Showing papers on "Nonlinear system published in 1990"

Identification and control of dynamical systems using neural networks

Abstract: It is demonstrated that neural networks can be used effectively for the identification and control of nonlinear dynamical systems. The emphasis is on models for both identification and control. Static and dynamic backpropagation methods for the adjustment of parameters are discussed. In the models that are introduced, multilayer and recurrent networks are interconnected in novel configurations, and hence there is a real need to study them in a unified fashion. Simulation results reveal that the identification and adaptive control schemes suggested are practically feasible. Basic concepts and definitions are introduced throughout, and theoretical questions that have to be addressed are also described. >

Numerical methods for conservation laws

Abstract: I Mathematical Theory- 1 Introduction- 11 Conservation laws- 12 Applications- 13 Mathematical difficulties- 14 Numerical difficulties- 15 Some references- 2 The Derivation of Conservation Laws- 21 Integral and differential forms- 22 Scalar equations- 23 Diffusion- 3 Scalar Conservation Laws- 31 The linear advection equation- 311 Domain of dependence- 312 Nonsmooth data- 32 Burgers' equation- 33 Shock formation- 34 Weak solutions- 35 The Riemann Problem- 36 Shock speed- 37 Manipulating conservation laws- 38 Entropy conditions- 381 Entropy functions- 4 Some Scalar Examples- 41 Traffic flow- 411 Characteristics and "sound speed"- 42 Two phase flow- 5 Some Nonlinear Systems- 51 The Euler equations- 511 Ideal gas- 512 Entropy- 52 Isentropic flow- 53 Isothermal flow- 54 The shallow water equations- 6 Linear Hyperbolic Systems 58- 61 Characteristic variables- 62 Simple waves- 63 The wave equation- 64 Linearization of nonlinear systems- 641 Sound waves- 65 The Riemann Problem- 651 The phase plane- 7 Shocks and the Hugoniot Locus- 71 The Hugoniot locus- 72 Solution of the Riemann problem- 721 Riemann problems with no solution- 73 Genuine nonlinearity- 74 The Lax entropy condition- 75 Linear degeneracy- 76 The Riemann problem- 8 Rarefaction Waves and Integral Curves- 81 Integral curves- 82 Rarefaction waves- 83 General solution of the Riemann problem- 84 Shock collisions- 9 The Riemann problem for the Euler equations- 91 Contact discontinuities- 92 Solution to the Riemann problem- II Numerical Methods- 10 Numerical Methods for Linear Equations- 101 The global error and convergence- 102 Norms- 103 Local truncation error- 104 Stability- 105 The Lax Equivalence Theorem- 106 The CFL condition- 107 Upwind methods- 11 Computing Discontinuous Solutions- 111 Modified equations- 1111 First order methods and diffusion- 1112 Second order methods and dispersion- 112 Accuracy- 12 Conservative Methods for Nonlinear Problems- 121 Conservative methods- 122 Consistency- 123 Discrete conservation- 124 The Lax-Wendroff Theorem- 125 The entropy condition- 13 Godunov's Method- 131 The Courant-Isaacson-Rees method- 132 Godunov's method- 133 Linear systems- 134 The entropy condition- 135 Scalar conservation laws- 14 Approximate Riemann Solvers- 141 General theory- 1411 The entropy condition- 1412 Modified conservation laws- 142 Roe's approximate Riemann solver- 1421 The numerical flux function for Roe's solver- 1422 A sonic entropy fix- 1423 The scalar case- 1424 A Roe matrix for isothermal flow- 15 Nonlinear Stability- 151 Convergence notions- 152 Compactness- 153 Total variation stability- 154 Total variation diminishing methods- 155 Monotonicity preserving methods- 156 l1-contracting numerical methods- 157 Monotone methods- 16 High Resolution Methods- 161 Artificial Viscosity- 162 Flux-limiter methods- 1621 Linear systems- 163 Slope-limiter methods- 1631 Linear Systems- 1632 Nonlinear scalar equations- 1633 Nonlinear Systems- 17 Semi-discrete Methods- 171 Evolution equations for the cell averages- 172 Spatial accuracy- 173 Reconstruction by primitive functions- 174 ENO schemes- 18 Multidimensional Problems- 181 Semi-discrete methods- 182 Splitting methods- 183 TVD Methods- 184 Multidimensional approaches

