<p>The numerical path integration method for solving stochastic differential equations is extended to solve systems up to six spatial dimensions, angular variables, and highly nonlinear systems - including systems that results in discontinuities in the response probability density function of the system. Novel methods to stabilize the numerical method and increase computation speed are presented and discussed. This includes the use of the fast Fourier transform (FFT) and some new spline interpolation methods. Some sufficient criteria for the path integration theory to be applicable is also presented. The development of complex numerical code is made possible through automatic code generation by scripting. The resulting code is applied to chaotic dynamical systems by adding a Gaussian noise term to the deterministic equation. Various methods and approximations to compute the largest Lyapunov exponent of these systems are presented and illustrated, and the results are compared. Finally, it is shown that the location and size of the additive noise term affects the results, and it is shown that additive noise for specific systems could make a non-chaotic system chaotic, and a chaotic system non-chaotic.</p>

The benefits of intentional friction damping to reduce the occurrence of wear and premature failure of turbomachinery bladed-disk assemblies are well known and many studies on this topic have focused on the analysis and prediction of the complicated nonlinear forced response exhibited by these structures. In this research, extensions of the recently introduced multi-harmonic Hybrid Frequency-Time method are proposed for the efficient analysis of the response of realistic structures featuring displacement-dependent nonlinearities, such as the friction and impact phenomena that may occur in the presence of friction dampers or when two parts of the same structure periodically contact each other. These theoretical extensions are adapted to the study of large scale, industrial bladed-disk structures that may feature cyclic symmetry or mistuning. Two analysis techniques are developed for the modeling of displacement-dependent nonlinearities. In the first technique friction dampers are modeled as nonlinear operators representing the contact forces acting on the blades, from the simple case of monodirectional friction with constant normal load to the more complex case of three dimensional contact with variable normal load. The analysis of the forced response of several nonlinear systems illustrates the capabilities of this approach as well as the complexity of the typical behavior exhibited by friction damped structures. The second technique introduced helps analyze structures experiencing intermittent contact or friction between two parts or sub-components of the same assembly. This method is applied to the study of the forced response of several simple systems and is used with great efficiency to predict the nonlinear behavior of a beam with a crack. This approach also allows the dampers to be modeled realistically as stand-alone components appended to the bladed disk assembly. In this case the bladed disk assembly as well as the friction dampers are modeled as independent structures that interact at their contacting interfaces. This allows the use of detailed finite element models of dampers rather than having to make simplifying assumptions regarding their geometry. These two methods are applied to the study of the nonlinear forced response a realistic bladed-disk assembly featuring a wedge damper model and a structure-like damper model.

<p>Phonocardiographic signals contain bioacoustic information reflecting the operation of the heart. Normally there are two heart sounds, and additional sounds indicate disease. If a third heart sound is present it could be a sign of heart failure whereas a murmur indicates defective valves or an orifice in the septal wall. The primary aim of this thesis is to use signal processing tools to improve the diagnostic value of this information. More specifically, three different methods have been developed:</p><p>• A nonlinear change detection method has been applied to automatically detect heart sounds. The first and the second heart sounds can be found using recurrence times of the first kind while the third heart sound can be found using recurrence times of the second kind. Most third heart sound occurrences were detected (98 %), but the amount of false extra detections was rather high (7 % of the heart cycles).</p><p>• Heart sounds obscure the interpretation of lung sounds. A new method based on nonlinear prediction has been developed to remove this undesired disturbance. High similarity was obtained when comparing actual lung sounds with lung sounds after removal of heart sounds.</p><p>• Analysis methods such as Shannon energy, wavelets and recurrence quantification analysis were used to extract information from the phonocardiographic signal. The most prominent features, determined by a feature selection method, were used to create a new feature set for heart murmur classification. The classification result was 86 % when separating patients with aortic stenosis, mitral insufficiency and physiological murmurs.</p><p>The derived methods give reasonable results, and they all provide a step forward in the quest for an intelligent stethoscope, a universal phonocardiography tool able to enhance auscultation by improving sound quality, emphasizing abnormal events in the heart cycle and distinguishing different heart murmurs.</p>

This paper offers empirical evidence on the power of Sornette et al's [2001] model of bubbles and crashes regarding the German stock market between 1960 and 2009. We identify relevant time periods and describe them with the function given by Sornette et al's model. Our results show some evidence in predicting crashes with the understanding of logarithmic periodic structures that are hidden in the stock price trajectories. It was shown that for the DAX most of the relevant parameters determining the shape of the logarithmic periodic structures are lying in the expected interval researched by Sornette et al. Further more the paper implicitly shows that the point of time of former crashes can be predicted with the presented formula. We conclude that the concept of financial time series conceived as purely random objects should be generalised as to admit complexity.

