Judėjimas yra pagrindinis žmogaus veiklos komponentas. Buvo atlikta daug mėginimų atskleisti jo principus pasitelkiant fiziką bei dinamiką. Kartu kompiuterinė grafika ir robotų technika plėtoja šias pastangas, tačiau daug problemų lieka neišspręstų, netgi ir aprašant paprasčiausią atvejį: linijinį, tiesiaeigį, ritmišką ėjimą. Taigi netiesinių sistemų tyrimo tikslas yra surasti tvarką chaose; surasti įrodymų, kad nereguliari elgsena yra valdoma nedidelės deterministinių lygčių sistemos, pritaikant ją eksperimentiniams signalams laike. Tokio tyrimo sėkmės galima tikėtis, nustačius, kad šios sistemos būsenos kintamieji yra tvirtai suporuoti tarpusavyje. Chaotiškų sistemų tyrimų tikslas yra nustatyti dvi pagrindines jų savybes: dimensiją ir entropijos spektrą. Paprastai kalbant, dimensija yra dydis, parodantis diferencialinių lygčių skaičių, reikalingą aprašyti sistemai, o entropija yra dydis, parodantis informacijos apie sistemos būseną praradimą laiko bėgyje. Teigiama baigtinė entropija yra chaoso egzistavimo įrodymas. Šio darbo tikslas yra sukurti chaotiško signalo analizės sistemą, kuri leistų ištirti elementarius monotoniškus žmogaus rankos judesius dvimatėje plokštumoje. / Since we encounter many phenomena with irregular motion, e.g. the weather, turbulence, carbon resistor noise, chemical reactions and biological signals (human motion), we are tempted to investigate whether we could model the dynamics with nonlinear differential equations. Our aim is to find order within the chaos; to find evidence that the irregular behavior is governed by a small set of deterministic equations, using experimental time series. We might be successful in particular when the state variables of the system are strongly coupled. In this report, we will restrict ourselves to the determination of several properties that describe a chaotic system, including the dimension and entropy spectra. Loosely speaking, the dimension is a measure for the number of differential equations needed to describe the system, while the entropy is a measure for the loss of information about the state of the system in the course of time. Positive but finite entropy is a hall-mark of chaos. In this paper, we will describe few experiments that were performed on a portion of human motion data, and compare the results to theoretical model of system for signal analysis.

Σκοπός της παρούσας διπλωματικής εργασίας ήταν η μελέτη ηλεκτροεγκεφαλογραφικών σημάτων επιληψίας με χρήση συνδεδεμένων χαοτικών συστημάτων. Στο πλαίσιο της εργασίας παρουσιάζονται αρχικά μέθοδοι συμφωνίας και οι διάφορες μορφές της συνάρτησης μεταφοράς πληροφορίας. Οι τεχνικές που έχουν χρησιμοποιηθεί κυρίως μέχρι σήμερα είναι γραμμικές ωστόσο έχει φανεί σε αρκετές περιπτώσεις ότι η γραμμικότητα δεν επαρκή για την εξήγηση των σημάτων ΗΕΓ και της σύζευξης μεταξύ τους. Για τον λόγο αυτόν αναπτύσσεται η μη γραμμική μέθοδος υπολογισμού της συνάρτησης μερικής κατευθυνόμενης συμφωνίας η οποία μελετά τα καταγεγραμμένα σήματα στον χώρο φάσης-κατάστασης. Βάση αυτής διερευνάται η μη γραμμική συνδεσιμότητα περιοχών του εγκεφάλου. Η εργασία αποτελείται από πέντε μέρη. Το πρώτο μέρος περιλαμβάνει τη παρουσίαση της φυσιολογίας του ανθρώπινου εγκεφάλου. Στο δεύτεροο μέρος παρουσιάζεται η ασθένεια της επιληψίας καθώς και οι διάφοροι τύποι της. Στο τρίτο μέρος παρουσιάζεται η μέθοδος της μη γραμμικής ανάλυσης χρονοσειρών μέσω της μερικής κατευθυνόμενης συμφωνίας που περιλαμβάνει την ανακατασκευή των σημάτων στον φασικό χώρο. Ακολουθεί ο υπολογισμός των κατάλληλων παραμέτρων για την σωστή ανακατασκευή του φασικού χώρου των σημάτων. Τέλος, στο πέμπτο κεφάλαιο παρουσιάζονται αποτελέσματα από εφαρμογή της μεθόδου σε προσομοιωμένα δεδομένα,καθώς και σε πραγματικά δεδομένα από ασθενή με επιληψία, τα οποία έχουν ληφθεί από το νοσοκομείο Ευαγγελισμός. / This diploma thesis aim at studying Electroencephalografic (EEG) Signal Recordings by adopting methodologies able to analyse and observe coupling of chaotic systems. Although linear methods seems to be inadequate for the analysis of EEG signals, the most commonly used methodologies today are linear. In this thesis, a non-linear partial directed coherence method is adopted to compute the transfer function of EEG signals in the phase-state space and is used to estimate the non-linear connectivity among brain areas. This thesis consists of five chapters. In the first and second chapter, an introduction to the brain's physiology and epilepsy pathophysiology is presented. In the third chapter, a methodology for the non-linear analysis of time series is presented based on the PDC method, which reconstructs attractors in the phase-state. In the fourth chapter, the parameters for the phase-state reconstruction of the attractors are properly selected. In the fifth chapter, the proposed method is applied on simulated and real epilepsy EEG data and the obtained results are presented and discussed.

