Effect of wave frequency on the nonlinear interaction between Görtler vortices and three-dimensional Tollmien-Schlichting waves

The nonlinear interaction between Görtler vortices (GV) and three-dimensional Tollmien-Schlichting (TS) waves nonlinear interaction is studied with a spatial, nonparallel model based on the Parabolized Stability Equations (PSE). In this investigation the effect of TS wave frequency on the nonlinear interaction is studied. As verified in previous investigations using the same numerical model, the relative amplitudes and growth rates are the dominant parameters in GV/TS wave interaction. In this sense, the wave frequency influence is important in defining the streamwise distance traveled by the disturbances in the unstable region of the stability diagram and in defining the amplification rates that they go through.

On the numerical integration of rigid body nonlinear dynamics in presence of parameters singularities

One of the main complexities in the simulation of the nonlinear dynamics of rigid bodies consists in describing properly the finite rotations that they may undergo. It is well known that, to avoid singularities in the representation of the SO(3) rotation group, at least four parameters must be used. However, it is computationally expensive to use a four-parameters representation since, as only three of the parameters are independent, one needs to introduce constraint equations in the model, leading to differential-algebraic equations instead of ordinary differential ones. Three-parameter representations are numerically more efficient. Therefore, the objective of this paper is to evaluate numerically the influence of the parametrization and its singularities on the simulation of the dynamics of a rigid body. This is done through the analysis of a heavy top with a fixed point, using two three-parameter systems, Euler's angles and rotation vector. Theoretical results were used to guide the numerical simulation and to assure that all possible cases were analyzed. The two parametrizations were compared using several integrators. The results show that Euler's angles lead to faster integration compared to the rotation vector. An Euler's angles singular case, where representation approaches a theoretical singular point, was analyzed in detail. It is shown that on the contrary of what may be expected, 1) the numerical integration is very efficient, even more than for any other case, and 2) in spite of the uncertainty on the Euler's angles themselves, the body motion is well represented.

Estimating Attractor Dimension on the Nonlinear Pendulum Time Series

Chaotic dynamical systems exhibit trajectories in their phase space that converges to a strange attractor. The strangeness of the chaotic attractor is associated with its dimension in which instance it is described by a noninteger dimension. This contribution presents an overview of the main definitions of dimension discussing their evaluation from time series employing the correlation and the generalized dimension. The investigation is applied to the nonlinear pendulum where signals are generated by numerical integration of the mathematical model, selecting a single variable of the system as a time series. In order to simulate experimental data sets, a random noise is introduced in the time series. State space reconstruction and the determination of attractor dimensions are carried out regarding periodic and chaotic signals. Results obtained from time series analyses are compared with a reference value obtained from the analysis of mathematical model, estimating noise sensitivity. This procedure allows one to identify the best techniques to be applied in the analysis of experimental data.

Dynamic Analysis Model of Spherical Roller Bearings with Defects

Rolling element bearings are essential components of rotating machinery. The spherical roller bearing (SRB) is one variant seeing increasing use, because it is self-aligning and can support high loads. It is becoming increasingly important to understand how the SRB responds dynamically under a variety of conditions. This doctoral dissertation introduces a computationally efficient, three-degree-of-freedom, SRB model that was developed to predict the transient dynamic behaviors of a rotor-SRB system. In the model, bearing forces and deflections were calculated as a function of contact deformation and bearing geometry parameters according to nonlinear Hertzian contact theory. The results reveal how some of the more important parameters; such as diametral clearance, the number of rollers, and osculation number; influence ultimate bearing performance. Distributed defects, such as the waviness of the inner and outer ring, and localized defects, such as inner and outer ring defects, are taken into consideration in the proposed model. Simulation results were verified with results obtained by applying the formula for the spherical roller bearing radial deflection and the commercial bearing analysis software. Following model verification, a numerical simulation was carried out successfully for a full rotor-bearing system to demonstrate the application of this newly developed SRB model in a typical real world analysis. Accuracy of the model was verified by comparing measured to predicted behaviors for equivalent systems.

