Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

ERROR ESTIMATES OF GAUSS-TURAN QUADRATURES

A survey of our recent results on the error of Gauss-Tur´an quadrature formulae for functions which are analytic on a neighborhood of the set of integration is given. In particular, a computable upper bound of the error is presented which is valid for arbitrary weight functions. A comparison is made with the exact error and number of numerical examples, for arbitrary weight functions, are given which show the advantages of using such rules as well as the sharpness of the error bound. Asymptotic error estimates when the number of nodes in the quadrature increases are presented. A couple of numerical examples are included

QUADRATURE FORMULAS FOR THE FOURIER-CHEBYSHEV COEFFICIENTS

We consider the well known Micchelli-Rivlin quadrature formula, of highest algebraic degree of precision, for the Fourier-Chebyshev coefficients. For analytic functions the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points ∓1 and a sum of semiaxes ρ > 1, for the quoted quadrature formula. Starting from the explicit expression of the kernel, we determine the locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective L ∞- error bounds for this quadrature formula. Complex-variable methods are used to obtain expansions of the error in the Micchelli-Rivlin quadrature formula over the interval [−1, 1]. Finally, effective L 1 -error bounds are also derived for this quadrature formul

Advanced analyses of physiological signals and their role in Neonatal Intensive Care

Preterm infants admitted to the neonatal intensive care unit (NICU) face an array of life-threatening diseases requiring procedures such as resuscitation and invasive monitoring, and other risks related to exposure to the hospital environment, all of which may have lifelong implications. This thesis examined a range of applications for advanced signal analyses in the NICU, from identifying of physiological patterns associated with neonatal outcomes, to evaluating the impact of certain treatments on physiological variability. Firstly, the thesis examined the potential to identify infants at risk of developing intraventricular haemorrhage, often interrelated with factors leading to preterm birth, mechanical ventilation, hypoxia and prolonged apnoeas. This thesis then characterised the cardiovascular impact of caffeine therapy which is often administered to prevent and treat apnoea of prematurity, finding greater pulse pressure variability and enhanced responsiveness of the autonomic nervous system. Cerebral autoregulation maintains cerebral blood flow despite fluctuations in arterial blood pressure and is an important consideration for preterm infants who are especially vulnerable to brain injury. Using various time and frequency domain correlation techniques, the thesis found acute changes in cerebral autoregulation of preterm infants following caffeine therapy. Nutrition in early life may also affect neurodevelopment and morbidity in later life. This thesis developed models for identifying malnutrition risk using anthropometry and near-infrared interactance features. This thesis has presented a range of ways in which advanced analyses including time series analysis, feature selection and model development can be applied to neonatal intensive care. There is a clear role for such analyses in early detection of clinical outcomes, characterising the effects of relevant treatments or pathologies and identifying infants at risk of later morbidity

Nonlinear Systems

The editors of this book have incorporated contributions from a diverse group of leading researchers in the field of nonlinear systems. To enrich the scope of the content, this book contains a valuable selection of works on fractional differential equations.The book aims to provide an overview of the current knowledge on nonlinear systems and some aspects of fractional calculus. The main subject areas are divided into two theoretical and applied sections. Nonlinear systems are useful for researchers in mathematics, applied mathematics, and physics, as well as graduate students who are studying these systems with reference to their theory and application. This book is also an ideal complement to the specific literature on engineering, biology, health science, and other applied science areas. The opportunity given by IntechOpen to offer this book under the open access system contributes to disseminating the field of nonlinear systems to a wide range of researchers

Experimental study of feedback-induced dynamics in semiconductor lasers : from symbolic analysis to subwavelength position sensing