Nonlinear Dynamical Control Systems

Abstract: Contents: Introduction.- Manifolds, Vectorfields, Lie Brackets, Distributions.- Controllability and Observability, Local Decompositions.- Input-Output Representations.- State Space Transformation and Feedback.- Feedback Linearization of Nonlinear Systems.- Controlled Invariant Distribution and the Disturbance Decoupling Problem.- The Input-Output Decoupling Problem: Geometric Considerations.- Local Stability and Stabilization of Nonlinear Systems.- Controlled Invariant Submanifolds and Nonlinear Zero Dynamics.- Mechanical Nonlinear Control Systems.- Controlled Invariance and Decoupling for General Nonlinear Systems.- Discrete-Time Nonlinear Control Systems.- Subject Index.

Variational methods: Applications to nonlinear partial differential equations and Hamiltonian systems

Abstract: Variational problems are part of our classical cultural heritage. The book gives an introduction to variational methods and presents on overview of areas of current research in this field. Particular topics included are the direct methods including lower semi-continuity results, the compensated compactness method, the concentration compactness method, Ekeland's variational principle, and duality methods or minimax methods, including the mountain pass theorems, index theory, perturbation theory, linking and extensions of these techniques to non-differentiable functionals and functionals defined on convex sets - and limit cases. All results are illustrated by specific examples, involving Hamiltonian systems, non-linear elliptic equations and systems, and non-linear evolution problems. These examples often represent the current state of the art in their fields and open perspective for further research. Special emphasis is laid on limit cases of the Palais-Smale condition.

Nonlinear Mixed Effects Models for Repeated Measures Data

Abstract: We propose a general, nonlinear mixed effects model for repeated measures data and define estimators for its parameters. The proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood (or restricted maximum likelihood) estimators for linear mixed effects models. We implement Newton-Raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models. Two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed.

Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series

Abstract: An approach is presented for making short-term predictions about the trajectories of chaotic dynamical systems. The method is applied to data on measles, chickenpox, and marine phytoplankton populations, to show how apparent noise associated with deterministic chaos can be distinguished from sampling error and other sources of externally induced environmental noise.

Output regulation of nonlinear systems

Abstract: The problem of controlling a fixed nonlinear plant in order to have its output track (or reject) a family of reference (or disturbance) signal produced by some external generator is discussed. It is shown that, under standard assumptions, this problem is solvable if and only if a certain nonlinear partial differential equation is solvable. Once a solution of this equation is available, a feedback law which solves the problem can easily be constructed. The theory developed incorporates previously published results established for linear systems. >

Application of Zinc Oxide Varistors

Abstract: This paper deals with the application of ZnO varistors—an area which has not been treated in a systematic way in the literature. The paper starts with a brief description of the fundamental properties comprising the electrical behavior as well as the physics, chemistry, and microstructure of the varistor. These properties then form the basis for defining the application parameters that are directly related to the nonlinear current-voltage characteristics of the varistor. This paper provides a detailed description of these parameters and their relation to microstructure and the processing of the varistor. Finally, a discussion is presented on the reliability of the varistor by considering a grain-boundary defect model which explains both the instability and the stability under use conditions.

Improving the learning speed of 2-layer neural networks by choosing initial values of the adaptive weights

Abstract: The authors describe how a two-layer neural network can approximate any nonlinear function by forming a union of piecewise linear segments. A method is given for picking initial weights for the network to decrease training time. The authors have used the method to initialize adaptive weights over a large number of different training problems and have achieved major improvements in learning speed in every case. The improvement is best when a large number of hidden units is used with a complicated desired response. The authors have used the method to train the truck-backer-upper and were able to decrease the training time from about two days to four hours

