TYPE-ZERO SADDLE-NODE BIFURCATIONS AND STABILITY REGION ESTIMATION OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEMS

A complete characterization of the stability boundary of a class of nonlinear dynamical systems that admit energy functions is developed in this paper. This characterization generalizes the existing results by allowing the type-zero saddle-node nonhyperbolic equilibrium points on the stability boundary. Conceptual algorithms to obtain optimal estimates of the stability region (basin of attraction) in the form of level sets of a given family of energy functions are derived. The behavior of the stability region and the corresponding estimates are investigated for parameter variation in the neighborhood of a type-zero saddle-node bifurcation value.

Tricritical-like behavior of the nonlinear optical refraction at the nematic-isotropic transition in the E7 thermotropic liquid crystal

We use Z-scan technique to investigate the nonlinear optical response of the thermotropic liquid crystal E7 in the neighborhood of the nematic-isotropic phase transition. The analysis of the data for the nonlinear optical birefringence is compatible with an effective critical exponent of the order parameter, beta = 0.28 +/- 0.03, which is close to the classical value, beta = 0.25, for a tricritical point. The nonlinear optical absorption in the nematic range depends on the geometrical configuration of the nematic director with respect to the polarization beam, and vanishes in the isotropic phase.

A new approach for modeling and control of nonlinear systems via norm-bounded linear differential inclusions

A systematic approach to model nonlinear systems using norm-bounded linear differential inclusions (NLDIs) is proposed in this paper. The resulting NLDI model is suitable for the application of linear control design techniques and, therefore, it is possible to fulfill certain specifications for the underlying nonlinear system, within an operating region of interest in the state-space, using a linear controller designed for this NLDI model. Hence, a procedure to design a dynamic output feedback controller for the NLDI model is also proposed in this paper. One of the main contributions of the proposed modeling and control approach is the use of the mean-value theorem to represent the nonlinear system by a linear parameter-varying model, which is then mapped into a polytopic linear differential inclusion (PLDI) within the region of interest. To avoid the combinatorial problem that is inherent of polytopic models for medium- and large-sized systems, the PLDI is transformed into an NLDI, and the whole process is carried out ensuring that all trajectories of the underlying nonlinear system are also trajectories of the resulting NLDI within the operating region of interest. Furthermore, it is also possible to choose a particular structure for the NLDI parameters to reduce the conservatism in the representation of the nonlinear system by the NLDI model, and this feature is also one important contribution of this paper. Once the NLDI representation of the nonlinear system is obtained, the paper proposes the application of a linear control design method to this representation. The design is based on quadratic Lyapunov functions and formulated as search problem over a set of bilinear matrix inequalities (BMIs), which is solved using a two-step separation procedure that maps the BMIs into a set of corresponding linear matrix inequalities. Two numerical examples are given to demonstrate the effectiveness of the proposed approach.

Einfluß von Topologie und Verscherung auf Dynamik und mechanische Parameter bei Homopolymeren

