La question de la propulsion spatiale a été un enjeu politique au coeur de la guerre froide et reste un enjeu stratégique de nos jours. La technologie chimique déjà en place sur les moteurs fusées s'avère être limitée par la vitesse d'éjection et la durée de vie des appareils. La propulsion électrique et plus particulièrement le moteur à effet Hall apparait ainsi comme la technologie la plus performante et la plus utilisée pour diriger un satellite dans l'espace. Cependant, la physique à l'intérieur d'un propulseur étant complexe, de par les champs électromagnétiques ou les processus de collisions importants, toutes les particularités de fonctionnement du moteur ne sont pas parfaitement expliquées. Au bout de centaines d'heures d'essais, certains prototypes voient leur paroi s'éroder de façon anormale et des instabilités électromagnétiques se développent au sein de la chambre d'ionisation. La mobilité des électrons mesurée est en contradiction avec les modèles analytiques et soulèvent des problématiques sur la physique du plasma à l'intérieur de ces moteurs. Par conséquent, le code AVIP a été développé afin de proposer un code 3D massivement parallèle et non-structuré à Safran Aircraft Engines modélisant le plasma instationnaire à l'intérieur du propulseur. Des méthodes lagrangiennes et eulériennes sont utilisées et intégrées dans le code et mon travail s'est concentré sur le développement d'un modèle fluide, étant plus rapide et donc mieux adapté à la conception et au design industriel. Le modèle fluide est basé sur un modèle aux moments avec une expression rigoureuse des termes de collisions et une description précise des conditions limites pour les gaines. Ce modèle a été implémenté numériquement dans un formalisme non structuré et optimisé de façon à être performant sur les nouvelles architectures de calcul. La modélisation retenue et les efforts d'optimisation ont permis de réaliser un calcul réel de moteur à effet Hall afin de retrouver les propriétés globales de fonctionnement telles que l'accélération des ions ou encore la localisation de la zone d'ionisation. Un second cas d'application a finalement reproduit avec succès les instabilités azimutales dans le propulseur avec un modèle fluide et a justifié le rôle de ces instabilités dans le transport anormal des électrons et l'érosion des parois / The space propulsion has been a political issue in the midst of the Cold War and remains nowadays a strategic and industrial issue. The chemical propulsion on rocket engines is limited by its ejection velocity and its lifetime. Electric propulsion and more particularly Hall effect thrusters appear then as the most powerful and used technology for space satellite operation. The physic inside a thruster is complex because of the electromagnetic fields and important collision processes. Therefore, all specificities of the engine operation are not perfectly understood. After hundreds of hours of tests, thruster walls are curiously eroded and electromagnetic instabilities are developping within the ionization chamber. The measured electron mobility is in contradiction with the analytical models and raises issues on the plasma behavior inside the discharg chamber. As a result, the AVIP code was developed to provide a massively parallel and unstructured 3D code to Safran Aircraft Engines modeling unsteady plasma inside the thruster. Lagrangian and Eulerian methods are used and integrated in the solver and my work has focused on the development of a fluid model which is faster and therefore better suited to industrial conception. The model is based on a set of equations for neutrals, ions and electrons without drift-diffusion hypothesis, combined with a Poisson equation to describe the electric potential. A rigorous expression of collision terms and a precise description of the boundary conditions for sheaths have been established. This model has been implemented numerically in an unstructured formalism and optimized to obtain good performances on new computing architectures. The model and the numerical implementation allow us to perform a real Hall effect thruster simulation. Overall operating properties such as the acceleration of the ions or the location of the ionization zone are captured. Finally, a second application has successfully reproduced azimuthal instabilities in the Hall thruster with the fluid model and justified the role of these instabilities in the anomalous electron transport and in theerosion of the walls

<p>  </p> <p>High altitude test facilities are needed for ground testing of upper stage rocket engines or small satellite thrusters with high expansion ratio nozzles to ensure full-flowing nozzle conditions. Rocket exhaust diffusers and ejector systems are essential components of these facilities and are frequently used to set desired simulated altitude/low pressure conditions and pump out rocket exhaust products. </p> <p>This dissertation combined experimental and numerical efforts on diffuser-ejector systems. The experimental efforts included the development of a Second Throat Exhaust Diffuser (STED) to aid with the qualification of space thrusters in the Purdue Altitude Chamber Facility. While performing these experiments, we characterized the single and two-stage ejector systems operating in conjunction with the diffuser to obtain and maintain specific simulated altitudes. </p> <p>The concurrent numerical effort focused on validating a Computational Fluid