Le thème central de cette thèse est l'étude des résonances pour le problème intérieur en élastodynamique (géométries elliptique et sphéroïdale), et pour le problème de diffusion en acoustique (géométrie elliptique). On s'intéresse en particulier à la levée de dégénérescence des résonances liée à la brisure de symétrie de l'objet lors de la transition du disque circulaire vers le disque elliptique (2D), et de la sphère vers le sphéroïde (3D). Ce phénomène est étudié et interprété d'un point de vue théorique en prenant en compte les symétries de l'objet à l'aide de la théorie des groupes. Cette approche est complétée par une modélisation numérique et une partie expérimentale. En 2D, nous étudions le problème intérieur pour un disque elliptique élastique (étude des modes résonants) et le problème de la diffusion acoustique par des cylindres elliptiques élastiques. Ils sont traités à partir du formalisme modal combiné à la théorie des groupes dans le contexte vectoriel de l'élastodynamique. La levée de dégénérescence est observée théoriquement mais aussi expérimentalement en diffusion. La méthode simplifie considérablement le traitement numérique des problèmes étudiés, fournit une classification des résonances selon les 4 représentations irréductibles du groupe de symétrie C2v (associé à la géométrie elliptique) et donne une interprétation physique de la levée de dégénérescence en termes de brisure de symétrie. Une partie expérimentale en spectroscopie ultrasonore vient compléter l'étude théorique du problème de diffusion. Une série d'expériences en cuve est menée dans le cas de cylindres elliptiques de différentes excentricités en aluminium immergés dans l'eau, dans la bande de fréquence 0 ≤ kr ≤ 50, où kr est le nombre d'onde réduit dans le fluide. Les résultats expérimentaux présentent un très bon accord avec les résultats théoriques, la levée de dégénérescence est observée expérimentalement sur des fonctions de forme et mise en évidence sur des diagrammes angulaires. Le problème intérieur en 3D est traité expérimentalement à partir de la génération et la détection optiques d'ondes élastiques. Une série d'expérimentations sur des objets tridimensionnels (sphère, sphéroïdes oblates et prolates de différentes excentricités) en aluminium est réalisée. Ils sont mis en vibration par impacts laser et les mesures de vitesse et de fréquence s'effectuent par vibrométrie laser. On réalise ainsi une comparaison qualitative entre la théorie 2D et l'expérience 3D. Les mesures sont menées à la fois dans les domaines temporel et fréquentiel pour mettre en évidence la levée de dégénérescence d'une part, et l'onde de Rayleigh qui se propage sur la surface des objets d'autre part. Nous identifions deux trajets pour cette onde en géométrie sphéroïdale, l'un circulaire et l'autre elliptique.Enfin, dans le cadre des problèmes intérieurs 2D et 3D, on donne une interprétation en termes de rayons à travers la dualité entre le spectre des résonances et le spectre des longueurs des orbites périodiques (OPs), avec la mise en évidence du phénomène de conversion de mode et l'identification de l'onde de Rayleigh. Un phénomène, nouveau à notre connaissance, vient s'ajouter au phénomène de bifurcation de certaines orbites. Au cours de la déformation vers le disque elliptique, les orbites avec conversion de mode du disque circulaire se séparent en deux orbites dont les longueurs sont associées aux trajets minimal et maximal qu'elles parcourent. Cette observation s'interprète comme une conséquence du théorème de Fermat. Dans le cas du sphéroïde, on retrouve les orbites du disque circulaire dans le plan équatorial et celles du disque elliptique dans le plan méridien. Nous mettons également en évidence les pics associés aux deux trajets parcourus par l'onde de Rayleigh sur le spectre des OPs. / Resonances for the interior problem in elastodynamics and the acoustic scattering problem are considered in elliptical and spheroidal geometries. Interest is focused on the splitting up of resonances which occurs when the symmetry is broken in the transition from the circular disc to the elliptical one (2D), and from the sphere to the spheroid (3D). From the theoretical point of view, this physical phenomenon is studied and interpreted taking into account the symmetries of the object with the help of group theory. This approach is completed by a numerical modeling and an experimental part. As far as the two dimensional problems are concerned, the interior problem for an elliptical elastic disc (study of resonant modes) and the acoustic scattering problem for infinite elliptical elastic cylinders are studied combining modal formalism and