The study of experimental recording of dynamical systems often consists in the analysis of signals produced by that system. Time series analysis consists of a wide range of methodologies ultimately aiming at characterizing the signals and, eventually, gaining insights on the underlying processes that govern the evolution of the system. A standard way to tackle this issue is spectrum analysis, which uses Fourier or Laplace transforms to convert time-domain data into a more useful frequency space. These analytical methods allow to highlight periodic patterns in the signal and to reveal essential characteristics of linear systems. Most experimental signals, however, exhibit strange and apparently unpredictable behavior which require more sophisticated analytical tools in order to gain insights into the nature of the underlying processes generating those signals. This is the case when nonlinearity enters into the dynamics of a system. Nonlinearity gives rise to unexpected and fascinating behavior, among which the emergence of deterministic chaos. In the last decades, chaos theory has become a thriving field of research for its potential to explain complex and seemingly inexplicable natural phenomena. The peculiarity of chaotic systems is that, despite being created by deterministic principles, their evolution shows unpredictable behavior and a lack of regularity. These characteristics make standard techniques, like spectrum analysis, ineffective when trying to study said systems. Furthermore, the irregular behavior gives the appearance of these signals being governed by stochastic processes, even more so when dealing with experimental signals that are inevitably affected by noise. Nonlinear time series analysis comprises a set of methods which aim at overcoming the strange and irregular evolution of these systems, by measuring some characteristic invariant quantities that describe the nature of the underlying dynamics. Among those quantities, the most notable are possibly the Lyapunov ex- ponents, that quantify the unpredictability of the system, and measure of dimension, like correlation dimension, that unravel the peculiar geometry of a chaotic system’s state space. These methods are ultimately analytical techniques, which can often be exactly estimated in the case of simulated systems, where the differential equations governing the system’s evolution are known, but can nonetheless prove difficult or even impossible to compute on experimental recordings. A different approach to signal analysis is provided by information theory. Despite being initially developed in the context of communication theory, by the seminal work of Claude Shannon in 1948, information theory has since become a multidisciplinary field, finding applications in biology and neuroscience, as well as in social sciences and economics. From the physical point of view, the most phenomenal contribution from Shannon’s work was to discover that entropy is a measure of information and that computing the entropy of a sequence, or a signal, can answer to the question of how much information is contained in the sequence. Or, alternatively, considering the source, i.e. the system, that generates the sequence, entropy gives an estimate of how much information the source is able to produce. Information theory comprehends a set of techniques which can be applied to study, among others, dynamical systems, offering a complementary framework to the standard signal analysis techniques. The concept of entropy, however, was not new in physics, since it had actually been defined first in the deeply physical context of heat exchange in thermodynamics in the 19th century. Half a century later, in the context of statistical mechanics, Boltzmann reveals the probabilistic nature of entropy, expressing it in terms of statistical properties of the particles’ motion in a thermodynamic system. A first link between entropy and the dynamical evolution of a system is made. In the coming years, following Shannon’s works, the concept of entropy has been further developed through the works of, to only cite a few, Von Neumann and Kolmogorov, being used as a tool for computer science and complexity theory. It is in particular in Kolmogorov’s work, that information theory and entropy are revisited from an algorithmic perspective: given an input sequence and a universal Turing machine, Kolmogorov found that the length of the shortest set of instructions, i.e. the program, that enables the machine to compute the input sequence was related to the sequence’s entropy. This definition of the complexity of a sequence already gives hint of the differences between random and deterministic signals, in the fact that a truly random sequence would require as many instructions for the machine as the size of the input sequence to compute, as there is no other option than programming the machine to copy the sequence point by point. On the other hand, a sequence generated by a deterministic system would simply require knowing the rules governing its evolution, for example the equations of motion in the case of a dynamical system. It is therefore through the work of Kolmogorov, and also independently by Sinai, that entropy is directly applied to the study of dynamical systems and, in particular, deterministic chaos. The so-called Kolmogorov-Sinai entropy, in fact, is a well-established measure of how complex and unpredictable a dynamical system can be, based on the analysis of trajectories in its state space. In the last decades, the use of information theory on signal analysis has contributed to the elaboration of many entropy-based measures, such as sample entropy, transfer entropy, mutual information and permutation entropy, among others. These quantities allow to characterize not only single dynamical systems, but also highlight the correlations between systems and even more complex interactions like synchronization and chaos transfer. The wide spectrum of applications of these methods, as well as the need for theoretical studies to provide them a sound mathematical background, make information theory still a