Rozprawy doktorskie na temat „Nonlinear”

Discrete optical systems are a subgroup of periodic structures in which the evolution of a continuous electromagnetic field can be described by a discrete model. In this model, the total field is the sum of localized, discrete modes. Weakly coupled arrays of single mode channel waveguides have been known to fall into this class of systems since the late 1960's. Nonlinear discrete optics has received a considerable amount of interest in the last few years, triggered by the experimental realization of discrete solitons in a Kerr nonlinear AlGaAs waveguide array by H. Eisenberg and coworkers in 1998. In this work a detailed experimental investigation of discrete nonlinear wave propagation and the interactions between beams, including discrete solitons, in discrete systems is reported for the case of a strong Kerr nonlinearity. The possibility to completely overcome "discrete" diffraction and create highly localized solitons, in a scalar or vector geometry, as well as the limiting factors in the formation of such nonlinear waves is discussed. The reversal of the sign of diffraction over a range of propagation angles leads to the stability of plane waves in a material with positive nonlinearity. This behavior can not be found in continuous self-focusing materials where plane waves are unstable against perturbations. The stability of plane waves in the anomalous diffraction region, even at highest powers, has been experimentally verified. The interaction of high power beams and discrete solitons in arrays has been studied in detail. Of particular interest is the experimental verification of a theoretically predicted unique, all optical switching scheme, based on the interaction of a so called "blocker" soliton with a second beam. This switching method has been experimentally realized for both the coherent and incoherent case. Limitations of such schemes due to nonlinear losses at the required high powers are shown.
Ph.D.
Other
Optics and Photonics
Optics

Dans cette thèse, nous sommes intéressés par des problèmes de préservation des applications non-linéaires entre deux algèbres de Banach complexes unitaires A et B. En général, ces problèmes demandent la caractérisation des applications φ : A → B non nécessairement linéaires, qui laissent invariant une propriété, une relation ou un sous-ensemble. Dans le Chapitre 3, la description des applications surjectives φ de B(X) sur B(Y), qui satisfont c(φ(S)±φ(T)) = c(S ± T), (S, T ∈ B(X)), est donnée, où c(·) représente soit le module minimal, ou le module de surjectivité ou le module maximal et B(X) (resp. B(Y)) dénote l’algèbre de tous les opérateurs linéaires et bornés sur X (resp. sur Y). Dans le Chapitre 4, une question similaire pour la conorme des opérateurs, est considérée. La caractérisation des applications bicontinues et bijectives φ deB(X) surB(Y), qui satisfont γ(φ(S ± φ(T)) = γ(S ± T), (S, T ∈ B(X)), est obtenue. Le Chapitre 5 est consacré à la description des applications surjectives φ1, φ2 d’une algèbre de Banach semisimple A sur une algèbre de Banach B avec un socle essentiel, qui satisfont σ(φ1(a)φ2(b)) = σ(ab), (a, b ∈ A). Aussi, la caractérisation des applications φ de A sur B, sous les mêmes hypothèses sur A et B, qui satisfont σ(φ(a)φ(b)φ(a)) = σ(aba), (a, b ∈ A), est donnée. Comme conséquences, nous incluons les résultats obtenus au cas des algèbres B(X) et B(Y).
In this thesis, we are interested in nonlinear preserver problems. In a general formulation, these demand the characterization of a map φ : A → B, which is not supposed to be linear and leaves a certain property, particular relation, or even a subset invariant, where A and B are complex Banach algebras with unit. In Chapter 3, the description of maps φ from B(X) onto B(Y) satisfying c(φ(S)±φ(T)) = c(S ± T), (S, T ∈ B(X)), is given, where c(·) stands either for the minimum modulus, or the surjectivity modulus, or the maximum modulus and B(X) (resp. B(Y)) denotes the algebra of all bounded linear operators on a Banach space X (resp. on Y). In Chapter 4, a similar question for the reduced minimum modulus of operators, is considered. The characterization of bijective bicontinuous maps φ from B(X) to B(Y) satisfying γ(φ(S ± φ(T)) = γ(S ± T), (S, T ∈ B(X)), is obtained. Chapter 5 is devoted to description of maps φ1, φ2 from a semisimple Banach algebra A onto a Banach algebra B with an essential socle, that satisfy σ(φ1(a)φ2(b)) = σ(ab), (a, b ∈ A). Also, the characterization of maps φ from A onto B, under the same assumptions on A and B, satisfying σ(φ(a)φ(b)φ(a)) = σ(aba), (a, b ∈ A), is given. The corollaries for algebras B(X) and B(Y), that follow immediately from the results, are included.

A coupled process one-dimensional model with two-region transport, two-domain nonlinear sorption, and nonlinear biodegradation is formulated in this research. A numerical code is developed for this complex system with two sets of initial/boundary conditions. The second order upwind method is used to solve PDEs of the system, and the Adam-Bashforth three step method is used to solve ODEs of the system. By nondimensionalizing the governing equations for transport and nonlinear biodegradation, we show that biodegradation is controlled by three characteristic combined factors: the effective maximum specific growth rate, the relative half-saturation constant, and the relative substrate-utilization coefficient. A diagram with type curves was constructed based on the three characteristic factors to show the conditions under which complete and incomplete biodegradation is observed, and the conditions for which the linear, first-order approximation is valid for representing biodegradation. Analytical and numerical approaches were used to study the effect of substrate boundary concentration on biodegradation in a coupled-process system. For a system with fixed biotic and abiotic properties, substrate input concentration could be positively or negatively correlated to the magnitude of substrate degradation, depending on the time scale of the process. The relative scale of substrate concentration and its half-saturation constant is very important for the success and efficiency of bioremediation. It is found that bioremediation can be more efficient for higher concentration contaminant under certain conditions. The impact of biodegradation on solute transport with linear or nonlinear, equilibrium sorption was studied by using moments analysis. Computation results show that linear biodegradation has no impact on spatial moments of transport with linear instantaneous sorption. Conversely, it has an impact when sorption is nonlinear, since nonlinear sorption is enhanced by biodegradation. Nonlinear biodegradation causes preferential non-uniform substrate degradation and, therefore, affects spatial moments of transport with linear or nonlinear sorption. The oxygen constraint decreases the degree of nonlinear biodegradation and increases the degree of preferential degradation, thus it also impacts spatial moments of transport with linear or nonlinear sorption.