group theory in the vectorial context of elastodynamics. The splitting up of resonances due to the circular symmetry breaking is not only theoretically observed but also experimentally for the scattering problem. The method significantly simplifies the numerical treatment of the problems studied, provides a full classification of resonances over the 4 irreducible representations of the symmetry group C2v (associated with the elliptical geometry) and gives a physical interpretation of the splitting up in terms of symmetry breaking of the symmetry group O(2) (invariance under rotation). An experimental part based on ultrasonic spectroscopy complements the theoretical study of the scattering problem. A series of tank experiments is carried out in the case of aluminum elliptical cylinders immersed in water, in the frequency range 0 ≤ kr ≤ 50, where kr is the reduced wave number in the fluid. The experimental results provide a very good agreement with the theoretical ones, the splitting up is observed on experimental form functions and the split resonant modes are identified on angular diagrams. The interior problem in 3D is studied by means of an experimental approach based on the optical generation and detection of elastic waves. A series of experiments is performed on three-dimensional objects in aluminium. These objects (sphere, prolate and oblate spheroids of various eccentricity) are excited by laser impacts, and the velocity and frequency measurements are carried out by laser vibrometry. Theory and experiments are qualitatively compared. The measurements are performed in both the frequency and time domains to highlight the splitting up of resonances on one hand, and the Rayleigh wave propagating on the surface of the objects on the other hand. We emphasize two paths for this surface wave in spheroidal geometry: a circular one in the equatorial plane and an elliptical one in the meridian plane. Finally, in the context of the interior problems in 2D and 3D, a physical interpretation of resonances in terms of geometrical paths is provided. Mode conversion is highlighted and the Rayleigh wave is identified on the periodic orbits lengths spectrum.In addition to the bifurcations of some periodic orbits, a phenomenon, new to our knowledge, appears. The orbits with mode conversion of the circular disc split in two orbits when the transition to the elliptic disc occurs. The lengths of these orbits are associated with the minimal and maximal travel paths. This observation is interpreted from Fermat's theorem.For the spheroid, orbits of the circular disc and those of the elliptical disc are recovered in the equatorial and meridian planes respectively. We also emphasize the peaks associated with the travel paths of Rayleigh wave in spheroidal geometry appearing on the periodic orbits spectrum.

Made available in DSpace on 2017-07-21T19:26:04Z (GMT). No. of bitstreams: 1 Rodrigo Frehse Pereira.pdf: 4666622 bytes, checksum: b2dcf2959eef9f7fd82301c2e45ac87f (MD5) Previous issue date: 2013-03-01 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Unstable dimension variability (UDV) is an extreme form of nonhyperbolicity. It is a structurally stable phenomenon, typical for high dimensional chaotic systems, which implies severe restrictions to shadowing of perturbed solutions. Perturbations are unavoidable in modelling Physical phenomena, since no system can be made completely isolated, states and parameters cannot be determined without uncertainties and any numeric approach to such models is affected by truncation and/or roundoff errors. Thus, the lack of shadowability in systems exhibiting UDV presents a challenge for modelling. Aiming to unveil the effect of perturbations a class of nonhyperbolic systems is studied. These systems present transversal unstable dimension variability (TUDV), which means the dynamics can be split in a skew direct product form, i. e. the phase space is decomposed in two components: a hyperbolic chaotic one, called longitudinal, and a nonhyperbolic transversal one. Moreover, in the absence of perturbations, the longitudinal component is a global attractor of the system. A prototype composed of two coupled piecewise-linear chaotic maps is presented in order to study the TUDV effects. This system has an invariant subspace S which characterizes the complete chaos synchronization and UDV, when present, is transversal to it. Taking advantage of (piecewise) linearity of the equations, an analytical method for unstable periodic orbits’ computation is presented. The set of all unstable periodic orbits (UPOs) is one of the building block of chaotic dynamics and its properties provide