thriving topic of research. In this thesis, I will approach the use of information theory on dynamical systems starting from fundamental issues, such as estimating the uncertainty of Shannon’s entropy measures on a sequence of data, in the case of an underlying memoryless stochastic process. This result, beside giving insights on sensitive and still-unsolved aspects when using entropy-based measures, provides a relation between the maximum uncertainty on Shannon’s entropy estimations and the size of the available sequences, thus serving as a practical rule for experiment design. Furthermore, I will investigate the relation between entropy and some characteristic quantities in nonlinear time series analysis, namely Lyapunov exponents. Some examples of this analysis on recordings of a nonlinear chaotic system are also provided. Finally, I will discuss other entropy-based measures, among them mutual information, and how they compare to analytical techniques aimed at characterizing nonlinear correlations between experimental recordings. In particular, the complementarity between information-theoretical tools and analytical ones is shown on experimental data from the field of neuroscience, namely magnetoencefalography and electroencephalography recordings, as well as mete- orological data.

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Game theory is the study of tactical interactions involving conflicts and cooperations among multiple decision makers called players with applications in diverse disciplines such as economics, biology, management, communication networks, electric power systems and control. This dissertation studies a statistical differential game problem where finite N players optimize their system performance by shaping the distribution of their cost function through cost cumulants. This research integrates game theory with statistical optimal control theory and considers a statistical Nash non-cooperative nonzero-sum game for a nonlinear dynamic system with nonquadratic cost functions. The objective of the statistical Nash game is to find the equilibrium solution where no player has the incentive to deviate once other players maintain their equilibrium strategy. The necessary condition for the existence of the Nash equilibrium solution is given for the m-th cumulant cost optimization using the Hamilton-Jacobi-Bellman (HJB) equations. In addition, the sufficient condition which is the verification theorem for the existence of Nash equilibrium solution is given for the m-th cumulant cost optimization using the Hamilton-Jacobi-Bellman (HJB) equations. However, solving the HJB equations even for relatively low dimensional game problem is not trivial, we propose to use neural network approximate method to find the solution of the HJB partial differential equations for the statistical game problem. Convergence proof of the neural network approximate method solution to exact solution is given. In addition, numerical examples are provided for the statistical game to demonstrate the applicability of the proposed theoretical developments. / Electrical and Computer Engineering

Automatic control techniques have been widely employed in industry to increase efficiency and profitability of the processes. However, reliability on automation increases the susceptibility of the system to faults in major control equipment such as actuators and sensors. This realization has motivated design of frameworks for fault detection and isolation (FDI) and fault tolerant control (FTC). The success of these FDI and FTC mechanisms is contingent on their ability to handle complexities associated with process systems such as nonlinearity, uncertainty, high dimensionality and the resulting effects of the existence of complexity in system structure such as faults that cannot be isolated. Motivated by the above considerations, this thesis considers the problem of fault diagnosis and fault tolerant control for complex process systems. First, an FDI framework is designed that can detect and confine possible locations for faults that cannot be isolated. Next, the problem of simultaneous actuator and sensor fault diagnosis for nonlinear uncertain systems. The key idea is to design FDI filters in a way they account for the impact of uncertainty explicitly. This work then considers the problem of simultaneous fault diagnosis in nonlinear uncertain networked systems. FDI is achieved using a distributed architecture, comprised of a bank of local FDI (LFDI) schemes that communicate with each other. The efficacy of the proposed FDI methodologies is shown via application to a number of chemical process examples. Finally, an integrated framework is proposed for fault diagnosis and fault tolerant control of variable air volume (VAV) boxes, a common component of heating, ventilation and air conditioning (HVAC) systems as an industrial case study of complex systems. The advantages of the proposed framework are diagnosing multiple faults and handling faults in stuck dampers using a safe parking strategy with energy saving capability. / Thesis / Doctor of Philosophy (PhD) / Automation is the key to increase efficiency and profitability of the processes. However, as the level of automation increases, major control equipment are more prone to faults. Thus, fault detection and isolation (FDI) and fault tolerant control (FTC) frameworks are required for fault handling. Fault handling, however, can only be efficiently achieved if the designed FDI and FTC frameworks are able to deal with complexities arising in process systems such as nonlinearity, uncertainty, high dimensionality and the resulting effects of the existence of complexity in system structure such as faults that cannot be isolated. This motivates design of FDI and FTC frameworks for complex process systems. First, FDI frameworks are presented that can diagnose faults in the presence of complexities mentioned above. Then, an integrated framework is designed for diagnosing and handling faults of heating, ventilation and air conditioning (HVAC) systems as an industrial case study of complex process systems.

