Stickiness is a ubiquitous property of dynamical systems. However, recognizing whether an orbit is temporarily `stuck' (and therefore very nearly quasiperiodic) is hard to detect. Outlined in this thesis is an approach to quantifying stickiness in area-preserving maps based on a tool called recurrence plots that is not very commonly used. With the analyses presented herein it is shown that recurrence plot methods can give very close estimates to stickiness exponents that were previously calculated using Poincare recurrence and other methods. To capture the dynamics, RP methods require shorter data series than more conventional methods and are able to represent a more-global analysis of recurrence. A description of stickiness of the standard map for a wide array of parameter strengths is presented and a start at analyzing the standard nontwist map is presented. / text

Made available in DSpace on 2017-07-21T19:25:57Z (GMT). No. of bitstreams: 1 ADRIANOPEREIRA.pdf: 5814518 bytes, checksum: fd0af0d5e0a77b57e48974063eac7e4f (MD5) Previous issue date: 2008-07-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Recurrence analysis has been extensively used in approaching problems that deal with transitions between regular and chaotic behaviors, identi¯cation of structure of dynamic systems, such as frequencies and correlations hard to detect by linear methods, for example. Among the main tools of this analysis are the Recurrence Plots (RP) and the Quantitative Recurrence Analysis (RQA), which are constantly used in the analysis of time series supposedly proceeding from non-linear and even non-stationary dynamical systems. These tools have been applied in a wide range of phenomena, since the study of cardiac arrhythmia until the greater phenomena of nature, such as sunspots. Recently, many economic and ¯financial séries are being investigated under this perspective, as exchange rates, the financial crashes" and the behavior of some stock index. In this work we employ the RP and RQA for the study of a long time series of the returns of the Bovespa Index (Ibovespa), where we carefully studied the obtention of the parameters for the phase space reconstruction of the supposed system which created the time series, we analyze the patterns formed in RP as well as the values of the quantities of RQA, comparing the results obtained with the original and randomized series. We search, from these results,to establish whether there is some sort of deterministic component in the studied system, and what its intensity. Our investigations suggest that the real financial market dynamics is a combination of deterministic chaos and stochastic behavior. / A análise da recorrência vem sendo muito usada na abordagem de problemas que tratam das transicões entre comportamentos regulares e caoticos, na identicação da estrutura de sistemas dinamicos, como frequências e correlacõess dificeis de detectar por metodos lineares, por exemplo. Dentre as principais ferramentas desta analise destacam-se os Gráficos de Recorrência (GR) e a Análise Quantitativa de Recorrência (AQR), que são constantemente empregadas na análise de séries temporais supostamente provenientes de sistemas dinâmicos não-lineares e até não-estacionários. Tais ferramentas vêm sendo aplicadas em uma grande gama de fenômenos, desde o estudo da arritmia cardíaca até os maiores fenômenos da natureza, como as manchas solares. Recentemente, muitas séries econômicas e financeiraso estão sendo investigadas sob esta ótica, como as taxas de cãmbio, os grandes crashes" financeiros e o comportamento de alguns índices de acões. Neste trabalho nós empregamos os GR e a AQR para o estudo de uma longa série temporal dos retornos do índice Bovespa (Ibovespa), onde estudamos cuidadosamente a obtencão dos parâmetros para a reconstrucão do suposto espaço de fase do sistema que gerou a série temporal, analisamos os padrões formados nos GR bem como os valores das quantidades da AQR, comparando os resultados obtidos com as séries originais e embaralhadas. Procuramos, a partir de tais resultados, estabelecer se existe algum tipo de componente determinística no sistema analisado, e qual sua intensidade. Nossas investigações sugeriram que a dinâmica real do mercado financeiro e uma combinacão de caos determinístico e comportamento estocástico.

In this paper we present an approach to recover the dynamics from recurrences of a system and then generate (multivariate) twin surrogate (TS) trajectories. In contrast to other approaches, such as the linear-like surrogates, this technique produces surrogates which correspond to an independent copy of the underlying system, i. e. they induce a trajectory of the underlying system visiting the attractor in a different way. We show that these surrogates are well suited to test for complex synchronization, which makes it possible to systematically assess the reliability of synchronization analyses. We then apply the TS to study binocular fixational movements and find strong indications that the fixational movements of the left and right eye are phase synchronized. This result indicates that there might be one centre only in the brain that produces the fixational movements in both eyes or a close link between two centres.