Dynamics (CFD) approach based on Reynolds-averaged Navier–Stokes equations flow simulations. After validating the ejector CFD, we used it to derive a corrective coefficient of a lumped parameter ejector model (LPM) developed previously for the ejectors used in the Purdue Altitude Facility. We created variable coefficient maps for the stages of the two-stage ejector system using the same LPM and the test data from one of our experiments. </p> <p>We designed, manufactured, and then validated a STED for altitude testing of a ~50 lbf hypergolic hybrid motor as a part of a NASA JPL project. The designed STED enabled the operation of the hybrid motor for the full duration of the test firing (about 2 seconds) at a simulated altitude of 102,000 feet, slightly above the targeted altitude of 100,000 feet. We also validated our diffuser CFD approach by creating a simulation using the measured diffuser back pressure and the average motor chamber pressure. </p> <p>We then devised an experiment to investigate several diffuser–ejector system configurations using cold gas thrusters with conical and bell nozzles. The main aim of that experiment was to explore the effects of different thruster nozzle geometries, diffuser geometries, and thruster/ejector operational parameters on the performance of a diffuser–ejector system. For all the configurations tested, we reported on the minimum starting and operating pressure ratios and corresponding correction factors on the normal shock method. The large hysteresis regions obtained mostly with bell nozzles having a high initial expansion angle provided an opportunity to economize the facility resources. In some cases which were later found to violate STED second throat contraction limits, we experienced a choking flow at the second throat. Then, we studied the second throat contraction limits in detail using CFD in addition to the experimental data and explored minimum diffuser second throats enabling diffuser starting and improving aerodynamic efficiency. </p> <p>Finally, we machined a larger scale cold gas thruster with different nozzle geometries (having throat diameters in the range of 0.367 – 0.52 inches) from acrylic rods to study possible flow separation and gas condensation events that could occur during tests in the altitude chamber. The main difference here with the previous experiment was that the diffuser (JPL STED) was fixed, and the two-stage ejector system was used to create the necessary back pressure. With the experiments performed at varying axial gaps between the nozzle exit and diffuser inlet, we were able to investigate the effect of that on the diffuser performance. The experimental data collected in this work and the complementary numerical efforts served to generate the operating envelope of the Purdue Altitude Chamber Facility.  </p>

Measuring the thrust of cold gas micro-thrusters, due to the small magnitude of the generated force, is a challenging task, and to obtain satisfying quality of results usually careful adjustments of the setup has to be performed. The work presented in this paper aims at improving the quality of such measurements at the GomSpace Sweden company. The work consists of two parts: In the first part the author focuses on improving current thrust measurements setup implemented in the company, which consists of a vacuum compatible laboratory scale. It was observed that the results obtained from this setup were approximately 25‒35% lower than expected. A test campaign performed within the work found that this was associated mainly with backflow of plume expansion gases. Covering the scale and the tested propulsion unit with a box having openings only for thrusters was proved to be sufficient setup upgrade to mitigate the issue. In the second part of the project an alternative setup utilising a hanging pendulum-based solution with strain measurement-based force sensor is proposed and its design is thoroughly described. Results of initial tests of a 3D printed prototype of this setup are presented, proving its potential to produce results with better quality of output than the laboratory scale. / Att mäta dragkraften från mikrodysor baserade på kallgas är på grund av den låga genererade kraften en utmanande uppgift. För att få tillfredsställande kvalitet på resultatet måste vanligtvis noggranna justeringar av inställningarna utföras. Arbetet som presenteras i denna rapport syftar till att förbättra kvaliteten på sådana mätningar vid GomSpace Sweden AB. Arbetet består av två delar: I den första delen fokuserar författaren på att förbättra den nuvarande uppställningen för mätning av dragkraft vid GomSpace, bestående av en vakuumkompatibel laboratorievåg. Det observerades att resultaten från denna mätupställning var cirka 25‒35% lägre än förväntat. En testkampanj som utfördes inom ramen för detta arbete visade att detta främst var förknippat med ett tillbakaflöde av gaser från den expanderande bränsleplymen. Att täcka över vågen och den testade framdrivningsenheten med en låda med öppningar endast för dysan visade sig vara tillräckligt för att minska problemet. I den andra delen föreslås ett alternativt upplägg baserat på en hängande pendel med en töjningsmätningsbaserad kraftsensor, dess utformming beskrivs utförligt. Resultaten av de första testerna av en 3D-utskriven prototyp av denna uppställning presenteras, vilket visar på dess potential att producera mätningar med bättre kvalitet än laboratorievågen.