Yes / In this paper, a performance enhancement scheme has been investigated for a class of stochastic nonlinear systems via set-point adjustment. Considering the practical industrial processes, the multi-layer systematic structure has been adopted to achieve the control design requirements subjected to random noise. The basic loop control is given by PID design while the parameters have been fixed after the design phase. Alternatively, we can consider that there exists an unadjustable loop control. Then, the additional loop is designed for performance enhancement in terms of the tracking accuracy. In particular, a novel approach has been presented to dynamically adjust the set-points using the estimated states of the systems through extended Kalman filter (EKF). Minimising the entropy criterion, the parameters of the set-point adjustment controller can be optimised which will enhance the performance of the entire closed-loop systems. Based upon the presented scheme, the stochastic stability analysis has been given to demonstrate that the closed-loop tracking errors are bounded in probability one. To indicate the effectiveness of the presented control scheme, the numerical examples have been given and the simulation results imply that the designed systems are bounded and the tracking performance can be enhanced simultaneously. In summary, a new framework for system performance enhancement has been presented even if the loop control is unadjustable which forms the main contribution of this paper.

The analysis of signals stemming from a physical system is crucial for the experimental investigation of the underlying dynamics that drives the system itself. The field of time series analysis comprises a wide variety of techniques developed with the purpose of characterizing signals and, ultimately, of providing insights on the phenomena that govern the temporal evolution of the generating system. A renowned example in this field is given by spectral analysis: the use of Fourier or Laplace transforms to bring time-domain signals into the more convenient frequency space allows to disclose the key features of linear systems. A more complex scenario turns up when nonlinearity intervenes within a system's dynamics. Nonlinear coupling between a system's degrees of freedom brings about interesting dynamical regimes, such as self-sustained periodic (though anharmonic) oscillations ("limit cycles"), or quasi-periodic evolutions that exhibit sharp spectral lines while lacking strict periodicity ("limit tori"). Among the consequences of nonlinearity, the onset of chaos is definitely the most fascinating one. Chaos is a dynamical regime characterized by unpredictability and lack of periodicity, despite being generated by deterministic laws. Signals generated by chaotic dynamical systems appear as irregular: the corresponding spectra are broad and flat, prediction of future values is challenging, and evolutions within the systems' state spaces converge to strange attractor sets with noninteger dimensionality. Because of these properties, chaotic signals can be mistakenly classified as noise if linear techniques such as spectral analysis are used. The identification of chaos and its characterization require the assessment of dynamical invariants that quantify the complex features of a chaotic system's evolution. For example, Lyapunov exponents provide a marker of unpredictability; the estimation of attractor dimensions, on the other hand, highlights the unconventional geometry of a chaotic system's state space. Nonlinear time series analysis techniques act directly within the state space of the system under investigation. However, experimentally, full access to a system's state space is not always available. Often, only a scalar signal stemming from the dynamical system can be recorded, thus providing, upon sampling, a scalar sequence. Nevertheless, by virtue of a fundamental theorem by Takens, it is possible to reconstruct a proxy of the original state space evolution out of a single, scalar sequence. This reconstruction is carried out by means of the so-called embedding procedure: m-dimensional vectors are built by picking successive elements of the scalar sequence delayed by a lag L. On the other hand, besides posing some necessary conditions on the integer embedding parameters m and L, Takens' theorem does not provide any clue on how to choose them correctly. Although many optimal embedding criteria were proposed, a general answer to the problem is still lacking. As a matter of fact, conventional methods for optimal embedding are flawed by several drawbacks, the most relevant being the need for a subjective evaluation of the outcomes of applied