Showing papers on "Nonlinear system published in 1993"

Pattern formation outside of equilibrium

Abstract: A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics

Abstract: Contents: General results and concepts on invariant sets and attractors.- Elements of functional analysis.- Attractors of the dissipative evolution equation of the first order in time: reaction-diffusion equations.- Fluid mechanics and pattern formation equations.- Attractors of dissipative wave equations.- Lyapunov exponents and dimensions of attractors.- Explicit bounds on the number of degrees of freedom and the dimension of attractors of some physical systems.- Non-well-posed problems, unstable manifolds. lyapunov functions, and lower bounds on dimensions.- The cone and squeezing properties.- Inertial manifolds.- New chapters: Inertial manifolds and slow manifolds the nonselfadjoint case.

Nonlinear Parabolic and Elliptic Equations

Abstract: In response to the growing use of reaction diffusion problems in many fields, this monograph gives a systematic treatment of a class of nonlinear parabolic and elliptic differential equations and their applications these problems. It is an important reference for mathematicians and engineers, as well as a practical text for graduate students.

Modelling nonlinear economic relationships

Abstract: Basic concepts general models and tools for analysis nonlinear models in economic theory particular nonlinear multivariate models long memory models linearity testing building nonlinear models forecasting, aggression and non-symmetry applications strategies for nonlinear modelling.

Variable structure control of nonlinear systems: a new approach

Abstract: A new approach for the design of variable structure control (VSC) of nonlinear systems is presented. It is based on a new method called the reaching law method, and is complemented by a sliding-mode equivalence technique. They facilitate the design of the system dynamics in all three modes of a VSC system including the sliding, reaching, and steady-state modes. Invariance and robustness properties are discussed. The approach is applied to a robot manipulator to demonstrate its effectiveness. >

Robust receding horizon control of constrained nonlinear systems

Abstract: We present a method for the construction of a robust dual-mode, receding horizon controller which can be employed for a wide class of nonlinear systems with state and control constraints and model error. The controller is dual-mode. In a neighborhood of the origin, the control action is generated by a linear feedback controller designed for the linearized system. Outside this neighborhood, receding horizon control is employed. Existing receding horizon controllers for nonlinear, continuous time systems, which are guaranteed to stabilize the nonlinear system to which they are applied, require the exact solution, at every instant, of an optimal control problem with terminal equality constraints. These requirements are considerably relaxed in the dual-mode receding horizon controller presented in this paper. Stability is achieved by imposing a terminal inequality, rather than an equality, constraint. Only approximate minimization is required. A variable time horizon is permitted. Robustness is achieved by employing conservative state and stability constraint sets, thereby permitting a margin of error. The resultant dual-mode controller requires considerably less online computation than existing receding horizon controllers for nonlinear, constrained systems. >

Global Behavior of Nonlinear Difference Equations of Higher Order with Applications

Abstract: Preface. 1. Introduction and Preliminaries. 2. Global Stability Results. 3. Rational Recursive Sequences. 4. Applications. 5. Periodic Cycles. 6. Open Problems and Conjectures. Appendix: A. The Riccati Difference Equation. B. A Generalized Contraction Principle. C. Global Behaviour of Systems of Nonlinear Difference Equations. Bibliography. Subject Index. Author Index.

Alternative form of Boussinesq equations for nearshore wave propagation

Abstract: Boussinesq‐type equations can be used to model the nonlinear transformation of surface waves in shallow water due to the effects of shoaling, refraction, diffraction, and reflection. Different linear dispersion relations can be obtained by expressing the equations in different velocity variables. In this paper, a new form of the Boussinesq equations is derived using the velocity at an arbitrary distance from the still water level as the velocity variable instead of the commonly used depth‐averaged velocity. This significantly improves the linear dispersion properties of the Boussinesq equations, making them applicable to a wider range of water depths. A finite difference method is used to solve the equations. Numerical and experimental results are compared for the propagation of regular and irregular waves on a constant slope beach. The results demonstrate that the new form of the equations can reasonably simulate several nonlinear effects that occur in the shoaling of surface waves from deep to shallow w...

An introduction to partial differential equations

Abstract: Introduction* Characteristics* Classification of Characteristics * Conservation Laws and Shocks* Maximum Principles* Distributions* Function Spaces* Sobolev Spaces * Operator Theory * Linear Elliptic Equations * Nonlinear Elliptic Equations * Energy Methods for Evolution Problems * Semigroup Methods * References * Index

Nonstandard Finite Difference Models of Differential Equations

Abstract: Numerical instabilities non-standard finite difference schemes first order ordinary differential equations second order, nonlinear oscillator equations Schrodinger type ordinary differential equations two first order, coupled ordinary differential equations partial differential equations summary and discussion linear difference equations linear stability analysis.