Efficient Optimization Algorithms for Nonlinear Data Analysis

Identification of low-dimensional structures and main sources of variation from multivariate data are fundamental tasks in data analysis. Many methods aimed at these tasks involve solution of an optimization problem. Thus, the objective of this thesis is to develop computationally efficient and theoretically justified methods for solving such problems. Most of the thesis is based on a statistical model, where ridges of the density estimated from the data are considered as relevant features. Finding ridges, that are generalized maxima, necessitates development of advanced optimization methods. An efficient and convergent trust region Newton method for projecting a point onto a ridge of the underlying density is developed for this purpose. The method is utilized in a differential equation-based approach for tracing ridges and computing projection coordinates along them. The density estimation is done nonparametrically by using Gaussian kernels. This allows application of ridge-based methods with only mild assumptions on the underlying structure of the data. The statistical model and the ridge finding methods are adapted to two different applications. The first one is extraction of curvilinear structures from noisy data mixed with background clutter. The second one is a novel nonlinear generalization of principal component analysis (PCA) and its extension to time series data. The methods have a wide range of potential applications, where most of the earlier approaches are inadequate. Examples include identification of faults from seismic data and identification of filaments from cosmological data. Applicability of the nonlinear PCA to climate analysis and reconstruction of periodic patterns from noisy time series data are also demonstrated. Other contributions of the thesis include development of an efficient semidefinite optimization method for embedding graphs into the Euclidean space. The method produces structure-preserving embeddings that maximize interpoint distances. It is primarily developed for dimensionality reduction, but has also potential applications in graph theory and various areas of physics, chemistry and engineering. Asymptotic behaviour of ridges and maxima of Gaussian kernel densities is also investigated when the kernel bandwidth approaches infinity. The results are applied to the nonlinear PCA and to finding significant maxima of such densities, which is a typical problem in visual object tracking.

A new model for the population pharmacokinetics of didanosine in healthy subjects

Didanosine (ddI) is a component of highly active antiretroviral therapy drug combinations, used especially in resource-limited settings and in zidovudine-resistant patients. The population pharmacokinetics of ddI was evaluated in 48 healthy volunteers enrolled in two bioequivalence studies. These data, along with a set of co-variates, were the subject of a nonlinear mixed-effect modeling analysis using the NONMEM program. A two-compartment model with first order absorption (ADVAN3 TRANS3) was fitted to the serum ddI concentration data. Final pharmacokinetic parameters, expressed as functions of the co-variates gender and creatinine clearance (CL CR), were: oral clearance (CL = 55.1 + 240 x CL CR + 16.6 L/h for males and CL = 55.1 + 240 x CL CR for females), central volume (V2 = 9.8 L), intercompartmental clearance (Q = 40.9 L/h), peripheral volume (V3 = 62.7 + 22.9 L for males and V3 = 62.7 L for females), absorption rate constant (Ka = 1.51/h), and dissolution time of the tablet (D = 0.43 h). The intraindividual (residual) variability expressed as coefficient of variation was 13.0%, whereas the interindividual variability of CL, Q, V3, Ka, and D was 20.1, 75.8, 20.6, 18.9, and 38.2%, respectively. The relatively high (>30%) interindividual variability for some of these parameters, observed under the controlled experimental settings of bioequivalence trials in healthy volunteers, may result from genetic variability of the processes involved in ddI absorption and disposition.