This thesis describes new developments in nonlinear controllers for industrial applications. It first introduces the Nonlinear Generalised Minimum Variance (NGMV) control algorithm, for Linear Parameter Varying systems (LPV). This combines the benefits of the basic NGMV algorithm in dealing with nonlinearities, where a black box input model can be used, and adds an option to also approximate a nonlinear system with an LPV output subsystem. The models can therefore represent LPV systems and characteristics including saturation, discontinuities and time-varying dynamics. The next major contribution is in the nonlinear predictive control algorithms proposed that are also using the LPV model structure. The simplest is the Nonlinear Generalized Predictive Control (NGPC) algorithm that relates to the best known model predictive control law for linear systems. The final predictive control solution is one that may be specialized to either the NGMV or NGPC cases and is therefore the most general. This is referred to as a Nonlinear Predictive Generalized Minimum Variance Controller (NPGMV). When the algorithms use only the LPV structure to approximate the nonlinear system the solutions are particularly simple in unconstrained and constrained versions, and are relatively light computationally for implementation. Three representative industrial design examples have been chosen to validate the algorithms for different Bandwidth (BW) and nonlinear characteristics. All three examples were based on real application problems with company interest. In the first example (small BW) the basic state-space and LPV versions of the algorithm are used for the auto-manoeuvring and dynamic positioning of marine vessel. In this application the parameter variations were representative of wave disturbance changes with sea state, rather than due to approximating nonlinear behaviour. Actuator constraints were considered in the design. In the second industrial example (medium BW) the LPV-NPGMV was implemented for controlling the blade pitch and generator torque of a 5MW offshore wind turbine. The main objective here was to maintain the power produced at the rated value which requires compensation against wind disturbances, so that wind speed is the varying parameter. The LPV-NPGMV controller produced here used a parameterised system model involving the wind speed so that the controller performance changed with wind conditions. Actuator constraints were included and statistical performance assessed. The third example (fast BW) explores the stabilisation of a 2-axis gyroscopic electro-optical turret used in surveillance applications. This application was designed and employed on a real system. Because of the limitations imposed by BW requirements and the memory of the digital controller, only the basic state-space version of the algorithm was possible to implement. The main objective in this problem was to improve the tracking performance around the NADIR singularity point (a discontinuity) in the trajectory. In all three examples the NGMV controllers showed notable improvement in comparison to the baseline controllers without the need for scheduled gains or re-configuration when moving across different operating points.

In this thesis I investigate two methods for generating ultrashort laser pulses in spectral regions which are ordinarily difficult to achieve by the existing techniques. These pulses are specially attractive in the study of ultrafast (few femtosecond) atomic and molecular dynamics. The first involves Optical Parametric Amplification (OPA) mediated by four-wave-mixing in gas and supports the generation of ultrashort pulses in the Near-InfraRed (NIR) to the Mid-InfraRed (MIR) spectral region. By combining pulses at a centre wavelength of 800 nm and their second harmonic in an argon-filled hollow-core fibre, we demonstrate near-infrared pulses, peaked at 1.4 µm, with 5 µJ energy and 45 fs duration at the fibre output. The four-wave-mixing process involved in the OPA is expected to lead carrier-envelope phase stable pulses which is of great importance for applications in extreme nonlinear optics. These NIR to MIR pulses can be used directly for nonlinear light-matter interactions making use of its long-wavelength characteristics. The second method allows the compression of intense femtosecond pulses in the ultraviolet (UV) region by sum-frequency mixing two bandwidth limited NIR pulses in a noncollinear phasematching geometry under particular conditions of group-velocity mismatch. Specifically, the crystal has to be chosen such that the group velocities of the NIR pump pulses, v1 and v2 , and of the sum-frequency generated pulse, vSF, meet the following condition, v1 < vSF < v2. In case of strong energy exchange and an appropriate pre-delay between the pump waves, the leading edge of the faster pump pulse and the trailing edge of the slower one are depleted. This way the temporal overlap region of the pump pulses remains narrow resulting in the shortening of the upconverted pulse. The noncollinear beam geometry allows to control the relative group velocities while maintaining the phasematching condition. To ensure parallel wavefronts inside the crystal and that the sum-frequency generated pulses emerge untilted, pre-compensation of the NIR pulse-front tilts is essential. I show that these pulse-front tilts can be achieved using a very compact setup based on transmission gratings and a more complex setup based on prisms combined with telescopes. UV pulses as short as 32 fs (25 fs) have been generated by noncollinear nonlinear pulse compression in a type II phasematching BBO crystal, starting with NIR pulses of 74 fs (46 fs) duration. This is of interest, because there is no crystal that can be used for nonlinear pulse compression at wavelengths near 800 nm in a collinear geometry. Compared to state-of-the-art compression techniques based on self-phase modulation, pulse compression by sum-frequency generation is free of aperture limitation, and thus scalable in energy. Such femtosecond pulses in the visible and in the ultraviolet are strongly desired for studying ultrafast dynamics of a variety of (bio)molecular systems.
En esta tesis he investigado dos métodos para generar pulsos láser ultracortos en regiones espectrales que son típicamente difíciles de lograr con las técnicas existentes. Estos pulsos son especialmente atractivos en el estudio de la dinámica ultrarrápida (pocos femtosegundos) en átomos y moléculas. La primera técnica implica Amplificación Paramétrica Óptica (OPA) mediante mezcla de cuatro ondas en fase gaseosa y soporta la generación de pulsos ultracortos desde el Infrarrojo-Cercano (NIR) hasta la región espectral del Infrarrojo-Medio (MIR). Mediante la combinación de pulsos centrados a una longitud de onda de 800 nm y su segundo armónico en una fibra hueca rellena de argón, hemos demostrado a la salida de la fibra la generación de pulsos en el NIR, centrados a 1.4 µm, con 5 µJ de energía y 45 fs de duración. Se espera que el proceso de mezcla de cuatro ondas involucrado en el OPA lleve a pulsos con fase de la envolvente de la portadora estables, ya que es de gran importancia para aplicaciones en óptica extrema no lineal. Estos pulsos desde el NIR hasta el MIR se pueden utilizar directamente en interacciones no-lineales materia-radiación, haciendo uso de sus características de longitud de onda largas. El segundo método permite la compresión de pulsos intensos de femtosegundos en la región del ultravioleta (UV) mediante la mezcla de suma de frecuencias de dos pulsos en el NIR limitados en el ancho de banda en una geometría de ajuste de fases no-colineal bajo condiciones particulares de discrepancia de velocidades de grupo. Específicamente, el cristal debe ser elegido de tal manera que las velocidades de grupo de los pulsos de bombeo del NIR, v1 y v2, y la del pulso suma-de-frecuencias generado, vSF, cumplan la siguiente condición, v1 < vSF < v2. En el caso de un fuerte intercambio de energía y un pre-retardo adecuado entre las ondas de bombeo, el borde delantero del pulso de bombeo más rápido y el borde trasero del más lento se agotan. De esta manera la región de solapamiento temporal de los impulsos de bombeo permanece estrecha, resultando en el acortamiento del impulso generado. La geometría de haces no-colineales permite controlar las velocidades de grupo relativas mientras mantiene la condición de ajuste de fase. Para asegurar frentes de onda paralelos dentro del cristal y que los pulsos generados por suma de frecuencias se generen sin inclinación, es esencial la pre-compensación de la inclinación de los frente de onda de los pulsos NIR. En esta tesis se muestra que estas inclinaciones de los frentes de onda se pueden lograr utilizando una configuración muy compacta basada en rejillas de transmisión y una configuración más compleja basada en prismas combinados con telescopios. Pulsos en el UV tan cortos como 32 fs (25 fs) se han generado mediante compresión de pulsos no-lineal no-colineal en un cristal BBO de ajuste de fase tipo II, comenzando con pulsos en el NIR de 74 fs (46 fs) de duración. El interés de este método radica en la inexistencia de cristales que se puedan utilizar para la compresión de impulsos no-lineal a longitudes de onda entorno a 800 nm en una geometría colineal. En comparación con las técnicas de última generación de compresión basadas en la automodulación de fase, la compresión de pulsos por suma de frecuencias esta libre de restricciones en la apertura de los pulsos, y por lo tanto es expandible en energía. Tales pulsos de femtosegundos en el visible y en el ultravioleta son fuertemente deseados en el estudio de dinámica ultrarrápida de una gran variedad de sistemas (bio)moleculares.