Gegenstand dieser Arbeit ist die Untersuchung universeller Skalengesetze, die in gekoppelten chaotischen Systemen beobachtet werden. Ergebnisse werden erzielt durch das Ersetzen der chaotischen Fluktuationen in der Störungsdynamik durch stochastische Prozesse. <br /> <br /> Zunächst wird ein zeitkontinuierliches stochastisches Modell fürschwach gekoppelte chaotische Systeme eingeführt, um die Skalierung der Lyapunov-Exponenten mit der Kopplungsstärke (coupling sensitivity of chaos) zu untersuchen. Mit Hilfe der Fokker-Planck-Gleichung werden Skalengesetze hergeleitet, die von Ergebnissen numerischer Simulationen bestätigt werden. <br /> <br /> Anschließend wird der neuartige Effekt der vermiedenen Kreuzung von Lyapunov-Exponenten schwach gekoppelter ungeordneter chaotischer Systeme beschrieben, der qualitativ der Abstoßung zwischen Energieniveaus in Quantensystemen ähnelt. Unter Benutzung der für die coupling sensitivity of chaos gewonnenen Skalengesetze wird ein asymptotischer Ausdruck für die Verteilungsfunktion kleiner Abstände zwischen Lyapunov-Exponenten hergeleitet und mit Ergebnissen numerischer Simulationen verglichen.<br /> <br /> Schließlich wird gezeigt, dass der Synchronisationsübergang in starkgekoppelten räumlich ausgedehnten chaotischen Systemen einem kontinuierlichen Phasenübergang entspricht, mit der Kopplungsstärke und dem Synchronisationsfehler als Kontroll- beziehungsweise Ordnungsparameter. Unter Benutzung von Ergebnissen numerischer Simulationen sowie theoretischen Überlegungen anhand einer partiellen Differentialgleichung mit multiplikativem Rauschen werden die Universalitätsklassen der zwei beobachteten Übergangsarten bestimmt (Kardar-Parisi-Zhang-Gleichung mit Sättigungsterm, gerichtete Perkolation). / Subject of this work is the investigation of universal scaling laws which are observed in coupled chaotic systems. Progress is made by replacing the chaotic fluctuations in the perturbation dynamics by stochastic processes. <br /> <br /> First, a continuous-time stochastic model for weakly coupled chaotic systems is introduced to study the scaling of the Lyapunov exponents with the coupling strength (coupling sensitivity of chaos). By means of the the Fokker-Planck equation scaling relations are derived, which are confirmed by results of numerical simulations. <br /> <br /> Next, the new effect of avoided crossing of Lyapunov exponents of weakly coupled disordered chaotic systems is described, which is qualitatively similar to the energy level repulsion in quantum systems. Using the scaling relations obtained for the coupling sensitivity of chaos, an asymptotic expression for the distribution function of small spacings between Lyapunov exponents is derived and compared with results of numerical simulations. <br /> <br /> Finally, the synchronization transition in strongly coupled spatially extended chaotic systems is shown to resemble a continuous phase transition, with the coupling strength and the synchronization error as control and order parameter, respectively. Using results of numerical simulations and theoretical considerations in terms of a multiplicative noise partial differential equation, the universality classes of the observed two types of transition are determined (Kardar-Parisi-Zhang equation with saturating term, directed percolation).