algorithms. Tackling the issue of optimally selecting embedding parameters makes up the core topic of this thesis work. In particular, I will discuss a novel approach that was pursued by our research group and that led to the development of a new method for the identification of suitable embedding parameters. Rather than most conventional approaches, which seek a single optimal value for m and L to embed an input sequence, our approach provides a set of embedding choices that are equivalently suitable to reconstruct the dynamics. The suitability of each embedding choice m, L is assessed by relying on statistical testing, thus providing a criterion that does not require a subjective evaluation of outcomes. The starting point of our method are embedding-dependent correlation integrals, i.e. cumulative distributions of embedding vector distances, built out of an input scalar sequence. In the case of Gaussian white noise, an analytical expression for correlation integrals is available, and, by exploiting this expression, a gauge transformation of distances is introduced to provide a more convenient representation of correlation integrals. Under this new gauge, it is possible to test—in a computationally undemanding way—whether an input sequence is compatible with Gaussian white noise and, subsequently, whether the sequence is compatible with the hypothesis of an underlying chaotic system. These two statistical tests allow ruling out embedding choices that are unsuitable to reconstruct the dynamics. The estimation of correlation dimension, carried out by means of a newly devised estimator, makes up the third stage of the method: sets of embedding choices that provide uniform estimates of this dynamical invariant are deemed to be suitable to embed the sequence.The method was successfully applied to synthetic and experimental sequences, providing new insight into the longstanding issue of optimal embedding. For example, the relevance of the embedding window (m-1)L, i.e. the time span covered by each embedding vector, is naturally highlighted by our approach. In addition, our method provides some information on the adequacy of the sampling period used to record the input sequence.The method correctly distinguishes a chaotic sequence from surrogate ones generated out of it and having the same power spectrum. The technique of surrogate generation, which I also addressed during my Ph. D. work to develop new dedicated algorithms and to analyze brain signals, allows to estimate significance levels in situations where standard analytical algorithms are unapplicable. The novel embedding approach being able to tell apart an original sequence from surrogate ones shows its capability to distinguish signals beyond their spectral—or autocorrelation—similarities.One of the possible applications of the new approach concerns another longstanding issue, namely that of distinguishing noise from chaos. To this purpose, complementary information is provided by analyzing the asymptotic (long-time) behaviour of the so-called time-dependent divergence exponent. This embedding-dependent metric is commonly used to estimate—by processing its short-time linearly growing region—the maximum Lyapunov exponent out of a scalar sequence. However, insights on the kind of source generating the sequence can be extracted from the—usually overlooked—asymptotic behaviour of the divergence exponent. Moreover, in the case of chaotic sources, this analysis also provides a precise estimate of the system's correlation dimension. Besides describing the results concerning the discrimination of chaotic systems from noise sources, I will also discuss the possibility of using the related correlation dimension estimates to improve the third stage of the method introduced above for the identification of suitable embedding parameters. The discovery of chaos as a possible dynamical regime for nonlinear systems led to the search of chaotic behaviour in experimental recordings. In some fields, this search gave plenty of positive results: for example, chaotic dynamics was successfully identified and tamed in electronic circuits and laser-based optical setups. These two families of experimental chaotic systems eventually became versatile tools to study chaos and its possible applications. On the other hand, chaotic behaviour is also looked for in climate science, biology, neuroscience, and even economics. In these fields, nonlinearity is widespread: many smaller units interact nonlinearly, yielding a collective motion that can be described by means of few, nonlinearly coupled effective degrees of freedom. The corresponding recorded signals exhibit, in many cases, an irregular and complex evolution. A possible