This thesis analyzes the process of deblurring of degraded images generated by space-variant nonlinear image systems with Gaussian observation noise. The restoration of blurred images is performed by using two methods
a modified version of the Optimum Decoding Based Smoothing Algorithm and the Bootstrap Filter Algorithm which is a version of Particle Filtering methods. A computer software called MATLAB is used for performing the simulations of image estimation. The results of some simulations for various observation and image models are presented.

Twenty different materials have been successfully deposited as Langmuir- Blodgett monolayer films. All exhibit second harmonic generation (SHG) when irradiated with laser light at 1064 nm. E-1-docosyl-4-{2-(4-dimethylami nophenyl)ethenyl}quinolinium bromide (C22H45QHBr) and E-1-docosyl-4-{2-(4-dimethy laminonaphthyl)ethenyl}quinolinium bromide (C22H45QNBr) have been deposited separately as multilayer films. They form Y-type structures when deposition is alternated with the material N-docosyl-4- methylquinolinium bromide. The nonlinear responses are quadratic up to 20 and 10 bilayers respectively and the response from the thick films is only 2 orders less than that produced by a Y-cut quartz plate. Similar results were obtained with C22H45QHBr when interleaved with 4,4'-dioctadecyl-3,5,3', 5'- tetra me thyldipyrrylmethenehydrobromide. Ellipsometry studies of the 10 bilayer film of C H45QNBr indicate that the structure is interdigitated. This explains the stability of the film which gave the same SH response up to 6 months after deposition. A 10 bilayer films has also been fabricated using E-1- docosyl-4-{2-(4-{2-(4-dimethylaminophenyl)ethenyl}benzyl)ethenyl}pyridinium bromide (C22H45PBHBr) alternated with E-1-docosy1-4-{2-(4-methylphenyl)ethenyl}pyridinium bromide (C22H45PT). E-1-octadecyl-4-{2-(4-methyloxyphenyl)ethenyl}pyridinium iodide and E-1- methyl-4-{2-(4-octadecyloxyphenyl)ethenyl}pyridinium iodide have been fabricated into monolayer films that are transparent at 1064 and 532 nm, therefore resonant enhancement does not contribute to their nonlinear response which is attributed solely to charge transfer in the molecule. Mixed solutions of E-1-octadecyl-4-{2-(4-methyloxyphenyl)ethenyl}pyridinium iodide and sodium octadecylsulphate (C1SH37OSO3 Na+) have been deposited as very stable monolayers. The nonlinear response from the mixed film offers a significant improvement upon the performance of the film containing pure hemicyanine. Novel zwitterionic materials have been fabricated as LB monolayers that also exhibit SHG.

Noise or interference is often assumed to be a random process. Conventional linear filtering, control or prediction techniques are used to cancel or reduce the noise. However, some noise processes have been shown to be nonlinear and deterministic. These nonlinear deterministic noise processes appear to be random when analysed with second order statistics. As nonlinear processes are widespread in nature it may be beneficial to exploit the coherence of the nonlinear deterministic noise with nonlinear filtering techniques. The nonlinear deterministic noise processes used in this thesis are generated from nonlinear difference or differential equations which are derived from real world scenarios. Analysis tools from the theory of nonlinear dynamics are used to determine an appropriate sampling rate of the nonlinear deterministic noise processes and their embedding dimensions. Nonlinear models, such as the Volterra series filter and the radial basis function network are trained to model or predict the nonlinear deterministic noise process in order to reduce the noise in a system. The nonlinear models exploit the structure and determinism and, therefore, perform better than conventional linear techniques. These nonlinear techniques are applied to cancel broadband nonlinear deterministic noise which corrupts a narrowband signal. An existing filter method is investigated and compared with standard linear techniques. A new filter method is devised to overcome the restrictions of the existing filter method. This method combines standard signal processing concepts (filterbanks and multirate sampling) with linear and nonlinear modelling techniques. It overcomes the restrictions associated with linear techniques and hence produces better performance. Other schemes for cancelling broadband noise are devised and investigated using quantisers and cascaded radial basis function networks. Finally, a scheme is devised which enables the detection of a signal of interest buried in heavy chaotic noise. Active noise control is another application where the acoustic noise may be assumed to be a nonlinear deterministic process. One of the problems in active noise control is the inversion process of the transfer function of the loudspeaker. This transfer function may be nonminimum phase. Linear controllers only perform sub-optimally in modelling the noncausal inverse transfer function. To overcome this problem in conjunction with the assumption that the acoustic noise is nonlinear and deterministic a combined linear and nonlinear controller is devised. A mathematical expression for the combined controller is derived which consists of a linear system identification part and a nonlinear prediction part. The traditional filtered-x least mean squares scheme in active noise control does not allow the implementation of a nonlinear controller. Therefore, a control scheme is devised to allow a nonlinear controller in conjunction with an adaptive block least squares algorithm. Simulations demonstrate that the combined linear and nonlinear controller outperforms the conventional linear controller.