The use of lateral divert thrusters on hypersonic vehicles would allow for fine-tuned attitude control at high Mach numbers. However, the jet interaction effects of lateral thrusters on the hypersonic flow field have not been investigated thoroughly. Computational Fluid Dynamics (CFD) can provide preliminary modeling of the jet interaction, but several variables such as vehicle geometry, velocity, and altitude, result in computationally expensive modeling or loss in accuracy of the results. Therefore, the goal of chemical kinetics testing and chemical model verification is to enhance the fidelity of the jet interaction effects, specifically the plume reaction with air and the plume interaction with vehicle instrumentation.

Um propulsor do tipo Hall é um mecanismo que utiliza predominantemente uma configuração de campos eletromagnéticos Hall, um campo elétrico perpendicular a um campo magnético, para confinar elétrons e acelerar íons. Os elétrons são confinados dentro de um canal de aceleração onde os campos eletromagnéticos estão presentes. Um gás neutro é lançado dentro desse canal de aceleração de forma que os elétrons confinados podem colidir com os átomos do gás e os ionizar. Os íons gerados dessas colisões, elétrons-gás, são fortemente repelidos para fora do canal de aceleração pelo campo elétrico. A expulsão desses íons é o fator responsável pela propulsão. Nesses propulsores é importante que os elétrons estejam confinados dentro do canal de aceleração e que sejam capazes de produzir o maior número possível de íons. Visando determinar quais são os parâmetros de controle– intensidade dos campos eletromagnéticos– que propiciam uma dinâmica eletrônica com essas características, derivamos, via formalismo Hamiltoniano, as equações de movimento de um elétron e as analisamos. Dessas equações de movimento encontramos funções analíticas que indicam os limites geométricos atingidos pelo elétron dentro do sistema propulsor para cada conjunto de parâmetros de controle. Essas funções constituem o critério de confinamento eletrônico utilizado nesse trabalho. Além disso, a partir das equações de movimento, mostramos quais as configurações de campos eletromagnéticos que teoricamente incrementam o desempenho dos propulsores Hall. Verificamos que nas configurações de maior desempenho a dinâmica eletrônica é caótica. Neste trabalho, o caos é determinado com o auxílio dos mapas de Poincaré e dos expoentes de Lyapunov. / A Hall thruster is a system that utilizes an electromagnetic fields configuration predominantly like Hall, an electric field which lies perpendicular to a magnetic field, to confine electrons and to accelerate ions. The electrons are confined within an acceleration chamber where the electromagnetic fields are present. A neutral gas is released within this acceleration chamber so that the confined electrons can collide with the gas and ionize it. The ions generated from these collisions, the electron-gas, are strongly repelled by the electric field system. The expulsion of these ions generate the propulsion. In these thrusters it is very important that the electrons are confined within the acceleration chamber and are able to produce the largest possible number of ions. In order to determine the control parameters, that is, the electromagnetic fields intensity which provides an electronic dynamic with these characteristics; we derived, via Hamiltonian formalism, the motion equations for an electron and we analyzed them. From these motion equations, we found functions that indicate the electron geometric boundaries within these thrusters, for each set of control parameters. In this work, these functions indicate the electronic confinement. Moreover, from the motion equations, we showed the electromagnetic fields settings which theoretically improve the Hall thruster’s performance. We found that, in these higher performance settings, the electron dynamics is chaotic. In this work, the chaos is determined by Poincaré maps and by Lyapunov exponents.

