Showing papers on "Nonlinear system published in 1995"

Principles of Nonlinear Optical Spectroscopy

Abstract: 1. Introduction 2. Quantum Dynamics in Hilbert Space 3. The Density Operator and Quantum Dynamics in Liouville Space 4. Quantum Electrodynamics, Optical Polarization, and Nonlinear Spectroscopy 5. Nonlinear Response Functions and Optical Susceptibilities 6. The Optical Response Functions of a Multilevel System with Relaxation 7. Semiclassical Simulation of the Optical Response Functions 8. The Cumulant Expansion and the Multimode Brownian Oscillator Model 9. Fluorescence, Spontaneous-Raman and Coherent-Raman Spectroscopy 10. Selective Elimination of Inhomogeneous Broadening Photon Echoes 11. Resonant Gratings, Pump-Probe, and Hole Burning Spectroscopy 12. Wavepacket Dynamics in Liouville Space The Wigner Representation 13. Wavepacket Analysis of Nonimpulsive Measurements 14. Off-Resonance Raman Scattering 15. Polarization Spectroscopy Birefringence and Dichroism 16. Nonlinear Response of Molecular Assemblies The Local-Field Approximation 17. Many Body and Cooperative Effects in the Nonlinear Response

Flatness and defect of non-linear systems: introductory theory and examples

Abstract: We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra where flatness- and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of plane curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strategy is proposed such that the averaged systems become flat.

Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering

Abstract: Preface 1. Overview 1.0 Chaos, Fractals, and Dynamics 1.1 Capsule History of Dynamics 1.2 The Importance of Being Nonlinear 1.3 A Dynamical View of the World PART I. ONE-DIMENSIONAL FLOWS 2. Flows on the Line 2.0 Introduction 2.1 A Geometric Way of Thinking 2.2 Fixed Points and Stability 2.3 Population Growth 2.4 Linear Stability Analysis 2.5 Existence and Uniqueness 2.6 Impossibility of Oscillations 2.7 Potentials 2.8 Solving Equations on the Computer Exercises 3. Bifurcations 3.0 Introduction 3.1 Saddle-Node Bifurcation 3.2 Transcritical Bifurcation 3.3 Laser Threshold 3.4 Pitchfork Bifurcation 3.5 Overdamped Bead on a Rotating Hoop 3.6 Imperfect Bifurcations and Catastrophes 3.7 Insect Outbreak Exercises 4. Flows on the Circle 4.0 Introduction 4.1 Examples and Definitions 4.2 Uniform Oscillator 4.3 Nonuniform Oscillator 4.4 Overdamped Pendulum 4.5 Fireflies 4.6 Superconducting Josephson Junctions Exercises PART II. TWO-DIMENSIONAL FLOWS 5. Linear Systems 5.0 Introduction 5.1 Definitions and Examples 5.2 Classification of Linear Systems 5.3 Love Affairs Exercises 6. Phase Plane 6.0 Introduction 6.1 Phase Portraits 6.2 Existence, Uniqueness, and Topological Consequences 6.3 Fixed Points and Linearization 6.4 Rabbits versus Sheep 6.5 Conservative Systems 6.6 Reversible Systems 6.7 Pendulum 6.8 Index Theory Exercises 7. Limit Cycles 7.0 Introduction 7.1 Examples 7.2 Ruling Out Closed Orbits 7.3 Poincare-Bendixson Theorem 7.4 Lienard Systems 7.5 Relaxation Oscillators 7.6 Weakly Nonlinear Oscillators Exercises 8. Bifurcations Revisited 8.0 Introduction 8.1 Saddle-Node, Transcritical, and Pitchfork Bifurcations 8.2 Hopf Bifurcations 8.3 Oscillating Chemical Reactions 8.4 Global Bifurcations of Cycles 8.5 Hysteresis in the Driven Pendulum and Josephson Junction 8.6 Coupled Oscillators and Quasiperiodicity 8.7 Poincare Maps Exercises PART III. CHAOS 9. Lorenz Equations 9.0 Introduction 9.1 A Chaotic Waterwheel 9.2 Simple Properties of the Lorenz Equations 9.3 Chaos on a Strange Attractor 9.4 Lorenz Map 9.5 Exploring Parameter Space 9.6 Using Chaos to Send Secret Messages Exercises 10. One-Dimensional Maps 10.0 Introduction 10.1 Fixed Points and Cobwebs 10.2 Logistic Map: Numerics 10.3 Logistic Map: Analysis 10.4 Periodic Windows 10.5 Liapunov Exponent 10.6 Universality and Experiments 10.7 Renormalization Exercises 11. Fractals 11.0 Introduction 11.1 Countable and Uncountable Sets 11.2 Cantor Set 11.3 Dimension of Self-Similar Fractals 11.4 Box Dimension 11.5 Pointwise and Correlation Dimensions Exercises 12. Strange Attractors 12.0 Introductions 12.1 The Simplest Examples 12.2 Henon Map 12.3 Rossler System 12.4 Chemical Chaos and Attractor Reconstruction 12.5 Forced Double-Well Oscillator Exercises Answers to Selected Exercises References Author Index Subject Index

