Since publication of the first edition of this book in 1988, the study of dynamical systems of infinite dimension has been a very active area in pure and applied mathematics; new results include the study of the existence of attractors for a large number of systems in mathematical physics and mechanics; lower and upper estimates on the dimension of the attractors; approximation of attractors; inertial manifolds and their approximation. The study of multilevel numerical methods stemming from dyna

Preface to the Second Edition
Preface to the First Edition
GENERAL INTRODUCTION.
The User‘s Guide
Introduction
1. Mechanism and Description of Chaos. The Finite-Dimensional Case
2. Mechanism and Description of Chaos. The Infinite-Dimensional Case
3. The Global Attractor. Reduction to Finite Dimension
4. Remarks on the Computational Aspect
5. The User‘s Guide
CHAPTER ⅠGeneral Results and Concepts on Invariant Sets and Attractors
Introduction
1. Semigroups， Invariant Sets， and Attractors
2. Examples in Ordinary Differential Equations
3. Fractal Interpolation and Attractors
CHAPTERⅡ Elements of Functional Analysis
Introduction
1. Function Spaces
2. Linear Operators
3. Linear Evolution Equations of the First Order in Time
4. Linear Evolution Equations of the Second Order in Time
CHAPTER Ⅲ Attractors of the Dissipative Evolution Equation of the First Order in Time: Reaction-Diffusion Equations. Fluid Mechanics and Pattern Formation Equations Introduction
1. Reaction-Diffusion Equations
2. Navier-Stokes Equations（n=2）
3. Other Equations in Fluid Mechanics
4. Some Pattern Formation Equations
5. Semilinear Equations
6. Backward Uniqueness
CHAPTER Ⅳ Attractors of Dissipative Wave Equations
Introduction
1. Linear Equations: Summary and Additional Results
2. The Sine-Gordon Equation
3. A Nonlinear Wave Equation of Relativistic Quantum Mechanics
4. An Abstract Wave Equation
5. The Ginzburg-Landau Equation
6. Weakly Dissipative Equations. I. The Nonlinear Schr6dinger Equation
7. Weakly Dissipative Equations II. The Korteweg-De Vries Equation
8. Unbounded Case: The Lack of Compactness
9. Regularity of Attractors
10. Stability of Attractors
CHAPTER Ⅴ Lyapunov Exponents and Dimension of Attractors
Introduction
1. Linear and Multilinear Algebra
2. Lyapunov Exponents and Lyapunov Numbers
3. Hausdorff and Fractal Dimensions of Attractors
CHAPTER Ⅵ Explicit Bounds on the Number of Degrees of Freedom and the Dimension of Attractors of Some Physical Systems
Introduction
1. The Lorenz Attractor
2. Reaction-Diffusion Equations
3. Navier-Stokes Equations（n=2）
4. Other Equations in Fluid Mechanic
5. Pattern Formation Equations
6. Dissipative Wave Equations
7. The Ginzburg-Landau Equation
8. Differentiability of the Semigroup
CHAPTER Ⅶ Non-Well-Posed Problems, Unstable Manifolds, Lyapunov Functions, and Lower Bounds on Dimensions
Introduction
PART A: NoN-WELL-POSED PROBLEMS
1. Dissipativity and Well Posedness
2. Estimate of Dimension for Non-Well-Posed Problems: Examples in Fluid Dynamics
PART B: UNSTABLE MANIFOLDS, LYAPUNOV FUNCTIONS, AND LOWER BOUNDS ON DIMENSIONS
3. Stable and Unstable Manifolds
4. The Attractor of a Semigroup with a Lyapunov Function
5. Lower Bounds on Dimensions of Attractors: An Example
CHAPTER Ⅷ The Cone and Squeezing Properties. Inertial Manifolds
Introduction
1. The Cone Property
2. Construction of an Inertial Manifold: DeScription of the Method
3. Existence of an Inertial Manifold
4. Examples
5. Approximation and Stability of the Inertial Manifold with Respect to Perturbations
CHAPTER Ⅸ Inertial Manifolds and Slow Manifolds. The Non-Self-Adjoint Case
Introduction
1. The Functional Setting
2. The Main Result Lipschitz Case
3. Complements and Applications
4. Inertial Manifolds and Slow Manifolds
CHAPTER Ⅹ Approximation of Attractors and Inertial Manifolds. Convergent Families of Approximate Inertial Manifolds
Introduction
1. Construction of the Manifolds
2. Approximation of Attractors
3. Convergent Families of Approximate Inertial Manifolds
APPENDIX Collective Sobolev Inequalities
Introduction
1. Notations and Hypotheses
2. Spectral Estimates for Schrodinger-Type Operators
3. Generalization of the Sobolev-Lieb-Thirring Inequality Ⅰ
4. Generalization of the Sobolev-Lieb-Thirring Inequality Ⅱ
5. Examples
Bibliography
Index