This thesis consists in the study of a periodically forced slow-fast system in both its excitable and oscillatory regimes. The slow-fast system under consideration is the FitzHugh-Nagumo model, and the periodic forcing consists of a train of Gaussian-shaped pulses, the width of which is much shorter than the action potential duration. This system is a qualitative model for both an excitable cell and a spontaneously beating cell submitted to periodic electrical stimulation. Such a configuration has often been studied in cardiac electrophysiology, due to the fact that it constitutes a simplified model of the situation of a cardiac cell in the intact heart, and might therefore contribute to the understanding of cardiac arrhythmias. Using continuation methods (AUTO software), we compute periodic-solution branches for the periodically forced system, taking the stimulation period as bifurcation parameter. We then study the evolution of the resulting bifurcation diagram as the stimulation amplitude is raised. In both the excitable and the oscillatory regimes, we find that a critical amplitude of stimulation exists below which the behaviour of the system is trivial: in the excitable case, the bifurcation diagram is restricted to a stable subthreshold period-1 branch, and in the oscillatory case, all the stable periodic solutions belong to isolated loops (i.e., to distinct closed solution branches). Due to the slow-fast nature of the system, the changes that take place in the bifurcation diagram as the critical amplitude is crossed are drastic, while the way the bifurcation diagram re-simplifies above some second amplitude is much more gentle. In the oscillatory case, we show that the critical amplitude is also the amplitude at which the topology of phase-resetting changes type. We explain the origin of this coincidence by considering a one-dimensional discrete map of the circle derived from the phase-resetting curve of the oscillator (the phase-resetting map), map which constitutes a good approximation of the original differential equations under certain conditions. We show that the bifurcation diagram of any such circle map, where the bifurcation parameter appears only in an additive fashion, is always characterized by the period-1 solutions belonging to isolated loops when the topological degree of the map is one, while these period-1 solutions belong to a unique branch when the topological degree of the map is zero.

Relaxation oscillations are highly non-linear oscillations, which appear to feature many important biological phenomena such as heartbeat, neuronal activity, and population cycles of predator-prey type. They are characterized by repeated switching of slow and fast motions and occur naturally in singularly perturbed ordinary differential equations, which exhibit dynamics on different time scales. Traditionally, slow-fast systems and the related oscillatory phenomena -- such as relaxation oscillations -- have been studied by the method of the matched asymptotic expansions, techniques from non-standard analysis, and recently a more qualitative approach known as geometric singular perturbation theory. It turns out that relaxation oscillations can be found in a more general setting; in particular, in slow-fast systems, which are not written in the standard form. Systems in which separation into slow and fast variables is not given a priori, arise frequently in applications. Many of these systems include additionally various parameters of different orders of magnitude and complicated (non-polynomial) non-linearities. This poses several mathematical challenges, since the application of singular perturbation arguments is not at all straightforward. For that reason most of such systems have been studied only numerically guided by phase-space analysis arguments or analyzed in a rather non-rigorous way. It turns out that the main idea of singular perturbation approach can also be applied in such non-standard cases. This thesis is concerned with the application of concepts from geometric singular perturbation theory and geometric desingularization based on the blow-up method to the study of relaxation oscillations in slow-fast systems beyond the standard form. A detailed geometric analysis of oscillatory mechanisms in three mathematical models describing biochemical processes is presented. In all the three cases the aim is to detect the presence of an isolated periodic movement represented by a limit cycle. By using geometric arguments from the perspective of dynamical systems theory and geometric desingularization based on the blow-up method analytic proofs of the existence of limit cycles in the models are provided. This work shows -- in the context of non-trivial applications -- that the geometric approach, in particular the blow-up method, is valuable for the understanding of the dynamics of systems with no explicit splitting into slow and fast variables, and for systems depending singularly on several parameters.

