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- Author: Jack K. Hale
- Pages: 198
- Language: English
- ISBN/ASIN: 821849344
- ISBN13: 9780821849347
- Upload date: 26-12-2016, 15:08
- Category: Mathematics

This book is directed at researchers in nonlinear ordinary and partial differential equations and at those who apply these topics to other fields of science. About one third of the book focuses on the existence and properties of the flow on the global attractor for a discrete or continuous dynamical system. The author presents a detailed discussion of abstract properties and examples of asymptotically smooth maps and semigroups. He also covers some of the continuity properties of the global attractor under perturbation, its capacity and Hausdorff dimension, and the stability of the flow on the global attractor under perturbation. The remainder of the book deals with particular equations occurring in applications and especially emphasizes delay equations, reaction-diffusion equations, and the damped wave equations. In each of the examples presented, the author shows how to verify the existence of a global attractor, and, for several examples, he discusses some properties of the flow on the global attractor. Table of Contents: Introduction. Discrete dynamical systems: Limit sets Stability of invariant sets and asymptotically smooth maps Examples of asymptotically smooth maps Dissipativeness and global attractors Dependence on parameters Fixed point theorems Stability relative to the global attractor and Morse-Smale maps Dimension of the global attractor Dissipativeness in two spaces Notes and remarks. Continuous dynamical systems: Limit sets Asymptotically smooth and $alpha$-contracting semigroups Stability of invariant sets Dissipativeness and global attractors Dependence on parameters Periodic processes Skew product flows Gradient flows Dissipativeness in two spaces Properties of the flow on the global attractor Notes and remarks. Applications: Retarded functional differential equations Sectorial evolutionary equations A scalar parabolic equation The Navier-Stokes equation Neutral functional differential equations Some abstract evolutionary equations A one-dimensional damped wave equation A three-dimensional damped wave equation Remarks on other applications Dependence on parameters and approximation of the attractor. Appendix. Stable and unstable manifolds. References. Index. This is a reprint of the 1988 original. Review from Zentralblatt MATH: This monograph reports the advances that have been made in the area by the author and many other mathematicians it is an important source of ideas for the researchers interested in the subject. Review from Mathematical Reviews: Although advanced, this book is a very good introduction to the subject, and the reading of the abstract part, which is elegant, is pleasant...this monograph will be of valuable interest for those who aim to learn in the very rapidly growing subject of infinite-dimensional dissipative dynamical systems.

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Stability and Stabilization of Nonlinear Systems (Communications and Control Engineering) Uncertain Dynamical Systems: Stability and Motion Control (Chapman & Hall/CRC Pure and Applied Mathematics) Stochastic Flows and Stochastic Differential Equations Stochastic Differential Equations and Applications, Second Edition Simulating, Analyzing, and Animating Dynamical Systems: A Guide to Xppaut for Researchers and Students (Software, Environments, Tools) (Software, Environments and Tools) Semigroups Associated with Dissipative Systems (Chapman & Hall/CRC Research Notes in Mathematics Series) Ordinary Differential Equations: Introduction and Qualitative Theory, Third Edition Local And Global Analysis of Nonlinear Dispersive And Wave Equations Lectures on Fractal Geometry and Dynamical Systems (Student Mathematical Library) Invariant Manifolds in Discrete and Continuous Dynamical Systems (EMS Tracts in Mathematics) Introduction to Smooth Ergodic Theory Handbook of Functional Equations: Stability Theory (Springer Optimization and Its Applications) Global Existence and Uniqueness of Nonlinear Evolutionary Fluid Equations (Frontiers in Mathematics) Evolution Semigroups in Dynamical Systems and Differential Equations Elements of Mathematical Theory of Evolutionary Equations in Banach Spaces (World Scientific Series on Nonlinear Science, Series a) (World Scientific Series in Nonlinear Science, Series a) Dynamical Systems:Differential Equations, Maps and Chaotic Behavior (Chapman Hall/Crc Mathematics Series) Dynamical Systems II: Ergodic Theory with Applications to Dynamical Systems and Statistical Mechanics (Encyclopaedia of Mathematical Sciences) Differential Dynamical Systems (Monographs on Mathematical Modeling and Computation) Bifurcation Theory: An Introduction with Applications to Partial Differential Equations (Applied Mathematical Sciences) Beginning Partial Differential Equations, 3rd Edition An Introduction to Dynamical Systems and Chaos A Basic Course in Partial Differential Equations (Graduate Studies in Mathematics) Exponentially Convergent Algorithms for Abstract Differential Equations (Frontiers in Mathematics) Hyperbolic Chaos: A Physicist's View

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