Analytic Semigroups and Optimal Regularity in Parabolic Problems

Abstract: Introduction.- 0 Preliminary material: spaces of continuous and Holder continuous functions.- 1 Interpolation theory.- Analytic semigroups and intermediate spaces.- 3 Generation of analytic semigroups by elliptic operators.- 4 Nonhomogeneous equations.- 5 Linear parabolic problems.- 6 Linear nonautonomous equations.- 7 Semilinear equations.- 8 Fully nonlinear equations.- 9 Asymptotic behavior in fully nonlinear equations.- Appendix: Spectrum and resolvent.- Bibliography.- Index.

A new approach for filtering nonlinear systems

Abstract: In this paper we describe a new recursive linear estimator for filtering systems with nonlinear process and observation models. This method uses a new parameterisation of the mean and covariance which can be transformed directly by the system equations to give predictions of the transformed mean and covariance. We show that this technique is more accurate and far easier to implement than an extended Kalman filter. Specifically, we present empirical results for the application of the new filter to the highly nonlinear kinematics of maneuvering vehicles.

Fully Nonlinear Elliptic Equations

Abstract: Introduction Preliminaries Viscosity solutions of elliptic equations Alexandroff estimate and maximum principle Harnack inequality Uniqueness of solutions Concave equations $W^{2,p}$ regularity Holder regularity The Dirichlet problem for concave equations Bibliography Index.

Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications

Abstract: Introduction 1. Convex Analysis 2. Fundamentals of Nonlinear Operations 3. Duality Theory 4. Mixed-Integer Linear Optimization 5. Mixed-Integer Nonlinear Optimization 6. Process Synthesis 7. Heat Exchanger Network Synthesis 8. Distillation-Based Separation Systems Synthesis 9. Synthesis of Reactor Networks and Reactor-Separator-Recycle Systems

Applied nonlinear dynamics : analytical, computational, and experimental methods

Abstract: Perturbation Methods Dynamical Systems and Equilibrium Solutions Dynamic Solutions Tools to Characterize Different Motions Two-to-One Internal Resonances Combination Internal Resonances Three-to-One Internal Resonances Combination Internal Resonances Systems with Quadratic and Cubic Nonlinearities Gyroscopic Systems Systems with More than One Internal Resonance Random Excitations

Positive Mathematical Programming

Abstract: A method for calibrating models of agricultural production and resource use using nonlinear yield or cost functions is developed. The nonlinear parameters are shown to be implicit in the observed land allocation decisions at a regional or farm level. The method is implemented in three stages and initiated by a constrained linear program. The procedure automatically calibrates the model in terms of output, input use, objective function values and dual values on model constraints. The resulting nonlinear models show smooth responses to parameterization and satisfy the Hicksian conditions for competitive firms.

Nonlinear image recovery with half-quadratic regularization

Abstract: One popular method for the recovery of an ideal intensity image from corrupted or indirect measurements is regularization: minimize an objective function that enforces a roughness penalty in addition to coherence with the data. Linear estimates are relatively easy to compute but generally introduce systematic errors; for example, they are incapable of recovering discontinuities and other important image attributes. In contrast, nonlinear estimates are more accurate but are often far less accessible. This is particularly true when the objective function is nonconvex, and the distribution of each data component depends on many image components through a linear operator with broad support. Our approach is based on an auxiliary array and an extended objective function in which the original variables appear quadratically and the auxiliary variables are decoupled. Minimizing over the auxiliary array alone yields the original function so that the original image estimate can be obtained by joint minimization. This can be done efficiently by Monte Carlo methods, for example by FFT-based annealing using a Markov chain that alternates between (global) transitions from one array to the other. Experiments are reported in optical astronomy, with space telescope data, and computed tomography. >

The relaxation schemes for systems of conservation laws in arbitrary space dimensions

Abstract: We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with a small dissipative correction. The new system can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solvers temporally. Numerical results for 1-D and 2-D problems are presented. The second-order schemes are shown to be total variation diminishing (TVD) in the zero relaxation limit for scalar equations. ©1995 John Wiley & Sons, Inc.