Le travail développé dans ce manuscrit porte sur la synchronisation des systèmes chaotiques. Il est articulé autour de deux axes principaux: la synthèse d'observateur et la détection de mode. Dans un premier temps, quelques rappels sur le chaos et les principales architectures de systèmes de chiffrement chaotiques sont effectués. Ensuite, nous montrons comment les systèmes chaotiques à non linéarité polynomiale ou affines à commutation peuvent se réécrire sous forme LPV polytopique. Une revue des principaux résultats sur la synthèse d'observateurs LPV polytopiques reposant sur l'utilisation des LMI est faite. Une extension des résultats aux observateurs polytopiques à entrées inconnues, à la fois dans le cas déterministe, bruité ou incertain est proposée. Ces observateurs assurent la synchronisation du chaos et donc le déchiffrement dans les systèmes de chiffrement "modulation paramétrique", "commutation chaotique", "transmission à deux canaux" et "chiffrement par inclusion". Pour les systèmes affines à commutation utilisés en tant que générateur du chaos, le cas où l'état discret n'est pas accessible est considéré. Une présentation unifiée des méthodes fondées sur les espaces de parité, proposées dans la littérature pour les systèmes linéaires et affines à commutation à temps discret, est réalisée. Le problème de discernabilité fait l'objet d'une étude approfondie. Une approche pour estimer les retards variables des systèmes affines et affines à commutation à temps discret, formulée en termes de détection de mode, est proposée en tant que solution à l'estimation de retard pour le chiffrement par injection de retard / The work developed in this manuscript addresses the synchronization of chaotic systems. It is organized around two main axes: the observer synthesis and the mode detection. In a first step, we recall the main architectures of chaotic encryption systems and show how chaotic systems with polynomial nonlinearities or switched affine dynamics can be rewritten in a polytopic LPV form. A review of the main LMI based results for polytopic LPV observers synthesis is made. An extension to polytopic unknown input observers, both in the deterministic case and noisy or uncertain case, is proposed. These observers ensure chaos synchronization and information recovering in the framework of the following encryption systems: "parametric modulation", "chaotic switching", "two channels transmission" and "inclusion encryption". For affine switched systems used as a generator of chaos, the case where the discrete state is not available is considered. A unified presentation of mode detection methods based on parity spaces proposed in the literature for linear and affine switched discrete time systems is proposed. The problem of discernibility is the subject of a complete study. An approach to estimate time varying delays for affine switched discrete time systems, formulated in terms of mode detection, is proposed as a solution for delay injection encryption

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This thesis deals with the study of two problems in the area of nonlinear dynamics using the method of multiple scales. Accordingly, it consists of two parts. In the first part of the thesis, we explore the asymptotic stability of a planar two-degree- of-freedom robot with two rotary (R) joints following a desired trajectory under feedback control. Although such robots have been extensively studied and there exists stability and other results for position control, there are no analytical results for asymptotic stability when the end of the robot or its joints are made to follow a time dependent trajectory. The nonlinear dynamics of a 2R planar robot, under a proportional plus derivative (PD) and a model based computed torque control, is studied. The method of multiple scales is applied to the two nonlinear second-order ordinary deferential equations which describes the dynamics of the feedback controlled 2R robot. Amplitude modulation equations, as a set of four first order equations, are derived. At a fixed point, the Routh-Hurwitz criterion is used to obtain positive values of proportional and derivative gains at which the controller is asymptotically stable or indeterminate. For the model based control, a parameter representing model mismatch is incorporated and again controller gains are obtained, for a chosen mismatch parameter value, where the controller results in asymptotic stability or is indeterminate. From numerical simulations with gain values in the indeterminate region, it is shown that for some values and ranges of the gains, the non- linear dynamical equations are chaotic and hence the 2R robot cannot follow the desired trajectory and be asymptotically stable. The second part of the thesis deals with the study of the nonlinear dynamics of a rotating flexible link, modeled as a one dimensional beam, undergoing large deformation and with geometric nonlinearities. The partial deferential equation of motion is discretized using a finite element approach to yield four nonlinear, non-autonomous and coupled ordinary deferential equations. The equations are non-dimensional zed using two characteristic velocities – the speed of sound in the material and a speed associated with