underlying chaotic evolution—as opposed to a stochastic one—would be of interest both to reveal the presence of determinism and to predict the system's future states. While some claims concerning the existence of chaos in these fields have been made, most results are debated or inconclusive. Nonstationarity, low signal-to-noise ratio, external perturbations and poor reproducibility are just few among the issues that hinder the search of chaos in natural systems. In the final part of this work, I will briefly discuss the problem of chasing chaos in experimental recordings by considering two example sequences, the first one generated by an electronic circuit and the second one corresponding to recordings of brain activity. The present thesis is organized as follows. The core concepts of time series analysis, including the key features of chaotic dynamics, are presented in Chapter 1. A brief review of the search for chaos in experimental systems is also provided; the difficulties concerning this quest in some research fields are also highlighted. Chapter 2 describes the embedding procedure and the issue of optimally choosing the related parameters. Thereupon, existing methods to carry out the embedding choice are reviewed and their limitations are pointed out. In addition, two embedding-dependent nonlinear techniques that are ordinarily used to characterize chaos, namely the estimation of correlation dimension by means of correlation integrals and the assessment of maximum Lyapunov exponent, are presented. The new approach for the identification of suitable embedding parameters, which makes up the core topic of the present thesis work, is the subject of Chapter 3 and 4. While Chapter 3 contains the theoretical outline of the approach, as well as its implementation details, Chapter 4 discusses the application of the approach to benchmark synthetic and experimental sequences, thus illustrating its perks and its limitations. The study of the asymptotic behaviour of the time-dependent divergent exponent is presented in Chapter 5. The alternative estimator of correlation dimension, which relies on this asymptotic metric, is discussed as a possible improvement to the approach described in Chapters 3, 4. The search for chaos out of experimental data is discussed in Chapter 6 by means of two examples of real-world recordings. Concluding remarks are finally drawn in Chapter 7.

Esta tesis considera un enfoque basado en Lyapunov para el análisis y control de sistemas no lineales cuyas ecuaciones dinámicas son reescritas como un modelo Takagi-Sugeno o uno polinomial convexo. Estas estructuras permiten resolver problemas de control mediante técnicas de optimización convexa, más concretamente desigualdades matriciales lineales y suma de cuadrados, que son eficientes herramientas desde un punto de vista computacional. Después de proporcionar una visión general básica del estado actual en el campo de los modelos Takagi-Sugeno, esta tesis aborda cuestiones sobre las funciones de Lyapunov por trozos, dependiente de parámetros e integral de línea, con las siguientes contribuciones: Un algoritmo mejorado para estimaciones del dominio de atracción de sistemas no lineales para sistemas de tiempo continuo. Los resultados se basan en funciones de Lyapunov por trozos, desigualdades matriciales lineales y argumentaciones geométricas; enfoques basados en conjuntos de nivel en la literatura previa se han mejorado significativamente. Una función Lyapunov generalizada dependiente de parámetros para la síntesis de controladores para sistemas Takagi-Sugeno. El enfoque propone una ley de control multi-índice que retroalimenta la derivada del tiempo de las funciones de membresía del modelo Takagi-Sugeno para anular los términos que causan localidad a priori en el análisis de Lyapunov. Una nueva función integral de Lyapunov para el análisis de estabilidad de sistemas no lineales. Estos resultados generalizan aquellos basados en funciones de Lyapunov integral de línea al marco polinomial; resulta que los requisitos de independencia del camino pueden ser anulados por una definición adecuada de una función Lyapunov con términos integrales. / This thesis considers a Lyapunov-based approach for analysis and control of nonlinear systems whose dynamical equations are rewritten as a Takagi-Sugeno model or a convex polynomial one. These structures allow solving control problems via convex optimisation techniques, more specifically linear matrix inequalities and sum-of-squares, which are efficient tools from the computational point of view. After providing a basic overview of the state of the art in the field of Takagi-Sugeno models, this thesis address issues on piecewise, parameter-dependent