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Nuestro trabajo se ha concentrado en la simulación y el análisis de modelos no-lineales aplicados a las curvas de luz ópticas de estrellas variables de largo periodo. Cuando las estrellas de masa pequeña o intermedia (de 1 a 11 masas solares) llegan a las fases finales de su evolución, tales como la Rama Asintótica de las (Super-)Gigantes, presentan oscilaciones regulares e irregulares de larga duración (i.e. estrellas Miras). Las pulsaciones regulares e irregulares facilitan los episodios de pérdida de masa y hacen que estas estrellas jueguen un papel crucial en la evolución química de las galaxias. Los modelos hidrodinámicos detallados existentes en la literatura proporcionan resultados muy completos, pero a menudo son difíciles de interpretar. Por lo tanto se ha creado la necesidad de entender, a un nivel más básico, los procesos que llevan a pulsaciones irregulares o comportamientos en contradicción con las observaciones. En este sentido, hemos considerado la estrella variable como constituida por un nucleo compacto y una envoltura cuyo movimiento forzado está dado por las ondas de presión originadas en el interior de la estrella. En el presente enfoque, aproximamos la envoltura mediante una sola capa y por lo tanto, los modelos basados en esta aproximación llevan el nombre de modelos de una capa. Entre ellos,los que hemos estudiado a lo largo de los últimos tres años dan cuenta de dichas irregularidades a través de un sistema de ecuaciones diferenciales que incluye, en las versiones más sencillas, la ecuación del movimiento y la conservación de masa, a las cuales hemos añadido a lo largo del estudio, la ecuación de la variación y transporte de energia en la envoltura estelar. En la presente tesis presentamos los resultados obtenidos, asi como una discusión sobre el estado de la modelización de estrellas variables en el contexto de los modelos de una capa. En el primer capítulo hemos introducido los tipos de estrellas variables y los mecanismos que llevan a su variabilidad, mientras que en el segundo capítulo hemos hecho un breve repaso de los modelos de una capa existentes en la literatura. Los modelos mencionados han sido seleccionados por su relación con los modelos que proponemos. En el primer estudio realizado que describimos en el tercer capítulo, hemos utilizado como punto de partida los resultados obtenidos por Icke et al., 1992, A&A, 258, 341, sobre un oscilador forzado que contiene la mínima dinámica necesaria para describir las oscilaciones estelares: la ecuación del movimiento y la conservación de masa. Aunque ideado para la variabilidad de estrellas de masa pequeña, nosotros hemos extendido el modelo para describir también las estrellas variables más masivas, en la fase de la Rama Asintótica de las Super-Gigantes. Hemos llevado a cabo un estudio paramétrico concienzudo a fin de investigar los tipos de com portamiento proporcionados por el modelo, identificando las bifurcaciones que producen dichos comportamientos y los rangos de los parámetros asociados a ellas. Para ello, hemos integrado en nuestro análisis los métodos característicos de la teoria de las bifurcaciones y del análisis tiempo-frecuencia, con el objetivo de determinar los posibles escenarios de transición al caos. Desde un punto de vista matemático, el análisis ha supuesto el estudio del mapa de Poincaré asociado a nuestro sistema, que, como se ha mencionado, está caracterizado por una perturbación periódica. Los resultados incluyen una serie de bifurcaciones locales y globales, entre las cuales la más importante es una triplicación. Entre las consecuencias de esta bifurcación mencionamos la adquisición por parte del mapa de Poincaré de la propiedad de nontwist que conlleva unas características del mapa de Poincaré típicas de los mapas nontwist (e.g. reconexión, meandros). Debido a la particular forma de la perturbación, la dinámica del sistema se diferencia de la del mapa cubico de Hénon, que se considera el prototipo de los mapas nontwist, y hacen del sistema investigado un ejemplo de dinámica prevista teóricamente, pero para la cual no se conocía ningún ejemplo. Desde un punto de vista astrofísico, la comparación con los resultados obtenidos por Icke et al. (1992) nos ha llevado a concluir que las estrellas variables más masivas presentan pulsaciones más irregulares que las estrella de masa pequeña, en acuerdo con las observaciones. El comportamiento irregular lleva a una pronunciada pérdida de masa, resultado comprobado por los datos observacionales. En el cuarto capítulo hemos extendido el modelo descrito anteriormente, añadiendole la ecuación del transporte de energia que permite una mejor comparación con las observaciones. Los resultados proporcionados por el modelo ampliado, entre los cuales destacamos las series temporales asociadas a la fluctuación de la luminosidad estelar, presentan sorprendentes similitudes con algunas de las más estudiadas y peculiares estrellas variables de largo periodo (las estrellas Miras). Adicionalmente, la dinámica encontrada conduce a series temporales en forma de pulsos energéticos a intervalos de tiempo del orden de mil años, que se pueden relacionar con las periodicidades encontradas en las capas circumestelares que rodean a ciertas nebulosas planetarias. En el quinto capítulo de la presente tesis hemos introducido el acoplamiento entre la convección y la pulsación estelar, proceso que se considera en la literatura como indispensable para una correcta modelización de la evolución de estas estrellas. Para ello hemos utilizado el modelo convectivo de una capa introducido en Stellingwerf, R.F., 1986, ApJ, 303, 119. Los resultados de dicho artículo nos han llamado la atención por algunas discrepancias relacionadas con el análisis de la morfologia de las curvas de luz y velocidad para casos en los cuales no existían ciclos límite. Por consiguiente, nuestra investigación se ha concentrado en el estudio parametrico del sistema con la intención de identificar las condiciones necesarias para la existencia de ciclos límite. Una vez identificadas la regiones de ciclos límite en el espacio parametrico, la morfologia de las curvas de luz asociadas ha revelado la existencia de una banda de inestabilidad semejante a la banda de inestabilidad de las variables Cefeidas. Adicionalmente, hemos ampliado el modelo considerando una forma más realista para los factores geometricos que describen el estado evolutivo de la estrella. Los resultados de esta ampliación indican una progresión de las curvas de luz - o para ser más exacto, de sus amplitudes y periodos - que recuerda la llamada Progresión Hertzsprung de las Cefeidas de tipo Bump. Hemos identificado también los tipos de contrapartidas observacionales que son susceptibles de ajustarse al modelo estudiado. Aunque existen muchos trabajos en la literatura basados en el modelo de Stellingwerf et al. (1986), consideramos que todavía muchas facetas e ideas quedan por aclarar y desarrollar.Para concluir, hemos comentado las implicaciones de nuestros resultados y estudios en el contexto de los modelos no lineales de una capa evidenciando sus capacidades y sus límites.

