Few-mode fibers (FMFs) and multimode fibers (MMFs) can provide much higher data-carrying capacities compared with single-mode fibers. But in order to achieve this goal, one must address the challenge of intermodal coupling and dispersion. Therefore the ability to accurately control the optical signal propagation in FMFs/MMFs can play a pivotal role in FMF/MMF applications. This thesis demonstrates the ability to excite, in FMFs and MMFs, the desired linearly polarized (LP) modes as well as their superpositions through adaptive optics (AO). Specifically, in the case of step-index FMFs, a phase-only spatial light modulator (SLM) is employed to manipulate the light at the fiber input end, driven by the feedback signal provided by the correlation between the charge coupled device (CCD) camera captured images at the fiber output end and the target light intensity profile. Through such an adaptive optical system, any arbitrarily selected LP modes can be excited at the distal end of the four-mode and seventeen-mode fibers, respectively. For a graded-index MMF with a uniform Bragg grating, we use a deformable mirror (DM) to perform the wavefront modulation at the fiber input end, where the feedback is based on the ratio of the grating-reflected signal power to the transmitted signal power. At the Bragg grating position of this highly multimode fiber, any desired principal mode groups can be successfully chosen. These experimental results suggest that adaptive control of optical wavefront in FMFs/MMFs is indeed feasible. / Master of Science / Optical fibers, in terms of the number of modes they support, can be generally divided into single-mode fibers (SMFs), and few-mode fibers/multimode fibers (FMFs/MMFs). FMFs/MMFs can provide much higher data-carrying capacities than SMFs. For example, an FMF/MMF that supports M modes can ideally increase the data transmission rate by a factor of M, where each mode can serve as a distinct communication channel. However, in order to achieve good performance, one must accurately control signal propagation in FMFs/MMFs, which are often degraded due to the multiple-mode nature. This thesis demonstrates the ability, using adaptive optics (AO), to control signal propagation in FMFs and a highly MMF, respectively. Specifically, in the case of FMFs, a phase-only spatial light modulator (SLM) is employed to manipulate the light at the fiber input, driven by AO feedback signal provided by the similarity between the real-time fiber output image and the target mode profile. Through such an adaptive optical system, any desired linearly-polarized (LP) modes can be excited at the output of the four-mode and seventeen-mode fibers, respectively. For the highly MMF with uniform Bragg grating, we use a deformable mirror (DM) to perform the wavefront modulation at the fiber input, where AO feedback is provided by the fiber Bragg grating (FBG) reflectivity. At the FBG position, any desired principal mode groups can be successfully chosen. These experimental results suggest that adaptive control of optical wavefront in FMFs/MMFs is indeed feasible, and may find a large number of applications in optical communication, sensing, and imaging.

Text plays a pivotal role in our daily lives, encompassing various forms such as social media posts, news articles, books, reports, and more. Consequently, Natural Language Processing (NLP) has garnered widespread attention. This technology empowers us to undertake tasks like text classification, entity recognition, and even crafting responses within a dialogue context. However, despite the expansive utility of NLP, it frequently necessitates a critical decision: whether to place trust in a model's predictions. To illustrate, consider a state-of-the-art (SOTA) model entrusted with diagnosing a disease or assessing the veracity of a rumor. An incorrect prediction in such scenarios can have dire consequences, impacting individuals' health or tarnishing their reputation. Consequently, it becomes imperative to establish a reliable method for evaluating the reliability of an NLP model's predictions, which is our focus-uncertainty estimation on NLP. Though many works have researched uncertainty estimation or NLP, the combination of these two domains is rare. This is because most NLP research emphasizes model prediction performance but tends to overlook the reliability of NLP model predictions. Additionally, current uncertainty estimation models may not be suitable for NLP due to the unique characteristics of NLP tasks, such as the need for more fine-grained information in named entity recognition. Therefore, this dissertation proposes novel uncertainty estimation methods for different NLP tasks by considering the NLP task's distinct characteristics. The NLP tasks are categorized into natural language understanding (NLU) and natural language generation (NLG, such as text summarization). Among the NLU tasks, the understanding could be on two views, global-view (e.g. text classification at document level) and local-view (e.g. natural language inference at sentence level and named entity recognition at token level). As a result, we research uncertainty estimation on three tasks: text classification, named entity recognition, and text summarization. Besides, because few-shot text classification has captured much attention recently, we also research the uncertainty estimation on few-shot text classification. For the first topic, uncertainty estimation on text classification, few uncertainty models focus on improving the performance of text classification where human resources are involved. In