Showing papers on "Nonlinear system published in 2000"

A Posteriori Error Estimation in Finite Element Analysis

Abstract: This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

Stability and Transition in Shear Flows

Abstract: 1 Introduction and General Results.- 1.1 Introduction.- 1.2 Nonlinear Disturbance Equations.- 1.3 Definition of Stability and Critical Reynolds Numbers.- 1.3.1 Definition of Stability.- 1.3.2 Critical Reynolds Numbers.- 1.3.3 Spatial Evolution of Disturbances.- 1.4 The Reynolds-Orr Equation.- 1.4.1 Derivation of the Reynolds-Orr Equation.- 1.4.2 The Need for Linear Growth Mechanisms.- I Temporal Stability of Parallel Shear Flows.- 2 Linear Inviscid Analysis.- 2.1 Inviscid Linear Stability Equations.- 2.2 Modal Solutions.- 2.2.1 General Results.- 2.2.2 Dispersive Effects and Wave Packets.- 2.3 Initial Value Problem.- 2.3.1 The Inviscid Initial Value Problem.- 2.3.2 Laplace Transform Solution.- 2.3.3 Solutions to the Normal Vorticity Equation.- 2.3.4 Example: Couette Flow.- 2.3.5 Localized Disturbances.- 3 Eigensolutions to the Viscous Problem.- 3.1 Viscous Linear Stability Equations.- 3.1.1 The Velocity-Vorticity Formulation.- 3.1.2 The Orr-Sommerfeld and Squire Equations.- 3.1.3 Squire's Transformation and Squire's Theorem.- 3.1.4 Vector Modes.- 3.1.5 Pipe Flow.- 3.2 Spectra and Eigenfunctions.- 3.2.1 Discrete Spectrum.- 3.2.2 Neutral Curves.- 3.2.3 Continuous Spectrum.- 3.2.4 Asymptotic Results.- 3.3 Further Results on Spectra and Eigenfunctions.- 3.3.1 Adjoint Problem and Bi-Orthogonality Condition.- 3.3.2 Sensitivity of Eigenvalues.- 3.3.3 Pseudo-Eigenvalues.- 3.3.4 Bounds on Eigenvalues.- 3.3.5 Dispersive Effects and Wave Packets.- 4 The Viscous Initial Value Problem.- 4.1 The Viscous Initial Value Problem.- 4.1.1 Motivation.- 4.1.2 Derivation of the Disturbance Equations.- 4.1.3 Disturbance Measure.- 4.2 The Forced Squire Equation and Transient Growth.- 4.2.1 Eigenfunction Expansion.- 4.2.2 Blasius Boundary Layer Flow.- 4.3 The Complete Solution to the Initial Value Problem.- 4.3.1 Continuous Formulation.- 4.3.2 Discrete Formulation.- 4.4 Optimal Growth.- 4.4.1 The Matrix Exponential.- 4.4.2 Maximum Amplification.- 4.4.3 Optimal Disturbances.- 4.4.4 Reynolds Number Dependence of Optimal Growth.- 4.5 Optimal Response and Optimal Growth Rate.- 4.5.1 The Forced Problem and the Resolvent.- 4.5.2 Maximum Growth Rate.- 4.5.3 Response to Stochastic Excitation.- 4.6 Estimates of Growth.- 4.6.1 Bounds on Matrix Exponential.- 4.6.2 Conditions for No Growth.- 4.7 Localized Disturbances.- 4.7.1 Choice of Initial Disturbances.- 4.7.2 Examples.- 4.7.3 Asymptotic Behavior.- 5 Nonlinear Stability.- 5.1 Motivation.- 5.1.1 Introduction.- 5.1.2 A Model Problem.- 5.2 Nonlinear Initial Value Problem.- 5.2.1 The Velocity-Vorticity Equations.- 5.3 Weakly Nonlinear Expansion.- 5.3.1 Multiple-Scale Analysis.- 5.3.2 The Landau Equation.- 5.4 Three-Wave Interactions.- 5.4.1 Resonance Conditions.- 5.4.2 Derivation of a Dynamical System.- 5.4.3 Triad Interactions.- 5.5 Solutions to the Nonlinear Initial Value Problem.- 5.5.1 Formal Solutions to the Nonlinear Initial Value Problem.- 5.5.2 Weakly Nonlinear Solutions and the Center Manifold.- 5.5.3 Nonlinear Equilibrium States.- 5.5.4 Numerical Solutions for Localized Disturbances.- 5.6 Energy Theory.- 5.6.1 The Energy Stability Problem.- 5.6.2 Additional Constraints.- II Stability of Complex Flows and Transition.- 6 Temporal Stability of Complex Flows.- 6.1 Effect of Pressure Gradient and Crossflow.- 6.1.1 Falkner-Skan (FS) Boundary Layers.- 6.1.2 Falkner-Skan-Cooke (FSC) Boundary layers.- 6.2 Effect of Rotation and Curvature.- 6.2.1 Curved Channel Flow.- 6.2.2 Rotating Channel Flow.- 6.2.3 Combined Effect of Curvature and Rotation.