PERIODIC PERTURBATION of QUADRATIC SYSTEMS WITH TWO INFINITE HETEROCLINIC CYCLES

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Quadratic perturbations of a quadratic reversible Lotka-Volterra System

"Vegeu el resum a l'inici del document del fitxer adjunt."

Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity

We consider solutions to the second-harmonic generation equations in two-and three-dimensional dispersive media in the form of solitons localized in space and time. As is known, collapse does not take place in these models, which is why the solitons may be stable. The general solution is obtained in an approximate analytical form by means of a variational approach, which also allows the stability of the solutions to be predicted. Then, we directly simulate the two-dimensional case, taking the initial configuration as suggested by the variational approximation. We thus demonstrate that spatiotemporal solitons indeed exist and are stable. Furthermore, they are not, in the general case, equivalent to the previously known cylindrical spatial solitons. Direct simulations generate solitons with some internal oscillations. However, these oscillations neither grow nor do they exhibit any significant radiative damping. Numerical solutions of the stationary version of the equations produce the same solitons in their unperturbed form, i.e., without internal oscillations. Strictly stable solitons exist only if the system has anomalous dispersion at both the fundamental harmonic and second harmonic (SH), including the case of zero dispersion at SH. Quasistationary solitons, decaying extremely slowly into radiation, are found in the presence of weak normal dispersion at the second-harmonic frequency.

An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index

The shifted Legendre orthogonal polynomials are used for the numerical solution of a new formulation for the multi-dimensional fractional optimal control problem (M-DFOCP) with a quadratic performance index. The fractional derivatives are described in the Caputo sense. The Lagrange multiplier method for the constrained extremum and the operational matrix of fractional integrals are used together with the help of the properties of the shifted Legendre orthonormal polynomials. The method reduces the M-DFOCP to a simpler problem that consists of solving a system of algebraic equations. For confirming the efficiency and accuracy of the proposed scheme, some test problems are implemented with their approximate solutions.

The third order Melnikov function of a quadratic center under quadratic perturbation

We study quadratic perturbations of the integrable system (1+x)dH; where H =(x²+y²)=2: We prove that the first three Melnikov functions associated to the perturbed system give rise at most to three limit cycles.

Do Rates and Splitting of Phosphogypsum Applications Influence the Soil and Annual Crops in a No-Tillage System?

ABSTRACT Applications of phosphogypsum (PG) provide nutrients to the soil and reduce Al3+ activity, favoring soil fertility and root growth, but allow Mg2+ mobilization through the soil profile, resulting in variations in the PG rate required to achieve the optimum crop yield. This study evaluated the effect of application rates and splitting of PG on soil fertility of a Typic Hapludox, as well as the influence on annual crops under no-tillage. Using a (4 × 3) + 1 factorial structure, the treatments consisted of four PG rates (3, 6, 9, and 12 Mg ha-1) and three split applications (P1 = 100 % in 2009; P2 = 50+50 % in 2009 and 2010; P3 = 33+33+33 % in 2009, 2010 and 2011), plus a control without PG. The soil was sampled six months after the last PG application, in stratified layers to a depth of 0.8 m. Corn, wheat and soybean were sown between November 2011 and December 2012, and leaf samples were collected for analysis when at least 50 % of the plants showed reproductive structures. The application of PG increased Ca2+ concentrations in all sampled soil layers and the soil pH between 0.2 and 0.8 m, and reduced the concentrations of Al3+ in all layers and of Mg2+ to a depth of 0.6 m, without any effect of splitting the applications. The soil Ca/Mg ratio increased linearly to a depth of 0.6 m with the rates and were found to be higher in the 0.0-0.1 m layer of the P2 and P3 treatments than without splitting (P1). Sulfur concentrations increased linearly by application rates to a depth of 0.8 m, decreasing in the order P3>P2>P1 to a depth of 0.4 m and were higher in the treatments P3 and P2 than P1 between 0.4-0.6 m, whereas no differences were observed in the 0.6-0.8 m layer. No effect was recorded for K, P and potential acidity (H+Al). The leaf Ca and S concentration increased, while Mg decreased for all crops treated with PG, and there was no effect of splitting the application. The yield response of corn to PG rates was quadratic, with the maximum technical efficiency achieved at 6.38 Mg ha-1 of PG, while wheat yield increased linearly in a growing season with a drought period. Soybean yield was not affected by the PG rate, and splitting had no effect on the yield of any of the crops. Phosphogypsum improved soil fertility in the profile, however, Mg2+ migrated downwards, regardless of application splitting. Splitting the PG application induced a higher Ca/Mg ratio in the 0.0-0.1 m layer and less S leaching, but did not affect the crop yield. The application rates had no effect on soybean yield, but were beneficial for corn and, especially, for wheat, which was affected by a drought period during growth.