A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves

Abstract: Fully nonlinear extensions of Boussinesq equations are derived to simulate surface wave propagation in coastal regions. By using the velocity at a certain depth as a dependent variable (Nwogu 1993), the resulting equations have significantly improved linear dispersion properties in intermediate water depths when compared to standard Boussinesq approximations. Since no assumption of small nonlinearity is made, the equations can be applied to simulate strong wave interactions prior to wave breaking. A high-order numerical model based on the equations is developed and applied to the study of two canonical problems: solitary wave shoaling on slopes and undular bore propagation over a horizontal bed. Results of the Boussinesq model with and without strong nonlinearity are compared in detail to those of a boundary element solution of the fully nonlinear potential flow problem developed by Grilli et al. (1989). The fully nonlinear variant of the Boussinesq model is found to predict wave heights, phase speeds and particle kinematics more accurately than the standard approximation.

Chaos in a fractional order Chua's system

Abstract: This brief studies the effects of fractional dynamics in chaotic systems. In particular, Chua's system is modified to include fractional order elements. By varying the total system order incrementally from 3.6 to 3.7, it is demonstrated that systems of "order" less than three can exhibit chaos as well as other nonlinear behavior. This effectively forces a clarification of the definition of order which can no longer be considered only by the total number of differentiations or by the highest power of the Laplace variable. >

Understanding Nonlinear Dynamics

Abstract: 1 Finite-Difference Equations.- 1.1 A Mythical Field.- 1.2 The Linear Finite-Difference Equation.- 1.3 Methods of Iteration.- 1.4 Nonlinear Finite-Difference Equations.- 1.5 Steady States and Their Stability.- 1.6 Cycles and Their Stability.- 1.7 Chaos.- 1.8 Quasiperiodicity.- 1 Chaos in Periodically Stimulated Heart Cells.- Sources and Notes.- Exercises.- Computer Projects.- 2 Boolean Networks and Cellular Automata.- 2.1 Elements and Networks.- 2.2 Boolean Variables, Functions, and Networks.- 2 A Lambda Bacteriophage Model.- 3 Locomotion in Salamanders.- 2.3 Boolean Functions and Biochemistry.- 2.4 Random Boolean Networks.- 2.5 Cellular Automata.- 4 Spiral Waves in Chemistry and Biology.- 2.6 Advanced Topic: Evolution and Computation.- Sources and Notes.- Exercises.- Computer Projects.- 3 Self-Similarity and Fractal Geometry.- 3.1 Describing a Tree.- 3.2 Fractals.- 3.3 Dimension.- 5 The Box-Counting Dimension.- 3.4 Statistical Self-Similarity.- 6 Self-Similarity in Time.- 3.5 Fractals and Dynamics.- 7 Random Walks and Levy Walks.- 8 Fractal Growth.- Sources and Notes.- Exercises.- Computer Projects.- 4 One-Dimensional Differential Equations.- 4.1 Basic Definitions.- 4.2 Growth and Decay.- 9 Traffic on the Internet.- 10 Open Time Histograms in Patch Clamp Experiments.- 11 Gompertz Growth of Tumors.- 4.3 Multiple Fixed Points.- 4.4 Geometrical Analysis of One-Dimensional Nonlinear Ordinary Differential Equations.- 4.5 Algebraic Analysis of Fixed Points.- 4.6 Differential Equations versus Finite-Difference Equations.- 4.7 Differential Equations with Inputs.- 12 Heart Rate Response to Sinusoid Inputs.- 4.8 Advanced Topic: Time Delays and Chaos.- 13 Nicholson's Blowflies.- Sources and Notes.- Exercises.- Computer Projects.- 5 Two-Dimensional Differential Equations.- 5.1 The Harmonic Oscillator.- 5.2 Solutions, Trajectories, and Flows.- 5.3 The Two-Dimensional Linear Ordinary Differential Equation.- 5.4 Coupled First-Order Linear Equations.- 14 Metastasis of Malignant Tumors.- 5.5 The Phase Plane.- 5.6 Local Stability Analysis of Two-Dimensional, Nonlinear Differential Equations.- 5.7 Limit Cycles and the van der Pol Oscillator.- 5.8 Finding Solutions to Nonlinear Differential Equations.- 15 Action Potentials in Nerve Cells.- 5.9 Advanced Topic: Dynamics in Three or More Dimensions.- 5.10 Advanced Topic: Poincare Index Theorem.- Sources and Notes.- Exercises.- Computer Projects.- 6 Time-Series Analysis.- 6.1 Starting with Data.- 6.2 Dynamics, Measurements, and Noise.- 16 Fluctuations in Marine Populations.- 6.3 The Mean and Standard Deviation.- 6.4 Linear Correlations.- 6.5 Power Spectrum Analysis.- 17 Daily Oscillations in Zooplankton.- 6.6 Nonlinear Dynamics and Data Analysis.- 18 Reconstructing Nerve Cell Dynamics.- 6.7 Characterizing Chaos.- 19 Predicting the Next Ice Age.- 6.8 Detecting Chaos and Nonlinearity.- 6.9 Algorithms and Answers.- Sources and Notes.- Exercises.- Computer Projects.- Appendix A A Multi-Functional Appendix.- A.1 The Straight Line.- A.2 The Quadratic Function.- A.3 The Cubic and Higher-Order Polynomials.- A.4 The Exponential Function.- A.5 Sigmoidal Functions.- A.6 The Sine and Cosine Functions.- A.7 The Gaussian (or "Normal") Distribution.- A.8 The Ellipse.- A.9 The Hyperbola.- Exercises.- Appendix B A Note on Computer Notation.- Solutions to Selected Exercises.