Nonlinear differential equations and dynamical systems

Abstract: 1 Introduction.- 1.1 Definitions and notation.- 1.2 Existence and uniqueness.- 1.3 Gronwall's inequality.- 2 Autonomous equations.- 2.1 Phase-space, orbits.- 2.2 Critical points and linearisation.- 2.3 Periodic solutions.- 2.4 First integrals and integral manifolds.- 2.5 Evolution of a volume element, Liouville's theorem.- 2.6 Exercises.- 3 Critical points.- 3.1 Two-dimensional linear systems.- 3.2 Remarks on three-dimensional linear systems.- 3.3 Critical points of nonlinear equations.- 3.4 Exercises.- 4 Periodic solutions.- 4.1 Bendixson's criterion.- 4.2 Geometric auxiliaries, preparation for the Poincare-Bendixson theorem.- 4.3 The Poincare-Bendixson theorem.- 4.4 Applications of the Poincare-Bendixson theorem.- 4.5 Periodic solutions in ?n.- 4.6 Exercises.- 5 Introduction to the theory of stability.- 5.1 Simple examples.- 5.2 Stability of equilibrium solutions.- 5.3 Stability of periodic solutions.- 5.4 Linearisation.- 5.5 Exercises.- 6 Linear Equations.- 6.1 Equations with constant coefficients.- 6.2 Equations with coefficients which have a limit.- 6.3 Equations with periodic coefficients.- 6.4 Exercises.- 7 Stability by linearisation.- 7.1 Asymptotic stability of the trivial solution.- 7.2 Instability of the trivial solution.- 7.3 Stability of periodic solutions of autonomous equations.- 7.4 Exercises.- 8 Stability analysis by the direct method.- 8.1 Introduction.- 8.2 Lyapunov functions.- 8.3 Hamiltonian systems and systems with first integrals.- 8.4 Applications and examples.- 8.5 Exercises.- 9 Introduction to perturbation theory.- 9.1 Background and elementary examples.- 9.2 Basic material.- 9.3 Naive expansion.- 9.4 The Poincare expansion theorem.- 9.5 Exercises.- 10 The Poincare-Lindstedt method.- 10.1 Periodic solutions of autonomous second-order equations.- 10.2 Approximation of periodic solutions on arbitrary long time-scales.- 10.3 Periodic solutions of equations with forcing terms.- 10.4 The existence of periodic solutions.- 10.5 Exercises.- 11 The method of averaging.- 11.1 Introduction.- 11.2 The Lagrange standard form.- 11.3 Averaging in the periodic case.- 11.4 Averaging in the general case.- 11.5 Adiabatic invariants.- 11.6 Averaging over one angle, resonance manifolds.- 11.7 Averaging over more than one angle, an introduction.- 11.8 Periodic solutions.- 11.9 Exercises.- 12 Relaxation Oscillations.- 12.1 Introduction.- 12.2 Mechanical systems with large friction.- 12.3 The van der Pol-equation.- 12.4 The Volterra-Lotka equations.- 12.5 Exercises.- 13 Bifurcation Theory.- 13.1 Introduction.- 13.2 Normalisation.- 13.3 Averaging and normalisation.- 13.4 Centre manifolds.- 13.5 Bifurcation of equilibrium solutions and Hopf bifurcation.- 13.6 Exercises.- 14 Chaos.- 14.1 Introduction and historical context.- 14.2 The Lorenz-equations.- 14.3 Maps associated with the Lorenz-equations.- 14.4 One-dimensional dynamics.- 14.5 One-dimensional chaos: the quadratic map.- 14.6 One-dimensional chaos: the tent map.- 14.7 Fractal sets.- 14.8 Dynamical characterisations of fractal sets.- 14.9 Lyapunov exponents.- 14.10 Ideas and references to the literature.- 15 Hamiltonian systems.- 15.1 Introduction.- 15.2 A nonlinear example with two degrees of freedom.- 15.3 Birkhoff-normalisation.- 15.4 The phenomenon of recurrence.- 15.5 Periodic solutions.- 15.6 Invariant tori and chaos.- 15.7 The KAM theorem.- 15.8 Exercises.- Appendix 1: The Morse lemma.- Appendix 2: Linear periodic equations with a small parameter.- Appendix 3: Trigonometric formulas and averages.- Appendix 4: A sketch of Cotton's proof of the stable and unstable manifold theorem 3.3.- Appendix 5: Bifurcations of self-excited oscillations.- Appendix 6: Normal forms of Hamiltonian systems near equilibria.- Answers and hints to the exercises.- References.