Linear-quadratic optimal control of descriptor systems

In recent years the analysis and synthesis of (mechanical) control systems in descriptor form has been established. This general description of dynamical systems is important for many applications in mechanics and mechatronics, in electrical and electronic engineering, and in chemical engineering as well. This contribution deals with linear mechanical descriptor systems and its control design with respect to a quadratic performance criterion. Here, the notion of properness plays an important role whether the standard Riccati approach can be applied as usual or not. Properness and non-properness distinguish between the cases if the descriptor system is exclusively governed by the control input or by its higher-order time-derivatives additionally. In the unusual case of non-proper systems a quite different problem of optimal control design has to be considered. Both cases will be solved completely.

Control of Structures with Cubic and Quadratic Non-Linearities with Time Delay Consideration

This paper studies the effect of time delay on the active non-linear control of dynamically loaded flexible structures. The behavior of non-linear systems under state feedback control, considering a fixed time delay for the control force, is investigated. A control method based on non-linear optimal control, using a tensorial formulation and state feedback control is used. The state equations and the control forces are expressed in polynomial form and a performance index, quadratic in both state vector and control forces, is used. General polynomial representations of the non-linear control law are obtained and implemented for control algorithms up to the fifth order. This methodology is applied to systems with quadratic and cubic non-linearities. Strongly non-linear systems are tested and the effectiveness of the control system including a delay for the application of control forces is discussed. Numerical results indicate that the adopted control algorithm can be efficient for non-linear systems, chiefly in the presence of strong non-linearities but increasing time delay reduces the efficiency of the control system. Numerical results emphasize the importance of considering time delay in the project of active structural control systems.

Quadratic Predictor based Differential Encoding and Decoding of Speech Signals

Modeling nonlinear systems using Volterra series is a century old method but practical realizations were hampered by inadequate hardware to handle the increased computational complexity stemming from its use. But interest is renewed recently, in designing and implementing filters which can model much of the polynomial nonlinearities inherent in practical systems. The key advantage in resorting to Volterra power series for this purpose is that nonlinear filters so designed can be made to work in parallel with the existing LTI systems, yielding improved performance. This paper describes the inclusion of a quadratic predictor (with nonlinearity order 2) with a linear predictor in an analog source coding system. Analog coding schemes generally ignore the source generation mechanisms but focuses on high fidelity reconstruction at the receiver. The widely used method of differential pnlse code modulation (DPCM) for speech transmission uses a linear predictor to estimate the next possible value of the input speech signal. But this linear system do not account for the inherent nonlinearities in speech signals arising out of multiple reflections in the vocal tract. So a quadratic predictor is designed and implemented in parallel with the linear predictor to yield improved mean square error performance. The augmented speech coder is tested on speech signals transmitted over an additive white gaussian noise (AWGN) channel.

Design, Development and Realization of Quadratic Volterra Filters for Selected Applications

The basic concepts of digital signal processing are taught to the students in engineering and science. The focus of the course is on linear, time invariant systems. The question as to what happens when the system is governed by a quadratic or cubic equation remains unanswered in the vast majority of literature on signal processing. Light has been shed on this problem when John V Mathews and Giovanni L Sicuranza published the book Polynomial Signal Processing. This book opened up an unseen vista of polynomial systems for signal and image processing. The book presented the theory and implementations of both adaptive and non-adaptive FIR and IIR quadratic systems which offer improved performance than conventional linear systems. The theory of quadratic systems presents a pristine and virgin area of research that offers computationally intensive work. Once the area of research is selected, the next issue is the choice of the software tool to carry out the work. Conventional languages like C and C++ are easily eliminated as they are not interpreted and lack good quality plotting libraries. MATLAB is proved to be very slow and so do SCILAB and Octave. The search for a language for scientific computing that was as fast as C, but with a good quality plotting library, ended up in Python, a distant relative of LISP. It proved to be ideal for scientific computing. An account of the use of Python, its scientific computing package scipy and the plotting library pylab is given in the appendix Initially, work is focused on designing predictors that exploit the polynomial nonlinearities inherent in speech generation mechanisms. Soon, the work got diverted into medical image processing which offered more potential to exploit by the use of quadratic methods. The major focus in this area is on quadratic edge detection methods for retinal images and fingerprints as well as de-noising raw MRI signals