L’objectif de ce travail de thèse est d’étudier des lasers semi-conducteur issus de la plateforme d’intégration III-V sur Si et présentant un faible bruit d’intensité relatif (RIN) pour le transport de signaux RADAR par voie optique. Nous cherchons à obtenir des lasers de comportement dynamique dit de «Classe A», i.e. avec une réponse dynamique sans oscillations de relaxation. Dans ce cas, il a été précédemment montré qu’un tel comportement dynamique présente un RIN limité au bruit de grenaille sur une large bande de fréquences et est obtenu quand la durée de vie des photons dans la cavité est grande devant la durée de vie des porteurs dans la zone active. La plateforme photonique sur silicium est alors intéressante car elle permet de réaliser des cavités longues grâce aux guides optiques offrant de faibles pertes de propagation, i.e. de l’ordre du dB/cm. En première approche, nous avons étudié des lasers dont la cavité de longueur centimétrique est composée d’une partie active fournissant le gain optique et d’une partie passive composée de guide en silicium à faibles pertes de propagation. Nous avons proposé différentes optimisations des pertes optiques intra-cavité ainsi que différentes solutions de filtrage spectral à grande finesse nécessaire à une oscillation laser monomode. La seconde approcheétudiée repose sur le filtrage du RIN d’un laser hybride de longueur millimétrique en exploitant les effets de saturation du gain optique dans un amplificateur optique à SC (SOA). Nous avons présenté un modèle décrivant les différents mécanismes altérant le bruit du laser amplifié par un SOA. Une étudeexpérimentale a permis de mettre en évidence la réduction du RIN d’un laser hybride III-V sur silicium, allant jusqu’à 15 dB pour des fréquences allant jusqu’à quelques GHz. La dernière approche explorée dans cette thèse repose sur la conception de lasers DFB hybride III-V sur silicium à très haut facteur de qualité. L’utilisation d’un réseau de Bragg à pas variable permet de réduire les pertes radiatives, usuellement importantes dans les lasers DFB, et d’obtenir une cavité de facteur de qualité de l’ordre de quelques millions. Un premier composant réalisé présente un facteur de qualité de 65 000. / The objective of the present thesis is to investigate new laser architectures with low Relative Intensity Noise (RIN) using the Silicon Photonics integration platform. We intend to reach “class-A” dynamics, in which relaxation oscillations are eliminated. In this conditions, lasers with class-A dynamics exhibit shot-noise limited RIN over a wide frequency bandwidth, typically from 100 MHz to 20 GHz. Such behaviour can be obtained with high-Q laser cavities, i.e with long cavities or with ultra-low losses cavities. The silicon photonics platform is a good candidate for the desired dynamical behaviour as it makes possible the implementation of long cavities (ten’s of cm) based on low losses silicon waveguides (dB/cm). Three different approaches have been considered in the present work. In the first approach, we have developed centimetre long lasers, consisting of an active section providing the optical gain coupled to a passive section made with low losses silicon waveguides. We proposed different approaches to optimize the intra-cavity optical losses, and different architectures of high finesse optical filters allowing simultaneously single-mode operation and high side mode suppression. The second approach consists on filtering the laser RIN by taking advantage of the coherent population oscillations effects in a SC Optical Amplifier (SOA). We proposed a model for describing the different mechanisms altering the RIN of the amplified laser. We demonstrated 15 dB RIN reduction for frequencies up to a few GHz, using a hybrid III-V on Si laser and a “classical” SOA. The last approach explored in the present thesis is based on the use of hybrid III-V on silicon DFB lasers with a high quality factor. Using Silicon Bragg grating with a variable pitch can reduce the radiative losses, usually important in DFB lasers. In this case, we can obtain optical cavities with few millions quality factor, leading to few ns photon lifetime. We realize a first design of Si Bragg grating with a Q factor of 65 000.

In modern low pressure plasma discharges, the electron mean free path often exceeds the device dimensions. Under such conditions the electron velocity distribution function may significantly deviate from Maxwellian, which strongly affects the discharge properties. The description of such plasmas has to be kinetic and often requires the use of numerical methods. This thesis presents the study of kinetic effects in inductively coupled plasmas and Hall thrusters carried out by means of particle-in-cell simulations. The important result and the essential part of the research is the development of particle-in-cell codes. <p>An advective electromagnetic 1d3v particle-in-cell code is developed for modelling the inductively coupled plasmas. An electrostatic direct implicit 1d3v particle-in-cell code EDIPIC is developed for plane geometry simulations of Hall thruster plasmas. The EDIPIC code includes several physical effects important for Hall thrusters: collisions with neutral atoms, turbulence, and secondary electron emission. In addition, the narrow sheath regions crucial for plasma-wall interaction are resolved in simulations. The code is parallelized to achieve fast run times. <p>Inductively coupled plasmas sustained by the external RF electromagnetic field are widely used in material processing reactors and electrodeless lighting sources. In a low pressure inductive discharge, the collisionless electron motion strongly affects the absorption of the external electromagnetic waves and, via the ponderomotive force, the density profile. The linear theory of the anomalous skin effect based on the linear electron trajectories predicts a strong decrease of the ponderomotive force for warm plasmas. Particle-in-cell simulations show that the nonlinear modification of electron trajectories by the RF magnetic field partially compensates the effects of electron thermal motion. As a result, the ponderomotive force in warm collisionless plasmas is stronger than predicted by linear kinetic theory. <p>Hall thrusters, where plasma is maintained by the DC electric field crossed with the stationary magnetic field, are efficient low-thrust devices for spacecraft propulsion. The energy exchange between the plasma and the wall in Hall thrusters is enhanced by the secondary electron emission, which strongly affects electron temperature and, subsequently, thruster operation. Particle-in-cell simulations show that the effect of secondary electron emission on electron cooling in Hall thrusters is quite different from predictions of previous fluid studies. Collisionless electron motion results in a strongly anisotropic, nonmonotonic electron velocity distribution function, which is depleted in the loss cone, subsequently reducing the electron wall losses compared to Maxwellian plasmas. Secondary electrons form two beams propagating between the walls of a thruster channel in opposite radial directions. The secondary electron beams acquire additional energy in the crossed external electric and magnetic fields. The energy increment depends on both the field magnitudes and the electron flight time between the walls. <p>A new model of secondary electron emission in a bounded plasma slab, allowing for emission due to the counter-propagating secondary electron beams, is developed. It is shown that in bounded plasmas the average energy of plasma bulk electrons is far less important for the space charge saturation of the sheath than it is in purely Maxwellian plasmas. A new regime with relaxation oscillations of the sheath has been identified in simulations. Recent experimental studies of Hall thrusters indirectly support the simulation results with respect to the electron temperature saturation and the channel width effect on the thruster discharge.