valuable informations about the asymptotic behaviour of the system as, for instance, the invariant natural measure. Therefore, the TUDV’s intensity is analytically studied by computing the contrast measure, which quantifies the difference between the statistical weights associated to UPOs with different unstable dimension. The effect of perturbations is modelled by the introduction of a small parameter mismatch, instead of noise addition, in order to keep the model’s determinism. Consequently, the characterization of dynamics by means of UPOs is still possible. It is shown the existence of a dense set G of UPOs outside the invariant subspace consistent with a chaotic repeller. When perturbation takes place, G merges with the set H of UPOs previously in S, given rise to a new nonhyperbolic stationary state. The analysis of G ∪H provides a topological explanation to the behaviour of systems with TUDV under perturbations. Moreover, the relation between the set of UPOs embedded in a chaotic attractor and its natural measure, proven only for hyperbolic systems, is successfully applied to this system: the error between the natural measure estimated both numerically and by means of UPOs is shown to be decreasing with p, the considered UPOs’ period. It is conjectured the coincidence between both in limit. Hence, a positive answer to reliability of numerical estimation to natural measure in nonhyperbolic systems via unstable dimension variability is presented. / A variabilidade da dimensão instável (VDI) é uma forma extrema de não-hiperbolicidade. É um fenômeno estruturalmente estável, típico para sistemas caóticos de alta dimensionalidade, que implica restrições severas ao sombreamento de soluções perturbadas. As perturbações¸ s são inevitáveis na modelagem de fenômenos fíısicos, uma vez que nenhum sistema pode ser isolado completamente, os estados e os parâmetros não podem ser determinados sem incertezas e qualquer abordagem numérica dos modelos é afetada por erros de arredondamento e/ou truncamento. Portanto, a falta da sombreabilidade em sistemas exibindo VDI apresenta um desafio à modelagem. Visando revelar os efeitos das perturbações, uma classe desses sistemas não hiperbó licos é estudada. Esses sistemas apresentam variabilidade da dimensão instável transversal (VDIT), significando que a dinâmica pode ser decomposta na forma de um produto direto assimétrico, i. e. o espação de fase é dividido em dois componentes: um hiperbólico e caótico, dito longitudinal, e um transversal e não-hiperbólico. Mais ainda, na ausência de perturbações, o componente longitudinal é um atrator global do sistema. Um protótipo composto de dois mapas ca´oticos lineares por partes acoplados é apresentado para o estudo dos efeitos da VDIT. Esse sistema possui um subespaço invariante S que caracteriza a sincronização completa de caos e a VDI, quando presente, é transversal a esse subespaço. Valendo-se da linearidade (por partes) das equações, um método analítico para o cálculo das órbitas periódicas instáveis é apresentado. O conjunto de todas as órbitas periódicas instáveis (OPIs) é um dos fundamentos da dinâmica caótica e suas propriedades fornecem informaões, valiosas sobre o comportamento assintótico do sistema como, por exemplo, a medida natural invariante. Assim, a intensidade da VDIT é estudada analiticamente pelo cálculo da medida de contraste, que quantifica a diferença entre o peso estatístico associado às OPIs com dimensão instável distintas. O efeito das perturbações é modelado pela introdução de um pequeno desvio nos parâmetros, ao invés da adição de ruído, a fim de manter o determinismo do modelo. Consequentemente, a caracterização da dinâmica em termos das OPIs ainda é possível. Demonstra-se a existência de um conjunto denso G de OPIs fora do subespaço invariante consistente com um repulsor caótico. Na presença de perturbações, G se funde com o conjunto H das OPIs previamente em S, dando origem a um novo estado estacionario não-hiperbólico. A análise de G ∪H fornece uma explicação topológica ao comportamento de sistemas com variabilidade da dimensão instável sob a açãoo de perturbações. Mais ainda, a relação entre o conjunto de OPIs imersas em um atrator caótico e sua medida natural, provada apenas para sistemas hiperbólicos, é aplicada com sucesso nesse sistema: mostra-se que o erro entre as medidas naturais estimadas numericamente e pelas OPIs é decrescente com p, o período das OPIs consideradas. Conjectura-se, portanto, a coincidência entre ambas no limite . Logo, apresenta-se uma resposta positiva à estimativa numérica da medida natural em sistemas não-hiperbólicos via variabilidade da dimensão instável.