The Elements of Nonlinear Optics

Abstract: Introduction 1. The constitutive relation 2. Review of quantum mechanics 3. The susceptibility tensors 4. Symmetry properties 5. Resonant nonlinearities 6. Wave propagation and processes in nonlinear media 7. Dynamic optical nonlinearities in semiconductors 8. The optical properties of artificial materials Bibliography Index.

Receding horizon control of nonlinear systems

Abstract: The receding horizon control strategy provides a relatively simple method for determining feedback control for linear or nonlinear systems. The method is especially useful for the control of slow nonlinear systems, such as chemical batch processes, where it is possible to solve, sequentially, open-loop fixed-horizon, optimal control problems online. The method has been shown to yield a stable closed-loop system when applied to time-invariant or time-varying linear systems. It is shown that the method also yields a stable closed-loop system when applied to nonlinear systems. >

Feature-oriented image enhancement using shock filters

Abstract: Shock filters for image enhancement are developed. The filters use new nonlinear time dependent partial differential equations and their discretizations. The evolution of the initial image $u_0 (x,y)$ as $t \to \infty $ into a steady state solution $u_\infty (x,y)$ through $u(x,y,t)$, $t > 0$, is the filtering process. The partial differential equations have solutions which satisfy a maximum principle. Moreover the total variation of the solution for any fixed $t > 0$ is the same as that of the initial data. The processed image is piecewise smooth, nonoscillatory, and the jumps occur across zeros of an elliptic operator (edge detector). The algorithm is relatively fast and easy to program.

Volterra Integral and Functional Equations

Abstract: Preface List of symbols 1. Introduction and overview Part I. Linear Theory: 2. Linear convolution integral equations 3. Linear integrodifferential convolution equations 4. Equations in weighted spaces 5. Completely monotone kernels 6. Nonintegrable kernels with integrable resolvents 7. Unbounded and unstable solutions 8. Volterra equations as semigroups 9. Linear nonconvolution equations 10. Linear nonconvolution equations with measure kernels Part II. General Nonlinear Theory: 11. Perturbed linear equations 12. Existence of solutions of nonlinear equations 13. Continuous dependence, differentiability and uniqueness 14. Lyapunov techniques 15. General asymptotics Part III. Frequency Domain and Monotonicity Techniques: 16. Convolution kernels of positive type 17. Frequency domain methods: basic results 18. Frequency domain methods: additional results 19. Combined Lyapunov and frequency domain methods 20. Monotonicity methods Bibliography Index.

Predicting the future: a connectionist approach

Abstract: We investigate the effectiveness of connectionist architectures for predicting the future behavior of nonlinear dynamical systems. We focus on real-world time series of limited record length. Two examples are analyzed: the benchmark sunspot series and chaotic data from a computational ecosystem. The problem of overfitting, particularly serious for short records of noisy data, is addressed both by using the statistical method of validation and by adding a complexity term to the cost function ("back-propagation with weight-elimination"). The dimension of the dynamics underlying the time series, its Liapunov coefficient, and its nonlinearity can be determined via the network. We also show why sigmoid units are superior in performance to radial basis functions for high-dimensional input spaces. Furthermore, since the ultimate goal is accuracy in the prediction, we find that sigmoid networks trained with the weight-elimination algorithm outperform traditional nonlinear statistical approaches.

Nonlinear Waves, Solitons and Chaos

Abstract: This revised and updated second edition of a highly successful book is the only text at this level to embrace a universal approach to three major developments in classical physics; namely nonlinear waves, solitons and chaos. The authors now include new material on biology and laser theory, and go on to discuss important recent developments such as soliton metamorphosis. A comprehensive treatment of basic plasma and fluid configurations and instabilities is followed by a study of the relevant nonlinear structures. Each chapter concludes with a set of problems. This text will be particularly valuable for students taking courses in nonlinear aspects of physics. In general, it will be of value to final year undergraduates and beginning graduate students studying fluid dynamics, plasma physics and applied mathematics.