Modeling Energy Futures Returns With Markov Regime Switching Model

Financial time series have a tendency of abruptly changing their behavior and maintain this behavior for several consecutive periods, and commodity futures returns are not an exception. This quality proposes that nonlinear models, as opposed to linear models, can more accurately describe returns and volatility. Markov regime switching models are able to match this behavior and have become a popular way to model financial time series. This study uses Markov regime switching model to describe the behavior of energy futures returns on a commodity level, because studies show that commodity futures are a heterogeneous asset class. The purpose of this thesis is twofold. First, determine how many regimes characterize individual energy commodities’ returns in different return frequencies. Second, study the characteristics of these regimes. We extent the previous studies on the subject in two ways: We allow for the possibility that the number of regimes may exceed two, as well as conduct the research on individual commodities rather than on commodity indices or subgroups of these indices. We use daily, weekly and monthly time series of Brent crude oil, WTI crude oil, natural gas, heating oil and gasoil futures returns over 1994–2014, where available, to carry out the study. We apply the likelihood ratio test to determine the sufficient number of regimes for each commodity and data frequency. Then the time series are modeled with Markov regime switching model to obtain the return distribution characteristics of each regime, as well as the transition probabilities of moving between regimes. The results for the number of regimes suggest that daily energy futures return series consist of three to six regimes, whereas weekly and monthly returns for all energy commodities display only two regimes. When the number of regimes exceeds two, there is a tendency for the time series of energy commodities to form groups of regimes. These groups are usually quite persistent as a whole because probability of a regime switch inside the group is high. However, individual regimes in these groups are not persistent and the process oscillates between these regimes frequently. Regimes that are not part of any group are generally persistent, but show low ergodic probability, i.e. rarely prevail in the market. This study also suggests that energy futures return series characterized with two regimes do not necessarily display persistent bull and bear regimes. In fact, for the majority of time series, bearish regime is considerably less persistent. Rahoituksen aikasarjoilla on taipumus arvaamattomasti muuttaa käyttäytymistään ja jatkaa tätä uutta käyttäytymistä useiden periodien ajan, eivätkä hyödykefutuurien tuotot tee tähän poikkeusta. Tämän ominaisuuden johdosta lineaaristen mallien sijasta epälineaariset mallit pystyvät tarkemmin kuvailemaan esimerkiksi tuottojen jakauman parametreja. Markov regiiminvaihtomallit pystyvät vangitsemaan tämän ominaisuuden ja siksi niistä on tullut suosittuja rahoituksen aikasarjojen mallintamisessa. Tämä tutkimus käyttää Markov regiiminvaihtomallia kuvaamaan yksittäisten energiafutuurien tuottojen käyttäytymistä, sillä tutkimukset osoittavat hyödykefutuurien olevan hyvin heterogeeninen omaisuusluokka. Tutkimuksen tarkoitus on selvittää, kuinka monta regiimiä tarvitaan kuvaamaan energiafutuurien tuottoja eri tuottofrekvensseillä ja mitkä ovat näiden regiimien ominaisuudet. Aiempaa tutkimusta aiheesta laajennetaan määrittämällä regiimien lukumäärä tilastotieteellisen testauksen menetelmin sekä tutkimalla energiafutuureja yksittäin; ei indeksi- tai alaindeksitasolla. Tutkimuksessa käytetään päivä-, viikko- ja kuukausiaikasarjoja Brent-raakaöljyn, WTI-raakaöljyn, maakaasun, lämmitysöljyn ja polttoöljyn tuotoista aikaväliltä 1994–2014, siltä osin kuin aineistoa on saatavilla. Likelihood ratio -testin avulla estimoidaan kaikille aikasarjoille regiimien määrä,jonka jälkeen Markov regiiminvaihtomallia hyödyntäen määritetään yksittäisten regiimientuottojakaumien ominaisuudet sekä regiimien välinen transitiomatriisi. Tulokset regiimien lukumäärän osalta osoittavat, että energiafutuurien päiväkohtaisten tuottojen aikasarjoissa regiimien lukumäärä vaihtelee kolmen ja kuuden välillä. Viikko- ja kuukausituottojen kohdalla kaikkien energiafutuurien prosesseissa regiimien lukumäärä on kaksi. Kun regiimejä on enemmän kuin kaksi, on prosessilla taipumus muodostaa regiimeistä koostuvia ryhmiä. Prosessi pysyy ryhmän sisällä yleensä pitkään, koska todennäköisyys siirtyä ryhmään kuuluvien regiimien välillä on suuri. Yksittäiset regiimit ryhmän sisällä eivät kuitenkaan ole kovin pysyviä. Näin ollen prosessi vaihtelee ryhmän sisäisten regiimien välillä tiuhaan. Regiimit, jotka eivät kuulu ryhmään, ovat yleensä pysyviä, mutta prosessi ajautuu niihin vain harvoin, sillä todennäköisyys siirtyä muista regiimeistä niihin on pieni. Tutkimuksen tulokset osoittavat myös, että prosesseissa, joita ohjaa kaksi regiimiä, nämä regiimit eivät välttämättä ole pysyvät bull- ja bear-markkinatilanteet. Tulokset osoittavat sen sijaan, että bear-markkinatilanne on energiafutuureissa selvästi vähemmän pysyvä.