response to this gap, our research focuses on enhancing the accuracy of uncertainty scores by bolstering the confidence associated with winning scores. we introduce MSD, a novel model comprising three distinct components: 'mix-up,' 'self-ensembling,' and 'distinctiveness score.' The primary objective of MSD is to refine the accuracy of uncertainty scores by mitigating the issue of overconfidence in winning scores while simultaneously considering various categories of uncertainty. seamlessly integrate with different Deep Neural Networks. Extensive experiments with ablation settings are conducted on four real-world datasets, resulting in consistently competitive improvements. Our second topic focuses on uncertainty estimation on few-shot text classification (UEFTC), which has few or even only one available support sample for each class. UEFTC represents an underexplored research domain where, due to limited data samples, a UEFTC model predicts an uncertainty score to assess the likelihood of classification errors. However, traditional uncertainty estimation models in text classification are ill-suited for UEFTC since they demand extensive training data, while UEFTC operates in a few-shot scenario, typically providing just a few support samples, or even just one, per class. To tackle this challenge, we introduce Contrastive Learning from Uncertainty Relations (CLUR) as a solution tailored for UEFTC. CLUR exhibits the unique capability to be effectively trained with only one support sample per class, aided by pseudo uncertainty scores. A distinguishing feature of CLUR is its autonomous learning of these pseudo uncertainty scores, in contrast to previous approaches that relied on manual specification. Our investigation of CLUR encompasses four model structures, allowing us to evaluate the performance of three commonly employed contrastive learning components in the context of UEFTC. Our findings highlight the effectiveness of two of these components. Our third topic focuses on uncertainty estimation on sequential labeling. Sequential labeling involves the task of assigning labels to individual tokens in a sequence, exemplified by Named Entity Recognition (NER). Despite significant advancements in enhancing NER performance in prior research, the realm of uncertainty estimation for NER (UE-NER) remains relatively uncharted but is of paramount importance. This topic focuses on UE-NER, seeking to gauge uncertainty scores for NER predictions. Previous models for uncertainty estimation often overlook two distinctive attributes of NER: the interrelation among entities (where the learning of one entity's embedding depends on others) and the challenges posed by incorrect span predictions in entity extraction. To address these issues, we introduce the Sequential Labeling Posterior Network (SLPN), designed to estimate uncertainty scores for the extracted entities while considering uncertainty propagation from other tokens. Additionally, we have devised an evaluation methodology tailored to the specific nuances of wrong-span cases. Our fourth topic focuses on an overlooked question that persists regarding the evaluation reliability of uncertainty estimation in text summarization (UE-TS). Text summarization, a key task in natural language generation (NLG), holds significant importance, particularly in domains where inaccuracies can have serious consequences, such as healthcare. UE-TS has garnered attention due to the potential risks associated with erroneous summaries. However, the reliability of evaluating UE-TS methods raises concerns, stemming from the interdependence between uncertainty model metrics and the wide array of NLG metrics. To address these concerns, we introduce a comprehensive UE-TS benchmark incorporating twenty-six NLG metrics across four dimensions. This benchmark evaluates the uncertainty estimation capabilities of two large language models and one pre-trained language model across two datasets. Additionally, it assesses the effectiveness of fourteen common uncertainty estimation methods. Our study underscores the necessity of utilizing diverse, uncorrelated NLG metrics and uncertainty estimation techniques for a robust evaluation of UE-TS methods. / Doctor of Philosophy / Text is integral to our daily activities, appearing in various forms such as social media posts, news articles, books, and reports. We rely on text for communication, information dissemination, and decision-making. Given its ubiquity, the ability to process and understand text through Natural Language Processing (NLP) has become increasingly important. NLP technology enables us to perform tasks like text classification, which involves categorizing text into predefined labels, and named entity recognition (NER), which identifies specific entities such as names, dates, and locations within text. Additionally, NLP facilitates generating coherent and contextually appropriate responses in conversational agents, enhancing human-computer interaction. However, the reliability of NLP models is crucial, especially in sensitive applications like medical diagnoses, where errors can have severe consequences. This dissertation focuses on uncertainty estimation in NLP, a less explored but essential area. Uncertainty estimation helps evaluate the confidence of NLP model predictions. We propose new methods tailored to various NLP tasks, acknowledging their unique needs. NLP tasks are divided into natural language understanding (NLU) and natural language generation (NLG). Within NLU, we look at tasks from two perspectives: a global view (e.g., document-level