfalls of each one are addressed in this work. <br /><br /> The damping torque of linearized models of power systems is studied in this thesis as another possible on-line security index. This index is estimated by means of proper system identification techniques applied to both power system transient response and ambient noise. The damping torque index is shown to address some of drawbacks of the SISI. <br /><br /> This thesis also demonstrates the connection between the second order statistical properties, including confidence intervals, of the estimated electromechanical modes and the variance of model parameters. These analyses show that Monte-Carlo type of experiments or simulations can be avoided, hence resulting in a significant reduction in the number of samples. <br /><br /> In these types of studies, the models available in simulation packages are extremely important due to their unquestionable impact on modal analysis results. Hence, in this thesis, the validity of generator subtransient model and a typical STATCOM transient stability (TS) model are also investigated by means of system identification, illustrating that under certain conditions the STATCOM TS model can yield results that are too optimistic, which can lead to errors in power system planning and operation. <br /><br /> In addition to several small test systems used throughout this thesis, the feasibility of the proposed indices are tested on a realistic system with 14,000 buses, demonstrating their usefulness in practice.

The dynamic response and bifurcations of a harmonic oscillator with a hysteretic restoring force and sinusoidal excitation are investigated. A multilinear model of hysteresis is presented. A hybrid system approach is used to formulate and study the problem. A novel method for obtaining exact transient and steady state response of the system is discussed. Simple periodic orbits of the system are analyzed using the KBM method and an analytic criterion for existence of bound and unbound resonance is derived. Results of KBM analysis are compared with those from numerical simulations. Stability and bifurcations of higher period orbits are studied using Poincar´e maps. The Poincar´e map for the system is constructed by composing the corresponding maps for the individual subsystems of the hybrid system. The novelty of this work lies in a.) the study of a multilinear model of hysteresis, and, b.) developing a methodology for obtaining the exact transient and steady state response of the system.