and line-integral Lyapunov functions, with the following contributions: An improved algorithm to estimate the domain of attraction of nonlinear systems for continuous-time systems. The results are based on piecewise Lyapunov functions, linear matrix inequalities, and geometrical argumentations; level-set approaches in prior literature are significantly improved. A generalised parameter-dependent Lyapunov function for synthesis of controllers for Takagi-Sugeno systems. The approach proposed a multi-index control law that feeds back the time derivative of the membership function of the Takagi-Sugeno model to cancel out the terms that cause a priori locality in the Lyapunov analysis. A new integral Lyapunov function for stability analysis of nonlinear systems. These results generalise those based on line-integral Lyapunov functions to the polynomial framework; it turns out path-independency requirements can be overridden by an adequate definition of a Lyapunov function with integral terms. / Aquesta tesi considera un enfocament basat en Lyapunov per a l'anàlisi i control de sistemes no lineals les equacions dinàmiques dels quals són reescrites com un model Takagi-Sugeno o un de polinomial convex. Aquestes estructures permeten resoldre problemes de control mitjançant tècniques d'optimització convexa, més concretament desigualtats matricials lineals i suma de quadrats, que són eines eficients des d'un punt de vista computacional. Després de proporcionar una visió general bàsica de l'estat actual en el camp dels models Takagi-Sugeno, aquesta tesi aborda qüestions sobre les funcions de Lyapunov per trossos, dependent de paràmetres i integral de línia, amb les següents contribucions: Un algoritme millorat per a estimar el domini d'atracció de sistemes no lineals per a sistemes de temps continu. Els resultats es basen en funcions de Lyapunov per trossos, desigualtats matricials lineals i argumentacions geomètriques; enfocaments basats en conjunts de nivell en la literatura prèvia s'han millorat significativament. Una funció Lyapunov generalitzada dependent de paràmetres per a la síntesi de controladors per a sistemes Takagi-Sugeno. L'enfocament proposa una llei de control multi-índex que retroalimenta la derivada del temps de les funcions de membres del model Takagi-Sugeno per anul·lar els termes que causen localitat a priori en l'anàlisi de Lyapunov. Una nova funció integral de Lyapunov per a l'anàlisi d'estabilitat de sistemes no lineals. Aquests resultats generalitzen aquells basats en funcions de Lyapunov integral de línia al marc polinomial; resulta que els requisits d'independència del camí poden ser anul·lats per una definició adequada d'una funció Lyapunov amb termes integrals. / González Germán, IT. (2018). Contributions to analysis and control of Takagi-Sugeno systems via piecewise, parameter-dependent, and integral Lyapunov functions [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/101282 / TESIS

Cette thèse porte sur l'analyse de la synchronisation des grands réseaux d'oscillateurs non linéaires et hétérogènes à l'aide d'outils et de méthodes issues de la théorie du contrôle. Nous considérons deux modèles de réseaux; à savoir, le modèle de Kuramoto qui considère seulement les coordonnées de phase des oscillateurs et des réseaux composés d'oscillateurs non linéaires de Stuart-Landau connectés par un couplage linéaire.Pour le modèle de Kuramoto nous construisons un système linéaire qui conserve les informations sur les fréquences naturelles et sur les gains d'interconnexion du modèle original de Kuramoto. Nous montrons en suite que l'existence de solutions à verrouillage de phase du modèle de Kuramoto est équivalente à l'existence d'un tel système linéaire avec certaines propriétés. Ce système est utilisé pour formuler les conditions d'existence de solutions à verrouillage de phase et de leur stabilité pour des structures particulières de l'interconnexion. Ensuite, cette analyse s'est étendue au cas où des interactions attractives et répulsives sont présentes dans le réseau. Nous considérons cette situation lorsque les gains d'interconnexion peuvent être à la fois positif et négatif. Dans le cadre de réseaux d'oscillateurs de Stuart-Landau, nous présentons une nouvelle transformation de coordonnées du réseau qui permet de réécrire le modèle du réseau en deux parties: une décrivant le comportement de l'oscillateur « moyenne » du réseau et la seconde partie présentant les dynamiques des erreurs de synchronisation par rapport à cet oscillateur « moyenne ». Cette transformation nous permet de caractériser les propriétés du réseau en termes de la stabilité des erreurs de synchronisation et du cycle limite de l'oscillateur « moyenne ». Pour ce faire, nous reformulons ce problème en un problème de stabilité de deux ensembles compacts et nous utilisons des outils issus de la stabilité de Lyapunov pour montrer la stabilité pratique de ces derniers pour des valeurs suffisamment grandes du gain d'interconnexion. / This thesis is devoted to the analysis of synchronization in large networks of heterogeneous nonlinear oscillators using tools and methods issued from control theory. We consider two models of networks; namely, the Kuramoto model which takes into account only phase coordinates of the oscillators and networks composed of nonlinear Stuart-Landau oscillators interconnected by linear coupling. For the Kuramoto model we construct an auxiliary linear system that preserves information on the natural frequencies and interconnection gains of the original Kuramoto model. We show next that existence of phase locked solutions of the Kuramoto model is equivalent to the existence of such a linear system with certain properties. This system is used to formulate conditions that ensure existence of phase-locked solutions and their stability for particular structures of network interconnections. Next, this analysis is extended to the case where both attractive and repulsive interactions are present in the network that is we consider the situation where some of the interconnection gains are allowed to be negative. In the context of networks of Stuart-Landau oscillators, we present a new coordinate transformation of the network which allows to split the network model into two parts, one describing behaviour of an "averaged" network oscillator and the second one, describing dynamics of the synchronization errors relative to this "averaged" oscillator. This transformation allows us to characterize properties of the network in terms of stability of synchronization errors and limit cycle of the "averaged" oscillator. To do so, we recast this problem as a problem of stability of compact sets and use Lyapunov stability tools to ensure practical stability of both sets for sufficiently large values of the coupling strength.

Este trabalho teve por objetivo construir e avaliar ferramentas de quantificação e análise da variabilidade da freqüência cardíaca segundo o paradigma não extensivo de Tsallis na modelagem do sistema de regulação da freqüência cardíaca, e na discriminação de situação de normalidade e cardiopatias e apoio ao diagnóstico de cardiopatias. O sistema de regulação da freqüência cardíaca é reconhecidamente não linear. Este estudo explora esta característica, quantificando a complexidade através de uma família de entropias condicionais não extensivas e outras medidas de avaliação usualmente utilizadas na análise da variabilidade cardíaca. Foram utilizados dados reais de 15 indivíduos saudáveis, 23 indivíduos chagásicos e 19 indivíduos hipertensos, além de dados simulados computacionalmente para uma avaliação controlada das ferramentas estatísticas estudadas. Durante a avaliação foram gerados dados substitutos (surrogate data) para os testes de validade e intensidade da hipótese de não linearidade das séries. Os resultados mostraram que o parâmetro q introduz uma forma diferente de quantificação da complexidade do sinal. Com o auxílio dos dados substitutos, foi possível identificar, para alguns sinais, a região de valores de q onde o comportamento não linear é mais evidente. Os resultados obtidos indicam que a abordagem merece estudos mais aprofundados. / This study aimed to construct and evaluate tools for quantification and analysis of heart rate variability under Tsallis non-extensive statistical paradigm to model heart rate regulation system, and discrimination in situations of healthy conditions and support diagnosis of heart diseases. The heart rate regulation system is known to be nonlinear. This study explores this feature by quantifying the signal complexity through a family of non extensive conditional entropy and other evaluation measures commonly used in the analysis of heart rate variability. We used real data from 15 healthy subjects, 23 individuals with Chagas disease and 19 hypertensive individuals, in addition to simulated data to make a computationally controlled evaluation of the statistical tools studied. During the evaluation were generated surrogate data for tests of validity and strength of the time series non-linearity hypothesis. The results have shown that the q parameter introduces a different way of quantifying the complexity of the signal. With the support of surrogate data, it was possible to identify a range of q values that nonlinear behavior is more evident for some signals. The results indicate that the approach deserves further study.