text classification) and a local view (e.g., sentence-level inference and token-level entity recognition). Our research spans text classification, named entity recognition (NER), and text summarization, with a special focus on few-shot text classification due to its recent prominence. For text classification, we introduce the MSD model, which includes three components to enhance uncertainty score accuracy and address overconfidence issues. This model integrates seamlessly with different neural networks and shows consistent improvements in experiments. For few-shot text classification, we develop Contrastive Learning from Uncertainty Relations (CLUR), designed to work effectively with minimal support samples per class. CLUR autonomously learns pseudo uncertainty scores, demonstrating effectiveness with various contrastive learning components. In NER, we address the unique challenges of entity interrelation and span prediction errors. We propose the Sequential Labeling Posterior Network (SLPN) to estimate uncertainty scores while considering uncertainty propagation from other tokens. For text summarization, we create a benchmark with tens of metrics to evaluate uncertainty estimation methods across two datasets. This benchmark helps assess the reliability of these methods, highlighting the need for diverse, uncorrelated metrics. Overall, our work advances the understanding and implementation of uncertainty estimation in NLP, providing more reliable and accurate predictions across different tasks.

Making information stored in databases more accessible to users inexperienced in structured query language (SQL) by converting natural language to SQL queries has long been a prominent research area in both the database and natural language processing (NLP) communities. There have been numerous approaches proposed for this task, such as encoder-decoder frameworks, semantic grammars, and more recently with the use of large language models (LLMs). When training LLMs to successfully generate SQL queries from natural language questions there are three notable methods used, pretraining, transfer learning and in-context learning (ICL). ICL is particularly advantageous in scenarios where the hardware at hand is limited, time is of concern and large amounts of task specific labled data is nonexistent. This study seeks to evaluate two strategies in ICL, namely zero-shot and few-shot scenarios using the Mistral-7B-Instruct LLM. Evaluation of the few-shot scenarios was conducted using two techniques, random selection and Jaccard Similarity. The zero-shot scenarios served as a baseline for the few-shot scenarios to overcome, which ended as anticipated, with the few-shot scenarios using Jaccard similarity outperforming the other two methods, followed by few-shot scenarios using random selection coming in at second best, and the zero-shot scenarios performing the worst. Evaluation results acquired based on execution accuracy and exact matching accuracy confirm that leveraging similarity in demonstrating examples when prompting the LLM will enhance the models knowledge about the database schema and table names which is used during the inference phase leadning to more accurately generated SQL queries than leveraging diversity in demonstrating examples.

Doctor of Philosophy / Department of Physics / Brett D. Esry / The Efimov effect plays a central role in few-body systems at ultracold temperature and has thus accelerated a lot of studies on its manifestation in the collisional stability of the quantum degenerate gases. Near broad Feshbach resonances, Efimov physics has been studied both theoretically and experimentally through the zero-energy scattering observables. We have extended the theoretical studies of Efimov physics to a much broader extent. In particular, we have investigated the three-body Efimov physics near narrow Feshbach resonances and have also identified the Efimov features beyond the zero temperature limit. We have found, near a narrow Feshbach resonance, the non-trivial contribution from both of the resonance width and the short-range physics to the three-body recombination and vibrational dimer relaxation. Remarkably, the collisional stability of the Feshbach molecules are found to be opposite to that near the broad resonances: an increased stability for molecules made by bosons and a decreased stability for those made by fermions. The universal physics observed near the narrow Feshbach resonances is further found not to be limited to the zero temperature observables. We have found that the general features of Efimov physics and those pertaining to a narrow resonance are manifested in different energy ranges above zero temperature. This opens the opportunity to observe Efimov physics by changing the collisional energy while keeping the atomic interaction fixed. The landscape of the universal Efimov physics is thus delineated in both of the interaction and the energy domain. We have also investigated Efimov physics in heteronuclear four-body systems where the complexity can be reduced by approximations. In particular, we have proposed ways for controllable production of the Efimov tri-atomic molecules by three-body or four-body recombinations involving four atoms. We have also confirmed the existence of four-body Efimov effect in a system of three heavy particles and one light particle, which has resolved a decade-long controversy on this topic. Finally, we have studied the collisional properties of four identical bosons in 1D, which is important to the experiments on the quantum gases confined in the 1D optical lattices.