Ziel dieser Arbeit ist die Überwindung einer Differenz, die zwischen der Theorie der Phase bzw. der Phasendynamik und ihrer Anwendung in der Zeitreihenanalyse besteht: Während die theoretische Phase eindeutig bestimmt und invariant unter Koordinatentransformationen bzw. gegenüber der jeweils gewählten Observable ist, führen die Standardmethoden zur Abschätzung der Phase aus gegebenen Zeitreihen zu Resultaten, die einerseits von den gewählten Observablen abhängen und so andererseits das jeweilige System keineswegs in eindeutiger und invarianter Weise beschreiben. Um diese Differenz deutlich zu machen, wird die terminologische Unterscheidung von Phase und Protophase eingeführt: Der Terminus Phase wird nur für Variablen verwendet, die dem theoretischen Konzept der Phase entsprechen und daher das jeweilige System in invarianter Weise charakterisieren, während die observablen-abhängigen Abschätzungen der Phase aus Zeitreihen als Protophasen bezeichnet werden. Der zentrale Gegenstand dieser Arbeit ist die Entwicklung einer deterministischen Transformation, die von jeder Protophase eines selbsterhaltenden Oszillators zur eindeutig bestimmten Phase führt. Dies ermöglicht dann die invariante Beschreibung gekoppelter Oszillatoren und ihrer Wechselwirkung. Die Anwendung der Transformation bzw. ihr Effekt wird sowohl an numerischen Beispielen demonstriert - insbesondere wird die Phasentransformation in einem Beispiel auf den Fall von drei gekoppelten Oszillatoren erweitert - als auch an multivariaten Messungen des EKGs, des Pulses und der Atmung, aus denen Phasenmodelle der kardiorespiratorischen Wechselwirkung rekonstruiert werden. Abschließend wird die Phasentransformation für autonome Oszillatoren auf den Fall einer nicht vernachlässigbaren Amplitudenabhängigkeit der Protophase erweitert, was beispielsweise die numerischen Bestimmung der Isochronen des chaotischen Rössler Systems ermöglicht. / The aim of this work is to bridge the gap between the theoretical description of the phase dynamics of coupled oscillators and the application of the theory to model reconstruction from time series analysis. In the theory, the phase of a self-sustained oscillator is defined in an unambiguous way, whereas the standard techniques used to estimate phases from given time series provide observabledependent results, so that generally these estimates deviate from the true phase. To stress this crucial issue, we term the observable-dependent phase-like variables as protophases. The main goal of this work is to develop a deterministic transformation from arbitrary protophases to the true, unique phase of the selfsustained oscillator. This approach allows us to obtain an invariant description of coupled oscillators and of their interaction. The application of the transformation and its efficiency are illustrated by means of numerical examples, as well as by the reconstruction of phase models of the cardiorespiratory interaction from multivariate time series of ECG, pulse and respiration. Next, the transformation from protophases to phases is extended for the case of three coupled oscillators. Finally, we go beyond the phase approximation and extend the phase transformation for autonomous oscillators to the case when the amplitude dynamics cannot be neglected. This technique for example allows us to compute numerically the isochrones of the chaotic Roessler system.