Μελετάµε χωρικά εντοπισµένες και χρονικά περιοδικές λύσεις σε διακριτά συστήµατα που εκτείνονται σε µία χωρική διάσταση. Αυτού του είδους οι λύσεις είναι γνωστές µε τον όρο discrete breathers (DB) ή intrinsic localized modes (ILM). Στην ελληνική ϐιϐλιογραϕία, έχουν ονοµαστεί ∆ιακριτές Πνοές. Απαραίτητα χαρακτηριστικά για την εµϕάνιση τέτοιων λύσεων είναι η ύπαρξη ενός άνω φράγµατος του γραµµικού φάσµατος καθώς και η µη γραµµικότητα των εξισώσεων κίνησης, χαρακτηριστικά που συναντάµε σε πολλά φυσικά συστήµατα. Συγκεκριμένα, ασχολούµαστε µε πλέγµατα τύπου Klein Gordon και παρουσιάσουµε μια αποδείξη ύπαρξης τέτοιων λύσεων καθώς και αριθµητικά αποτελέσµατα µελετώντας παράλληλα την ευστάθεια των περιοδικών αυτών λύσεων µέσω της ϑεωρίας Floquet. Πέραν του κλασικού µοντέλου, όπου έχουµε αλληλεπιδράσεις πλησιέστερων γειτόνων, εισάγουµε επίσης ένα νέο µοντέλο µε αλληλεπιδράσεις µακράς εµβέλειας η οποία ελέγχεται µέσω µιας παράµετρου α και µελετάµε τις επιπτώσεις που έχει η μεταβολή του εύρους αλληλεπίδρασης στον χωρικό εντοπισµό και την ευστάθεια ενός DB. / We study time-periodic and spatially localized solutions in discrete dynamical systems describing Hamiltonian lattices in one spatial dimension. These solutions are called discrete breathers (DBs) or intrinsic localized modes (ILM). Necessary conditions for their occurrence are the boundedness of the spectrum of linear oscillations of the system as well as the nonlinearity of the equations of motion. More specifically, we focus on a Klein Gordon lattice and present an existence proof for such solutions, as well as numerical results revealing the stability (or instability) of DBs using Floquet theory. Besides reporting on the classical Klein Gordon model with nearest neighbor interactions, we also introduce long range interactions in our model, which are controlled by a parameter α and study the effect of varying the range of interactions on the spatial localization and the stability of a DB.

The main problem evaluated in this manuscript is the stabilization of periodic orbits of non-linear dynamical systems by use of feedback control. The goal of the control methods proposed in this work is to achieve a stable periodic oscillation. These control methods are applied to systems that present unstable periodic orbits in the state space, and the latter are the orbits to be stabilized.The methods proposed here are such that the resulting stable oscillation is obtained with low control effort, and the control signal is designed to converge to zero when the trajectory tends to the stabilized orbit. Local stability of the periodic orbits is analyzed by studying the stability of some linear time-periodic systems, using the Floquet stability theory. These linear systems are obtained by linearizing the trajectories in the vicinity of the periodic orbits.The control methods used for stabilization of periodic orbits here are the proportional feedback control, the delayed feedback control and the prediction-based feedback control. These methods are applied to discrete and continuous-time systems with the necessary modifications. The main contributions of the thesis are related to these methods, proposing an alternative control gain design, a new control law and related results.

Η διδακτορική διατριβή επικεντρώθηκε στη μελέτη νέων τεχνικών κρυπτογραφίας και κρυπτανάλυσης, αλλά και στην ανάπτυξη νέων πρωτοκόλλων για την ασφαλή ηλεκτρονική συγκέντρωση δεδομένων. Το πρώτο πρόβλημα το οποίο διερεύνησε η διατριβή ήταν η δυνατότητα εφαρμογής των μεθόδων Υπολογιστικής Νοημοσύνης στην κρυπτολογία. Στόχος ήταν η ανίχνευση των κρίσιμων σημείων κατά την εφαρμογή των μεθόδων αυτών στον πολύ απαιτητικό αυτό τομέα προβλημάτων και η μελέτη της αποτελεσματικότητας και της αποδοτικότητάς τους σε διάφορα προβλήματα κρυπτολογίας. Συνοψίζοντας, τα αποτελέσματα της διατριβής για την εφαρμογή μεθόδων Υπολογιστικής Νοημοσύνης στην κρυπτολογία υποδεικνύουν ότι παρά το γεγονός ότι η κατασκευή των αντικειμενικών συναρτήσεων είναι πολύ κρίσιμη για την αποδοτικότητα των μεθόδων, η Υπολογιστική Νοημοσύνη μπορεί να προσφέρει σημαντικά πλεονεκτήματα στον κλάδο αυτό όπως είναι η αυτοματοποίηση κάποιων διαδικασιών κρυπτανάλυσης ή κρυπτογράφησης, ο γρήγορος έλεγχος της σθεναρότητας νέων κρυπτοσυστημάτων αλλά και ο συνδυασμός τους με τυπικές μεθόδους που χρησιμοποιούνται μέχρι σήμερα για την αξιοποίηση της απλότητας και της αποδοτικότητάς τους. Το δεύτερο πρόβλημα που μελετάται στην διατριβή είναι η εφαρμογή μεθόδων αντίστροφης πολυωνυμικής παρεμβολής για την εύρεση της τιμής του διακριτού λογαρίθμου αλλά και του λογαρίθμου του Lucas. Για την μελέτη αυτή χρησιμοποιήθηκαν δύο υπολογιστικές μέθοδοι αντίστροφης πολυωνυμικής παρεμβολής, οι μέθοδοι Aitken και Neville, οι οποίες είναι κατασκευαστικές και