Bound state systems consisting of three nonrelativistic particles are numerically studied. Calculations are performed employing two-body and three-body forces as input in the Hamiltonian in order to study the role or contribution of three-body forces to the binding in these systems. The resulting differential Faddeev equations are solved as three-dimensional equations in the two Jacobi coordinates and the angle between them, as opposed to the usual partial wave expansion approach. By expanding the wave function as a sum of the products of spline functions in each of the three coordinates, and using the orthogonal collocation procedure, the equations are transformed into an eigenvalue problem. The matrices in the aforementioned eigenvalue equations are generally of large order. In order to solve these matrix equations with modest and optimal computer memory and storage, we employ the iterative Restarted Arnoldi Algorithm in conjunction with the so-called tensor trick method. Furthermore, we incorporate a polynomial accelerator in the algorithm to obtain rapid convergence. We applied the method to obtain the binding energies of Triton, Carbon-12, and Ozone molecule. / Physics / M.Sc (Physics)

Na análise de estruturas é comum a consideração da concentricidade entre vigas e placas. Estudos numéricos em pavimentos de laje nervurada, onde os espaçamentos das vigas são pequenos, mostram que a consideração da excentricidade entre placa e a viga resultam em redução dos deslocamentos na estrutura, entretanto não existem estudos sobre a influência da excentricidade em pavimentos com poucas vigas. Este trabalho apresenta a análise numérica de pavimentos com dimensões usuais considerando ou não a excentricidade com o objetivo de verificar a influência da excentricidade no comportamento da estrutura. As análises mostraram diferenças de deslocamentos e tensões nas estruturas estudadas. O efeito de excentricidade pode ser incluído através da modificação da matriz de rigidez do elemento finito de viga. Todos os pavimentos foram analisados com o software ANSYS com elementos de placa e viga que representassem os modelos do trabalho. / It\'s usual practice on structural analysis of buildings slabs to assume the concentricity between plates and beams. Numerical studies in waffle-slabs, where the distance between beams is small, showed that the eccentricity of beam to plate results in reduction of stresses and displacements of the system, however there is not similar study about the influence of eccentricity in slabs with few beams. This work presents numerical analysis results of at slabs with usual dimensions considering or not plate-beam eccentricity, aiming at evaluating its influence in structural behavior. The analysis showed differences of displacements and stress in studied structures. The effects of eccentricity can be included by modifying the stiffness matrix of the beam finite element. All slabs were analyzed with ANSYS program with suitable beam and plate elements.