Hybrid Krylov methods for nonlinear systems of equations

Abstract: Several implementations of Newton-like iteration schemes based on Krylov subspace projection methods for solving nonlinear equations are considered. The simplest such class of methods is Newton's algorithm in which a (linear) Krylov method is used to solve the Jacobian system approximately. A method in this class is referred to as a Newton–Krylov algorithm. To improve the global convergence properties of these basic algorithms, hybrid methods based on Powell's dogleg strategy are proposed, as well as linesearch backtracking procedures. The main advantage of the class of methods considered in this paper is that the Jacobian matrix is never needed explicitly.

Weak convergence methods for nonlinear partial differential equations

Abstract: As working analysts we do not have to be reminded about the significance of understanding weak convergence. The routine of our work, the pattern of our daily lives, consists in searching for appropriate weak topologies or proving estimates suitable for them. Our motto, "Whatever is bounded converges," asks of us only to decide what "bounded" means and what "converges" means. Nonetheless, we should be mindful of what weak convergence means to our colleagues in other sciences. In this context, it is often associated with persistent oscillatory behavior, even as amplitudes decay, or the coagulation or concentration of singular behavior or defect behavior. This occurs in the treatment of almost all physical problems. The importance of understanding the role of weak convergence in these areas is fundamental to understanding the underlying physical phenomena. The primary difficulty is that there are so few weakly continuous functions. Knowing the limit of a sequence does not help us to know the limit of its square, for example. What can be new in weak convergence? Here we have a slim volume of some of the most useful and fruitful methods developed over the last fifteen years applied to questions in partial differential equations. It is so clear and well organized, we have bought copies of it for our engineer friends. We expect our students to master it. It is not a reference work, but contains most of the methods every working analyst in this field wants to know in a concise form, often explained by a careful choice of example. It is, in its own way, a sort of hight brow Cliff Notes that we can roll up and stick in our pockets. An AMS paperback, it is a rave bargain (along with all the other AMS/CBMS paperbacks). To understand the organization of the book, let us recall that one encounters a weakly convergent sequence when, for example, attempting to solve a nonlinear partial differential equation by approximation or when solving a variational principle by extracting a minimizing sequence. The weak convergence inhibits us from deciding immediately that the limit function is the solution of the limit problem. The methods for overcoming this difficulty are the subject of the monograph. Most commonly, there are three possible scenarios, all of which take advantage of the structure inherent in the particular problem. We may seek to show that the sequence converges strongly after all. Secondly, concentrations

Calculation of the nonlinear optical properties of molecules

Abstract: : A finite field method for the calculation of polarizabilities and hyperpolarizabilities is developed based on both an energy expansion and a dipole moment expansion. This procedure is implemented in the MOPAC semiempirical program. Values and components of the dipole moment (mu), polarizability (alpha), first hyperpolarizability (beta), and second hyperpolarizability (gamma) are calculated as an extension of the usual MOPAC run. Applications to benzene and substituted benzenes are shown as test cases utilizing both MNDO and AM1 Hamiltonians.

A general stochastic maximum principle for optimal control problems

Abstract: The maximum principle for nonlinear stochastic optimal control problems in the general case is proved. The control domain need not be convex, and the diffusion coefficient can contain a control variable.

Nonlinear Wave Equations

Abstract: Invariance Existence Singularities Solutions of small amplitude Scattering Stability of solitary waves Yang-Mills equations Vlasov-Maxwell equations.

Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis

Abstract: This paper surveys an area of nonlinear functional analysis and its applications. The main application is to the existence and multiplicity of periodic solutions of a possible mathematical models of nonlinearly supported bending beams, and their possible application to nonlinear behavior as observed in large-amplitude flexings in suspension bridges. A second area, periodic flexings in a floating beam, also nonlinearly supported, is covered at the end of the paper.

Iterative Functional Equations

Abstract: Preface Symbols and conventions 1. Introduction 2. Iteration 3. Linear equations and branching processes 4. Regularity of solutions of linear equations 5. Analytic and integrable solutions of linear equations 6. Theory of nonlinear equations 7. Equations of higher orders and systems of linear equations 8. Equations of infinite order and systems of nonlinear equations 9. On conjugacy 10. More on the Schroder and Abel equations 11. Characterization of functions 12. Iterative roots and invariant curves 13. Linear iterative functional inequalities References Author index Subject index.