This paper investigates whether the inherent non-stationarity of macroeconomic time series is entirely due to a random walk or also to non-linear components. Applying the numerical tools of the analysis of dynamical systems to long time series for the US, we reject the hypothesis that these series are generated solely by a linear stochastic process. Contrary to the Real Business Cycle theory that attributes the irregular behavior of the system to exogenous random factors, we maintain that the fluctuations in the time series we examined cannot be explained only by means of external shocks plugged into linear autoregressive models. A dynamical and non-linear explanation may be useful for the double aim of describing and forecasting more accurately the evolution of the system. Linear growth models that find empirical verification on linear econometric analysis, are therefore seriously called in question. Conversely non-linear dynamical models may enable us to achieve a more complete information about economic phenomena from the same data sets used in the empirical analysis which are in support of Real Business Cycle Theory. We conclude that Real Business Cycle theory and more in general the unit root autoregressive models are an inadequate device for a satisfactory understanding of economic time series. A theoretical approach grounded on non-linear metric methods, may however allow to identify non-linear structures that endogenously generate fluctuations in macroeconomic time series. (authors' abstract) / Series: Working Papers Series "Growth and Employment in Europe: Sustainability and Competitiveness"

The search for deterministic chaos in economic and financial time series has attracted much interest over the past decade. However, clear evidence of chaotic structures is usually prevented by large random components in the time series. In the first part of this paper we show that even if a sophisticated algorithm estimating and testing the positivity of the largest Lyapunov exponent is applied to time series generated by a stochastic dynamical system or a return series of a stock index, the results are difficult to interpret. We conclude that the notion of sensitive dependence on initial conditions as it has been developed for deterministic dynamics, can hardly be transfered into a stochastic context. Therefore, in the second part of the paper our starting point for measuring dependencies for stochastic dynamics is a distributional characterization of the dynamics, e.g. by heteroskedastic models for economic and financial time series. We adopt a sensitivity measure proposed in the literature which is an information-theoretic measure of the distance between probability density functions. This sensitivity measure is well defined for stochastic dynamics, and it can be calculated analytically for the classes of stochastic dynamics with conditional normal distributions of constant and state-dependent variance. In particular, heteroskedastic return series models such as ARCH and GARCH models are investigated. (author's abstract) / Series: Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