In der vorliegenden Arbeit wird die Wiederkehr im Phasenraum ausgenutzt. Dabei werden drei Hauptresultate besprochen. 1. Die Wiederkehr erlaubt die Vorhersagbarkeit des Systems zu quantifizieren. 2. Die Wiederkehr enthaelt (unter bestimmten Voraussetzungen) sämtliche relevante Information über die Dynamik im Phasenraum 3. Die Wiederkehr erlaubt die Erzeugung dynamischer Ersatzdaten. / Recurrence plots, a rather promising tool of data analysis, have been introduced by Eckman et al. in 1987. They visualise recurrences in phase space and give an overview about the system's dynamics. Two features have made the method rather popular. Firstly they are rather simple to compute and secondly they are putatively easy to interpret. However, the straightforward interpretation of recurrence plots for some systems yields rather surprising results. For example indications of low dimensional chaos have been reported for stock marked data, based on recurrence plots. In this work we exploit recurrences or ``naturally occurring analogues'' as they were termed by E. Lorenz, to obtain three key results. One of which is that the most striking structures which are found in recurrence plots are hinged to the correlation entropy and the correlation dimension of the underlying system. Even though an eventual embedding changes the structures in recurrence plots considerably these dynamical invariants can be estimated independently of the special parameters used for the computation. The second key result is that the attractor can be reconstructed from the recurrence plot. This means that it contains all topological information of the system under question in the limit of long time series. The graphical representation of the recurrences can also help to develop new algorithms and exploit specific structures. This feature has helped to obtain the third key result of this study. Based on recurrences to points which have the same ``recurrence structure'', it is possible to generate surrogates of the system which capture all relevant dynamical characteristics, such as entropies, dimensions and characteristic frequencies of the system. These so generated surrogates are shadowed by a trajectory of the system which starts at different initial conditions than the time series in question. They can be used then to test for complex synchronisation.

Die tägliche Erfahrung zeigt uns, daß bei vielen physikalischen Systemen kleine Änderungen in den Anfangsbedingungen auch zu kleinen Änderungen im Verhalten des Systems führen. Wenn man z.B. das Steuerrad beim Auto fahren nur ein wenig zur Seite dreht, unterscheidet sich die Richtung des Wagens auch nur wenig von der ursprünglichen Richtung. Aber es gibt auch Situationen, für die das Gegenteil dieser Regel zutrifft. Die Folge von Kopf und Zahl, die wir erhalten, wenn wir eine Münze werfen, zeigt ein irreguläres oder chaotisches Zeitverhalten, da winzig kleine Änderungen in den Anfangsbedingungen, die z.B. durch leichte Drehung der Hand hervorgebracht werden, zu vollkommen verschiedenen Resultaten führen. In den letzten Jahren hat man sehr viele nichtlineare Systeme mit schnellen Rechnern untersucht und festgestellt, daß eine sensitive Abhängigkeit von den Anfangsbedingungen, die zu einem chaotischen Verhalten führt, keinesfalls die Ausnahme darstellt, sondern eine typische Eigenschaft vieler Systeme ist. Obwohl chaotische Systeme kleinen Änderungen in den Anfangsbedingungen gegenüber sehr empfindlich reagieren, können sie synchronisieren wenn sie durch eine gemeinsame äußere Kraft getrieben werden, oder wenn sie miteinander gekoppelt sind. Das heißt, sie vergessen ihre Anfangsbedingungen und passen ihre Rhythmen aneinander. Diese Eigenschaft chaotischer Systeme hat viele Anwendungen, wie z.B. das Design von Kommunikationsgeräte und die verschlüsselte Übertragung von Mitteilungen. Abgesehen davon, findet man Synchronisation in natürlichen Systemen, wie z.B. das Herz-Atmungssystem, raumverteilte ökologische Systeme, die Magnetoenzephalographische Aktivität von Parkinson Patienten, etc. In solchen komplexen Systemen ist es nicht trivial Synchronisation zu detektieren und zu quantifizieren. Daher ist es notwendig, besondere mathematische Methoden zu entwickeln, die diese Aufgabe erledigen. Das ist das Ziel dieser Arbeit. Basierend auf dergrundlegenden Idee von Rekurrenzen (Wiederkehr) von Trajektorien dynamischer Systeme, sind verschiedene Maße entwickelt worden, die Synchronisation in chaotischen und komplexen Systemen detektieren. Das Wiederkehr von Trajektorien erlaubt uns Vorhersagen über den zukünftigen Zustand eines Systems zu treffen. Wenn man diese Eigenschaft der Wiederkehr von zwei interagierenden Systemen vergleicht, kann man Schlüsse über ihre dynamische Anpassung oder Synchronisation ziehen. Ein wichtiger Vorteil der Rekurrenzmaße für Synchronisation ist die Robustheit gegen Rauschen und Instationariät. Das erlaubt eine Synchronisationsanalyse in Systemen durchzuführen, die bisher nicht darauf untersucht werden konnten. / This work deals with the connection between two basic phenomena in Nonlinear Dynamics: synchronization of chaotic systems and recurrences in phase space. Synchronization takes place when two or more systems adapt (synchronize) some characteristic of their respective motions, due to an interaction between the systems or to a common external forcing. The appearence of synchronized dynamics in chaotic systems is rather universal but not trivial. In some sense, the possibility that two chaotic systems synchronize is counterintuitive: chaotic systems are characterized by the sensitivity ti different initial conditions. Hence, two identical chaotic systems starting at two slightly different initial conditions evolve in a different manner, and after a certain time, they become uncorrelated. Therefore, at a first glance, it does not seem to be plausible that two chaotic systems are able to synchronize. But as we will see later, synchronization of chaotic systems has been demonstrated. On one hand it is important to investigate the conditions under which synchronization of chaotic systems occurs, and on the other hand, to develop tests for the detection of synchronization. In this work, I have concentrated on the second task for the cases of phase synchronization (PS) and generalized synchronization (GS). Several measures have been proposed so far for the detection of PS and GS. However, difficulties arise with the detection of synchronization in systems subjected to rather large amounts of noise and/or instationarities, which are common when analyzing experimental data. The new measures proposed in the course of this thesis are rather robust with respect to these effects. They hence allow to be applied to data, which have evaded synchronization analysis so far. The proposed tests for synchronization in this work are based on the fundamental property of recurrences in phase space.