the trans- verse bending vibration of the beam. The method of multiple scales is used to perform a detailed study of the system. A set of four autonomous equations of the first-order are derived considering primary resonance of the external excitation with one of the natural frequencies of the model and one-to-one internal resonance between two different natural frequencies of the model. Numerical simulations show that for certain ranges of values of these characteristic velocities, the slow flow equations can exhibit chaotic motions. The numerical simulations and the results are related to a rotating wind turbine blade and the approach can be used for the study of the nonlinear dynamics of a single link flexible manipulator. The second part of the thesis also deals with the synchronization of chaos in the equations of motion of the flexible beam. A nonlinear control scheme via active nonlinear control and Lyapunov stability theory is proposed to synchronize the chaotic system. The proposed controller ensures that the error between the controlled and the original system asymptotically go to zero. A numerical example using parameters of a rotating power generating wind turbine blade is used to illustrate the theoretical approach.

Orientadores: Romis Ribeiro de Faissol Attux, Ricardo Suyama / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-19T01:05:07Z (GMT). No. of bitstreams: 1 Soriano_DiogoCoutinho_D.pdf: 11590205 bytes, checksum: 910b0e2b03527e1ad8b7a82694be170a (MD5) Previous issue date: 2011 / Resumo: A presente tese de doutoramento apresenta contribuições à análise de estabilidade, ao controle e ao processamento de sinais no contexto de sistemas dinâmicos não-lineares. No que se refere à análise de estabilidade, este trabalho apresenta um novo método para calcular o espectro de Lyapunov para soluções de sistemas dinâmicos a partir de cópias ("clones") das equações de estado com pequenas perturbações nas condições iniciais. A proposta se caracteriza por: não exigir linearizações das equações de movimento; possibilitar a estimação parcial do espectro de Lyapunov; viabilizar a estimação do espectro para sistemas dinâmicos não-suaves. Além disso, este procedimento de cálculo é utilizado para construir uma estratégia de controle de sistemas dinâmicos baseada no ajuste de parâmetros livres que levam a um determinado espectro de Lyapunov desejado, estratégia esta que é testada no âmbito do modelo neuronal de Hodgkin-Huxley. Como contribuição no contexto de processamento de sinais, este trabalho se dedica a apresentar uma nova metodologia para separação de sinais caóticos misturados com sinais estocásticos baseada na análise por quantificação de recorrência, assim como uma nova técnica de sombreamento e filtragem de sinais caóticos quando a estrutura das equações de estado está disponível / Abstract: This doctoral thesis presents contributions to the stability analysis, control and signal processing in the context of nonlinear dynamical systems. With regard to the stability analysis, this work presents a new method to calculate the Lyapunov spectrum of solutions for dynamical systems based on copies ("clones") of the state equations with small perturbations in the initial conditions. The proposal has the following key features: it does not require linearization of the motion equations; it allows the partial estimation of the Lyapunov spectrum; it allows the spectrum estimation for non-smooth dynamical systems. Moreover, this calculation procedure is used to construct a strategy of control of dynamical systems based on the selection of parameters that lead to a particular desired Lyapunov spectrum, which is tested for the neuronal model proposed by Hodgkin and Huxley. As a contribution in the context of signal processing, this work presents a new methodology for blind source separation of chaotic signals mixed with stochastic sources based on recurrence quantification analysis, as well as a new technique for shadowing and filtering chaotic signals when the structure of the state equations is available / Doutorado / Automação / Doutor em Engenharia Elétrica

Estudamos nesta dissertação o tempo de recorrência em sistemas dinâmicos, concentrando-nos na estatística do tempo de retorno. Calculamos numericamente a distribuição de tempo de retorno a uma região específica do espaço de fases de sistemas caóticos e comparamos com a distribuição binomial, deduzida para um processo aleatório. Os principais resultados obtidos foram: surgimento do efeito que denominamos memória de curto alcance, típico de sistemas determinísticos e associado à distribuição das órbitas periódicas instáveis; a distribuição de tempo de retorno caracteriza as principais propriedades temporais no caso de sistemas intermitentes. As conexões do tempo de retorno com regimes de transporte anômalo foram apresentadas, ressaltando suas limitações. O tempo de retorno foi utilizado ainda para analisar séries temporais, obtidas tanto de um modelo de mistura de um contaminante escalar passivo, como experimentalmente no plasma confinado magnéticamente. No primeiro caso constatamos que os retornos da série temporal assemelham-se às recorrências no espaço de fases do sistema dinâmico responsável pela mistura do contaminante: o mapa