Global stability for the SEIR model in epidemiology

Abstract: The SEIR model with nonlinear incidence rates in epidemiology is studied. Global stability of the endemic equilibrium is proved using a general criterion for the orbital stability of periodic orbits associated with higher-dimensional nonlinear autonomous systems as well as the theory of competitive systems of differential equations.

Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems

Abstract: The purpose of this paper is to investigate neural network capability systematically. The main results are: 1) every Tauber-Wiener function is qualified as an activation function in the hidden layer of a three-layered neural network; 2) for a continuous function in S'(R/sup 1/) to be a Tauber-Wiener function, the necessary and sufficient condition is that it is not a polynomial; 3) the capability of approximating nonlinear functionals defined on some compact set of a Banach space and nonlinear operators has been shown; and 4) the possibility by neural computation to approximate the output as a whole (not at a fixed point) of a dynamical system, thus identifying the system. >

Applied analysis of the Navier-Stokes equations

Abstract: The Navier–Stokes equations are a set of nonlinear partial differential equations comprising the fundamental dynamical description of fluid motion. They are applied routinely to problems in engineering, geophysics, astrophysics, and atmospheric science. This book is an introductory physical and mathematical presentation of the Navier–Stokes equations, focusing on unresolved questions of the regularity of solutions in three spatial dimensions, and the relation of these issues to the physical phenomenon of turbulent fluid motion. Intended for graduate students and researchers in applied mathematics and theoretical physics, results and techniques from nonlinear functional analysis are introduced as needed with an eye toward communicating the essential ideas behind the rigorous analyses.

Nonlinear adaptive control of active suspensions

Abstract: In this paper, a previously developed nonlinear "sliding" control law is applied to an electro-hydraulic suspension system. The controller relies on an accurate model of the suspension system. To reduce the error in the model, a standard parameter adaptation scheme, based on Lyapunov analysis, is introduced. A modified adaptation scheme, which enables the identification of parameters whose values change with regions of the state space, is then presented. These parameters are not restricted to being slowly time-varying as in the standard adaptation scheme; however, they are restricted to being constant or slowly time varying within regions of the state space. The adaptation algorithms are coupled with the control algorithm and the resulting system performance is analyzed experimentally. The performance is determined by the ability of the actuator output to track a specified force. The performance of the active system, with and without the adaptation, is analyzed. Simulation and experimental results show that the active system is better than a passive system in terms of improving the ride quality of the vehicle. Furthermore, both of the adaptive schemes improve performance, with the modified scheme giving the greater improvement in performance. >