In this work, some new results to exploit the recurrence properties of quasiperiodic dynamical systems are presented by means of a two dimensional visualization technique, Recurrence Plots(RPs). Quasiperiodicity is the simplest form of dynamics exhibiting nontrivial recurrences, which are common in many nonlinear systems. The concept of recurrence was introduced to study the restricted three body problem and it is very useful for the characterization of nonlinear systems. I have analyzed in detail the recurrence patterns of systems with quasiperiodic dynamics both analytically and numerically. Based on a theoretical analysis, I have proposed a new procedure to distinguish quasiperiodic dynamics from chaos. This algorithm is particular useful in the analysis of short time series. Furthermore, this approach demonstrates to be efficient in recognizing regular and chaotic trajectories of dynamical systems with mixed phase space. Regarding the application to real situations, I have shown the capability and validity of this method by analyzing time series from fluid experiments. / In dieser Arbeit stelle ich neue Resultate vor, welche zeigen, wie man Rekurrenzeigenschaften quasiperiodischer, dynamischer Systeme für eine Datenanalyse ausnutzen kann. Die vorgestellten Algorithmen basieren auf einer zweidimensionalen Darstellungsmethode, den Rekurrenz-Darstellungen. Quasiperiodizität ist die einfachste Dynamik, die nicht-triviale Rekurrenzen zeigt und tritt häufig in nichtlinearen Systemen auf. Nicht-triviale Rekurrenzen wurden im Zusammenhang mit dem eingeschränkten Dreikörper-problem eingeführt. In dieser Arbeit, habe ich mehrere Systeme mit quasiperiodischem Verhalten analytisch untersucht. Die erhaltenen Ergebnisse helfen die Wiederkehreigenschaften dieser Systeme im Detail zu verstehen. Basierend auf den analytischen Resultaten, schlage ich einen neuen Algorithmus vor, mit dessen Hilfe selbst in kurzen Zeitreihen zwischen chaotischem und quasiperiodischem Verhalten unterschieden werden kann. Die vorgeschlagene Methode ist besonders effizient zur Unterscheidung regulärer und chaotischer Trajektorien mischender dynamischer Systeme.Die praktische Anwendbarkeit der vorgeschlagenen Analyseverfahren auf Messdaten, habe ich gezeigt, indem ich erfolgreich Zeitreihen aus fluid-dynamischen Experimenten untersucht habe.

ε-Differential Privacy (DP) has been popularly used for anonymizing data to protect sensitive information and for machine learning (ML) tasks. However, there is a trade-off in balancing privacy and achieving ML accuracy since ε-DP reduces the model’s accuracy for classification tasks. Moreover, not many studies have applied DP to time series from sensors and Internet-of-Things (IoT) devices. In this work, we try to achieve the accuracy of ML models trained with ε-DP data to be as close to the ML models trained with non-anonymized data for two different physiological time series. We propose to transform time series into domain-specific 2D (image) representations such as scalograms, recurrence plots (RP), and their joint representation as inputs for training classifiers. The advantages of using these image representations render our proposed approach secure by preventing data leaks since these image transformations are irreversible. These images allow us to apply state-of-the-art image classifiers to obtain accuracy comparable to classifiers trained on non-anonymized data by ex- ploiting the additional information such as textured patterns from these images. In order to achieve classifier performance with anonymized data close to non-anonymized data, it is important to identify the value of ε and the input feature. Experimental results demonstrate that the performance of the ML models with scalograms and RP was comparable to ML models trained on their non-anonymized versions. Motivated by the promising results, an end-to-end IoT ML edge-cloud architecture capable of detecting input drifts is designed that employs our technique to train ML models on ε-DP physiological data. Our classification approach ensures the privacy of individuals while processing and analyzing the data at the edge securely and efficiently.