- 6.3 Effect of Surface Tension.- 6.3.1 Water Table Flow.- 6.3.2 Energy and the Choice of Norm.- 6.3.3 Results.- 6.4 Stability of Unsteady Flow.- 6.4.1 Oscillatory Flow.- 6.4.2 Arbitrary Time Dependence.- 6.5 Effect of Compressibility.- 6.5.1 The Compressible Initial Value Problem.- 6.5.2 Inviscid Instabilities and Rayleigh's Criterion.- 6.5.3 Viscous Instability.- 6.5.4 Nonmodal Growth.- 7 Growth of Disturbances in Space.- 7.1 Spatial Eigenvalue Analysis.- 7.1.1 Introduction.- 7.1.2 Spatial Spectra.- 7.1.3 Gaster's Transformation.- 7.1.4 Harmonic Point Source.- 7.2 Absolute Instability.- 7.2.1 The Concept of Absolute Instability.- 7.2.2 Briggs' Method.- 7.2.3 The Cusp Map.- 7.2.4 Stability of a Two-Dimensional Wake.- 7.2.5 Stability of Rotating Disk Flow.- 7.3 Spatial Initial Value Problem.- 7.3.1 Primitive Variable Formulation.- 7.3.2 Solution of the Spatial Initial Value Problem.- 7.3.3 The Vibrating Ribbon Problem.- 7.4 Nonparallel Effects.- 7.4.1 Asymptotic Methods.- 7.4.2 Parabolic Equations for Steady Disturbances.- 7.4.3 Parabolized Stability Equations (PSE).- 7.4.4 Spatial Optimal Disturbances.- 7.4.5 Global Instability.- 7.5 Nonlinear Effects.- 7.5.1 Nonlinear Wave Interactions.- 7.5.2 Nonlinear Parabolized Stability Equations.- 7.5.3 Examples.- 7.6 Disturbance Environment and Receptivity.- 7.6.1 Introduction.- 7.6.2 Nonlocalized and Localized Receptivity.- 7.6.3 An Adjoint Approach to Receptivity.- 7.6.4 Receptivity Using Parabolic Evolution Equations.- 8 Secondary Instability.- 8.1 Introduction.- 8.2 Secondary Instability of Two-Dimensional Waves.- 8.2.1 Derivation of the Equations.- 8.2.2 Numerical Results.- 8.2.3 Elliptical Instability.- 8.3 Secondary Instability of Vortices and Streaks.- 8.3.1 Governing Equations.- 8.3.2 Examples of Secondary Instability of Streaks and Vortices.- 8.4 Eckhaus Instability.- 8.4.1 Secondary Instability of Parallel Flows.- 8.4.2 Parabolic Equations for Spatial Eckhaus Instability.- 9 Transition to Turbulence.- 9.1 Transition Scenarios and Thresholds.- 9.1.1 Introduction.- 9.1.2 Three Transition Scenarios.- 9.1.3 The Most Likely Transition Scenario.- 9.1.4 Conclusions.- 9.2 Breakdown of Two-Dimensional Waves.- 9.2.1 The Zero Pressure Gradient Boundary Layer.- 9.2.2 Breakdown of Mixing Layers.- 9.3 Streak Breakdown.- 9.3.1 Streaks Forced by Blowing or Suction.- 9.3.2 Freestream Turbulence.- 9.4 Oblique Transition.- 9.4.1 Experiments and Simulations in Blasius Flow.- 9.4.2 Transition in a Separation Bubble.- 9.4.3 Compressible Oblique Transition.- 9.5 Transition of Vortex-Dominated Flows.- 9.5.1 Transition in Flows with Curvature.- 9.5.2 Direct Numerical Simulations of Secondary Instability of Crossflow Vortices.- 9.5.3 Experimental Investigations of Breakdown of Cross-flow Vortices.- 9.6 Breakdown of Localized Disturbances.- 9.6.1 Experimental Results for Boundary Layers.- 9.6.2 Direct Numerical Simulations in Boundary Layers.- 9.7 Transition Modeling.- 9.7.1 Low-Dimensional Models of Subcritical Transition.- 9.7.2 Traditional Transition Prediction Models.- 9.7.3 Transition Prediction Models Based on Nonmodal Growth.- 9.7.4 Nonlinear Transition Modeling.- III Appendix.- A Numerical Issues and Computer Programs.- A.1 Global versus Local Methods.- A.2 Runge-Kutta Methods.- A.3 Chebyshev Expansions.- A.4 Infinite Domain and Continuous Spectrum.- A.5 Chebyshev Discretization of the Orr-Sommerfeld Equation.- A.6 MATLAB Codes for Hydrodynamic Stability Calculations.- A.7 Eigenvalues of Parallel Shear Flows.- B Resonances and Degeneracies.- B.1 Resonances and Degeneracies.- B.2 Orr-Sommerfeld-Squire Resonance.- C Adjoint of the Linearized Boundary Layer Equation.- C.1 Adjoint of the Linearized Boundary Layer Equation.- D Selected Problems on Part I.

Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models

Abstract: 1 Introduction- I Optimization Techniques- 2 Introduction to Optimization- 3 Linear Optimization- 4 Nonlinear Local Optimization- 5 Nonlinear Global Optimization- 6 Unsupervised Learning Techniques- 7 Model Complexity Optimization- II Static Models- 9 Introduction to Static Models- 10 Linear, Polynomial, and Look-Up Table Models- 11 Neural Networks- 12 Fuzzy and Neuro-Fuzzy Models- 13 Local Linear Neuro-Fuzzy Models: Fundamentals- 14 Local Linear Neuro-Fuzzy Models: Advanced Aspects- III Dynamic Models- 16 Linear Dynamic System Identification- 17 Nonlinear Dynamic System Identification- 18 Classical Polynomial Approaches- 19 Dynamic Neural and Fuzzy Models- 20 Dynamic Local Linear Neuro-Fuzzy Models- 21 Neural Networks with Internal Dynamics- IV Applications- 22 Applications of Static Models- 23 Applications of Dynamic Models- 24 Applications of Advanced Methods- A Vectors and Matrices- A1 Vector and Matrix Derivatives- A2 Gradient, Hessian, and Jacobian- B Statistics- B1 Deterministic and Random Variables- B2 Probability Density Function (pdf)- B3 Stochastic Processes and Ergodicity- B4 Expectation- B5 Variance- B6 Correlation and Covariance- B7 Properties of Estimators- References

Phase noise in oscillators: a unifying theory and numerical methods for characterization

Abstract: Phase noise is a topic of theoretical and practical interest in electronic circuits, as well as in other fields, such as optics. Although progress has been made in understanding the phenomenon, there still remain significant gaps, both in its fundamental theory and in numerical techniques for its characterization. In this paper, we develop a solid foundation for phase noise that is valid for any oscillator, regardless of operating mechanism. We establish novel results about the dynamics of stable nonlinear oscillators in the presence of perturbations, both deterministic and random. We obtain an exact nonlinear equation for phase error, which we solve without approximations for random perturbations. This leads us to a precise characterization of timing jitter and spectral dispersion, for computing of which we have developed efficient numerical methods. We demonstrate our techniques on a variety of practical electrical oscillators and obtain good matches with measurements, even at frequencies close to the carrier, where previous techniques break down. Our methods are more than three orders of magnitude faster than the brute-force Monte Carlo approach, which is the only previously available technique that can predict phase noise correctly.

Fundamentals of Vibrations

Abstract: 1 Concepts from Vibrations 2 Response of Single-Degree-of-Freedom Systems to Initial Excitations 3 Response of Single-Degree-of-Freedom Systems to Harmonic and Periodic Excitations 4 Response of Single-Degree-of-Freedom Systems to Nonperiodic Excitations 5 Two-Degree-of-Freedom Systems 6 Elements of Analytical Dynamics 7 Multi-Degree-of-Freedom Systems 8 Distributed-Parameter Systems: Exact Solutions 9 Distributed-Parameter Systems: Approximate Mathods 10 The Finite Element Method 11 Nonlinear Oscilations 12 Random Vibrations Appendix A. Fourier Series Appendix B. Laplace Transformation Appendix C. Linear Algebra