επιτρέπουν την πρόσθεση νέων σημείων παρεμβολής για καλύτερη προσέγγιση του πολυωνύμου με μικρό υπολογιστικό κόστος. Η παρούσα μελέτη έδειξε ότι και με την προτεινόμενη μεθοδολογία το συνολικό κόστος υπολογισμού της τιμής των λογαρίθμων παραμένει υψηλό, ωστόσο η κατανομή των πολυωνύμων που έδωσαν την λύση των προβλημάτων δείχνει ότι η μεθοδολογία που χρησιμοποιήθηκε είτε εντόπισε την λύση στα πρώτα στάδια κατασκευής των πολυωνύμων είτε εντόπισε πολυώνυμα μικρού σχετικά βαθμού που προσεγγίζουν την αντίστοιχη λύση. Το τρίτο πρόβλημα που πραγματεύεται η παρούσα διατριβή είναι η δημιουργία νέων σθεναρών κρυπτοσυστημάτων με την χρήση μη-γραμμικών δυναμικών απεικονίσεων. Η αξιοποίηση των ιδιοτήτων του χάους στην κρυπτογραφία έχει αποτελέσει αντικείμενο μελέτης τα τελευταία χρόνια από τους ερευνητές λόγω της αποδεδειγμένης πολυπλοκότητας των συστημάτων του και των ιδιαίτερων στατιστικών ιδιοτήτων τους. Η διατριβή συνεισφέρει προτείνοντας ένα νέο συμμετρικό κρυπτοσύστημα που βασίζεται σε περιοδικές δυναμικές τροχιές και παρουσιάζει και τρεις τροποποιήσεις του που το καθιστούν ιδιαίτερα σθεναρό απέναντι στις συνήθεις κρυπταναλυτικές επιθέσεις. Δίνεται επίσης το υπολογιστικό κόστος κρυπτογράφησης και αποκρυπτογράφης του προτεινόμενου σχήματος και παρουσιάζονται πειραματικά αποτελέσματα που δείχνουν ότι η δομή των κρυπτογραφημάτων του κρυπτοσυστήματος δεν παρέχει πληροφορία για την ύπαρξη τυχόν μοτίβων στο αρχικό κείμενο. Τέλος, στην διατριβή αυτή προτείνονται δύο πρωτόκολλα για την ασφαλή ηλεκτρονική συγκέντρωση δεδομένων. Η συγκέντρωση δεδομένων από διαφορετικές βάσεις με ασφάλεια και ιδιωτικότητα θα ήταν σημαντική για την μελέτη των γνώσεων που ενυπάρχουν στα δεδομένα αυτά, με διάφορες μεθόδους εξόρυξης δεδομένων και ανάλυσης, καθώς οι γνώσεις αυτές ενδεχομένως δεν θα μπορούσαν να αποκαλυφθούν από την επιμέρους μελέτη των δεδομένων χωριστά από κάθε βάση. Τα δύο πρωτόκολλα που προτείνονται βασίζονται σε τροποποιήσεις πρωτοκόλλων ηλεκτρονικών εκλογών με τρόπο τέτοιο ώστε να ικανοποιούνται τα απαραίτητα κριτήρια ασφάλειας και ιδιωτικότητας που απαιτούνται για την συγκέντρωση των δεδομένων. Η βασική διαφορά των δύο πρωτοκόλλων είναι ότι στο ένα γίνεται χρήση έμπιστου τρίτου μέλους για την συγκέντρωση των δεδομένων, ενώ στο δεύτερο όχι. Και στις δύο περιπτώσεις, παρουσιάζεται ανάλυση της ασφάλειας των σχημάτων αλλά και της πολυπλοκότητάς τους αναφορικά με το υπολογιστικό τους κόστος. / In this PhD thesis we study problems of cryptography and cryptanalysis through Computational Intelligence methods and computational mathematics. Furthermore, we examine the establishment and security of new privacy preserving protocols for electronic data gathering. Part I is dedicated to the application of Computational Intelligence (CI) methods, namely Evolutionary Computation (EC) methods and Artificial Neural Networks (ANNs), for solving problems of cryptology. Initially, three problems of cryptanalysis are formulated as discrete optimization tasks and Evolutionary Computation methods are utilized to address them. The first conclusion derived by these experiments is that when EC methods are applied to cryptanalysis special attention must be paid to the design of the fitness function so as to include as much information as possible for the target problem. The second conclusion is that when EC methods (and CI methods in general) can be used as a quick practical assessment for the efficiency and the effectiveness of proposed cryptographic systems. We also apply EC methods for the cryptanalysis of Feistel ciphers and for designing strong Substitution boxes. The results show that the proposed methods are able to tackle theses problem efficiently and effectively with low cost and in automated way. Then, ANNs are employed for classical problems of cryptography as a measure of their robustness. The results show that although different topologies, training methods and formulation of the problems were tested, ANNs were able to obtain the solution of the problems at hand only for small values of their parameters. The performance of ANNs is also studied on the computation of a Boolean function derived from the use of elliptic curves in cryptographic applications. The results indicate that ANNs are able to adapt to the data presented with high accuracy, while their response to unknown data is slightly better than a random selection. Another important finding is that ANNs require a small amount of storage for the known patterns in contrast to the storage needed of the data itself. Finally, a theoretical study of the application of Ridge Polynomial Networks for the computation of the least significant bit of the discrete logarithm is presented. In Part II, computational mathematics are utilized for different cryptographic problems. Initially, we