The Shape of the Risk Premium: Evidence from a Semiparametric Garch Model.

We examine the relationship between the risk premium on the S&P 500 index return and its conditional variance. We use the SMEGARCH - Semiparametric-Mean EGARCH - model in which the conditional variance process is EGARCH while the conditional mean is an arbitrary function of the conditional variance. For monthly S&P 500 excess returns, the relationship between the two moments that we uncover is nonlinear and nonmonotonic. Moreover, we find considerable persistence in the conditional variance as well as a leverage effect, as documented by others. Moreover, the shape of these relationships seems to be relatively stable over time.

A new avalanche model for solar flares

Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal

Essays on numerically efficient inference in nonlinear and non-Gaussian state space models, and commodity market analysis.

The first two articles build procedures to simulate vector of univariate states and estimate parameters in nonlinear and non Gaussian state space models. We propose state space speci fications that offer more flexibility in modeling dynamic relationship with latent variables. Our procedures are extension of the HESSIAN method of McCausland[2012]. Thus, they use approximation of the posterior density of the vector of states that allow to : simulate directly from the state vector posterior distribution, to simulate the states vector in one bloc and jointly with the vector of parameters, and to not allow data augmentation. These properties allow to build posterior simulators with very high relative numerical efficiency. Generic, they open a new path in nonlinear and non Gaussian state space analysis with limited contribution of the modeler. The third article is an essay in commodity market analysis. Private firms coexist with farmers' cooperatives in commodity markets in subsaharan african countries. The private firms have the biggest market share while some theoretical models predict they disappearance once confronted to farmers cooperatives. Elsewhere, some empirical studies and observations link cooperative incidence in a region with interpersonal trust, and thus to farmers trust toward cooperatives. We propose a model that sustain these empirical facts. A model where the cooperative reputation is a leading factor determining the market equilibrium of a price competition between a cooperative and a private firm

STUDIES ON THE STRESS FLUCTUATIONS IN SHEARED STOKESIAN SUSPENSIONS USING CHAOS THEORY AND NONLINEAR DYNAMICS

The thesis report results obtained from a detailed analysis of the fluctuations of the rheological parameters viz. shear and normal stresses, simulated by means of the Stokesian Dynamics method, of a macroscopically homogeneous sheared suspension of neutrally buoyant non-Brownian suspension of identical spheres in the Couette gap between two parallel walls in the limit of vanishingly small Reynolds numbers using the tools of non-linear dynamics and chaos theory for a range of particle concentration and Couette gaps. The thesis used the tools of nonlinear dynamics and chaos theory viz. average mutual information, space-time separation plots, visual recurrence analysis, principal component analysis, false nearest-neighbor technique, correlation integrals, computation of Lyapunov exponents for a range of area fraction of particles and for different Couette gaps. The thesis observed that one stress component can be predicted using another stress component at the same area fraction. This implies a type of synchronization of one stress component with another stress component. This finding suggests us to further analysis of the synchronization of stress components with another stress component at the same or different area fraction of particles. The different model equations of stress components for different area fraction of particles hints at the possible existence a general formula for stress fluctuations with area fraction of particle as a parameter

Nonlinear Dynamics of Nd:YAG lasers: Hopf bifurcation, Multistability and Chaotic Synchronization

In this thesis we have presented some aspects of the nonlinear dynamics of Nd:YAG lasers including synchronization, Hopf bifurcation, chaos control and delay induced multistability.We have chosen diode pumped Nd:YAG laser with intracavity KTP crystal operating with two mode and three mode output as our model system.Different types of orientation for the laser cavity modes were considered to carry out the studies. For laser operating with two mode output we have chosen the modes as having parallel polarization and perpendicular polarization. For laser having three mode output, we have chosen them as two modes polarized parallel to each other while the third mode polarized orthogonal to them.