Diese Arbeit beschäftigt sich mit der Carrier-Envelope Phasendrift modengekoppelter Laser sowie Maßnahmen zu deren aktiver und passiver Stabilisierung. Das restliche Phasenrauschen wird in verschiedenen Messaufbauten im Hinblick auf physikalische Ursachen und mögliche Optimierungen untersucht. Dazu werden einführend verschiedene Interferometeranordnungen zur Messung der CEP Drift von Ti:Saphir Oszillatoren systematisch auf ihr Eigenrauschen hin untersucht. In einem Vergleichsexperiment wird die Überlegenheit kompakter Interferometeraufbauten demonstriert. Zusätzlich wird eine einfache Anordnung vorgestellt, welche die Bestimmung der Phase eines spektralen Interferenzmusters. Das analoge Verfahren wird zur Einzelschussanalyse der CEP Drift von Verstärkerimpulsen verwendet. Hiermit werden erstmals schnelle Rauschbeiträge aufgedeckt. Ergänzend wird ein Konzept zur orthogonalen Kontrolle der CEP Drift mithilfe einer speziellen Kompensatoreinheit diskutiert. Der Kompensator besteht aus zwei Keilprismen, hergestellt aus unterschiedlichen optischen Materialien, die als Einheit verschoben werden. Durch geeignete Wahl der Materialien werden Effekte auf die Gruppenlaufzeit oder deren Dispersion gleichzeitig eliminiert. Darüber hinaus wird ebenfalls erstmalig ein lineares Messverfahren demonstriert, das die CEP Drift auch für ps-Oszillatoren erschließt. Zur linearen Detektion wird die spektrale Interferenz aufeinander folgender Impulse eines Impulszuges aufgelöst, die mit einem Ringresonator überlagert werden. Abschließend wird ein neues Verfahren präsentiert, das einen Impulszug generiert, dessen Einzelimpulse eine identische Feldstruktur aufweisen. Dazu das Messsignal direkt an ein externes Rückstellelement übergeben ohne eine zusätzliche Regelschleife zu benötigen. Auf diese Weise können Bandbreitenprobleme der Regelschleife vermieden werden. Da Rückstellelement und Laser voneinander getrennt sind, bleibt der Laser in seinem Betrieb ungestört. / This work discusses the carrier-envelope phase drift of mode-locked lasers as well as techniques for its active and passive stabilization. In order to reveal the physical origin of the drift, the investigations focus on the analysis of residual phase noise. From the analysis, potential improvements are developed. For this purpose, in a first experimental approach, different interferometer configurations for CEP drift detection of Ti:sapphire oscillators are compared. Comparative studies clearly reveal the superiority of compact interferometer set-ups in terms of noise. In a second series of experiments, a simple assembly is introduced for the direct extraction of the phase from spectral interference patterns. The analog method is demonstrated with single-shot measurements and utilized for stabilization of the CEP drift of an amplifier system, thereby enabling monitoring of additional fast noise contributions for the first time. Moreover, a concept for orthogonal control of the CEP drift by a specially designed compensator assembly is discussed. This assembly consists of two thin wedge prisms made from different optical materials. By choosing an appropriate material group delay and its dispersion are eliminated synchronously. Furthermore, a linear method is presented that provides access to the CEP drift of ps-oscillators for the first time. The newly introduced much more general linear approach relies on resolving the spectral interference of subsequent pulses from a pulse train, which are superimposed utilizing a ring resonator. Finally, a technique is demonstrated that generates a pulse train consisting of single pulses with identical field structure. Particularly, the heterodyne signal is directly fed forward to the external feedback element, replacing the classical servo loop. Therefore, servo bandwidth limitations are eliminated. Since feedback element and laser oscillator are decoupled, the laser performance is not corrupted by side effects from the feedback.

No presente trabalho, estudamos alguns aspectos da teoria de estruturas complexas em espaços de Banach. Demonstramos que se um espaço de Banach real $X$ tem a propriedade $P$, então todas as estruturas complexas em $X$ também satisfazem $P$, quando $P$ é qualquer uma das seguintes propriedades: propriedade de aproximação limitada, \\emph{G.L-l.u.st}, ser injetivo e ser complementado num espaço dual. Abordamos o problema da unicidade de estruturas complexas em espaços de Banach com base subsimétrica, provando que um espaço de Banach real $E$ com base subsimétrica e isomorfo ao espaço de sequências $E[E]$ admite estrutura complexa única. Por outro lado, apresentamos um exemplo de espaço de Banach com exatamente $\\omega$ estruturas complexas distintas. Também usamos a teoria de estruturas complexas para estudar o clássico problema dos hiperplanos no espaço $Z_2$ de Kalton-Peck. Com o propósito de distinguir $Z_2$ de seus hiperplanos nos perguntamos se os hiperplanos admitem estrutura complexa. Nesse sentido, provamos que os hiperplanos de $Z_2$ contendo a cópia canônica de $\\ell_2$ não admitem estruturas complexas que sejam extensões de estruturas complexas em $\\ell_2$. Também construímos uma estrutura complexa em $\\ell_2$ que não pode-se estender a nenhum operador em $Z_2$. / In this work, we study some aspects of the theory of complex structures in Banach spaces. We show that if a real Banach space $X$ has the property $P$, then all its complex structures also satisfy $P$, where $P$ is any of the following properties: bounded approximation property, \\emph{G.L-l.u.st}, being injective and being complemented in a dual space. We address the problem of uniqueness of complex structures in Banach spaces with subsymmetric basis by proving that a real Banach space $E$ with subsymmetric basis and isomorphic to the space of sequences $E [E]$ admits a unique complex structure. On the other hand, we show an example of Banach space with exactly $\\omega$ different complex structures. We also use the theory of complex structures to study the classical problem of hyperplanes in the Kalton-Peck space $Z_2$. In order to distinguish between $Z_2$ and its hyperplanes we wonder whether the hyperplanes admit complex structures. In this sense we prove that no complex structure on $\\ell_2$ can be extended to a complex structure on the hyperplanes of $Z_2$ containing the canonical copy $l_2$. We also constructed a complex structure on $l_2$ that can not be extended to any operator in $Z_2$.