Path-ensemble averages in systems driven far from equilibrium

Abstract: The Kawasaki nonlinear response relation, the transient fluctuation theorem, and the Jarzynski nonequilibrium work relation are all expressions that describe the behavior of a system that has been driven from equilibrium by an external perturbation In contrast to linear response theory, these expressions are exact no matter the strength of the perturbation, or how far the system has been driven away from equilibrium In this paper, I show that these three relations (and several other closely related results) can all be considered special cases of a single theorem This expression is explicitly derived for discrete time and space Markovian dynamics, with the additional assumptions that the unperturbed dynamics preserve the appropriate equilibrium ensemble, and that the energy of the system remains finite

The Development of Discontinuous Galerkin Methods

Abstract: In this paper, we present an overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments. We show how these methods made their way into the main stream of computational fluid dynamics and how they are quickly finding use in a wide variety of applications. We review the theoretical and algorithmic aspects of these methods as well as their applications to equations including nonlinear conservation laws, the compressible Navier-Stokes equations, and Hamilton-Jacobi-like equations.

A MATLAB differentiation matrix suite

Abstract: A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sinc interpolants. Auxiliary functions are included for incorporating boundary conditions, performing interpolation using barycentric formulas, and computing roots of orthogonal polynomials. It is demonstrated how to use the package for solving eigenvalue, boundary value, and initial value problems arising in the fields of special functions, quantum mechanics, nonlinear waves, and hydrodynamic stability.

Difference Equations: An Introduction with Applications

Abstract: Introduction. The Difference Calculus. Linear Difference Equations. Stability Theory. Asymptotic Methods. The Self-Adjoint Second Order Linear Equation. The Sturm-Liouville Problem. Discrete Calculus of Variations. Boundary Value Problems for Nonlinear Equations. Partial Difference Equations.

An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows

Abstract: Some new developments of explicit algebraic Reynolds stress turbulence models (EARSM) are presented. The new developments include a new near-wall treatment ensuring realizability for the individual stress components, a formulation for compressible flows, and a suggestion for a possible approximation of diffusion terms in the anisotropy transport equation. Recent developments in this area are assessed and collected into a model for both incompressible and compressible three-dimensional wall-bounded turbulent flows. This model represents a solution of the implicit ARSM equations, where the production to dissipation ratio is obtained as a solution to a nonlinear algebraic relation. Three-dimensionality is fully accounted for in the mean flow description of the stress anisotropy. The resulting EARSM has been found to be well suited to integration to the wall and all individual Reynolds stresses can be well predicted by introducing wall damping functions derived from the van Driest damping function. The platform for the model consists of the transport equations for the kinetic energy and an auxiliary quantity. The proposed model can be used with any such platform, and examples are shown for two different choices of the auxiliary quantity.

A characterization of integral input-to-state stability

Abstract: The notion of input-to-state stability (ISS) is now recognized as a central concept in nonlinear systems analysis. It provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite L/sup 2/ gains. It plays a central role in recursive design, coprime factorizations, controllers for nonminimum phase systems, and many other areas. In this paper, a newer notion, that of integral input-to-state stability (iISS), is studied. The notion of iISS generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states, in that sense generalizing the linear "H/sup 2/" theory. It allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance. We obtain several necessary and sufficient characterizations of the iISS property, expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations.

A theoretical characterization of nonlinear distortion effects in OFDM systems

Abstract: This paper presents a theoretical characterization of nonlinear distortion effects in orthogonal frequency division multiplexing (OFDM) transmission systems. In the theoretical framework developed, it is shown that the effects on the decision variables of the in-band distortion introduced by a bandpass memoryless nonlinearity can be described by means of a complex gain and an additive Gaussian term with zero mean and suitable variance; analytical expressions for gain and variance are given. The conditions which allow this description are emphasized and discussed. As a consequence, a completely analytical procedure to evaluate error probability is also obtained and illustrated using OFDM/discrete multitone modulation (DMT) systems with rectangular pulse shaping; for the soft-envelope limiter nonlinearity, a closed form is derived. A comparison with simulation results is carried out to verify the accuracy of this method.