consider the Aitken and Neville inverse interpolation methods for a discrete exponential function and the Lucas logarithm function. The results indicate that the computational cost for addressing the problems through this approach is high; however interesting features regarding the degree of the resulting interpolation polynomials are reported. Next, a new symmetric key cryptosystem that exploits the idea of nonlinear mappings and their fixed points to encrypt information is presented. Furthermore, a measure of the quality of the keys used is introduced. The experimental results indicate that the proposed cryptosystem is efficient and secure to ciphertext-only attacks. Finally, three modifications of the basic cryptosystem that render it more robust are presented and efficiency issues are discussed. Finally, at Part III of the thesis, two protocols for privacy preserving electronic data gathering are proposed. The security requirements that must be met for data gathering with privacy are presented and then two protocols, based on electronic voting protocols, are analytically described. Security and complexity issues are also discussed.

<p> </p> <p>In the coming decades, numerous missions plan to exploit multi-body orbits for operations. Given the complex nature of multi-body systems, trajectory designers must possess effective tools that leverage aspects of the dynamical environment to streamline the design process and enable these missions. In this investigation, a particular class of dynamical structures, quasi-periodic orbits, are examined. This work summarizes a computational framework to construct quasi-periodic orbits and a design framework to leverage quasi-periodic motion within the path planning process. First, quasi-periodic orbit computation in the Circular Restricted Three-Body Problem (CR3BP) and the Bicircular Restricted Four-Body Problem (BCR4BP) is summarized. The CR3BP and BCR4BP serve as preliminary models to capture fundamental motion that is leveraged for end-to-end designs. Additionally, the relationship between the Earth-Moon CR3BP and the BCR4BP is explored to provide insight into the effect of solar acceleration on multi-body structures in the lunar vicinity. Characterization of families of quasi-periodic orbits in the CR3BP and BCR4BP is also summarized. Families of quasi-periodic orbits prove to be particularly insightful in the BCR4BP, where periodic orbits only exist as isolated solutions. Computation of three-dimensional quasi-periodic tori is also summarized to demonstrate the extensibility of the computational framework to higher-dimensional quasi-periodic orbits. Lastly, a design framework to incorporate quasi-periodic orbits into the trajectory design process is demonstrated through a series of applications. First, several applications were examined for transfer design in the vicinity of the Moon. The first application leverages a single quasi-periodic trajectory arc as an initial guess to transfer between two periodic orbits. Next, several quasi-periodic arcs are leveraged to construct transfer between a planar periodic orbit and a spatial periodic orbit. Lastly, transfers between two quasi-periodic orbits are demonstrated by leveraging heteroclinic connections between orbits at the same energy. These transfer applications are all constructed in the CR3BP and validated in a higher-fidelity ephemeris model to ensure the geometry persists. Applications to ballistic lunar transfers are also constructed by leveraging quasi-periodic motion in the BCR4BP. Stable manifold trajectories of four-body quasi-periodic orbits supply an initial guess to generate families of ballistic lunar transfers to a single quasi-periodic orbit. Poincare mapping techniques are used to isolate transfer solutions that possess a low time of flight or an outbound lunar flyby. Additionally, impulsive maneuvers are introduced to expand the solution space. This strategy is extended to additional orbits in a single family to demonstrate "corridors" of transfers exist to reach a type of destination motion. To ensure these transfers exist in a higher fidelity model, several solutions are transitioned to a Sun-Earth-Moon ephemeris model using a differential corrections process to show that the geometries persist.</p>