No presente trabalho, estudamos alguns aspectos da teoria de estruturas complexas em espaços de Banach. Demonstramos que se um espaço de Banach real $X$ tem a propriedade $P$, então todas as estruturas complexas em $X$ também satisfazem $P$, quando $P$ é qualquer uma das seguintes propriedades: propriedade de aproximação limitada, \\emph{G.L-l.u.st}, ser injetivo e ser complementado num espaço dual. Abordamos o problema da unicidade de estruturas complexas em espaços de Banach com base subsimétrica, provando que um espaço de Banach real $E$ com base subsimétrica e isomorfo ao espaço de sequências $E[E]$ admite estrutura complexa única. Por outro lado, apresentamos um exemplo de espaço de Banach com exatamente $\\omega$ estruturas complexas distintas. Também usamos a teoria de estruturas complexas para estudar o clássico problema dos hiperplanos no espaço $Z_2$ de Kalton-Peck. Com o propósito de distinguir $Z_2$ de seus hiperplanos nos perguntamos se os hiperplanos admitem estrutura complexa. Nesse sentido, provamos que os hiperplanos de $Z_2$ contendo a cópia canônica de $\\ell_2$ não admitem estruturas complexas que sejam extensões de estruturas complexas em $\\ell_2$. Também construímos uma estrutura complexa em $\\ell_2$ que não pode-se estender a nenhum operador em $Z_2$. / In this work, we study some aspects of the theory of complex structures in Banach spaces. We show that if a real Banach space $X$ has the property $P$, then all its complex structures also satisfy $P$, where $P$ is any of the following properties: bounded approximation property, \\emph{G.L-l.u.st}, being injective and being complemented in a dual space. We address the problem of uniqueness of complex structures in Banach spaces with subsymmetric basis by proving that a real Banach space $E$ with subsymmetric basis and isomorphic to the space of sequences $E [E]$ admits a unique complex structure. On the other hand, we show an example of Banach space with exactly $\\omega$ different complex structures. We also use the theory of complex structures to study the classical problem of hyperplanes in the Kalton-Peck space $Z_2$. In order to distinguish between $Z_2$ and its hyperplanes we wonder whether the hyperplanes admit complex structures. In this sense we prove that no complex structure on $\\ell_2$ can be extended to a complex structure on the hyperplanes of $Z_2$ containing the canonical copy $l_2$. We also constructed a complex structure on $l_2$ that can not be extended to any operator in $Z_2$.