Die FT-Rheologie wird zur Unterscheidung verschiedener Kamm-Topologien in Polystyrollösungen und –schmelzen angewendet. Die Polystyrole werden in Abhängigkeit der Deborahzahl De unter LAOS-Bedingungen vermessen. Die Meßergebnisse zeigen, daß der Schritt von wohldefinierten Systemen (lineare Ketten, Sterne) zu solchen mit statistischer Verteilung wie in Kämmen zu großen Veränderungen sowohl im linearen als auch im nichtlinearen Bereich der rheologischen Messungen führt. Sowohl die Masterkurven als auch die Intenstiäten I3/1 und Phasen Phi3 der Nichtlinearitäten der einzelnen Proben weisen jeweils deutliche Unterschiede untereinander auf. Diese sind durch die bisherigen Ergebnisse noch nicht vollständig mit topologischen Merkmalen in Verbindung zu bringen. Die Messungen wurden mit dem von McLeish eingeführten Pom-pom Modell und daraus weiterentwickelten double convected-Pom-pom Modell (DCPP) simuliert und lieferten gute Übereinstimmung sowie auch Vorhersagen über den experimentell nicht mehr zugänglichen Bereich. Zur Untersuchung des Einflusses von mechanischer Scherung auf die lokale, molekulare Dynamik wird das LAOS-Experiment in situ mit dielektrischer Spektroskopie kombiniert. Dazu wurde eine Apparatur entwickelt, die das hochsensitive ARES-Rheometer mit dem hochauflösenden dielektrischen ALPHA-Analyzer verbindet. Mit dieser Apparatur wurde das Typ-A Polymer 1,4-cis-Polyisopren, mit einem Dipolmoment entlang des Rückgrats, bei oszillatorischer Scherung unter gleichzeitiger Aufnahme eines dielektrischen Spektrums vermessen. Es konnte gezeigt werden, daß die oszillatorische Verscherung weder die charakteristische Relaxationszeit noch die Form des Normal Mode Peaks beeinflußt, wohl aber die dielektrische Stärke Delta epsilon. Diese entspricht der Fläche unter dem e“-Peak und kann mit einer Debye- und einer Cole/Davidson-Funktion angepasst werden. Die Abnahme der dielektrischen Stärke mit zunehmender Scheramplitude kann mit der Orientierungsverteilung der End-zu-End-Vektoren in der Probe erklärt werden.

Parameter estimation in a growth model for a biological population

The motivating problem concerns the estimation of the growth curve of solitary corals that follow the nonlinear Von Bertalanffy Growth Function (VBGF). The most common parameterization of the VBGF for corals is based on two parameters: the ultimate length L∞ and the growth rate k. One aim was to find a more reliable method for estimating these parameters, which can capture the influence of environmental covariates. The main issue with current methods is that they force the linearization of VBGF and neglect intra-individual variability. The idea was to use the hierarchical nonlinear model which has the appealing features of taking into account the influence of collection sites, possible intra-site measurement correlation and variance heterogeneity, and that can handle the influence of environmental factors and all the reliable information that might influence coral growth. This method was used on two databases of different solitary corals i.e. Balanophyllia europaea and Leptopsammia pruvoti, collected in six different sites in different environmental conditions, which introduced a decisive improvement in the results. Nevertheless, the theory of the energy balance in growth ascertains the linear correlation of the two parameters and the independence of the ultimate length L∞ from the influence of environmental covariates, so a further aim of the thesis was to propose a new parameterization based on the ultimate length and parameter c which explicitly describes the part of growth ascribable to site-specific conditions such as environmental factors. We explored the possibility of estimating these parameters characterizing the VBGF new parameterization via the nonlinear hierarchical model. Again there was a general improvement with respect to traditional methods. The results of the two parameterizations were similar, although a very slight improvement was observed in the new one. This is, nevertheless, more suitable from a theoretical point of view when considering environmental covariates.

NLGA based Active Control in Aerospace: fault tolerability, disturbance rejection, and parameter estimation

A new control scheme has been presented in this thesis. Based on the NonLinear Geometric Approach, the proposed Active Control System represents a new way to see the reconfigurable controllers for aerospace applications. The presence of the Diagnosis module (providing the estimation of generic signals which, based on the case, can be faults, disturbances or system parameters), mean feature of the depicted Active Control System, is a characteristic shared by three well known control systems: the Active Fault Tolerant Controls, the Indirect Adaptive Controls and the Active Disturbance Rejection Controls. The standard NonLinear Geometric Approach (NLGA) has been accurately investigated and than improved to extend its applicability to more complex models. The standard NLGA procedure has been modified to take account of feasible and estimable sets of unknown signals. Furthermore the application of the Singular Perturbations approximation has led to the solution of Detection and Isolation problems in scenarios too complex to be solved by the standard NLGA. Also the estimation process has been improved, where multiple redundant measuremtent are available, by the introduction of a new algorithm, here called "Least Squares - Sliding Mode". It guarantees optimality, in the sense of the least squares, and finite estimation time, in the sense of the sliding mode. The Active Control System concept has been formalized in two controller: a nonlinear backstepping controller and a nonlinear composite controller. Particularly interesting is the integration, in the controller design, of the estimations coming from the Diagnosis module. Stability proofs are provided for both the control schemes. Finally, different applications in aerospace have been provided to show the applicability and the effectiveness of the proposed NLGA-based Active Control System.