The sigma-delta modulator is a popular signal amplitude quantization error (or noise) shaper used in oversampling analogue-to-digital and digital-to-analogue converter systems. The shaping of the noise frequency spectrum is performed by feeding back the quantization errors through a time delay element filter and feedback loop in the circuit, and by the addition of a possible stochastic dither signal at the quantizer. The aim in audio systems is to limit audible noise and distortions in the reconverted analogue signal. The formulation of the sigma-delta modulator as a discrete dynamical system provides a useful framework for the mathematical analysis of such a complex nonlinear system, as well as a unifying basis from which to consider other systems, from pseudorandom number generators to stochastic resonance processes, that yield equivalent formulations. The study of chaos and other complementary aspects of internal dynamical behaviour in previous research has left important issues unresolved. Advancement of this study is naturally facilitated by the dynamical systems approach. In this thesis, the general order feedback/feedforward sigma-delta modulator with multi-bit quantizer (no overload) and general input, is modelled and studied mathematically as a dynamical system. This study employs pertinent topological methods and relationships, which follow centrally from the symmetry of the circle map interpretation of the error state space dynamcis. The main approach taken is to reduce the nonlinear system into local or special case linear systems. Systems of sufficient structure are shown to often possess structured random, or random-like behaviour. An adaptation of Devaney's definition of chaos is applied to the model, and an extensive investigation of the conditions under which the associated chaos conditions hold or do not hold is carried out. This seeks, in part, to address the unresolved research issues. Chaos is shown to hold if all zeros of the noise transfer function lie outside the unit circle of radius two, provided the input is either periodic or persistently random (mod delta). When the filter satisfies a certain continuity condition, the conditions for chaos are extended, and more clear cut classifications emerge. Other specific chaos classifications are established. A study of the statistical properties of the error in dithered quantizers and sigma-delta modulators is pursued using the same state space model. A general treatment of the steady state error probability distribution is introduced, and results for predicting uniform steady state errors under various conditions are found. The uniformity results are applied to RPDF dithered systems to give conditions for a steady state error variance of delta squared over six. Numerical simulations support predictions of the analysis for the first-order case with constant input. An analysis of conditions on the model to obtain bounded internal stability or instability is conducted. The overall investigation of this thesis provides a theoretical approach upon which to orient future work, and initial steps of inquiry that can be advanced more extensively in the future.

We propose a dynamical systems approach to the study of weak turbulence(spatiotemporal chaos) based on the periodic orbit theory, emphasizing the role of recurrent patterns and coherent structures. After a brief review of the periodic orbit theory and its application to low-dimensional dynamics, we discuss its possible extension to study dynamics of spatially extended systems. The discussion is three-fold. First, we introduce a novel variational scheme for finding periodic orbits in high-dimensional systems. Second, we prove rigorously the existence of periodic structures (modulated amplitude waves) near the first instability of the complex Ginzburg-Landau equation, and check their role in pattern formation. Third, we present the extensive numerical exploration of the Kuramoto-Sivashinsky system in the chaotic regime: structure of the equilibrium solutions, our search for the shortest periodic orbits, description of the chaotic invariant set in terms of intrinsic coordinates and return maps on the Poincare section.

The detection for phase locking in a forced oscillator with dual attractors and ill-defined phase structure is hard until a quantitative approach was constructed for detecting phase locking via stroboscopic method. We study the route to weak phase locking in a chaotic system ¡§Chua oscillator¡¨ with complex attractor structure by analyzing the stroboscopic points. The onset of weak phase locking detected by using this statistical approach and the critical coupling strength calculated by conditional Lyapunov exponent are matched well. Detailed structure of phase locking intensity is described by the Arnold tongue diagram. Moreover, we apply this approach on three hyper-chaotic systems with multi-scroll attractor, including hyper-chaotic Rössler system, hyper-chaotic Lorenz system, and modified MCK oscillator. The weak phase locking between hyper-chaotic system and a periodic or a chaotic driving force is observable following the condition of stroboscopic point concentration.

Linear vibration measurement and analysis techniques have appeared to be sufficient with most vibration problems. This is partially due to the lack of proper identification of physical nonlinear dynamic responses. Therefore, as an example, a vehicle driveshaft exhibits a nonlinear super harmonic jump due to nonconstant velocity, NCV, joint excitation. Previously documented measurements or analytical predictions of vehicle driveshaft systems do not indicate nonlinear jump as a typical vibration mode. The nonlinear jump was both measured on a driveshaft test rig and simulated with a correlated model reproduced the jump. Subsequent development of the applied moments and simplified equations of motion provided the basis for nonlinear analysis. The nonlinear analyses included bifurcation, Poincare, Lyapunov Exponent, and identification of multiple solutions. Previous analytical models of driveshafts incorporating NCV joints are typically simple lumped parameter models. Complexity of models produce significant processing costs to completing significant analysis, and therefore large DOF systems incorporating significant flexibility are not analyzed. Therefore, a generalized method for creating simplified equations of motion while retaining integrity of the base system was developed. This method includes modal coupling, modal modification, and modal truncation techniques applied with nonlinear constraint conditions. Correlation of resonances and simulation results to operating results were accomplished. Previous NCV joint analyses address only the torsional degree of freedom. Limited background on lateral excitations and vibrations exist, and primarily focus on friction in the NCV joint or significant applied load. Therefore, the secondary moment was developed from the NCV joint excitation for application to the driveshaft system. This derivation provides detailed understanding of the vibration harmonic excitations due to NCV joints operating at misalignment angles. The model provides a basis for completing nonlinear analysis studying the system in more detail. Bifurcation analysis identified ranges of misalignment angles and speeds that produced nonlinear responses. Lyapunov Exponent analysis identified that these ranges were chaotic in nature. In addition, these analyses isolated the nonlinear response to the addition of the ITD nonlinear stiffness. In summary, the system and analysis show how an ITD installed to attenuate unwanted vibrations can cause other objectionable nonlinear response characteristics.