Στην παρούσα διπλωματική εργασία μελετάμε την κλασική δυναμική ατόμου υδρογόνου σε γενικευμένο δυναμικό Van der Waals. Το πρόβλημα ανήκει στην ευρύτερη κατηγορία των μη γραμμικών Χαμιλτώνιων δυναμικών συστημάτων. Σκοπός της μελέτης είναι η ανάλυση των ιδιομορφιών της κανονικής και χαοτικής κίνησης του συστήματος στο μιγαδικό πεδίο του χρόνου και η εξαγωγή συμπερασμάτων σχετικά με την ολοκληρωσιμότητα και επιλυσιμότητά του. Εκείνο που θέλουμε να κατανοήσουμε, επίσης, είναι τον ρόλο που παίζει η εμφάνιση ιδιομορφιών σε κάποια σημεία του χώρου των φάσεων και κατά πόσο μπορούν αυτές να επηρεάσουν συνολικά τις ιδιότητες των λύσεων. Για πρώτη φορά, επίσης, στην διπλωματική αυτή εργασία εφαρμόζεται σε ένα Χαμιλτώνιο δυναμικό σύστημα μία νέα αριθμητική μέθοδος διάκρισης μεταξύ κανονικής και χαοτικής συμπεριφοράς σε διαφορετικές περιοχές του χώρου φάσεων, η μέθοδος των Μικρότερων Δεικτών Ευθυγράμμισης (Smaller Alignment Indices method ή SALI). Η μέθοδος αυτή έχει χρησιμοποιηθεί κατά το πρόσφατο παρελθόν σε απεικονίσεις δύο, τεσσάρων και έξι διαστάσεων με πολύ ενδιαφέροντα αποτελέσματα. Χαρακτηριστικά της είναι η αποτελεσματικότητα και η δυνατότητα εξαγωγής χρήσιμων συμπερασμάτων ως προς την κανονική και χαοτική φύση των τροχιών ενός δυναμικού συστήματος με μεγαλύτερη ταχύτητα και αξιοπιστία από την μέθοδο των χαρακτηριστικών εκθετών Lyapunov καθώς και άλλων νεότερων μεθόδων στην σύγχρονη βιβλιογραφία. Εδώ θα παρουσιασθεί η μέθοδος αυτή με ορισμένες βελτιώσεις ώστε να μπορεί να εφαρμοσθεί σε συστήματα μη γραμμικών διαφορικών εξισώσεων οποιασδήποτε διάστασης ελέγχοντας συστηματικά ένα όσο πυκνό πλέγμα αρχικών συνθηκών του χώρου φάσεων επιθυμούμε, αντιστοιχώντας σε κάθε μία από αυτές ένα χρώμα. Κάθε χρώμα αντιστοιχεί και σε ένα διαφορετικό εύρος τάξεων του SALI δημιουργώντας έτσι μία συνολική εικόνα στο χώρο φάσεων που μας επιτρέπει να γνωρίζουμε τη φύση της τροχιάς κάθε συγκεκριμένης αρχικής συνθήκης. Σχηματίζεται με αυτόν τον τρόπο το "μωσαϊκό" του χώρου φάσεων και αποκαλύπτονται περιοχές κανονικής κίνησης, χαοτικής κίνησης καθώς και νησίδες ή περιοχές στις οποίες δεν αντιστοιχεί καθόλου κίνηση. / In this master thesis we study the classical dynamics of hydrogen atoms in a generalized Van der Waals potential. The problem belongs to the class of non linear Hamiltonian systems. Our aim is the singularity analysis of the ordered and chaotic motion of the system in the complex plain of time and the extraction of valuable conclusions concerning its integrability and solvability. What we want to understand, also, is the role of the emergence of singularities in some points of the phase space of the aforementioned system and how the singularities can affect globally the properties of the solutions. For the first time, in this master thesis, we introduce and apply in a Hamiltonian system a new numerical method for the fast and efficient discrimination between ordered and chaotic motion in different parts of phase space, namely the method of the Smaller Alignment Index (SALI). The method has been introduced and applied recently in mappings of two, four and six dimensions with very satisfactory results. Its main characteristics are the effectiveness and the ability of extracting valuable conclusions about the ordered and chaotic nature of trajectories of a dynamical system faster than the traditional method of Lyapunov exponents as well as of other indices in the bibliography. We will introduce SALI with appropriate modifications that help using it in non linear systems of differential equations of arbitrary dimensions checking systematically a dense grid of initial conditions and corresponding in every orbit a different color. Every color corresponds to a different range of SALI values creating by that way a global picture of the phase space that allows us to know the dynamic nature of initial conditions. By that way, we construct a “mosaic” of the phase space and reveal parts of ordered motion as well as parts of chaotic motion and islands of stability.