<div>As destinations of missions in both human and robotic spaceflight become more exotic, a foundational understanding the dynamical structures in the gravitational environments enable more informed mission trajectory designs. One particular type of structure, quasi-periodic orbits, are examined in this investigation. Specifically, efficient computation of quasi-periodic orbits and leveraging quasi-periodic orbits as trajectory design alternatives in the Earth-Moon and Sun-Earth systems. First, periodic orbits and their associated center manifold are discussed to provide the background for the existence of quasi-periodic motion on n-dimensional invariant tori, where n corresponds to the number of fundamental frequencies that define the motion. Single and multiple shooting differential corrections strategies are summarized to compute families 2-dimensional tori in the Circular Restricted Three-Body Problem (CR3BP) using a stroboscopic mapping technique, originally developed by Howell and Olikara. Three types of quasi-periodic orbit families are presented: constant energy, constant frequency ratio, and constant mapping time families. Stability of quasi-periodic orbits is summarized and characterized with a single stability index quantity. For unstable quasi-periodic orbits, hyperbolic manifolds are computed from the differential of a discretized invariant curve. The use of quasi-periodic orbits is also demonstrated for destination orbits and transfer trajectories. Quasi-DROs are examined in the CR3BP and the Sun-Earth-Moon ephemeris model to achieve constant line of sight with Earth and avoid lunar eclipsing by exploiting orbital resonance. Arcs from quasi-periodic orbits are leveraged to provide an initial guess for transfer trajectory design between a planar Lyapunov orbit and an unstable halo orbit in the Earth-Moon system. Additionally, quasi-periodic trajectory arcs are exploited for transfer trajectory initial guesses between nearly stable periodic orbits in the Earth-Moon system. Lastly, stable hyperbolic manifolds from a Sun-Earth L₁ quasi-vertical orbit are employed to design maneuver-free transfer from the LEO vicinity to a quasi-vertical orbit.</div>

Le travail de cette thèse a été réalisé dans le cadre d'une collaboration entre EDF R&D et le LMA de Marseille (CNRS). Le but était de développer des outils théoriques et numériques pour le calcul de modes non-linéaires de structures industrielles possédant des non-linéarités localisées à jeu. La méthode de calcul utilisée est une combinaison de la méthode d'équilibrage harmonique (EH) et de la méthode asymptotique numérique (MAN), appelée EHMAN. Elle est réputée pour sa robustesse sur les problèmes réguliers. L'enjeu de ce travail de thèse est de l'appliquer sur des problèmes non-réguliers régularisés de type butée à jeu pour lequel un grand nombre d'harmonique est nécessaire. Des améliorations ont été apportées à la méthode de base pour rendre effectif le traitement de modèles à "grand" nombre de degrés de liberté (DDL). Les développements réalisés pendant la thèse ont été capitalisés par la création de nouveaux opérateurs dans Code_Aster.Une étude approfondie d'un système à 2 degrés de liberté a permis de faire émerger quelques caractéristiques des systèmes non-linéaires à jeu. Celles-ci ont servi entre autre à établir une méthodologie pour l'étude de systèmes à grand nombre de DDL. Pour finir, la potentialité des modes non-linéaires comme outil de diagnostic vibratoire est démontrée avec l'étude d'un tube cintré de générateur de vapeur. Le calcul des modes non-linéaires a monté l'existence d'une interaction entre un mode hors-plan (basse fréquence) et un mode plan (haute fréquence) expliquant des régimes vibratoires non-standards. Ce résultat, impossible à obtenir avec les outils de l'analyse modale linéaire, est confirmé expérimentalement. / This work is a collaboration between EDF R&D and the Laboratory of Mechanics and Acoustics. The objective is to develop theoretical and numerical tools to compute nonlinear normal modes (NNMs) of structures with localized nonlinearities.We use an approach combining the harmonic balance and the asymptotic numerical methods, known for its robustness principally for smooth systems. Regularization techniques are used to apply this approach for the study of nonsmooth problems. Moreover, several aspects of the method are improved to allow the computation of NNMs for systems with a high number of degrees of freedom (DOF). Finally, the method is implemented in Code_Aster, an open-source finite element solver developed by EDF R&D.The nonlinear normal modes of a two degrees-of-freedom system are studied and some original characteristics are observed. These observations are then used to develop a methodology for the study of systems with a high number of DOFs. The developed method is finally used to compute the NNMs for a model U-tube of a nuclear plant steam generator. The analysis of the NNMs reveals the presence of an interaction between an out-of-plane (low frequency) and an in-plane (high frequency) modes, a result also confirmed by the experiment. This modal interaction is not possible using linear modal analysis and confirms the interest of NNMs as a diagnostic tool in structural dynamics.