Regularization methods for the solution of a nonlinear least-squares problem in tomography

In this work we study a polyenergetic and multimaterial model for the breast image reconstruction in Digital Tomosynthesis, taking into consideration the variety of the materials forming the object and the polyenergetic nature of the X-rays beam. The modelling of the problem leads to the resolution of a high-dimensional nonlinear least-squares problem that, due to its nature of inverse ill-posed problem, needs some kind of regularization. We test two main classes of methods: the Levenberg-Marquardt method (together with the Conjugate Gradient method for the computation of the descent direction) and two limited-memory BFGS-like methods (L-BFGS). We perform some experiments for different values of the regularization parameter (constant or varying at each iteration), tolerances and stop conditions. Finally, we analyse the performance of the several methods comparing relative errors, iterations number, times and the qualities of the reconstructed images.

NONLINEAR TUBE-FITTING FOR THE ANALYSIS OF ANATOMICAL AND FUNCTIONAL STRUCTURES

We are concerned with the estimation of the exterior surface of tube-shaped anatomical structures. This interest is motivated by two distinct scientific goals, one dealing with the distribution of HIV microbicide in the colon and the other with measuring degradation in white-matter tracts in the brain. Our problem is posed as the estimation of the support of a distribution in three dimensions from a sample from that distribution, possibly measured with error. We propose a novel tube-fitting algorithm to construct such estimators. Further, we conduct a simulation study to aid in the choice of a key parameter of the algorithm, and we test our algorithm with validation study tailored to the motivating data sets. Finally, we apply the tube-fitting algorithm to a colon image produced by single photon emission computed tomography (SPECT)and to a white-matter tract image produced using diffusion tensor `imaging (DTI).

Parameter estimation for transformer modeling

Large Power transformers, an aging and vulnerable part of our energy infrastructure, are at choke points in the grid and are key to reliability and security. Damage or destruction due to vandalism, misoperation, or other unexpected events is of great concern, given replacement costs upward of $2M and lead time of 12 months. Transient overvoltages can cause great damage and there is much interest in improving computer simulation models to correctly predict and avoid the consequences. EMTP (the Electromagnetic Transients Program) has been developed for computer simulation of power system transients. Component models for most equipment have been developed and benchmarked. Power transformers would appear to be simple. However, due to their nonlinear and frequency-dependent behaviors, they can be one of the most complex system components to model. It is imperative that the applied models be appropriate for the range of frequencies and excitation levels that the system experiences. Thus, transformer modeling is not a mature field and newer improved models must be made available. In this work, improved topologically-correct duality-based models are developed for three-phase autotransformers having five-legged, three-legged, and shell-form cores. The main problem in the implementation of detailed models is the lack of complete and reliable data, as no international standard suggests how to measure and calculate parameters. Therefore, parameter estimation methods are developed here to determine the parameters of a given model in cases where available information is incomplete. The transformer nameplate data is required and relative physical dimensions of the core are estimated. The models include a separate representation of each segment of the core, including hysteresis of the core, λ-i saturation characteristic, capacitive effects, and frequency dependency of winding resistance and core loss. Steady-state excitation, and de-energization and re-energization transients are simulated and compared with an earlier-developed BCTRAN-based model. Black start energization cases are also simulated as a means of model evaluation and compared with actual event records. The simulated results using the model developed here are reasonable and more correct than those of the BCTRAN-based model. Simulation accuracy is dependent on the accuracy of the equipment model and its parameters. This work is significant in that it advances existing parameter estimation methods in cases where the available data and measurements are incomplete. The accuracy of EMTP simulation for power systems including three-phase autotransformers is thus enhanced. Theoretical results obtained from this work provide a sound foundation for development of transformer parameter estimation methods using engineering optimization. In addition, it should be possible to refine which information and measurement data are necessary for complete duality-based transformer models. To further refine and develop the models and transformer parameter estimation methods developed here, iterative full-scale laboratory tests using high-voltage and high-power three-phase transformer would be helpful.

Extended kalman filter for estimation of parameters in nonlinear state-space models of biochemical networks.