On the breakup of viscous liquid threads

Abstract: A one-dimensional model evolution equation is used to describe the nonlinear dynamics that can lead to the breakup of a cylindrical thread of Newtonian fluid when capillary forces drive the motion. The model is derived from the Stokes equations by use of rational asymptotic expansions and under a slender jet approximation. The equations are solved numerically and the jet radius is found to vanish after a finite time yielding breakup. The slender jet approximation is valid throughout the evolution leading to pinching. The model admits self-similar pinching solutions which yield symmetric shapes at breakup. These solutions are shown to be the ones selected by the initial boundary value problem, for general initial conditions. Further more, the terminal state of the model equation is shown to be identical to that predicted by a theory which looks for singular pinching solutions directly from the Stokes equations without invoking the slender jet approximation throughout the evolution. It is shown quantitatively, therefore, that the one-dimensional model gives a consistent terminal state with the jet shape being locally symmetric at breakup. The asymptotic expansion scheme is also extended to include unsteady and inertial forces in the momentum equations to derive an evolution system modelling the breakup of Navier-Stokes jets. The model is employed in extensive simulations to compute breakup times for different initial conditions; satellite drop formation is also supported by the model and the dependence of satellite drop volumes on initial conditions is studied.

Parallel distributed compensation of nonlinear systems by Takagi-Sugeno fuzzy model

Abstract: We present a design methodology for stabilization of a class of nonlinear systems. First, we approximate a nonlinear plant with a Takagi-Sugeno fuzzy model. Then a model-based fuzzy controller design utilizing the concept of so-called "parallel distributed compensation" is employed. The design procedure is conceptually simple and straightforward. The method is illustrated by application to the problem of balancing an inverted pendulum on a cart. >

Adaptive control of a class of nonlinear discrete-time systems using neural networks

Abstract: Layered neural networks are used in a nonlinear self-tuning adaptive control problem. The plant is an unknown feedback-linearizable discrete-time system, represented by an input-output model. To derive the linearizing-stabilizing feedback control, a (possibly nonminimal) state-space model of the plant is obtained. This model is used to define the zero dynamics, which are assumed to be stable, i.e., the system is assumed to be minimum phase. A linearizing feedback control is derived in terms of some unknown nonlinear functions. A layered neural network is used to model the unknown system and generate the feedback control. Based on the error between the plant output and the model output, the weights of the neural network are updated. A local convergence result is given. The result says that, for any bounded initial conditions of the plant, if the neural network model contains enough number of nonlinear hidden neurons and if the initial guess of the network weights is sufficiently close to the correct weights, then the tracking error between the plant output and the reference command will converge to a bounded ball, whose size is determined by a dead-zone nonlinearity. Computer simulations verify the theoretical result. >

Changing supply functions in input/state stable systems

Abstract: We consider the problem of characterizing possible supply functions for a given dissipative nonlinear system and provide a result which allows some freedom in the modification of such functions. >

Observation of two-dimensional spatial solitary waves in a quadratic medium.

Abstract: We report the first experimental demonstration of two-dimensional spatial solitary waves in second-order nonlinear optical material When an intense optical beam is focused into a phase-matchable second-order nonlinear material, the fundamental and generated second-harmonic fields are mutually trapped as a result of the strong nonlinear coupling which counteracts both diffraction and beam walkoff

A Level-set Approach Inverse Problems Involving Obstacles

Abstract: An approach for solving inverse problems involving obstacles is proposed. The approach uses a level-set method which has been shown to be effective in treating problems involving moving boundaries. We develop two computational methods based on this idea. One method results in a nonlinear time-dependent partial differential equation for the level-set function whose evolution minimizes the residual in the data fit. The second method is an optimization that generates a sequence of level-set functions that reduces the residual. The methods are illustrated in two applications: a deconvolution problem, and a diffraction screen reconstruction problem.

Discrete‐time reference governors and the nonlinear control of systems with state and control constraints

Abstract: Discrete-time, linear control systems with specified pointwise-in-time constraints, such as those imposed by actuator saturation, are considered. The constraints are enforced by the addition of a nonlinear ‘reference governor’ that attenuates, when necessary, the input commands. Because the constraints are satisfied, the control system remains linear and undesirable response effects such as instability due to saturation are avoided. The nonlinear action of the reference governor is defined in terms of a finitely determined maximal output admissible set and can be implemented on-line for systems of moderately high order. The main result is global in nature: if the input command converges to a statically admissible input and the initial state of the system belongs to the maximal output admissible set, the eventual action of the reference governor is a unit delay.