La modélisation des propulseurs ioniques de type SPT pose de nombreux <br />problèmes dans le domaine du transport des particules chargées. Nous nous <br />intéressons à deux de ces problèmes, à savoir le transport des électrons et <br />le calcul du potentiel électrique.<br /><br />Le transport des électrons résulte de l'influence conjuguée des champs <br />(électrique et magnétique) établis dans la cavité du propulseur et des <br />collisions des électrons (dans la cavité et avec la paroi limitant celle-ci). <br />Nous avons participé au développement d'un modèle SHE (Spherical Harmonics <br />Expansion) qui résulte d'une analyse asymptotique de l'équation de Boltzmann <br />munie de conditions de réflexion aux bords. Ce modèle permet d'approcher la <br />fonction de distribution en énergie des électrons en résolvant une <br />équation de diffusion dans un espace \{position, énergie\}. Plus précisément, <br />nous avons étendu une démarche existante au cas où les collisions en volume <br />(excitation, ionisation) et les collisions inélastiques à la paroi <br />(attachement et émission secondaire) sont prises en compte. Enfin, nous <br />avons écrit un code de résolution du modèle SHE, dont les résultats ont <br />été comparés avec ceux d'une méthode de Monte Carlo. <br /><br />\vspace*{1mm}<br />Dans un deuxième temps, nous avons étudié le calcul du potentiel électrique. <br />La présence du champ magnétique impose d'écrire le courant d'électrons sous <br />la forme ${\cal J}=\sigma \nabla W$<br /> où W est le potentiel électrique et le tenseur de conductivité $\sigma$<br />est fortement anisotrope compte tenu des grandeurs physiques en jeu dans <br />le SPT. Pour résoudre $\mbox{div }{\cal J}(x,y)=S(x,y)$, <br />nous avons implémenté une méthode de volumes finis <br />sur maillage cartésien permettant de résoudre ce problème elliptique <br />anisotrope, et nous avons vérifié qu'elle échouait lorsque le rapport <br />d'anisotropie devenait grand. Aussi nous avons développé une méthode de <br />paramétrisation, qui consiste à extrapoler la solution d'un problème <br />anisotrope à l'aide d'une suite de problèmes isotropes. Cette méthode a <br />donné des résultats encourageants pour de forts rapports d'anisotropie, <br />et devrait nous permettre d'atteindre des cas réels.

Um propulsor do tipo Hall é um mecanismo que utiliza predominantemente uma configuração de campos eletromagnéticos Hall, um campo elétrico perpendicular a um campo magnético, para confinar elétrons e acelerar íons. Os elétrons são confinados dentro de um canal de aceleração onde os campos eletromagnéticos estão presentes. Um gás neutro é lançado dentro desse canal de aceleração de forma que os elétrons confinados podem colidir com os átomos do gás e os ionizar. Os íons gerados dessas colisões, elétrons-gás, são fortemente repelidos para fora do canal de aceleração pelo campo elétrico. A expulsão desses íons é o fator responsável pela propulsão. Nesses propulsores é importante que os elétrons estejam confinados dentro do canal de aceleração e que sejam capazes de produzir o maior número possível de íons. Visando determinar quais são os parâmetros de controle– intensidade dos campos eletromagnéticos– que propiciam uma dinâmica eletrônica com essas características, derivamos, via formalismo Hamiltoniano, as equações de movimento de um elétron e as analisamos. Dessas equações de movimento encontramos funções analíticas que indicam os limites geométricos atingidos pelo elétron dentro do sistema propulsor para cada conjunto de parâmetros de controle. Essas funções constituem o critério de confinamento eletrônico utilizado nesse trabalho. Além disso, a partir das equações de movimento, mostramos quais as configurações de campos eletromagnéticos que teoricamente incrementam o desempenho dos propulsores Hall. Verificamos que nas configurações de maior desempenho a dinâmica eletrônica é caótica. Neste trabalho, o caos é determinado com o auxílio dos mapas de Poincaré e dos expoentes de Lyapunov. / A Hall thruster is a system that utilizes an electromagnetic fields configuration predominantly like Hall, an electric field which lies perpendicular to a magnetic field, to confine electrons and to accelerate ions. The electrons are confined within an acceleration chamber where the electromagnetic fields are present. A neutral gas is released within this acceleration chamber so that the confined electrons can collide with the gas and ionize it. The ions generated from these collisions, the electron-gas, are strongly repelled by the electric field system. The expulsion of these ions generate the propulsion. In these thrusters it is very important that the electrons are confined within the acceleration chamber and are able to produce the largest possible number of ions. In order to determine the control parameters, that is, the electromagnetic fields intensity which provides an electronic dynamic with these characteristics; we derived, via Hamiltonian formalism, the motion equations for an electron and we analyzed them. From these motion equations, we found functions that indicate the electron geometric boundaries within these thrusters, for each set of control parameters. In this work, these functions indicate the electronic confinement. Moreover, from the motion equations, we showed the electromagnetic fields settings which theoretically improve the Hall thruster’s performance. We found that, in these higher performance settings, the electron dynamics is chaotic. In this work, the chaos is determined by Poincaré maps and by Lyapunov exponents.