A hydraulically undersized control structure could result in water overtopping a dam or channel banks. To increase hydraulic capacity and reduce flooding risk, nonlinear spillways are frequently replacing linear weirs. This study investigates four subjects to further knowledge for two types of nonlinear weir, the piano key and labyrinth. Weir submergence is a condition when the downstream water level of a weir exceeds the weir crest elevation, and can influence the head-discharge relationship of the structure. The effects of submergence on laboratory-scale piano key weir head-discharge relationships were evaluated experimentally and compared to published submergence data for linear and labyrinth weirs. For relatively low levels of submergence, the piano key weir requires less upstream head relative to the labyrinth weir (<6%). This increase in efficiency was reversed at higher levels. Staged labyrinth weirs feature multiple weir segments with different crest elevations, which confine base flows and/or satisfy downstream discharge requirements. Head-discharge relationships for various laboratory-scale staged labyrinth weir configurations were established. The accuracy of a head-discharge predictive technique based upon superposition and traditional labyrinth weir empirical data was evaluated, and found to be generally within ?5%. The influence of linear, labyrinth, and staged labyrinth weir head-discharge characteristics on the outflow hydrograph behavior was evaluated by numerically routing various flood discharges through a fictitious reservoir; peak outflow, maximum water surface elevation, and required detention volume data are presented for each weir alternative. A staged labyrinth weir can be an effective alternative for decreasing the peak outflow hydrograph for frequent events, while increasing discharge for higher return period storm events. Approach flow perpendicular to the labyrinth weir centerline axis may not be possible in all situations. The head-discharge characteristics of a laboratory-scale labyrinth weir were evaluated with three different approach flow angles (0?, 15?, and 45?). For approach flow angles up to 15?, no measurable loss in discharge efficiency occurred. The discharge efficiency reduced as much as 11% for the 45? approach angle case. While all data presented are specific to the weir configurations and geometries tested, these data can be beneficial to the general understanding of nonlinear weirs.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Earth, Atmospheric, and Planetary Sciences, 1993.
Includes bibliographical references (p. 249-254).
by Joseph R. Materese.
Ph.D.

In many astrophysical problems magnetic reconnection plays a major role. Despite this reconnection theory remains incompletely understood, partly due to the strong non-linearity of the governing equations and the resulting difficulties in demonstrating analytical solutions. This thesis examines some fundamental aspects of reconnection theory: namely, the dynamics of driven and spontaneously reconnecting systems. We first consider the dynamics of a driven reconnecting system by numerically modelling a configuration consisting of two oppositely oriented flux systems with a variety of different boundary conditions and internal parameters. The results indicate that the rate of reconnection is chiefly dependent on the magnetic Reynolds number, but that the plasma flow is weakly dependent on this parameter, being more affected by the curvature of Incoming magnetic field. Scaling laws for the dimensions of the diffusion region are derived, and the existence of several reconnection regimes is shown. Using the same computer code we also simulate tearing modes in Cartesian geometry under different boundary conditions. By imposing a suitable perturbation a magnetic island is generated. The plasma flows show marked differences for the different boundary conditions implemented. Lastly, we examine some aspects of the coalescence instability. The usual flux function taken to represent a tearing node Island in the linear growth phase is shown to be erroneous, and we derive a correct expression. We show that under certain conditions there exists a threshold to coalescence that depends on the island wavenumbers and the associated perturbation.

We will present recent results on a special class of dynamical systems designated as nonlinear sonic vacua. These systems are non-linearizable, and have zero speed of sound (in the sense of classical acoustics). Accordingly, their dynamics and acoustics are highly degenerate and tunable with energy, enabling new and highly complex nonlinear phenomena. Two examples of sonic vacua will be discussed. The first is uncompressed ordered granular media, which, depending on their local state, behave either as strongly nonlinear and non-smooth dynamical systems (in the absence of strong local compression), or as almost linear coupled oscillators (under strong local compression, e.g., in the primary fronts of propagating solitary pulses) [1,2]. The second example concerns a spring-mass lattice in the plane. In the small energy limit this seemingly simple system is 'transformed' by geometric nonlinearity to a nonlinear sonic vacuum with surprising properties, such as strong nonlocality (despite o f only next-neighbor interactions in the lattice!), orthogonal nonlinear normal modes, and accelerating propagating fronts [3,4]. Interesting applications of nonlinear sonic vacua will be discussed, including intense energy cascading from low-to-high frequencies and long-to-short wavelengths, resembling “mechanical turbulence ”. We will discuss the implications of these findings on the design of dynamical systems for predictive passive energy management.

The concept of harvesting electrical energy from ambient vibration sources has been a popular topic of research in recent years. The motivation behind this research is largely due to recent advancements in microelectromechanical systems (MEMS) technology - specifically the construction of small low powered sensors which are capable of being placed in inaccessible or hostile environments. The main drawback with these devices is that they require an external power source. For example, if one considers large networks of low powered sensors (such as those which may be attached to a bridge as part of a structural health monitoring system) then one can envisage a scenario where energy harvesters are used to transfer the vibration energy of the bridge into electrical energy for the sensors. This would alleviate the need for batteries which, in this scenario, would be difficult to replace. Initial energy harvester designs suffered from a major flaw: they were only able to produce useful amounts of power if they were excited close to their resonant frequency. This narrow bandwidth of operation meant that they were poorly suited to harvesting energy from ambient vibration sources which are often broadband and have time dependent dominant frequencies. This led researchers to consider the concept of nonlinear energy harvesting - the hypothesis that the performance of energy harvesters could be improved via the deliberate introduction of dynamic nonlinearities. This forms the main focus of the work in this thesis. The first major part of this work is concerned with the development of an experimentally validated physical-law based model of an electromagnetic energy harvester with Duffing-type nonlinearities. To this end, a self-adaptive differential evolution vi (SADE) algorithm is used in conjunction with experimental data to estimate the parameters needed to accurately model the behaviour of the device. During this investigation it is found that the response of the energy harvesting device in question is very sensitive to the effects of friction. Consequently, a detailed study is undertaken with the aim of finding whether the model performance could be improved by accounting for this complex nonlinear phenomenon. After investigating several different friction models, a reliable and extensively validated digital model of a nonlinear energy harvesting device is realised. With the appropriate equations of motion identified, analytical approximation methods are used to analyse the response of the device to sinusoidal excitations. The motivation for the second main part of this work arises from the fact that ambient excitations are often stochastic in nature. As a result, much of the work in this section is directed towards gaining an understanding of how nonlinear energy harvesters respond to random excitations. This is an interesting problem because, as a result of the random excitation, it is impossible to say exactly how such a device will respond - the problem must be tackled using a probabilistic approach. To this end, the Fokker-Planck-Kolmogorov (FPK) equation is used to develop probability density functions describing how the nonlinear energy harvester in question responds to Gaussian white noise excitations. By conducting this analysis, previously unrecognised benefits of Duffing-type nonlinearities in energy harvesters are identified along with important findings with regards to device electrical optimisation. As for friction effects, the technique of equivalent linearisation is employed alongside known solutions of the FPK equation to develop expressions approximating the effect of friction on randomly excited energy harvesters. These results are then validated using Monte-Carlo methods thus revealing important results about the interaction between Duffing-type and friction nonlinearities. Having investigated sinusoidal and random excitations, the final part of this work focuses on the application of nonlinear energy harvesting techniques to real energy harvesting scenarios. Excitation data from human walking motion and bridge vibrations is used to excite digital models of a variety of recently proposed nonlinear energy harvesters. This analysis reveals important information with respect to how well energy harvesting solutions developed under the assumption of Gaussian white noise excitations can be extended to real world scenarios.