In the present work synchronization phenomena in complex dynamical systems exhibiting multiple time scales have been analyzed. Multiple time scales can be active in different manners. Three different systems have been analyzed with different methods from data analysis. The first system studied is a large heterogenous network of bursting neurons, that is a system with two predominant time scales, the fast firing of action potentials (spikes) and the burst of repetitive spikes followed by a quiescent phase. This system has been integrated numerically and analyzed with methods based on recurrence in phase space. An interesting result are the different transitions to synchrony found in the two distinct time scales. Moreover, an anomalous synchronization effect can be observed in the fast time scale, i.e. there is range of the coupling strength where desynchronization occurs. The second system analyzed, numerically as well as experimentally, is a pair of coupled CO₂ lasers in a chaotic bursting regime. This system is interesting due to its similarity with epidemic models. We explain the bursts by different time scales generated from unstable periodic orbits embedded in the chaotic attractor and perform a synchronization analysis of these different orbits utilizing the continuous wavelet transform. We find a diverse route to synchrony of these different observed time scales. The last system studied is a small network motif of limit cycle oscillators. Precisely, we have studied a hub motif, which serves as elementary building block for scale-free networks, a type of network found in many real world applications. These hubs are of special importance for communication and information transfer in complex networks. Here, a detailed study on the mechanism of synchronization in oscillatory networks with a broad frequency distribution has been carried out. In particular, we find a remote synchronization of nodes in the network which are not directly coupled. We also explain the responsible mechanism and its limitations and constraints. Further we derive an analytic expression for it and show that information transmission in pure phase oscillators, such as the Kuramoto type, is limited. In addition to the numerical and analytic analysis an experiment consisting of electrical circuits has been designed. The obtained results confirm the former findings. / In der vorliegenden Arbeit wurden Synchronisationsphänomene in komplexen Systemen mit mehreren Zeitskalen untersucht. Es gibt mehrere Möglichkeiten wie diese verschiedenen Zeitskalen vorkommen können. Drei verschiedene Systeme, jedes mit einer anderen Art von zeitlicher Multiskalität, wurden mit unterschiedlichen Methoden der Datenanalyse untersucht. Das erste untersuchte System ist ein ausgedehntes heterogenes Netzwerk von Neuronen mit zwei dominanten Zeitskalen, zum einen die schnelle Folge von Aktionspotenzialen und zum anderen einer abwechselnden Folge von einer Phase von Aktionspotenzialen und einer Ruhephase. Dieses System wurde numerisch integriert und mit Methoden der Phasenraumrekurrenz untersucht. Ein interessantes Ergebnis ist der unterschiedliche Übergang zur Synchronisation der Neuronen auf den beiden verschiedenen Zeitskalen. Des weiteren kann auf der schnellen Zeitskala eine anomale Synchronisation beobachtet werden, d.h. es gibt einen Bereich der Kopplungsstärke in dem es zu einer Desynchronisation kommt. Als zweites wurde, sowohl numerisch als auch experimentell, ein System von gekoppelten CO₂ Lasern untersucht, welche in einem chaotischen bursting Modus arbeiten. Dieses System ist auch durch seine Äquivalenz zu Epidemiemodellen interessant. Wir erklären die Bursts durch unterschiedliche Zeitskalen, welche durch in den chaotischen Attraktor eingebettete instabile periodische Orbits generiert werden. Wir führen eine Synchronisationsanalyse mit Hilfe der kontinuierlichen Wavelettransformation durch und finden einen unterschiedlichen Übergang zur Synchronisation auf den unterschiedlichen Zeitskalen. Das dritte analysierte System ist ein Netzwerkmotiv von Grenzzyklusoszillatoren. Genauer handelt es sich um ein Nabenmotiv, welches einen elementaren Grundbaustein von skalenfreien Netzwerken darstellt, das sind Netzwerke die eine bedeutende Rolle in vielen realen Anwendungen spielen. Diese Naben sind von besonderer Bedeutung für die Kommunikation und den Informationstransport in komplexen Netzwerken. Hierbei wurde eine detaillierte Untersuchung des Synchronisationsmechanismus in oszillatorischen Netzwerken mit einer breiten Frequenzverteilung durchgeführt. Insbesondere beobachten wir eine Fernsynchronisation von Netzwerkknoten, die nur indirekt über andere Oszillatoren miteinander gekoppelt sind. Wir erklären den zu Grunde liegenden Mechanismus und zeigen dessen Grenzen und Bedingungen auf. Des weiteren leiten wir einen analytischen Ausdruck für den Mechanismus her und zeigen, dass eine Informationsübertragung in reinen Phasenoszillatoren, wie beispielsweise vom Kuramototyp, eingeschränkt ist. Diese Ergebnisse konnten wir durch Experimente mit elektrischen Schaltkreisen bestätigen.

The numerical path integration method for solving stochastic differential equations is extended to solve systems up to six spatial dimensions, angular variables, and highly nonlinear systems - including systems that results in discontinuities in the response probability density function of the system. Novel methods to stabilize the numerical method and increase computation speed are presented and discussed. This includes the use of the fast Fourier transform (FFT) and some new spline interpolation methods. Some sufficient criteria for the path integration theory to be applicable is also presented. The development of complex numerical code is made possible through automatic code generation by scripting. The resulting code is applied to chaotic dynamical systems by adding a Gaussian noise term to the deterministic equation. Various methods and approximations to compute the largest Lyapunov exponent of these systems are presented and illustrated, and the results are compared. Finally, it is shown that the location and size of the additive noise term affects the results, and it is shown that additive noise for specific systems could make a non-chaotic system chaotic, and a chaotic system non-chaotic.