La cathode creuse est un élément clef des propulseurs à plasma. Dans un propulseur à plasma, un gaz propulsif est ionisé dans un canal de décharge puis accéléré hors de celui-ci afin de créer la poussée. Dans le propulseur de Hall en particulier, l'ionisation du gaz est provoquée par l'injection dans le canal de décharge d'un intense courant électronique (de quelques ampères à plus d'une centaine d'ampères). L'élément chargé de fournir le courant électronique de la décharge, la cathode creuse, est crucial dans le fonctionnement du propulseur. Or, celle-ci est souvent idéalisée dans les modèles de propulseur et n'est que rarement étudiée pour sa physique propre. Pourtant, le développement de propulseurs de Hall de haute puissance, destinés à terme à équiper l'ensemble des missions spatiales, requiert la mise au point de cathodes capable de délivrer un fort courant (jusqu'à plus de 100 A) sur des durées de l'ordre de la dizaine de milliers d'heures. Or, la mise au point de nouvelles cathodes s'est révélée difficile en raison de l'absence de modèle susceptible de prédire a priori les performances d'une cathode en fonction de sa conception. On se propose ici de mettre en place un modèle prédictif de cathode creuse capable de retranscrire la physique du fonctionnement de la cathode. L'objectif in fine est bien sûr d'utiliser ce modèle afin de faire le lien entre la conception de la cathode et son fonctionnement dans le but de guider le développement de futures cathodes. On présentera tout d'abord brièvement le contexte d'application des cathodes creuses, et on donnera un rapide aperçu du principe de fonctionnement global de la cathode. Ensuite, après avoir effectué un tour d'horizon des différents modèles numériques de cathode creuse préexistants dans la littérature, on détaillera le modèle de la cathode développé ici, qui incorpore une description fluide du plasma, ainsi que des transferts thermiques aux parois, qui conditionnent en grande partie le bon fonctionnement de la cathode. Un soin particulier sera apporté à la validation des résultats de simulation vis-à-vis des mesures expérimentales disponibles dans la littérature, ce qui nous permettra de perfectionner certains points du modèle afin de mieux traduire la réalité physique. En particulier, une modélisation spécifique de la région de transition entre la décharge interne de la cathode et la plume du propulseur sera réalisée. Ce modèle permettra de mettre en évidence certains phénomènes d'instabilité du plasma spécifiques de cette décharge, qui ont été jusqu'ici observés expérimentalement mais jamais pleinement intégrés aux modèles de cathode creuse. A l'aide du modèle validé, on procèdera à l'analyse physique de l'ensemble des phénomènes qui gouvernent le fonctionnement d'une cathode particulière, la cathode NSTAR développée par la NASA au Jet Propulsion Laboratory. Ensuite, on s'appuiera sur le modèle numérique pour comprendre l'impact sur le fonctionnement de la cathode des choix de conception au travers d'une étude paramétrique autour de la cathode NSTAR. Les tendances dégagées nous permettront de formuler des recommandations quant au développement de cathodes de haute puissance. Enfin, dans le but d'illustrer la versatilité du modèle développé, le comportement d'une cathode creuse employant une géométrie alternative à la cathode NSTAR sera également présenté. / A hollow cathode is a critical component of plasma thrusters. In a plasma thruster, a propellant gas is ionized in a discharge chamber and accelerated out of it so as to generate thrust. In Hall thrusters in particular, the ionization of the gas is caused by an intense electron current (from a few to hundred amps) which flows through the discharge chamber. The hollow cathode is the device which is responsible for providing the discharge current. This key element is often idealized in thruster numerical models and its physical behavior is rarely studied for its own sake. Yet, developing high power Hall thrusters, designed to propel in the long run every type of space mission, requires new hollow cathodes able to supply an intense electron current (over 100 A) over a duration on the order of ten thousand hours. So far, designing new cathodes proved difficult because of the lack of model capable of predicting the performance of a cathode based on its design. In this work, we build up a predictive model of a hollow cathode capable of simulating the physics relevant to the operation of the cathode. In the end, we aim at using this model to associate design characteristics of the cathode to key aspects of the cathode performance during operation. Our goal with this model is to guide the development of future high power hollow cathodes. We will first briefly describe the range of application of hollow cathodes related to space propulsion. Then we will give a brief