A hierarchy of vector fields (master symmetries) and homogeneous nonlinear Poisson structures associated with the Toda lattice are constructed and the various connections between them are investigated. Among their properties: new brackets are generated from old ones by using Lie-derivatives in the direction of certain vector fields; the infinite sequences obtained consist of compatible Poisson brackets in which the constants of motion for the Toda lattice are in involution. The vector fields in the construction are unique up to addition of a Hamiltonian vector field. Similarly the Poisson brackets are unique up to addition of a trivial Poisson bracket. These are Poisson tensors generated by wedge products of Hamiltonian vector fields. The non-trivial brackets may also be obtained by the use of r-matrices; we give formulas and prove this for the quadratic and cubic Toda brackets. We also indicate how these results can be generalized to other (semisimple) Toda flows and we give explicit formulas for the rank 2 Lie algebra of type B₂. The main tool in this calculation is Dirac's constraint bracket formula. Finally we study nonlinear Poisson brackets associated with orbits through nilpotent conjugacy classes in gl(n, R) and formulate some conjectures. We determine the degree of the transverse Poisson structure through such nilpotent elements in gl(n, R) for n ≤ 7. This is accomplished also by the use of Dirac's bracket formula.

In contrast to electromagnetic fields, matter-wave fields are intrinsically interacting due to the presence of atom-atom collisions. Hence, matter-wave optics becomes effectively nonlinear as soon as the atomic densities are high enough that collisions can no longer be ignored. The goal of this dissertation is to study selected aspects of atom optics under such conditions. Specifically, Chapter 2 studies the near-resonant dipole-dipole interaction between two atoms in tailored vacua. In contrast to spontaneous emission, whose rate is known to be influenced by the type of vacuum the atom interacts with, we find that the dipole-dipole potential is determined only by the free space vacuum and is not modified either by thermal or squeezed vacua. In addition in the far off-resonance regime we find that the squeezed vacuum results in an additional contribution to the effective potential governing the evolution of the atomic ground state. In the second part of the dissertation, which comprises Chapter 3, we then study several aspects of the many-body theory of atomic ultracold systems in situations where the nonlinearity arises due to the two-body dipole-dipole interaction. After a formal theoretical development we discuss the possibility of using atomic phase conjugation off Bose condensates as a diagnostic tool to access the spatial coherence properties and to measure the lifetime of the condensate. We argue that phase conjugation provides an attractive alternative to the optical methods of probing condensate proposed in the past. We further study the elementary excitations in a multicomponent Bose condensates and determine the quasi-particle frequency spectrum. We show that in that case interferences resulting from cross-coupling between the condensate components can lead to a reversal of the sign of the effective two-body interaction and to the onset of spatial instabilities.

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia Elétrica, Florianópolis, 2016.
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Abstract : Mixing phenomena in hyperspectral images depend on a variety of factors such as the resolution of observation devices, the properties of materials, and how these materials interact with incident light in the scene. Different parametric and nonparametric models have been considered to address hyperspectral unmixing problems. The simplest one is the linear mixing model. Nevertheless, it has been recognized that mixing phenomena can also be nonlinear. Kernel-based nonlinear mixing models have been applied to unmix spectral information of hyperspectral images when the type of mixing occurring in the scene is too complex or unknown. However, the corresponding nonlinear analysis techniques are necessarily more challenging and complex than those employed for linear unmixing. Within this context, it makes sense to search for different strategies to produce simpler and/or more accurate results. In this thesis, we tackle three distinct parts of the complete spectral unmixing (SU) problem. First, we propose a technique for detecting nonlinearly mixed pixels. The detection approach is based on the comparison of the reconstruction errors using both a Gaussian process regression model and a linear regression model. The two errors are combined into a detection test statistics for which a probability density function can be reasonably approximated. Second, we propose an iterative endmember extraction algorithm to be employed in combination with the detection algorithm. The proposed detect-then-unmix strategy, which consists of extracting endmembers, detecting nonlinearly mixed pixels and unmixing, is tested with synthetic and real images. Finally, we propose two methods for band selection (BS) in the reproducing kernel Hilbert space (RKHS), which lead to a significant reduction of the processing time required by nonlinear unmixing techniques. The first method employs the kernel k-means (KKM) algorithm to find clusters in the RKHS. Each cluster centroid is then associated to the closest mapped spectral vector. The second method is centralized, and it is based upon the coherence criterion, which sets the largest value allowed for correlations between the basis kernel functions characterizing the unmixing model. We show that the proposed BS approach is equivalent to solving a maximum clique problem (MCP), that is, to searching for the largest complete subgraph in a graph. Furthermore, we devise a strategy for selecting the coherence threshold and the Gaussian kernel bandwidth using coherence bounds for linearly independent bases. Simulation results illustrate the efficiency of the proposed method.