Orientador: Prof. Dr. Filipe Ieda Fazanaro / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Engenharia da Informação, 2018. / A detecção de atividade de voz é um problema importante em muitas aplicações de fala/áudio, incluindo codificação e reconhecimento automático de fala; vários algoritmos foram propostos na literatura explorando diferentes métricas de sinais (como a energia do sinal). Neste trabalho, é apresentada uma metodologia alternativa para detecção de atividade vocal (VAD) de um discurso ou sinal de áudio com base nas informações fornecidas pelos gráficos de recorrência do sinal. O método proposto foi capaz de classificar corretamente sinais limpos e com baixos níveis de ruído, apresentando desempenho próximo ao algoritmo incluído no codec G.729, que é comumente usado em aplicativos de Voz sobre IP (VoIP). / Voice activity detection is an important problem in many speech/audio applications, including coding and automatic speech recognition; several algorithms have been proposed in the literature to explore different signal metrics (such as signal energy). In this work, an alternative methodology for the Voice Activity Detection (VAD) of a discourse or audio signal is presented based on the information provided by the signals¿ recurrence plots. The proposed method was able to correctly classify clean signals and with low levels of noise, obtained performance similar to the algorithm included in the G.729 codec, which is commonly used in VoIP applications.

L'analyse de séries temporelles biomédicales chaotiques tirées de systèmes dynamiques non-linéaires est toujours un challenge difficile à relever puisque dans certains cas bien spécifiques les techniques existantes basées sur les multi-fractales, les entropies et les graphes de récurrence échouent. Pour contourner les limitations des invariants précédents, de nouveaux descripteurs peuvent être proposés. Dans ce travail de recherche nos contributions ont porté à la fois sur l’amélioration d’indicateurs multifractaux (basés sur une fonction de structure) et entropiques (approchées) mais aussi sur des indicateurs de récurrences (non biaisés). Ces différents indicateurs ont été développés avec pour objectif majeur d’améliorer la discrimination entre des signaux de complexité différente ou d’améliorer la détection de transitions ou de changements de régime du système étudié. Ces changements agissant directement sur l’irrégularité du signal, des mouvements browniens fractionnaires et des signaux tirés du système du Lorenz ont été testés. Ces nouveaux descripteurs ont aussi été validés pour discriminer des fœtus en souffrance de fœtus sains durant le troisième trimestre de grossesse. Des mesures statistiques telles que l’erreur relative, l’écart type, la spécificité, la sensibilité ou la précision ont été utilisées pour évaluer les performances de la détection ou de la classification. Le fort potentiel de ces nouveaux invariants nous laisse penser qu’ils pourraient constituer une forte valeur ajoutée dans l’aide au diagnostic s’ils étaient implémentés dans des logiciels de post-traitement ou dans des dispositifs biomédicaux. Enfin, bien que ces différentes méthodes aient été validées exclusivement sur des signaux fœtaux, une future étude incluant des signaux tirés d’autres systèmes dynamiques nonlinéaires sera réalisée pour confirmer leurs bonnes performances. / The analysis of biomedical time series derived from nonlinear dynamic systems is challenging due to the chaotic nature of these time series. Only few classical parameters can be detected by clinicians to opt the state of patients and fetuses. Though there exist valuable complexity invariants such as multi-fractal parameters, entropies and recurrence plot, they were unsatisfactory in certain cases. To overcome this limitation, we propose in this dissertation new entropy invariants, we contributed to multi-fractal analysis and we developed signal-based (unbiased) recurrence plots based on the dynamic transitions of time series. Principally, we aim to improve the discrimination between healthy and distressed biomedical systems, particularly fetuses by processing the time series using our techniques. These techniques were either validated on Lorenz system, logistic maps or fractional Brownian motions modeling chaotic and random time series. Then the techniques were applied to real fetus heart rate signals recorded in the third trimester of pregnancy. Statistical measures comprising the relative errors, standard deviation, sensitivity, specificity, precision or accuracy were employed to evaluate the performance of detection. Elevated discernment outcomes were realized by the high-order entropy invariants. Multi-fractal analysis using a structure function enhances the detection of medical fetal states. Unbiased cross-determinism invariant amended the discrimination process. The significance of our techniques lies behind their post-processing codes which could build up cutting-edge portable machines offering advanced discrimination and detection of Intrauterine Growth Restriction prior to fetal death. This work was devoted to Fetal Heart Rates but time series generated by alternative nonlinear dynamic systems should be further considered.