The aim of this thesis is the study of the dynamics induced by optical feedback in semiconductor lasers. This study aims, on the one hand, to improve our knowledge of stocahstic complex systems, and on the other hand, to use complex dynamics of semiconductor lasers to develop a protocol for subwavelength position sensing. The intensity of the light emitted by a semiconductor laser is stable, besides fluctuations due to spontaneous emission noise. When the light of the laser is reflected and part re-enters into the laser, the laser intensity can become unstable, displaying a broad range of dynamical behaviors. One of the dynamical regimes present in lasers with optical feedback is the low frequency fluctuations (LFF). This dynamics is characterized by sharp drops in the laser intensity (to almost switch the laser off), followed by gradual recoveries. The time intervals between two consecutive drops is irregular. The first part of this Thesis is focused on this dynamic regime, and a detailed experimental study has been performed to characterize it. A symbolic time series analysis has been used, based on the comparison of successive time intervals between dropouts. The dynamics of a semiconductor laser with feedback is governed by nonlinear light-matter interaction in the active medium of the laser, quantum noise due to spontaneous emission and time-delayed feedback. Therefore, the dropouts in the LFF regime can be noise-induced or triggered by deterministic processes. In this Thesis symbolic ordinal analysis has been used to statisticlly distinguish dropouts that can be noise-induced from those that have signatures of a deterministic origin. In this Thesis, the symbolic dynamics in the LFF regime has also been studied, and serial correlations have been found among several consecutive dropouts. It has been found a hierarchical and clustered structure of the symbolic dynamics. Moreover, a minimal iterative model has been found that, despite its simplicity, describes successfully the correlations found in the experiments. Because of the importance of external forcing in dynamical systems, the effect of current modulation on the symbolic dynamics of the LFFs has been studied. These experiments have allowed to characterize the effect of the modulation in the symbolic dynamics. The clusters of ordinal patterns formed without forcing remain under external periodic forcing. The minimal model has been verified, as it reproduces satisfactorily the symbolic dynamics of the experimental data. Also, in this Thesis a technique has been developed to detect displacements of two independent objects at subwavelength resolution (λ/160). With this purpose, a setup has been developed with a semiconductor laser with dual feedback. In addition to the high resolution, this protocol offers the advantage of sensing two objects by just measuring one variable.El objetivo de esta tesis es el estudio de la dinámica inducida por realimentación óptica en láseres de semiconductor. Dicho estudio persigue, por un lado, profundizar en el conocimiento de aspectos generales de los sistemas complejos, y por otro lado, utilizar dicha dinámica para crear un protocolo para medir desplazamientos en dos dimensiones con una resolución mucho menor que la longitud de onda del láser utilizado. La intensidad de la luz emitida por un láser de semiconductor es estable salvo fluctuaciones debidas al ruido de emisión espontánea. Sin embargo, cuando la luz del láser se refleja en una superficie y parte de esta luz vuelve a entrar en el láser, la intensidad de la luz emitida se puede desestabilizar y mostrar una amplia gama de comportamientos dinámicos. Uno de los regímenes dinámicos presentes en láseres con realimentación óptica es el de fluctuaciones de baja frecuencia (LFF de sus siglas en inglés). Esta dinámica se caracteriza por caídas abruptas de la intensidad del láser (hasta casi apagarse), seguidas de recuperaciones graduales, siendo la separación temporal entre dos caídas consecutivas irregular. La primera parte de esta tesis está centrada en este régimen dinámico, habiéndose realizado un detallado estudio experimental para caracterizarlo. Se ha utilizado un análisis simbólico de series de datos basado en patrones ordinales, definidos mediante la comparación de tiempos consecutivos entre caídas. En la dinámica del láser de semiconductor con realimentación intervienen varios factores: la interacción no lineal entre luz y materia en el medio activo del láser, el ruido cuántico debido a la emisión espontánea y la señal retardada de la realimentación. Por ello las caídas en el régimen de LFFs pueden ser inducidas tanto por ruido como por procesos deterministas. En esta tesis, mediante el análisis simbólico se ha conseguido distinguir estadísticamente, qué caídas pueden ser inducidas por ruido y cuales presentan una estadística que muestra señales de determinismo. En esta tesis también se ha estudiado la dinámica simbólica del régimen de LFF y se han encontrado correlaciones entre varias caídas consecutivas. También se ha encontrado una estructura jerárquica en la dinámica simbólica que incluye emparejamientos de las probabilidades de los patrones simbólicos. Además se ha encontrado un modelo simple a tiempo discreto (mapa) que describe adecuadamente la dinámica simbólica del régimen de LFF. Debido a la importancia de forzamientos externos en sistemas dinámicos, se han realizado experimentos incorporando modulación en la corriente de inyección del láser. Estos experimentos han permitido caracterizar el efecto de la amplitud de la modulación en la dinámica simbólica, encontrando cambios claros en la estructura simbólica, inducidos por la modulación, pero que se conservan los emparejamientos observados sin modulación. El modelo simple ha sido verificado ya que reproduce satisfactoriamente la dinámica simbólica encontrada en los datos experimentales. Asimismo, en esta tesis se ha demostrado experimentalmente un protocolo que permite detectar desplazamientos de dos objetos independientes en una escala muy inferior a la longitud de onda de la luz empleada (Λ/160). Para ello se ha diseñado un experimento donde el láser está sometido a realimentación de dos espejos que se mueven de manera independiente. Además de la alta resolución, otra ventaja de este protocolo reside en que únicamente es preciso medir una variable para calcular los dos desplazamientos

Some problems in combinatorial topology of flag complexes

In this work we study simplicial complexes associated to graphs and their homotopical and combinatorial properties. The main focus is on the family of flag complexes, which can be viewed as independence complexes and clique complexes of graphs. In the first part we study independence complexes of graphs using two cofibre sequences corresponding to vertex and edge removals. We give applications to the connectivity of independence complexes of chordal graphs and to extremal problems in topology and we answer open questions about the homotopy types of those spaces for particular families of graphs. We also study the independence complex as a space of configurations of particles in the so-called hard-core models on various lattices. We define, and investigate from an algorithmic perspective, a special family of combinatorially defined homology classes in independence complexes. This enables us to give algorithms as well as NP-hardness results for topological properties of some spaces. As a corollary we prove hardness of computing homology of simplicial complexes in general. We also view flag complexes as clique complexes of graphs. That leads to the study of various properties of Vietoris-Rips complexes of graphs. The last result is inspired by a problem in face enumeration. Using methods of extremal graph theory we classify flag triangulations of 3-manifolds with many edges. As a corollary we complete the classification of face vectors of flag simplicial homology 3-spheres