Imagem hiperespectral (HI) é uma imagem em que cada pixel contém centenas (ou até milhares) de bandas estreitas e contíguas amostradas num amplo domínio do espectro eletromagnético. Sensores hiperespectrais normalmente trocam resolução espacial por resolução espectral devido principalmente a fatores como a distância entre o instrumento e a cena alvo, e limitada capacidade de processamento, transmissão e armazenamento históricas, mas que se tornam cada vez menos problemáticas. Este tipo de imagem encontra ampla utilização em uma gama de aplicações em astronomia, agricultura, imagens biomédicas, geociências, física, vigilância e sensoriamento remoto. A usual baixa resolução espacial de sensores espectrais implica que o que se observa em cada pixel é normalmente uma mistura das assinaturas espectrais dos materiais presentes na cena correspondente (normalmente denominados de endmembers). Assim um pixel em uma imagem hiperespectral não pode mais ser determinado por um tom ou cor mas sim por uma assinatura espectral do material, ou materiais, que se encontram na região analisada. O modelo mais simples e amplamente utilizado em aplicações com imagens hiperespectrais é o modelo linear, no qual o pixel observado é modelado como uma combinação linear dos endmembers. No entanto, fortes evidências de múltiplas reflexões da radiação solar e/ou materiais intimamente misturados, i.e., misturados em nível microscópico, resultam em diversos modelos não-lineares dos quais destacam-se os modelos bilineares, modelos de pós não-linearidade, modelos de mistura íntima e modelos não-paramétricos. Define-se então o problema de desmistura espectral (ou em inglês spectral unmixing - SU), que consiste em determinar as assinaturas espectrais dos endmembers puros presentes em uma cena e suas proporções (denominadas de abundâncias) para cada pixel da imagem. SU é um problema inverso e por natureza cego uma vez que raramente estão disponíveis informações confiáveis sobre o número de endmembers, suas assinaturas espectrais e suas distribuições em uma dada cena. Este problema possui forte conexão com o problema de separação cega de fontes mas difere no fato de que no problema de SU a independência de fontes não pode ser considerada já que as abundâncias são de fato proporções e por isso dependentes (abundâncias são positivas e devem somar 1). A determinação dos endmembers é conhecida como extração de endmembers e a literatura apresenta uma gama de algoritmos com esse propósito. Esses algoritmos normalmente exploram a geometria convexa resultante do modelo linear e da restrições sobre as abundâncias. Quando os endmembers são considerados conhecidos, ou estimados em um passo anterior, o problema de SU torna-se um problema supervisionado, com pares de entrada (endmembers) e saída (pixels), reduzindo-se a uma etapa de inversão, ou regressão, para determinar as proporções dos endmembers em cada pixel. Quando modelos não-lineares são considerados, a literatura apresenta diversas técnicas que podem ser empregadas dependendo da disponibilidade de informações sobre os endmembers e sobre os modelos que regem a interação entre a luz e os materiais numa dada cena. No entanto, informações sobre o tipo de mistura presente em cenas reais são raramente disponíveis. Nesse contexto, métodos kernelizados, que assumem modelos não-paramétricos, têm sido especialmente bem sucedidos quando aplicados ao problema de SU. Dentre esses métodos destaca-se o SK-Hype, que emprega a teoria de mínimos quadrados-máquinas de vetores de suporte (LS-SVM), numa abordagem que considera um modelo linear com uma flutuação não-linear representada por uma função pertencente a um espaço de Hilbert de kernel reprodutivos (RKHS). Nesta tese de doutoramento diferentes problemas foram abordados dentro do processo de SU de imagens hiperespectrais não-lineares como um todo. Contribuições foram dadas para a detecção de misturas não-lineares, estimação de endmembers quando uma parte considerável da imagem possui misturas não-lineares, e seleção de bandas no espaço de Hilbert de kernels reprodutivos (RKHS). Todos os métodos foram testados através de simulações com dados sintéticos e reais, e considerando unmixing supervisionado e não-supervisionado. No Capítulo 4, um método semi-paramétrico de detecção de misturas não-lineares é apresentado para imagens hiperespectrais. Esse detector compara a performance de dois modelos: um linear paramétrico, usando mínimos-quadrados (LS), e um não-linear não-paramétrico usando processos Gaussianos. A idéia da utilização de modelos não-paramétricos se conecta com o fato de que na prática pouco se sabe sobre a real natureza da não-linearidade presente na cena. Os erros de ajuste desses modelos são então comparados em uma estatística de teste para a qual é possível aproximar a distribuição na hipótese de misturas lineares e, assim, estimar um limiar de detecção para uma dada probabilidade de falso-alarme. A performance do detector proposto foi estudada considerando problemas supervisionados e não-supervisionados, sendo mostrado que a melhoria obtida no desempenho SU utilizando o detector proposto é estatisticamente consistente. Além disso, um grau de não-linearidade baseado nas energias relativas das contribuições lineares e não-lineares do processo de mistura foi definido para quantificar a importância das parcelas linear e não-linear dos modelos. Tal definição é importante para uma correta avaliação dos desempenhos relativos de diferentes estratégias de detecção de misturas não-lineares. No Capítulo 5 um algoritmo iterativo foi proposto para a estimação de endmembers como uma etapa de pré-processamento para problemas SU não supervisionados. Esse algoritmo intercala etapas de detecção de misturas não-lineares e estimação de endmembers de forma iterativa, na qual uma etapa de estimação de endmembers é seguida por uma etapa de detecção, na qual uma parcela dos pixels mais não-lineares é descartada. Esse processo é repetido por um número máximo de execuções ou até um critério de parada ser atingido. Demonstra-se que o uso combinado do detector proposto com um algoritmo de estimação de endmembers leva a melhores resultados de SU quando comparado com soluções do estado da arte. Simulações utilizando diferentes cenários corroboram as conclusões. No Capítulo 6 dois métodos para SU não-linear de imagens hiperespectrais, que empregam seleção de bandas (BS) diretamente no espaço de Hilbert de kernels reprodutivos (RKHS), são apresentados. O primeiro método utiliza o algoritmo Kernel K-Means (KKM) para encontrar clusters diretamente no RKHS onde cada centroide é então associada ao vetor espectral mais próximo. O segundo método é centralizado e baseado no critério de coerência, que incorpora uma medida da qualidade do dicionário no RKHS para a SU não-linear. Essa abordagem centralizada é equivalente a resolver um problema de máximo clique (MCP). Contrariamente a outros métodos concorrentes que não incluem uma escolha eficiente dos parâmetros do modelo, o método proposto requer apenas uma estimativa inicial do número de bandas selecionadas. Os resultados das simulações empregando dados, tanto sintéticos como reais, ilustram a qualidade dos resultados de unmixing obtidos com os métodos de BS propostos. Ao utilizar o SK-Hype, para um número reduzido de bandas, são obtidas estimativas de abundância tão precisas quanto aquelas obtidas utilizando o método SK-Hype com todo o espectro disponível, mas com uma pequena fração do custo computacional.