Bayes quadratic unbiased estimator of spatial covariance parameters

This work presents Bayes invariant quadratic unbiased estimator, for short BAIQUE. Bayesian approach is used here to estimate the covariance functions of the regionalized variables which appear in the spatial covariance structure in mixed linear model. Firstly a brief review of spatial process, variance covariance components structure and Bayesian inference is given, since this project deals with these concepts. Then the linear equations model corresponding to BAIQUE in the general case is formulated. That Bayes estimator of variance components with too many unknown parameters is complicated to be solved analytically. Hence, in order to facilitate the handling with this system, BAIQUE of spatial covariance model with two parameters is considered. Bayesian estimation arises as a solution of a linear equations system which requires the linearity of the covariance functions in the parameters. Here the availability of prior information on the parameters is assumed. This information includes apriori distribution functions which enable to find the first and the second moments matrix. The Bayesian estimation suggested here depends only on the second moment of the prior distribution. The estimation appears as a quadratic form y'Ay , where y is the vector of filtered data observations. This quadratic estimator is used to estimate the linear function of unknown variance components. The matrix A of BAIQUE plays an important role. If such a symmetrical matrix exists, then Bayes risk becomes minimal and the unbiasedness conditions are fulfilled. Therefore, the symmetry of this matrix is elaborated in this work. Through dealing with the infinite series of matrices, a representation of the matrix A is obtained which shows the symmetry of A. In this context, the largest singular value of the decomposed matrix of the infinite series is considered to deal with the convergence condition and also it is connected with Gerschgorin Discs and Poincare theorem. Then the BAIQUE model for some experimental designs is computed and compared. The comparison deals with different aspects, such as the influence of the position of the design points in a fixed interval. The designs that are considered are those with their points distributed in the interval [0, 1]. These experimental structures are compared with respect to the Bayes risk and norms of the matrices corresponding to distances, covariance structures and matrices which have to satisfy the convergence condition. Also different types of the regression functions and distance measurements are handled. The influence of scaling on the design points is studied, moreover, the influence of the covariance structure on the best design is investigated and different covariance structures are considered. Finally, BAIQUE is applied for real data. The corresponding outcomes are compared with the results of other methods for the same data. Thereby, the special BAIQUE, which estimates the general variance of the data, achieves a very close result to the classical empirical variance.

Application of constrained predictive control on a 3D crane system

This paper describes the SIMULINK implementation of a constrained predictive control algorithm based on quadratic programming and linear state space models, and its application to a laboratory-scale 3D crane system. The algorithm is compatible with Real Time. Windows Target and, in the case of the crane system, it can be executed with a sampling period of 0.01 s and a prediction horizon of up to 300 samples, using a linear state space model with 3 inputs, 5 outputs and 13 states.

Apodisation, denoising and system identification techniques for THz transients in the wavelet domain

This work describes the use of a quadratic programming optimization procedure for designing asymmetric apodization windows to de-noise THz transient interferograms and compares these results to those obtained when wavelet signal processing algorithms are adopted. A systems identification technique in the wavelet domain is also proposed for the estimation of the complex insertion loss function. The proposed techniques can enhance the frequency dependent dynamic range of an experiment and should be of particular interest to the THz imaging and tomography community. Future advances in THz sources and detectors are likely to increase the signal-to-noise ratio of the recorded THz transients and high quality apodization techniques will become more important, and may set the limit on the achievable accuracy of the deduced spectrum.

Dynamic integrated system optimization and parameter estimation for discrete time optimal control of nonlinear systems

An algorithm for solving nonlinear discrete time optimal control problems with model-reality differences is presented. The technique uses Dynamic Integrated System Optimization and Parameter Estimation (DISOPE), which achieves the correct optimal solution in spite of deficiencies in the mathematical model employed in the optimization procedure. A version of the algorithm with a linear-quadratic model-based problem, implemented in the C+ + programming language, is developed and applied to illustrative simulation examples. An analysis of the optimality and convergence properties of the algorithm is also presented.