Le problème principalement étudié dans ce manuscrit est la stabilisation d’orbites périodiques de systèmes dynamiques non linéaires à l’aide d’une commande de rétroaction (feedback). Le but des méthodes de contrôle proposées ici est d’obtenir une oscillation périodique stable. Ces méthodes de contrôle sont appliquées à des systèmes présentant des orbites périodiques instables dans l’espace d’état, et ces dernières sont les orbites destinées à être stabilisées.Les méthodes proposées ici sont telles que l’oscillation stable qui en résulte est obtenue avec un effort de contrôle faible, et que la valeur de la commande tend vers zéro lorsque la trajectoire tend vers l’orbite stabilisée. La stabilité locale des orbites périodiques est analysée par l’étude de la stabilité des systèmes linéaires périodiques à l’aide de la théorie de Floquet. Ces systèmes linéaires sont obtenus par linéarisation des trajectoires au voisinage de l’orbite périodique.Les méthodes de contrôle utilisées ici pour la stabilisation des orbites périodiques sont une loi de commande proportionnelle, une loi de commande de rétroaction retardée et une loi de commande de rétroaction basée sur une prédiction. Ces méthodes sont appliquées aux systèmes en temps discret et aux systèmes en temps continu avec les modifications nécessaires. Les contributions principales de cette thèse sont associées à ces méthodes, proposant une méthode alternative de design de gain, une nouvelle loi de commande et des résultats associés. / The main problem evaluated in this manuscript is the stabilization of periodic orbits of non-linear dynamical systems by use of feedback control. The goal of the control methods proposed in this work is to achieve a stable periodic oscillation. These control methods are applied to systems that present unstable periodic orbits in the state space, and the latter are the orbits to be stabilized.The methods proposed here are such that the resulting stable oscillation is obtained with low control effort, and the control signal is designed to converge to zero when the trajectory tends to the stabilized orbit. Local stability of the periodic orbits is analyzed by studying the stability of some linear time-periodic systems, using the Floquet stability theory. These linear systems are obtained by linearizing the trajectories in the vicinity of the periodic orbits.The control methods used for stabilization of periodic orbits here are the proportional feedback control, the delayed feedback control and the prediction-based feedback control. These methods are applied to discrete and continuous-time systems with the necessary modifications. The main contributions of the thesis are related to these methods, proposing an alternative control gain design, a new control law and related results. / O principal problema avaliado neste manuscrito é a estabilização de órbitas periódicas em sistemas dinâmicos não-lineares utilizando controle por realimentação. O objetivo dos métodos de controle propostos neste trabalho é obter uma oscilação periódica estável. Estes métodos de controle são aplicados a sistemas que apresentam órbitas periódicas instáveis no espaço de estados, estas são as órbitas a serem estabilizadas.Os métodos propostos aqui são tais que a oscilação periódica estável resultante é obtida utilizando um baixo esforço de controle, e o sinal de controle é projetado de forma a convergir para zero quanto a trajetória tende à órbita estabilizada. A estabilidade local de órbitas periódicas é analisada através do estudo da estabilidade de alguns sistemas lineares periódicos no tempo, utilizando a teoria de estabilidade de Floquet. Estes sistemas lineares são obtidos por linearização das trajetórias na vizinhança da órbita periódica.Os métodos de controle utilizados aqui para estabilização de órbitas periódicas são o proportional feedback control, o delayed feedback control e o prediction-based feedback control (controle por realimentação baseado em predição). Estes métodos são aplicados a sistemas de tempo discreto e de tempo contínuo, com as modificações necessárias. As principais contribuições da tese são relacionadas a esses métodos, propondo um projeto de ganho de controle alternativo, uma nova lei de controle e resultados relacionados.