Novel Soliton Solutions of the Nonlinear Schrödinger Equation Model

Abstract: The methodology developed provides for a systematic way to find an infinite number of the novel stable bright and dark ``soliton islands'' in a ``sea of solitary waves'' of the nonlinear Schr\"odinger equation model with varying dispersion, nonlinearity, and gain or absorption It is shown that solitons exist only under certain conditions and the parameter functions describing dispersion, nonlinearity, and gain or absorption inhomogeneities cannot be chosen independently Fundamental soliton management regimes are discovered

On the Stability of Functional Equations and a Problem of Ulam

Abstract: In this paper, we study the stability of functional equations that has its origins with S. M. Ulam, who posed the fundamental problem 60 years ago and with D. H. Hyers, who gave the first significant partial solution in 1941. In particular, during the last two decades, the notion of stability of functional equations has evolved into an area of continuing research from both pure and applied viewpoints. Both classical results and current research are presented in a unified and self-contained fashion. In addition, related problems are investigated. Some of the applications deal with nonlinear equations in Banach spaces and complementarity theory.

Dissipative Systems Analysis and Control: Theory and Applications

Abstract: Dissipative Systems Analysis and Control (second edition) presents a fully revised and expanded treatment of dissipative systems theory, constituting a self-contained, advanced introduction for graduate students, researchers and practising engineers. It examines linear and nonlinear systems with examples of both in each chapter; some infinite-dimensional examples are also included. Throughout, emphasis is placed on the use of the dissipative properties of a system for the design of stable feedback control laws. The theory is substantiated by experimental results and by reference to its application in illustrative physical cases (Lagrangian and Hamiltonian systems and passivity-based and adaptive controllers are covered thoroughly). The second edition is substantially reorganized both to accommodate new material and to enhance its pedagogical properties. Some of the changes introduced are: * Complete proofs of the main theorems and lemmas. * The Kalman-Yakubovich-Popov Lemma for non-minimal realizations, singular systems, and discrete-time systems (linear and nonlinear). * Passivity of nonsmooth systems (differential inclusions, variational inequalities, Lagrangian systems with complementarity conditions). * Sections on optimal control and H-infinity theory. * An enlarged bibliography with more than 550 references, and an augmented index with more than 500 entries. * An improved appendix with introductions to viscosity solutions, Riccati equations and some useful matrix algebra.

Nonlinear limits to the information capacity of optical fiber communications

Abstract: The exponential growth in the rate at which information can be communicated through an optical fiber is a key element in the so called information revolution. However, like all exponential growth laws, there are physical limits to be considered. The nonlinear nature of the propagation of light in optical fiber has made these limits difficult to elucidate. Here we obtain basic insights into the limits to the information capacity of an optical fiber arising from these nonlinearities. The key simplification lies in relating the nonlinear channel to a linear channel with multiplicative noise, for which we are able to obtain analytical results. In fundamental distinction to the linear additive noise case, the capacity does not grow indefinitely with increasing signal power, but has a maximal value. The ideas presented here have broader implications for other nonlinear information channels, such as those involved in sensory transduction in neurobiology. These have been often examined using additive noise linear channel models, and as we show here, nonlinearities can change the picture qualitatively.

Robust adaptive control of a class of nonlinear systems with unknown backlash-like hysteresis

Abstract: Deals with adaptive control of a class of nonlinear dynamic systems preceded by unknown backlash-like hysteresis nonlinearities, where the hysteresis is modeled by a differential equation. By exploiting solution properties of the differential equation and combining those properties with adaptive control techniques, a robust adaptive control algorithm is developed without constructing a hysteresis inverse. The new control law ensures global stability of the adaptive system and achieves both stabilization and tracking to within a desired precision. Simulations performed on a nonlinear system illustrate and clarify the approach.

Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem

Abstract: We develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled Lagrangians. The procedure involves making structured modifications to the Lagrangian for the uncontrolled system, thereby constructing the controlled Lagrangian. The Euler-Lagrange equations derived from the controlled Lagrangian describe the closed-loop system, where new terms in these equations are identified with control forces. Since the controlled system is Lagrangian by construction, energy methods can be used to find control gains that yield closed-loop stability. We use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group. The procedure is demonstrated for several underactuated balance problems, including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane.

Robust backstepping control of nonlinear systems using neural networks

Abstract: A controller is proposed for the robust backstepping control of a class of general nonlinear systems using neural networks (NNs). A tuning scheme is proposed which can guarantee the boundedness of tracking error and weight updates. Compared with adaptive backstepping control schemes, we do not require the unknown parameters to be linear parametrizable. No regression matrices are needed, so no preliminary dynamical analysis is needed. One salient feature of our NN approach is that there is no need for the off-line learning phase. Three nonlinear systems, including a one-link robot, an induction motor, and a rigid-link flexible-joint robot, were used to demonstrate the effectiveness of the proposed scheme.