padrão com fase aleatória. Constatamos o surgimento de caudas de lei de potência na distribuição de tempo de retorno e calculamos sua dependência com o aumento da não linearidade e da aleatoriedade do sistema. Destacamos o efeito de múltiplas caudas de lei de potência, ausente no caso das distribuições obtidas no espaço de fases. Às séries obtidas em Tokamaks aplicamos o modelo de cascata log-normal para explicar sua função densidade de probabilidade. A distribuição de tempo de retorno destas séries mostrou estar diretamente relacionada com a correlação de curto e longo alcance presente na série. / We study the recurrence time in dynamical systems. The statistics of the recurrence time to a specific region of the phase space of chaotic dynamical systems were obtained numerically and compared with the binomial-like distribution, deduced for a random process. The main results are: the presence of the so called short time memory effect, typical for deterministic systems and related to the distribution of the unstable periodic orbits; the return time distribution captures the main temporal properties of intermittent systems. The possible connections of the recurrence time statistics to the anomalous transport were presented, with special attention to their limitations. The return time statistics was applied to analyze time series obtained from an Hamiltonian model and from magnetically confined plasma. In the first case we noticed that the recurrences of the series were similar to the recurrences obtained in the phase space of the Hamiltonian dynamical system: the standard map with a random phase. We analyze the dependence of the power-law tails of the distributions with the non-linearity and with the randomness of the system. One effect that appears only in the time series case is the multiple power law tails. We apply the log-normal cascade model to explain the probability density function of the series obtained in Tokamaks. The recurrence time statistics of the series is closely related to the short and long time correlation present on the series.

Les travaux effectués, dans le cadre de cette thèse, concernent l’analyse et la synthèse de systèmes dynamiques continus complexes de grande dimension. Pour la classe des systèmes étudiés, est mise en exergue en particulier l’importance du choix de la description des systèmes sur l’étendue des résultats pouvant être obtenus lorsque la méthode d’étude de la stabilité est fixée.L’utilisation des normes vectorielles comme fonction d’agrégation et du critère pratique de Borne et Gentina pour l’étude de la stabilité, associée à la description des systèmes par des matrices caractéristiques de forme en flèche, a permis l’élaboration de nouvelles conditions suffisantes de stabilisabilité de systèmes dynamiques continus non linéaires, monovariables et multivariables, formulées en théorèmes et corollaires.Ces résultats obtenus, pour une classe de processus, pouvant être caractérisés par des matrices instantanées de forme en flèche mince, ont été généralisés au cas des matrices quelconques, pouvant être mises sous forme en flèche mince généralisée ou en flèche épaisse.Les critères élaborés, soit pour l’analyse de la stabilité soit pour la synthèse d’une loi de commande stabilisante, sont ensuite exploités, avec succès, pour la formulation de nouvelles conditions suffisantes de vérification des propriétés de synchronisation, d’anti-synchronisation et de synchronisation hybride de systèmes chaotiques du type maître-esclave, d’un grand intérêt, en particulier, pour garantir une transmission sécurisée / This Thesis deals with the analysis and the synthesis of dynamic large scale continuous systems depending on the choice of the system description.Stability and stabilisability proposed studies are based on the use of vector norms as an aggregation function and of the practical Borne-Gentina criterion, associated to the description of the system by instantaneous characteristic matrix in arrow form.Practical stability conditions, easy to use, are obtained for both dynamic nonlinear continuous single input single output systems and multiple inputs multiple outputs ones, formulated by means of theorems and corollaries. These obtained results for thin arrow form, are generalized to the case of matrices, which can be putted under thin generalized arrow form or thick arrow form. The proposed stability and stabilisability criteria are afterwards, successfully, exploited to formulate new sufficient conditions, guaranteeing the synchronization, the anti-synchronization and the hybrid synchronization properties, for chaotic master-slave systems, having an increasing interest throughout their application in the secure communication field