A hydraulically undersized control structure could result in water overtopping a dam or channel banks. To increase hydraulic capacity and reduce flooding risk, nonlinear spillways are frequently replacing linear weirs. This study investigates four subjects to further knowledge for two types of nonlinear weir, the piano key and labyrinth. Weir submergence is a condition when the downstream water level of a weir exceeds the weir crest elevation, and can influence the head-discharge relationship of the structure. The effects of submergence on laboratory-scale piano key weir head-discharge relationships were evaluated experimentally and compared to published submergence data for linear and labyrinth weirs. For relatively low levels of submergence, the piano key weir requires less upstream head relative to the labyrinth weir (<6%). This increase in efficiency was reversed at higher levels. Staged labyrinth weirs feature multiple weir segments with different crest elevations, which confine base flows and/or satisfy downstream discharge requirements. Head-discharge relationships for various laboratory-scale staged labyrinth weir configurations were established. The accuracy of a head-discharge predictive technique based upon superposition and traditional labyrinth weir empirical data was evaluated, and found to be generally within ±5%. The influence of linear, labyrinth, and staged labyrinth weir head-discharge characteristics on the outflow hydrograph behavior was evaluated by numerically routing various flood discharges through a fictitious reservoir; peak outflow, maximum water surface elevation, and required detention volume data are presented for each weir alternative. A staged labyrinth weir can be an effective alternative for decreasing the peak outflow hydrograph for frequent events, while increasing discharge for higher return period storm events. Approach flow perpendicular to the labyrinth weir centerline axis may not be possible in all situations. The head-discharge characteristics of a laboratory-scale labyrinth weir were evaluated with three different approach flow angles (0°, 15°, and 45°). For approach flow angles up to 15°, no measurable loss in discharge efficiency occurred. The discharge efficiency reduced as much as 11% for the 45° approach angle case. While all data presented are specific to the weir configurations and geometries tested, these data can be beneficial to the general understanding of nonlinear weirs.

We study nonlinear fractional convection-diffusion equations with nonlocal and nonlinear fractional diffusion. By the idea of Kru\v{z}kov (1970), entropy sub- and supersolutions are defined in order to prove well-posedness under the assumption that the solutions are elements in $L^{\infty}(\mathbb{R}^d\times (0,T))\cap C([0,T];L_\text{loc}^1(\mathbb{R}^d))$. Based on the work of Alibaud (2007) and Cifani and Jakobsen (2011), a local contraction is obtained for this type of equations for a certain class of L\'evy measures. In the end, this leads to an existence proof for initial data in $L^{\infty}(\mathbb{R}^d)$

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 89-96).
In this thesis, an iterative nonlinear beam propagation method is introduced and applied to optical devices. This method is based on Hamiltonian ray tracing and the Wigner distribution function. First, wave propagation simulation using Hamiltonian ray tracing is illustrated and verified with different examples. Based on this, the iterative method is presented for beam propagation in nonlinear media, which is validated with common Kerr effect phenomena such as self-focusing and spatial solitons. As the application to the analysis of nonlinear optical devices, this method is applied to nonlinear Lineburg lens. It is found that the nonlinear Liineburg lens is able to compensate the focal shift caused by the diffraction of Gaussian illumination. The iterative nonlinear beam propagation method is computationally efficient and provides much physical insights into the wave propagation. Since it is based on Hamiltonian ray tracing, a ray diagram can be easily obtained which contains the evolution of generalized radiances. Besides bulk nonlinear media, this method provides a systematic approach to beam propagation problem in complex media such as nonlinear photonic crystals and metamaterials. Also, it is applicable to both coherent and partially coherent illumination. Therefore, this method has potential applications in the design and analysis of nonlinear optical devices and systems.
by Hanhong Gao.
S.M.

Nonlinear effects in optical fibers impose a capacity limit for optical communication systems. In this dissertation, the nonlinear Fourier transform (NFT) is investigated as a method to mitigate and compensate for those effects. This study consists of two parts: first a computational complexity analysis for the use of the NFT for nonlinear compensation in the normal dispersion regime, and second, an analysis of the robustness of the performance of the discrete spectrum modulation in the anomalous dispersion regime using the NFT. The first part investigates the computational complexity of the NFT based on the Zakharov-Shabat scattering problem as a nonlinear compensation technique for quadrature-phase-shift keyed (QPSK) signals with raised cosine frequency characteristic in optical fiber transmission systems with normal dispersion fibers. Results show that there are two primary sources of computational errors that arise from the use of the NFT: The computational error due to the finite eigenvalue resolution of the reflection spectra and the computational error due to the Born approximation used in the inverse NFT. In this scenario, computational costs become unacceptably large at data frame lengths and powers that are too small for this approach to be competitive with standard transmission methods. The second part investigates the robustness of a recently proposed nonlinear frequency-division multiplexing (NFDM) system comprised of two independent QPSK channels modulated in the discrete spectrum associated with two distinct eigenvalues. Among the many fiber impairments that may limit this system, we focus on determining the limits given by third-order dispersion, the Raman effect, amplified spontaneous emission (ASE) noise from erbium-doped fiber amplifiers (EDFAs), and lumped gain from EDFAs. Each of these impairments impact this system with discrete spectrum modulation and 1600 km of propagation distance in different ways: Third-order dispersion limits the maximum launch power to 13 dBm, the Raman effect limits the maximum launch power to 10.25 dBm, the ASE noise limits the maximum launch power to 9 dBm, while lumped gain limits the maximum launch power at 3.75 dBm. Additional studies are needed to investigate the effectiveness of the NFT for discrete spectrum modulation formats with three or more eigenvalues.

Thesis (Ph. D.)--University of Alabama at Birmingham, 2008.
Title from PDF title page (viewed Sept. 23, 2009). Additional advisors: Inmaculada Aban, Alexander Frenkel, Wenzhang Huang, Yanni Zeng. Includes bibliographical references.

This dissertation reports on the development of a nonlinear surface spectroscopy and the observation of nonlinear optical phenomena using sputtered zinc oxide waveguides. The first is known as Surface Coherent Raman Spectroscopy, or SCRS, and is capable of monolayer sensitivity. The second, discovered during the development of SCRS, is optical limiting and a previously unobserved form of optical switching based on an absorptive nonlinear coupling mechanism. Overviews of the theories of waveguiding, linear coupling, and SCRS are given. Experiments showing that the spectrum of a monolayer coverage of molecules on the surface of a metal oxide waveguide can be obtained using SCRS are reported. For this purpose ZnO waveguides were fabricated using rf magnetron sputtering; the details of which are presented. The results of the characterization of these films, using an optical loss technique, Rutherford backscattering, and X-ray diffraction, are also presented. Experiments are described and data are presented to show the existence of optical limiting and optical switching phenomena in ZnO waveguides. The experimental dependence of these phenomena on input pulse energy, wavelength, temporal pulse width, and type of distributed coupling mechanism is described. Existing nonlinear distributed coupler theory is extended to include the effect of an absorptive nonlinearity and the results of this theory are used to explain some of the characteristic features of the experimental results. A value of n₂ ≅ 2 x 10⁻¹⁶ m²/W for the nonlinear coefficient of sputtered ZnO films is obtained.