It is system dynamics that determines the function of cells, tissues and organisms. To develop mathematical models and estimate their parameters are an essential issue for studying dynamic behaviors of biological systems which include metabolic networks, genetic regulatory networks and signal transduction pathways, under perturbation of external stimuli. In general, biological dynamic systems are partially observed. Therefore, a natural way to model dynamic biological systems is to employ nonlinear state-space equations. Although statistical methods for parameter estimation of linear models in biological dynamic systems have been developed intensively in the recent years, the estimation of both states and parameters of nonlinear dynamic systems remains a challenging task. In this report, we apply extended Kalman Filter (EKF) to the estimation of both states and parameters of nonlinear state-space models. To evaluate the performance of the EKF for parameter estimation, we apply the EKF to a simulation dataset and two real datasets: JAK-STAT signal transduction pathway and Ras/Raf/MEK/ERK signaling transduction pathways datasets. The preliminary results show that EKF can accurately estimate the parameters and predict states in nonlinear state-space equations for modeling dynamic biochemical networks.

Finite Element Based Nonlinear Normalization of Human Lumbar Intervertebral Disc Stiffness to Account for Its Morphology

Disc degeneration, usually associated with low back pain and changes of intervertebral stiffness, represents a major health issue. As the intervertebral disc (IVD) morphology influences its stiffness, the link between mechanical properties and degenerative grade is partially lost without an efficient normalization of the stiffness with respect to the morphology. Moreover, although the behavior of soft tissues is highly nonlinear, only linear normalization protocols have been defined so far for the disc stiffness. Thus, the aim of this work is to propose a nonlinear normalization based on finite elements (FE) simulations and evaluate its impact on the stiffness of human anatomical specimens of lumbar IVD. First, a parameter study involving simulations of biomechanical tests (compression, flexion/extension, bilateral torsion and bending) on 20 FE models of IVDs with various dimensions was carried out to evaluate the effect of the disc's geometry on its compliance and establish stiffness/morphology relations necessary to the nonlinear normalization. The computed stiffness was then normalized by height (H), cross-sectional area (CSA), polar moment of inertia (J) or moments of inertia (Ixx, Iyy) to quantify the effect of both linear and nonlinear normalizations. In the second part of the study, T1-weighted MRI images were acquired to determine H, CSA, J, Ixx and Iyy of 14 human lumbar IVDs. Based on the measured morphology and pre-established relation with stiffness, linear and nonlinear normalization routines were then applied to the compliance of the specimens for each quasi-static biomechanical test. The variability of the stiffness prior to and after normalization was assessed via coefficient of variation (CV). The FE study confirmed that larger and thinner IVDs were stiffer while the normalization strongly attenuated the effect of the disc geometry on its stiffness. Yet, notwithstanding the results of the FE study, the experimental stiffness showed consistently higher CV after normalization. Assuming that geometry and material properties affect the mechanical response, they can also compensate for one another. Therefore, the larger CV after normalization can be interpreted as a strong variability of the material properties, previously hidden by the geometry's own influence. In conclusion, a new normalization protocol for the intervertebral disc stiffness in compression, flexion, extension, bilateral torsion and bending was proposed, with the possible use of MRI and FE to acquire the discs' anatomy and determine the nonlinear relations between stiffness and morphology. Such protocol may be useful to relate the disc's mechanical properties to its degree of degeneration.

NONLINEAR TIME SERIES/SYSTEM ANALYSIS (STATE-SPACE, DYNAMICS, INTERVENTION, RESILIENCE, KALMAN)