xix, 171 p. : ill. (some col.) A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / The field of quantum chaos investigates the quantum mechanical behavior of classically chaotic systems. This dissertation begins by describing an experiment conducted on an apparatus constructed to represent a three dimensional analog of a classically chaotic system. Patterns of reflected light are shown to produce fractals, and the behavior of the fractal dimension D F is shown to depend on the light's ability to escape the apparatus. The classically chaotic system is then used to investigate the conductance properties of semiconductor heterostructures engineered to produce a conducting plane relatively free of impurities and defects. Introducing walls that inhibit conduction to partition off sections considerably smaller than the mean distance between impurities defines devices called 'billiards'. Cooling to low temperatures enables the electrons traveling through the billiard to maintain quantum mechanical phase. Exposure to a changing electric or magnetic field alters the electron's phase, leading to fluctuations in the conductance through the billiard. Magnetoconductance fluctuations in billiards have previously been shown to be fractal. This behavior has been charted using an empirical parameter, Q , that is a measure of the resolution of the energy levels within the billiard. The relationship with Q is shown to extend beyond the ballistic regime into the 'quasi-ballistic' and 'diffusive' regimes, characterized by having defects within the conduction plane. A model analogous to the classically chaotic system is proposed as the origin of the fractal conductance fluctuations. This model is shown to be consistent with experiment and to account for changes of fine scale features in MCF known to occur when a billiard is brought to room temperature between low temperature measurements. An experiment is conducted in which fractal conductance fluctuations (FCF) are produced by exposing a billiard to a changing electric field. Comparison of D F values of FCF produced by electric fields is made to FCF produced by magnetic fields. FCF with high D F values are shown to de-correlate at smaller increments of field than the FCF with lower D F values. This indicates that FCF may be used as a novel sensor of external fields, so the response of FCF to high bias voltages is investigated. / Adviser: Stephen Kevan, Chairperson, Physics; Richard Taylor, Advisor, Physics; Robert Zimmerman, Member, Physics; Stephen Gregory, Member, Physics; David Johnson, Outside Member, Chemistry

Estudamos nesta dissertação o tempo de recorrência em sistemas dinâmicos, concentrando-nos na estatística do tempo de retorno. Calculamos numericamente a distribuição de tempo de retorno a uma região específica do espaço de fases de sistemas caóticos e comparamos com a distribuição binomial, deduzida para um processo aleatório. Os principais resultados obtidos foram: surgimento do efeito que denominamos memória de curto alcance, típico de sistemas determinísticos e associado à distribuição das órbitas periódicas instáveis; a distribuição de tempo de retorno caracteriza as principais propriedades temporais no caso de sistemas intermitentes. As conexões do tempo de retorno com regimes de transporte anômalo foram apresentadas, ressaltando suas limitações. O tempo de retorno foi utilizado ainda para analisar séries temporais, obtidas tanto de um modelo de mistura de um contaminante escalar passivo, como experimentalmente no plasma confinado magnéticamente. No primeiro caso constatamos que os retornos da série temporal assemelham-se às recorrências no espaço de fases do sistema dinâmico responsável pela mistura do contaminante: o mapa padrão com fase aleatória. Constatamos o surgimento de caudas de lei de potência na distribuição de tempo de retorno e calculamos sua dependência com o aumento da não linearidade e da aleatoriedade do sistema. Destacamos o efeito de múltiplas caudas de lei de potência, ausente no caso das distribuições obtidas no espaço de fases. Às séries obtidas em Tokamaks aplicamos o modelo de cascata log-normal para explicar sua função densidade de probabilidade. A distribuição de tempo de retorno destas séries mostrou estar diretamente relacionada com a correlação de curto e longo alcance presente na série. / We study the recurrence time in dynamical systems. The statistics of the recurrence time to a specific region of the phase space of chaotic dynamical systems were obtained numerically and compared with the binomial-like distribution, deduced for a random process. The main results are: the presence of the so called short time memory effect, typical for deterministic systems and related to the distribution of the unstable periodic orbits; the return time distribution captures the main temporal properties of intermittent systems. The possible connections of the recurrence time statistics to the anomalous transport were presented, with special attention to their limitations. The return time statistics was applied to analyze time series obtained from an Hamiltonian model and from magnetically confined plasma. In the first case we noticed that the recurrences of the series were similar to the recurrences obtained in the phase space of the Hamiltonian dynamical system: the standard map with a random phase. We analyze the dependence of the power-law tails of the distributions with the non-linearity and with the randomness of the system. One effect that appears only in the time series case is the multiple power law tails. We apply the log-normal cascade model to explain the probability density function of the series obtained in Tokamaks. The recurrence time statistics of the series is closely related to the short and long time correlation present on the series.