Calculating total electrostatic energies with the nonlinear Poisson-Boltzmann equation

Abstract: The Poisson-Boltzmann (PB) equation is enjoying a resurgence in popularity and usefulness in biophysics and biochemistry due to numerical advances which allow the equation to be rapidly solved for arbitrary geometries and nonuniform dielectrics. The great simplification of PB models is to use the mean electrostatic potential to give an estimate of the potential of mean force (PMF) governing the distribution of the mobile ions in the solvent. This approximation enables both the mean potential and mean ion distribution to be obtained directly from solutions to the PB equation without performing complex statistical mechanical integrations. The nonlinear form of the PB equation has greater accuracy and range of validity than the linear form, but the approximation of the PMF by the mean potential creates theoretical difficulties in defining the total electrostatic energy for the former. In this paper we use the calculus of variations to provide a unique definition of the total energy and to obtain expressions for the total electrostatic free energy for various forms of the PB equation. These expressions involve energy density integrals over the volume of the system. Various equivalent expressions for the total energy are given and the physical meaning of the different terms that appear is discussed. Numerical calculations are carried out to demonstrate the feasibility of our approach and to assess the magnitude of the various terms that arise in the theory. Both the more familiar charging integral and the energy density integral methods can be applied to the PB equation with equal accuracy, but the latter is much more efficient computationally. The energy density integral involves the integral of the excess osmotic pressure of the ion atmosphere. The various forms of the PB equation which have been most widely discussed to date because of the availability of analytical solutions are shown to be special cases where the osmotic term is absent.

Convergence of approximation schemes for fully nonlinear second order equations

Abstract: The convergence of a wide class of approximation schemes to the viscosity solution of fully nonlinear second-order elliptic or parabolic, possibly degenerate, partial differential equations is studied It is proved that any monotone, stable, and consistent scheme converges (to the correct solution), provided that there exists a comparison principle for the limiting equation Several examples are given where the result applies >

Abstract functional-differential equations and reaction-diffusion systems

Abstract: FUNCTIONAL DIFFERENTIAL EQUATIONS AND REACTION-DIFFUSION SYSTEMS R. H. MARTIN, JR. AND H. L. SMITH ABSTRACT. Several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a Banach space setting. The ideas are applied to reaction-diffusion systems that have time delays in the nonlinear reaction terms. The techniques presented here include differential inequalities, invariant sets, and Lyapunov functions, and therefore they provide for a wide range of applicability. The results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays. Several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a Banach space setting. The ideas are applied to reaction-diffusion systems that have time delays in the nonlinear reaction terms. The techniques presented here include differential inequalities, invariant sets, and Lyapunov functions, and therefore they provide for a wide range of applicability. The results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays. Suppose Q is a bounded domain in RN with aQ smooth and A is the Laplacian operator on Q. Also, let m be a positive integer, z a positive number, and f = (fi)m a continuous, bounded function from [0, xc] x Q x C([-z, 0])m into Rm where C([-T, 0]) is the space of continuous functions from [-z, 0] into R. The purpose of this paper is to apply abstract results for semilinear functional differential equations in Banach spaces to reactiondiffusion systems with time delays having the form a tu(x, t) = diAu' (x, t) + ?i(t, x, ut(x, *)), t>a, xeQ, i=1,...,m, (RDD) ai(x)ui(x, t) + au i(x, t) = ,81(x, t), t > a , x EaQ u'(x , a + 0) = X'(x , 0) 5 -T 0, di > 0, and ca:Q * [0,o ) is C' and ,i:Q x [0, oo) R is C2. Here an is the outward normal derivative on aQ and if di = 0 it is assumed that no boundary conditions are specified for this i. Also, tu'(x, t) denotes the partial with respect to t, whereas ut(x, *) denotes the member of C([-z, 0]) defined by 0 -u(x, t + 0) = (u'(x, t + 0))M. Our techniques provide basic existence criteria, but the main point is that they can also be effectively applied to obtain estimates for solutions, especially Received by the editors October 7, 1987 and, in revised form, October 7, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 35R10, 34K30.