Nonlinearity in Structural Dynamics: Detection, Identification and Modelling

Abstract: Summary chapter and guidelines on nonlinear procedures: Flow diagrams. Why not dynamical systems theory. Summary of linear system theory: Continuous-time. Discrete-time. Dynamic testing of linear and nonlinear structures: Simple procedures for detecting nonlinearity in dynamic testing. Sine, Chirp, Random, Impulse etc. Linearisation. Correlations-coherence. FRFs of linear and nonlinear systems: Harmonic balance. Averaging methods. Nyquist plots. Carpet plots. MDOF systems. Hilbert transform - a practical approach: Definition in terms of odd/even functions. Time-frequency domain definitions. Computation - fast method. Correction terms. Principal component analysis. Corehence. Damping estimation (Khalid's work). Linearisation from random testing. Spectral moments. Hilbert transform - a complex analytical approach: Contour integrals. Titchmarsh's theorem. Artificial noncausality. Correction for asymptotic behaviour. Viscous v. Hysteretic damping - exponential integrals. Pole-zero decomposition - estimation without truncation. Restoring force surfaces and direct parameter estimation: Masri/Caughey theory. Link models/Khalids approach. Application requirements - integration/differentiation of data. Least-squares estimation: Normal equations. Orthogonal estimator. SVD. Recursive LS. - forgetting factors. Discrete-time methods: NARMAX. AVD. Model validity. Functional series: Volterra series. Existence, uniqueness, convergence. Connection with Green's functions - calculation - symmetries. Fliess/Lamnabhi power series approach. Wiener series - high-dimensional correlations - Volterra limit. Higher order FRFs/Transfer functions: Harmonic probing. Interpretaton. SDOF/MDOF systems. Hypercurve fitting (S. Gifford). Convergence revisited (Dr Lee). Neural networks: Multi-layer perceptrons. Radial basis functions. Modelling nonlinear systems. Dynamics neurons. Networks as nonlinear dynamical systems. Classification of nonlinear systems: Pattern recognition/feature extraction. Wigner-Ville distribution. Wavelet transform. Neural networks. FRFs for classification SDOF/MDOF. Nonlinear least-squares: Piecewise-linear systems. Hysteretic systems. Yar/Hammond - GA parameter estimation. Gradient descent - shock absorber model.

Oscillation Theory for Difference and Functional Differential Equations

Abstract: Preface. 1. Oscillation of Difference Equations. 1.1. Introduction. 1.2. Oscillation of Scalar Difference Equations. 1.3. Oscillation of Orthogonal Polynomials. 1.4. Oscillation of Functions Recurrence Equations. 1.5. Oscillation in Ordered Sets. 1.6. Oscillation in Linear Spaces. 1.7. Oscillation in Archimedean Spaces. 1.8. Oscillation of Partial Recurrence Equations. 1.9. Oscillation of System of Equations. 1.10. Oscillation Between Sets. 1.11. Oscillation of Continuous-Discrete Recurrence Equations. 1.12. Second Order Quasilinear Difference Equations. 1.13. Oscillation of Even Order Difference Equations. 1.14. Oscillation of Odd Order Difference Equations. 1.15. Oscillation of Neutral Difference Equations. 1.16. Oscillation of Mixed Difference Equations. 1.17. Difference Equations Involving Quasi-differences. 1.18. Difference Equations with Distributed Deviating Arguments. 1.19. Oscillation of Systems of Higher Order Difference Equations. 1.20. Partial Difference Equations with Continuous Variables. 2. Oscillation of Functional Differential Equations. 2.1. Introduction. 2.2. Definitions, Notations and Preliminaries. 2.3. Ordinary Difference Equations. 2.4. Functional Difference Equations. 2.5. Comparison of Equations of the Same Form. 2.6. Comparison of Equations with Others of Lower Order. 2.7. Further Comparison Results. 2.8. Equations with Middle Term of Order (n - 1). 2.9. Forced Differential Equations. 2.10.Forced Equations with Middle Term of Order (n - 1). 2.11. Superlinear Forced Equations. 2.12. Sublinear Forced Equations. 2.13. Perturbed Functional Equations. 2.14. Comparison of Neutral Equations with Nonneutral Equations. 2.15 Comparison of Neutral Equations with Equations of the Same Form. 2.16. Neutral Differential Equations of Mixed Type. 2.17. Functional Differential Equations Involving Quasi-derivatives. 2.18. Neutral and Damped Functional Differential Equations Involving Quasi-derivatives. 2.19. Forced Functional Differential Equations Involving Quasi-derivatives. 2.20. Systems of Higher Order Functional Differential Equations. References. Subject Index.