The objective of this investigation is to measure and analyze surface temperature fluctuations in pool boiling. The surface temperature fluctuations were recorded on silicon surfaces with and without multi-walled carbon nanotubes (MWCNT). Novel Thin Film Thermocouples (TFT) are micro-fabricated on test substrates to measure surface temperatures. A dielectric liquid refrigerant (PF-5060) is used as test fluid. Both nucleate and lm boiling regimes are investigated for the silicon test substrates. Dynamics of nucleate boiling is investigated on the CNT coated substrates. High frequency temperature fluctuation data is analyzed for the presence of determinism using non-linear time series analysis techniques in TISEAN(copyright) software. The impact of subcooling and micro/nano-scale surface texturing using MWCNT coatings on the dynamics of pool boiling is assessed. Dynamic invariants such as correlation dimensions and Lyapunov spectrum are evaluated for the reconstructed attractor. A non-linear noise reduction scheme is employed to reduce the level of noise in the data. Previous investigations in pool boiling chaos, reported in literature were based on temperature measurements underneath the test surface consisting of single or few active nucleation sites. Previous studies have indicated the presence of low-dimensional behavior in nucleate boiling and high-dimensional behavior in CHF and film boiling. Currently, there is no study detailing the effects of multiple nucleation sites, subcooling and surface texturing on pool boiling dynamics. The investigation comprises of four parts: i) in situ micro-machining of Chromelalumel (K-type) TFT, ii) calibration of these sensors, iii) utilizing these sensors in pool boiling experiments iv) analysis of these fluctuations using techniques of nonlinear time series analysis. Ten TFT are fabricated on a rectangular silicon surface within an area of ~ 3.00 cm x 3.00 cm. The sensing junctions of the TFT measure 50 mm in width and 250 nm in depth. Surface temperature fluctuations of the order of i) 0.65-0.93 degrees C are observed near ONB ii) 2.3-6.5 degrees C in FDNB iii) 2.60-5.00 degrees C at CHF and iv) 2.3-3.5 degrees C in film boiling. Investigations show the possible presence of chaotic dynamics near CHF and in film-boiling in saturated and subcooled pool boiling. Fully-developed nucleate boiling (FDNB) is chaotic. No clear assessment of the dynamics could be made in the onset of nucleate boiling (ONB) and partial nucleate boiling (PNB) regimes due to the effects of noise. However, the frequency spectra in these regimes appear to have two independent frequencies and their integral combinations indicating a possible quasiperiodic bifurcation route to chaos. The dimensionality in FDNB, at CHF and in film-boiling is lower in saturated pool boiling as compared to values in corresponding regimes in subcooled pool boiling. Surface temperature fluctuations can damage electronic components and need to be carefully controlled. Understanding the nature of these fluctuations will aid in deciding the modeling approach for surface temperature transients on an electronic chip. Subsequently, the TFT signals can be employed in a suitable feedback control loop to prevent the occurrence of hotspots.

In this research we consider synchronization of oscillators. We use mechanical metronomes that are coupled through a mechanical medium. We investigate the problem for three different cases: 1) In passive coupling of two oscillators, the coupling medium is a one degree of freedom passive mechanical basis. The analysis of the system is supported by simulations of the proposed model and experimental results. 2) In another case, the oscillator is forced by an external input while the input is also affected by the oscillator. This feedback loop introduces dynamics to the whole system. For realization, we place the mechanical metronome on a one degree of freedom moving base. The movements of the base are a function of a feedback from the phase of the metronome. We study a family of functions for the reactions of the base and their impact on the behavior of the metronome. 3) We consider two metronomes located on a moving base. In this case the two metronomes oscillate and as the base is not freely moving, they are not directly coupled to each other. Now based on the feedbacks from the vision system, the base moves and hence the phases of the metronomes are affected by these movements. We study the space of possibilities for the movements of the base and consider impacts of the base movement on the synchronization of metronomes. We also show how such a system evolves in time.