Today's product development process should be rapid andcost-efficient, and should result in innovative and reliableproducts. A crucial factor is the dynamic behaviour of theproduct. This thesis focuses on theoretical, numerical andexperimental approaches to achieve a comprehensiveunderstanding of dynamical phenomena occurring in nonlinearproducts, especially in products with parts that includeimpacts. The aim is to show the usefulness of nonlineartheories to better understand and optimise the dynamicbehaviour of products and thereby account for nonlinearphenomena already in the product development process. This is achieved through an investigation of researchefforts in the field of nonlinear dynamics; identification ofimportant research directions; a study on the effect ofcouplings between nonlinear parts; a detailed study on thedynamic behaviour of a product component; investigations oflow-cost strategies for controlling the dynamics of a nonlinearsystem; and the design and implementation of experimentalset-ups of two studied products. The investigation of research efforts shows that nonlinearparts are frequently included in products. Most common areparts that are nonlinear due to impacts and friction. Twoimportant areas are identified; to study coupling effectsbetween nonlinear subsystems and to study how nonlinearanalysis can be used to improve existing designs. Considering the studied products; a pantograph on a trainand a Braille printer, it can be concluded that thecharacteristics of a part can largely affect the dynamicbehaviour of the product. Typical nonlinear behaviour, such ascoexisting solutions and irregular motions, do occur. Theanalysis of the pan- tograph motion shows important aspects toconsider in the modelling process; coupling effects. In thecase of the Braille printer it is shown possible to create alow-cost control, by taking advantage of an existingdiscontinuity, to achieve a desired motion. Altogether, this work contributes to improved understandingof the be- haviour of nonlinear parts in products, especiallythose including impacts, pro- viding greater knowledge aboutaspects to consider in the design process. Keywords:Nonlinear Dynamics, Impacts, Discontinuities,Subsystems, Chaos, Irregular Behaviour, Printer Dynamics,Suspensions, Coupled Systems, Control. / QC 20100621

This thesis develops methods to exploit static and dynamic instabilities in electrostatic MEMS to develop new MEMS devices, namely dynamically actuated micro switches and binary micro gas sensors. Models are developed for the devices under consideration where the structures are treated as elastic continua. The electrostatic force is treated as a nonlinear function of displacement derived under the assumption of parallel-plate theorem. The Galerkin method is used to discretize the distributed-parameter models, thus reducing the governing partial differential equations into sets of nonlinear ordinary-differential equations. The shooting method is used to numerically solve those equations to obtain the frequency-response curves of those devices and the Floquet theory is used to investigate their stability. To develop the dynamically actuated micro switches, we investigate the response of microswitches to a combination of DC and AC excitations. We find that dynamically actuated micro switches can realize significant energy savings, up to 60 %, over comparable switches traditionally actuated by pure DC voltage. We devise two dynamic actuation methods: a fixed-frequency method and a shifted-frequency method. While the fixed-frequency method is simpler to implement, the shifted-frequency method can minimize the switching time to the same order as that realized using traditional DC actuation. We also introduce a parameter identification technique to estimate the switch geometrical and material properties, namely thickness, modulus of elasticity, and residual stress. We also develop a new detection technique for micro mass sensors that does not require any readout electronics. We use this method to develop static and dynamic binary mass sensors. The sensors are composed of a cantilever beam connected to a rigid plate at its free end and electrostatically coupled to an electrode underneath it. Two versions of micro mass sensors are presented: static binary mass sensor and dynamic binary mass sensor. Sensitivity analysis shows that the sensitivity of our static mass sensor represents an upper bound for the sensitivity of comparable statically detected inertial mass sensors. It also shows that the dynamic binary mass sensors is three orders of magnitude more sensitive than the static binary mass sensor. We equip our mass sensor with a polymer detector, doped Polyaniline, to realize a formaldehyde vapor sensor and demonstrate its functionality experimentally. We find that while the static binary gas sensor is simpler to realize than the dynamic binary gas sensor, it is more susceptible to external disturbances.