Nonlinear Dynamics of Multiple Quantum Well Lasers: Chaos and Multistability

This thesis presents analytical and numerical results from studies based on the multiple quantum well laser rate equation model. We address the problem of controlling chaos produced by direct modulation of laser diodes. We consider the delay feedback control methods for this purpose and study their performance using numerical simulation. Besides the control of chaos, control of other nonlinear effects such as quasiperiodicity and bistability using delay feedback methods are also investigated.A number of secure communication schemes based on synchronization of chaos semiconductor lasers have been successfully demonstrated theoretically and experimentally. The current investigations in these field include the study of practical issues on the implementations of such encryption schemes. We theoretically study the issues such as channel delay, phase mismatch and frequency detuning on the synchronization of chaos in directly modulated laser diodes. It would be helpful for designing and implementing chaotic encryption schemes using synchronization of chaos in modulated semiconductor lasers.

Studies on Some Aspects of Light Beam Propagation Through Certain Nonlinear Media

This thesis deals with the study of light beam propagation through different nonlinear media. Analytical and numerical methods are used to show the formation of solitonS in these media. Basic experiments have also been performed to show the formation of a self-written waveguide in a photopolymer. The variational method is used for the analytical analysis throughout the thesis. Numerical method based on the finite-difference forms of the original partial differential equation is used for the numerical analysis.In Chapter 2, we have studied two kinds of solitons, the (2 + 1) D spatial solitons and the (3 + l)D spatio-temporal solitons in a cubic-quintic medium in the presence of multiphoton ionization.In Chapter 3, we have studied the evolution of light beam through a different kind of nonlinear media, the photorcfractive polymer. We study modulational instability and beam propagation through a photorefractive polymer in the presence of absorption losses. The one dimensional beam propagation through the nonlinear medium is studied using variational and numerical methods. Stable soliton propagation is observed both analytically and numerically.Chapter 4 deals with the study of modulational instability in a photorefractive crystal in the presence of wave mixing effects. Modulational instability in a photorefractive medium is studied in the presence of two wave mixing. We then propose and derive a model for forward four wave mixing in the photorefractive medium and investigate the modulational instability induced by four wave mixing effects. By using the standard linear stability analysis the instability gain is obtained.Chapter 5 deals with the study of self-written waveguides. Besides the usual analytical analysis, basic experiments were done showing the formation of self-written waveguide in a photopolymer system. The formation of a directional coupler in a photopolymer system is studied theoretically in Chapter 6. We propose and study, using the variational approximation as well as numerical simulation, the evolution of a probe beam through a directional coupler formed in a photopolymer system.

Development and Evaluation of Blind Identification Techniques for Nonlinear Systems

Identification and Control of Non‐linear dynamical systems are challenging problems to the control engineers.The topic is equally relevant in communication,weather prediction ,bio medical systems and even in social systems,where nonlinearity is an integral part of the system behavior.Most of the real world systems are nonlinear in nature and wide applications are there for nonlinear system identification/modeling.The basic approach in analyzing the nonlinear systems is to build a model from known behavior manifest in the form of system output.The problem of modeling boils down to computing a suitably parameterized model,representing the process.The parameters of the model are adjusted to optimize a performanace function,based on error between the given process output and identified process/model output.While the linear system identification is well established with many classical approaches,most of those methods cannot be directly applied for nonlinear system identification.The problem becomes more complex if the system is completely unknown but only the output time series is available.Blind recognition problem is the direct consequence of such a situation.The thesis concentrates on such problems.Capability of Artificial Neural Networks to approximate many nonlinear input-output maps makes it predominantly suitable for building a function for the identification of nonlinear systems,where only the time series is available.The literature is rich with a variety of algorithms to train the Neural Network model.A comprehensive study of the computation of the model parameters,using the different algorithms and the comparison among them to choose the best technique is still a demanding requirement from practical system designers,which is not available in a concise form in the literature.The thesis is thus an attempt to develop and evaluate some of the well known algorithms and propose some new techniques,in the context of Blind recognition of nonlinear systems.It also attempts to establish the relative merits and demerits of the different approaches.comprehensiveness is achieved in utilizing the benefits of well known evaluation techniques from statistics. The study concludes by providing the results of implementation of the currently available and modified versions and newly introduced techniques for nonlinear blind system modeling followed by a comparison of their performance.It is expected that,such comprehensive study and the comparison process can be of great relevance in many fields including chemical,electrical,biological,financial and weather data analysis.Further the results reported would be of immense help for practical system designers and analysts in selecting the most appropriate method based on the goodness of the model for the particular context.