Thesis (M. S.)--Civil and Environmental Engineering, Georgia Institute of Technology, 2006.
Kim, Jin-Yeon, Committee Member ; Qu, Jianmin, Committee Member ; Jacobs, Laurence, Committee Chair. Includes bibliographical references.

Thesis (M.S. in Mechanical Engineering) Naval Postgraduate School, December 1998.
"December 1998." Thesis advisor(s): Joshua H. Gordis. Includes bibliographical references (p. 85). Also available online.

This dissertation describes our recent results in the study of various types of photonic switches. Special attention is given to the devices with Fabry-Perot etalon or planar waveguide structures based on dispersive optical nonlinearities. Basic optical logic functions, such as digital pattern recognition, symbolic substitution, and all-optical compare-and-exchange operation are demonstrated using ZnS and ZnSe nonlinear interference filters. Differential gain, cascading, and optical latching circuits are demonstrated using GaAs/AlGaAs multiple-quantum-well nonlinear etalons that are compatible with diode-laser sources, and the relationship between differential gain and device response time is established through a thorough investigation of the switching dynamics. Preliminary results also indicate that optical fibers can be used as interconnects between optical logic gates. Picosecond all-optical switching with good (> 3:1) contrast is demonstrated for the first time in single-mode strip-loaded GaAs/AlGaAs nonlinear directional couplers (NLDC's). The anisotropy of quantum-well structure to light polarization is used to achieve polarization-dependent two-beam switching, and the optical Stark effect is used to demonstrate all-optical modulation in an NLDC with subpicosecond recovery time.

Riemann-Hilbert-Probleme sind Randwertaufgaben für im Einheitskreis $\mathbb D$ holomorphe Funktionen $w$, deren Randwerte $w(t)$ auf gewissen Kurven $M_t$ liegen sollen. Ein Teil der Untersuchungen ist dem Fall explizit gegebener Kurven gewidmet. Dabei werden bekannte Resultate über glatte Kurven auf stetige Restriktionskurven erweitert, und die Existenz von Lösungen in gewissen Hardy-Räumen gezeigt. Die Eindeutigkeitsfrage führt auf ein Gegenbeispiel, das zugleich eine Vermutung aus einer Dissertation von Belch widerlegt. Der andere Teil der Untersuchungen ist dem klassischen Fall geschlossener Restriktionskurven gewidmet. Hier steht statt der Abschwächung von Glattheitsvoraussetzungen die Formulierung geeigneter Nebenbedingungen im Mittelpunkt. Die Abhängigkeit der Lösung von Zusatzbedingungen erweist sich als Verallgemeinerung des Verhaltens von Blaschkeprodukten. Für drei Interpolationpunkte kann charakterisiert werden, wann durch sie eine Lösung mit Windungszahl 1 verläuft, durch $k$ Interpolationspunkte wird die Existenz einer Lösung mit Windungszahl $k-1$ gezeigt.

Packet-switched communication networks such as today's Internet are built with several interconnected core and distribution packet forwarding routers and several sender and sink transport agents. In order to maintain stability and avoid congestion collapse in the network, the sources control their rate behavior and voluntarily adjust their sending rates to accommodate other sources in the network. In this thesis, we study one class of sender rate control that is modeled using continuous first-order differential equation of the sending rates. In order to adjust the rates appropriately, the network sends continuous packet-loss feedback to the sources. We study a form of closed-loop feedback congestion controllers whose rate adjustments exhibit a nonlinear form. There are three dimensions to our work in this thesis. First, we study the network optimization problem in which sources choose utilities to maximize their underlying throughput. Each sender maximizes its utility proportional to the throughput achieved. In our model, sources choose a utility function to define their level of satisfaction of the underlying resource usages. The objective of this direction is to establish the properties of source utility functions using inequality constrained bounded sets and study the functional forms of utilities against a chosen rate differential equation. Second, stability of the network and tolerance to perturbation are two essential factors that keep communication networks operational around the equilibrium point. Our objective in this part of the thesis is to analytically understand the existence of local asymptotic stability of delayed-feedback systems under homogeneous network delays. Third, we propose a novel tangential controller for a generic maximization function and study its properties using nonlinear optimization techniques. We develop the necessary theoretical background and the properties of our controller to prove that it is a better rate adaptation algorithm for logarithmic utilities compared to the well-studied proportional controllers. We establish the asymptotic local stability of our controller with upper bounds on the increase / decrease gain parameters.

The aim of this thesis work has been to develop signal analysis methods for a computerized cardiac auscultation system, the intelligent stethoscope. In particular, the work focuses on classification and interpretation of features derived from the phonocardiographic (PCG) signal by using advanced signal processing techniques. The PCG signal is traditionally analyzed and characterized by morphological properties in the time domain, by spectral properties in the frequency domain or by nonstationary properties in a joint time-frequency domain. The main contribution of this thesis has been to introduce nonlinear analysis techniques based on dynamical systems theory to extract more information from the PCG signal. Especially, Takens' delay embedding theorem has been used to reconstruct the underlying system's state space based on the measured PCG signal. This processing step provides a geometrical interpretation of the dynamics of the signal, whose structure can be utilized for both system characterization and classification as well as for signal processing tasks such as detection and prediction. In this thesis, the PCG signal's structure in state space has been exploited in several applications. Change detection based on recurrence time statistics was used in combination with nonlinear prediction to remove obscuring heart sounds from lung sound recordings in healthy test subjects. Sample entropy and mutual information were used to assess the severity of aortic stenosis (AS) as well as mitral insufficiency (MI) in dogs. A large number of, partly nonlinear, features was extracted and used for distinguishing innocent murmurs from murmurs caused by AS or MI in patients with probable valve disease. Finally, novel work related to very accurate localization of the first heart sound by means of ECG-gated ensemble averaging was conducted. In general, the presented nonlinear processing techniques have shown considerably improved results in comparison with other PCG based techniques. In modern health care, auscultation has found its main role in primary or in home health care, when deciding if special care and more extensive examinations are required. Making a decision based on auscultation is however difficult, why a simple tool able to screen and assess murmurs would be both time- and cost-saving while relieving many patients from needless anxiety. In the emerging field of telemedicine and home care, an intelligent stethoscope with decision support abilities would be of great value.

We present a method of reconstructing the area of origin for blood droplets given the position and directions of impact stains. This method works for nonlinear trajectories, such as parabolic motion or motion with drag. A model for fitting ellipses to the stains, obtaining impact velocities, blood drop mass and drag coefficient from blood splatter image densities, impact angle and pattern is also provided. We also show how to use this extra data to aid with our estimation. We validate our method in several experiments involving blood splatters at varying velocities and angles.