A discussion of nonlinear dynamics, demonstrated by the familiar automobile, is followed by the development of a systematic method of analysis of a possibly nonlinear time series using difference equations in the general state-space format. This format allows recursive state-dependent parameter estimation after each observation thereby revealing the dynamics inherent in the system in combination with random external perturbations.^ The one-step ahead prediction errors at each time period, transformed to have constant variance, and the estimated parametric sequences provide the information to (1) formally test whether time series observations y(,t) are some linear function of random errors (ELEM)(,s), for some t and s, or whether the series would more appropriately be described by a nonlinear model such as bilinear, exponential, threshold, etc., (2) formally test whether a statistically significant change has occurred in structure/level either historically or as it occurs, (3) forecast nonlinear system with a new and innovative (but very old numerical) technique utilizing rational functions to extrapolate individual parameters as smooth functions of time which are then combined to obtain the forecast of y and (4) suggest a measure of resilience, i.e. how much perturbation a structure/level can tolerate, whether internal or external to the system, and remain statistically unchanged. Although similar to one-step control, this provides a less rigid way to think about changes affecting social systems.^ Applications consisting of the analysis of some familiar and some simulated series demonstrate the procedure. Empirical results suggest that this state-space or modified augmented Kalman filter may provide interesting ways to identify particular kinds of nonlinearities as they occur in structural change via the state trajectory.^ A computational flow-chart detailing computations and software input and output is provided in the body of the text. IBM Advanced BASIC program listings to accomplish most of the analysis are provided in the appendix. ^

Analysis of life expectancy using linear and nonlinear models

Life expectancy has consistently increased over the last 150 years due to improvements in nutrition, medicine, and public health. Several studies found that in many developed countries, life expectancy continued to rise following a nearly linear trend, which was contrary to a common belief that the rate of improvement in life expectancy would decelerate and was fit with an S-shaped curve. Using samples of countries that exhibited a wide range of economic development levels, we explored the change in life expectancy over time by employing both nonlinear and linear models. We then observed if there were any significant differences in estimates between linear models, assuming an auto-correlated error structure. When data did not have a sigmoidal shape, nonlinear growth models sometimes failed to provide meaningful parameter estimates. The existence of an inflection point and asymptotes in the growth models made them inflexible with life expectancy data. In linear models, there was no significant difference in the life expectancy growth rate and future estimates between ordinary least squares (OLS) and generalized least squares (GLS). However, the generalized least squares model was more robust because the data involved time-series variables and residuals were positively correlated. ^

Type-I intermittency with discontinuous reinjection probability density in a truncation model of the derivative nonlinear Schrödinger equation

In previous papers, the type-I intermittent phenomenon with continuous reinjection probability density (RPD) has been extensively studied. However, in this paper type-I intermittency considering discontinuous RPD function in one-dimensional maps is analyzed. To carry out the present study the analytic approximation presented by del Río and Elaskar (Int. J. Bifurc. Chaos 20:1185-1191, 2010) and Elaskar et al. (Physica A. 390:2759-2768, 2011) is extended to consider discontinuous RPD functions. The results of this analysis show that the characteristic relation only depends on the position of the lower bound of reinjection (LBR), therefore for the LBR below the tangent point the relation {Mathematical expression}, where {Mathematical expression} is the control parameter, remains robust regardless the form of the RPD, although the average of the laminar phases {Mathematical expression} can change. Finally, the study of discontinuous RPD for type-I intermittency which occurs in a three-wave truncation model for the derivative nonlinear Schrodinger equation is presented. In all tests the theoretical results properly verify the numerical data

Modelling relativistic solitary wave interactions in over-dense plasmas: a perturbed nonlinear Schröndinger equation framework

We investigate the dynamics of localized solutions of the relativistic cold-fluid plasma model in the small but finite amplitude limit, for slightly overcritical plasma density. Adopting a multiple scale analysis, we derive a perturbed nonlinear Schrödinger equation that describes the evolution of the envelope of circularly polarized electromagnetic field. Retaining terms up to fifth order in the small perturbation parameter, we derive a self-consistent framework for the description of the plasma response in the presence of localized electromagnetic field. The formalism is applied to standing electromagnetic soliton interactions and the results are validated by simulations of the full cold-fluid model. To lowest order, a cubic nonlinear Schrödinger equation with a focusing nonlinearity is recovered. Classical quasiparticle theory is used to obtain analytical estimates for the collision time and minimum distance of approach between solitons. For larger soliton amplitudes the inclusion of the fifth-order terms is essential for a qualitatively correct description of soliton interactions. The defocusing quintic nonlinearity leads to inelastic soliton collisions, while bound states of solitons do not persist under perturbations in the initial phase or amplitude