Classical equations for quantum systems

Abstract: The origin of the phenomenological deterministic laws that approximately govern the quasiclassical domain of familiar experience is considered in the context of the quantum mechanics of closed systems such as the universe as a whole. A formulation of quantum mechanics is used that predicts probabilities for the individual members of a set of alternative coarse-grained histories that decohere, which means that there is negligible quantum interference between the individual histories in the set. We investigate the requirements for coarse grainings to yield decoherent sets of histories that are quasiclassical, i.e., such that the individual histories obey, with high probability, effective classical equations of motion interrupted continually by small fluctuations and occasionally by large ones. We discuss these requirements generally but study them specifically for coarse grainings of the type that follows a distinguished subset of a complete set of variables while ignoring the rest. More coarse graining is needed to achieve decoherence than would be suggested by naive arguments based on the uncertainty principle. Even coarser graining is required in the distinguished variables for them to have the necessary inertia to approach classical predictability in the presence of the noise consisting of the fluctuations that typical mechanisms of decoherence produce. We describe the derivation of phenomenological equations of motion explicitly for a particular class of models. Those models assume configuration space and a fundamental Lagrangian that is the difference between a kinetic energy quadratic in the velocities and a potential energy. The distinguished variables are taken to be a fixed subset of coordinates of configuration space. The initial density matrix of the closed system is assumed to factor into a product of a density matrix in the distinguished subset and another in the rest of the coordinates. With these restrictions, we improve the derivation from quantum mechanics of the phenomenological equations of motion governing a quasiclassical domain in the following respects: Probabilities of the correlations in time that define equations of motion are explicitly considered. Fully nonlinear cases are studied. Methods are exhibited for finding the form of the phenomenological equations of motion even when these are only distantly related to those of the fundamental action. The demonstration of the connection between quantum-mechanical causality and causality in classical phenomenological equations of motion is generalized. The connections among decoherence, noise, dissipation, and the amount of coarse graining necessary to achieve classical predictability are investigated quantitatively. Routes to removing the restrictions on the models in order to deal with more realistic coarse grainings are described.

Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations

Abstract: This paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally Lipschitzian functions. For the directional derivative-based and the generalized Jacobian-based Newton methods, both the iterates and the corresponding function values are locally, superlinearly convergent. Globally, a limiting point of the iterate sequence generated by the damped, directional derivative-based Newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one, provided that the system is strongly BD-regular and semismooth at this point. In this case, the convergence is superlinear. A general attraction theorem is presented, which can be applied to two algorithms proposed by Han, Pang and Rangaraj. A hybrid method, which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent, is also presented.

Adaptive input-output linearizing control of induction motors

Abstract: A nonlinear adaptive state feedback input-output linearizing control is designed for a fifth-order model of an induction motor which includes both electrical and mechanical dynamics under the assumptions of linear magnetic circuits. The control algorithm contains a nonlinear identification scheme which asymptotically tracks the true values of the load torque and rotor resistance which are assumed to be constant but unknown. Once those parameters are identified, the two control goals of regulating rotor speed and rotor flux amplitude are decoupled, so that power efficiency can be improved without affecting speed regulation. Full state measurements are required. >

Sur les mesures de Wigner

Abstract: We study the properties of the Wigner transform for arbitrary functions in L2 or for hermitian kernels like the so-called density matrices. And we introduce some limits of these transforms for sequences of functions in L2, limits that correspond to the semi-classical limit in Quantum Mechanics. The measures we obtain in this way, that we call Wigner measures, have various mathematical properties that we establish. In particular, we prove they satisfy, in linear situations (Schrodinger equations) or nonlinear ones (time-dependent Hartree equations), transport equations of Liouville or Vlasov type.

Global adaptive output-feedback control of nonlinear systems. II. Nonlinear parameterization

Abstract: For pt.I, see ibid., p.17-32 (1993). The problem of designing global output-feedback robust stabilizing controls for a class of single-input single-output minimum-phase uncertain nonlinear systems with known and constant relative degree is addressed. They are assumed to be linear with respect to the input and nonlinear with respect to an unknown constant parameter vector. The nonlinearities depend on the output only. The nonlinearities may be uncertain and are only required to be bounded by known smooth functions. The order of the robust compensator is equal to the relative degree minus one and is static when the relative degree is one. A self-tuning version of the robust control capable of achieving set point regulation is developed in which the control gains are tuned by an output-feedback adaptive algorithm. When the parameter vector enters linearly, the self-tuning regulator does not require the knowledge of parameter bounds and guarantees set point regulation for the same class of systems considered in Part I. >

Analysis and control of nonlinear infinite dimensional systems

Abstract: Nonlinear operators of monotone type controlled elliptical variational inequalities nonlinear accretive differential equations optimal control of parabolic variational inequalities optimal control in real time.