In modern low pressure plasma discharges, the electron mean free path often exceeds the device dimensions. Under such conditions the electron velocity distribution function may significantly deviate from Maxwellian, which strongly affects the discharge properties. The description of such plasmas has to be kinetic and often requires the use of numerical methods. This thesis presents the study of kinetic effects in inductively coupled plasmas and Hall thrusters carried out by means of particle-in-cell simulations. The important result and the essential part of the research is the development of particle-in-cell codes. <p>An advective electromagnetic 1d3v particle-in-cell code is developed for modelling the inductively coupled plasmas. An electrostatic direct implicit 1d3v particle-in-cell code EDIPIC is developed for plane geometry simulations of Hall thruster plasmas. The EDIPIC code includes several physical effects important for Hall thrusters: collisions with neutral atoms, turbulence, and secondary electron emission. In addition, the narrow sheath regions crucial for plasma-wall interaction are resolved in simulations. The code is parallelized to achieve fast run times. <p>Inductively coupled plasmas sustained by the external RF electromagnetic field are widely used in material processing reactors and electrodeless lighting sources. In a low pressure inductive discharge, the collisionless electron motion strongly affects the absorption of the external electromagnetic waves and, via the ponderomotive force, the density profile. The linear theory of the anomalous skin effect based on the linear electron trajectories predicts a strong decrease of the ponderomotive force for warm plasmas. Particle-in-cell simulations show that the nonlinear modification of electron trajectories by the RF magnetic field partially compensates the effects of electron thermal motion. As a result, the ponderomotive force in warm collisionless plasmas is stronger than predicted by linear kinetic theory. <p>Hall thrusters, where plasma is maintained by the DC electric field crossed with the stationary magnetic field, are efficient low-thrust devices for spacecraft propulsion. The energy exchange between the plasma and the wall in Hall thrusters is enhanced by the secondary electron emission, which strongly affects electron temperature and, subsequently, thruster operation. Particle-in-cell simulations show that the effect of secondary electron emission on electron cooling in Hall thrusters is quite different from predictions of previous fluid studies. Collisionless electron motion results in a strongly anisotropic, nonmonotonic electron velocity distribution function, which is depleted in the loss cone, subsequently reducing the electron wall losses compared to Maxwellian plasmas. Secondary electrons form two beams propagating between the walls of a thruster channel in opposite radial directions. The secondary electron beams acquire additional energy in the crossed external electric and magnetic fields. The energy increment depends on both the field magnitudes and the electron flight time between the walls. <p>A new model of secondary electron emission in a bounded plasma slab, allowing for emission due to the counter-propagating secondary electron beams, is developed. It is shown that in bounded plasmas the average energy of plasma bulk electrons is far less important for the space charge saturation of the sheath than it is in purely Maxwellian plasmas. A new regime with relaxation oscillations of the sheath has been identified in simulations. Recent experimental studies of Hall thrusters indirectly support the simulation results with respect to the electron temperature saturation and the channel width effect on the thruster discharge.

Relaxation oscillations are highly non-linear oscillations, which appear to feature many important biological phenomena such as heartbeat, neuronal activity, and population cycles of predator-prey type. They are characterized by repeated switching of slow and fast motions and occur naturally in singularly perturbed ordinary differential equations, which exhibit dynamics on different time scales. Traditionally, slow-fast systems and the related oscillatory phenomena -- such as relaxation oscillations -- have been studied by the method of the matched asymptotic expansions, techniques from non-standard analysis, and recently a more qualitative approach known as geometric singular perturbation theory. It turns out that relaxation oscillations can be found in a more general setting; in particular, in slow-fast systems, which are not written in the standard form. Systems in which separation into slow and fast variables is not given a priori, arise frequently in applications. Many of these systems include additionally various parameters of different orders of magnitude and complicated (non-polynomial) non-linearities. This poses several mathematical challenges, since the application of singular perturbation arguments is not at all straightforward. For that reason most of such systems have been studied only numerically guided by phase-space analysis arguments or analyzed in a rather non-rigorous way. It turns out that the main idea of singular perturbation approach can also be applied in such non-standard cases. This thesis is concerned with the application of concepts from geometric singular perturbation theory and geometric desingularization based on the blow-up method to the study of relaxation oscillations in slow-fast systems beyond the standard form. A detailed geometric analysis of oscillatory mechanisms in three mathematical models describing biochemical processes is presented. In all the three cases the aim is to detect the presence of an isolated periodic movement represented by a limit cycle. By using geometric arguments from the perspective of dynamical systems theory and geometric desingularization based on the blow-up method analytic proofs of the existence of limit cycles in the models are provided. This work shows -- in the context of non-trivial applications -- that the geometric approach, in particular the blow-up method, is valuable for the understanding of the dynamics of systems with no explicit splitting into slow and fast variables, and for systems depending singularly on several parameters.

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