account of the working principles of the cathode and we will set the numerical models available in the literature prior to this one out. The numerical model developed in this work will then be described. It includes a fluid treatment of the plasma as well as an account of the heat fluxes to the walls which largely control the performance of the cathode. Simulation results will be thoroughly compared to experimental measurements available in the literature and specific aspects of the model will be refined to match up simulation results with the physical reality. For instance, a model that specifically represents the transition region between the internal plasma of the cathode and the plume of the cathode will be described. This model will enable us to highlight plasma instability phenomena which were so far observed experimentally, yet never properly included in hollow cathode models. Using the model just developed, we will analyze the physics of a particular hollow cathode which has been developed by NASA at the Jet Propulsion Laboratory, the NSTAR hollow cathode. Then, thanks to the numerical model, we will be able to carry out a parametric study revolving around the design of the NSTAR cathode. This will allow us to bring out the influence of the design on the cathode performance and we will eventually express recommendations regarding the design of future high power cathodes. To conclude, the versatility of the numerical model built up here will also be displayed through simulations of the behavior of a hollow cathode based on an alternate geometry.

Um propulsor do tipo Hall é um mecanismo que utiliza predominantemente uma configuração de campos eletromagnéticos Hall, um campo elétrico perpendicular a um campo magnético, para confinar elétrons e acelerar íons. Os elétrons são confinados dentro de um canal de aceleração onde os campos eletromagnéticos estão presentes. Um gás neutro é lançado dentro desse canal de aceleração de forma que os elétrons confinados podem colidir com os átomos do gás e os ionizar. Os íons gerados dessas colisões, elétrons-gás, são fortemente repelidos para fora do canal de aceleração pelo campo elétrico. A expulsão desses íons é o fator responsável pela propulsão. Nesses propulsores é importante que os elétrons estejam confinados dentro do canal de aceleração e que sejam capazes de produzir o maior número possível de íons. Visando determinar quais são os parâmetros de controle– intensidade dos campos eletromagnéticos– que propiciam uma dinâmica eletrônica com essas características, derivamos, via formalismo Hamiltoniano, as equações de movimento de um elétron e as analisamos. Dessas equações de movimento encontramos funções analíticas que indicam os limites geométricos atingidos pelo elétron dentro do sistema propulsor para cada conjunto de parâmetros de controle. Essas funções constituem o critério de confinamento eletrônico utilizado nesse trabalho. Além disso, a partir das equações de movimento, mostramos quais as configurações de campos eletromagnéticos que teoricamente incrementam o desempenho dos propulsores Hall. Verificamos que nas configurações de maior desempenho a dinâmica eletrônica é caótica. Neste trabalho, o caos é determinado com o auxílio dos mapas de Poincaré e dos expoentes de Lyapunov. / A Hall thruster is a system that utilizes an electromagnetic fields configuration predominantly like Hall, an electric field which lies perpendicular to a magnetic field, to confine electrons and to accelerate ions. The electrons are confined within an acceleration chamber where the electromagnetic fields are present. A neutral gas is released within this acceleration chamber so that the confined electrons can collide with the gas and ionize it. The ions generated from these collisions, the electron-gas, are strongly repelled by the electric field system. The expulsion of these ions generate the propulsion. In these thrusters it is very important that the electrons are confined within the acceleration chamber and are able to produce the largest possible number of ions. In order to determine the control parameters, that is, the electromagnetic fields intensity which provides an electronic dynamic with these characteristics; we derived, via Hamiltonian formalism, the motion equations for an electron and we analyzed them. From these motion equations, we found functions that indicate the electron geometric boundaries within these thrusters, for each set of control parameters. In this work, these functions indicate the electronic confinement. Moreover, from the motion equations, we showed the electromagnetic fields settings which theoretically improve the Hall thruster’s performance. We found that, in these higher performance settings, the electron dynamics is chaotic. In this work, the chaos is determined by Poincaré maps and by Lyapunov exponents.