A Luenberger-like observer for nonlinear systems

Abstract: A state observer is proposed for nonlinear continuous time systems which extends the well known Luenberger observer. In particular, on the basis of simple assumptions on the regularity of the system equations (observability and the global Holder condition for suitable functions), which are generally satisfied for physically meaningful dynamic systems, the global asymptotic convergence of the estimated state towards the true state is shown. Finally, some examples of applications are also reported showing the effectiveness of the proposed observer.

Balancing for nonlinear systems

Abstract: We present a method of balancing for nonlinear systems which is an extension of balancing for linear systems in the sense that it is basd on the input and output energy of a system. It is a local result, but gives `broader? results than we obtain by just linearizing the system. Furthermore, the relation with balancing of the linearization is dealt with. We propose to use the method as a tool for nonlinear model reduction and investigate some of the properties of the reduced system.

Qualitative theory of compartmental systems

Abstract: Dynamic models of many processes in the biological and physical sciences which depend on local mass balance conditions give rise to systems of ordinary differential equations, many nonlinear, that are called compartmental systems. In this paper, the authors define compartmental systems, specify their relations to other nonnegative systems, and discuss examples of applications.The authors review the qualitative results on linear and nonlinear compartmental systems, including their relation to cooperative systems. They review the results for linear compartmental systems and then integrate and expand the results on nonlinear compartmental systems, providing a framework for unifying them under a few general theorems. In the course of that they complete the solution of a problem posed by Bellman and show that closed nonlinear, autonomous, n-compartment systems can show the full gamut of possible behaviors of systems of ODES.Finally, to provide additional structure to this study, the authors show how to partiti...

Newton's method and periodic solutions of nonlinear wave equations

Abstract: We prove the existence of periodic solutions of the nonlinear wave equation satisfying either Dirichlet or periodic boundary conditions on the interval [O, π]. The coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation. Using a version of Newton's method, we show that this equation has solutions provided the nonlinearity g(x, u) satisfies certain generic conditions of nonresonance and genuine nonlinearity. © 1993 John Wiley & Sons, Inc.

Nonlinear dynamic load models with recovery for voltage stability studies

Abstract: Motivated by projects in Sweden on voltage stability analysis and associated load modeling, a simple dynamic load model is proposed which captures the usual nonlinear steady-state behavior plus load recovery and overshoot. The parameters of the model can be related to physical devices depending on the time zone following a disturbance. A simple but important dynamic voltage stability analysis is developed based on the model. >

Mathematical structures of epidemic systems

Abstract: Linear models.- Strongly nonlinear models.- Quasimonotone systems. Positive feedback systems. Cooperative systems.- Spatial heterogeneity.- Age structure.- Optimization problems.

On feedback control of chaotic continuous-time systems

Abstract: Extends the ideas and techniques developed previously by the present authors for controlling discrete-time chaotic dynamic systems using traditional feedback control strategies to continuous-time chaotic systems. The authors study how the conventional engineering approach using canonical feedback controllers can control the chaotic trajectory of a continuous-time nonlinear system to converge to its equilibrium points and, more significantly, to its multiperiodic orbits including unstable limit cycles. They describe an approach via a detailed investigation of the chaotic Duffing equation, with special emphasis on the control of its chaotic trajectory to one of its multiperiodic orbits. Finally, the authors provide a rigorous mathematical theory and some computer simulations to support and visualize such controllability of the Duffing equation. >

A direct nonlinear predictor-corrector primal-dual interior point algorithm for optimal power flows

Abstract: A new algorithm using the primal-dual interior point method with the predictor-corrector for solving nonlinear optimal power flow (OPF) problems is presented. The formulation and the solution technique are new. Both equalities and inequalities in the OPF are considered and simultaneously solved in a nonlinear manner based on the Karush-Kuhn-Tucker (KKT) conditions. The major computational effort of the algorithm is solving a symmetrical system of equations, whose sparsity structure is fixed. Therefore only one optimal ordering and one symbolic factorization are involved. Numerical results of several test systems ranging in size from 9 to 2423 buses are presented and comparisons are made with the pure primal-dual interior point algorithm. The results show that the predictor-corrector primal-dual interior point algorithm for OPF is computationally more attractive than the pure primal-dual interior point algorithm in terms of speed and iteration count. >

H/sup infinity / control for nonlinear systems with output feedback

Abstract: The basic question of nonlinear H/sup infinity / control theory is to decide, for a given two-port system, when feedback that makes the full system dissipative and internally stable exists. This problem can also be viewed as a question about circuits, and, after translation, also has a game-theoretic statement. Several necessary conditions for solutions to exist are presented, and sufficient conditions for a certain construction to lead to a solution are given. >