Originally published in Russian, this advanced mathematics textbook is being shared by the ACTA Scientific Publishing House. The new edition is from 2002.
I’ll let the editor of Introduction to the Theory of Infinite-Dimensional Dissipative Systems explain the textbook as it covers material far beyond what I studied. It strives to provide “an exhaustive introduction to the scope of main ideas and methods of the theory of infinite-dimensional dissipative dynamical systems which has been rapidly developing in recent years. In the examples systems generated by nonlinear partial differential equations arising in the different problems of modern mechanics of continua are considered. The main goal of the book is to help the reader to master the basic strategies used in the study of infinite-dimensional dissipative systems and to qualify him/her for an independent scientific research in the given branch. Experts in nonlinear dynamics will find many fundamental facts in the convenient and practical form in this book.”
The author taught courses at the Department of Mechanics and Mathematics at Kharkov University. The material covered in those courses are the basis for this textbook and accompanying exercises. Students should know the fundamentals of functional analysis and ordinary differential equations to be successful with this material.
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Table of Contents for Introduction to the Theory of Infinite-Dimensional Dissipative Systems
- Chapter 1. Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems
- 1. Notion of Dynamical System
- 2. Trajectories and Invariant Sets
- 3. Definition of Attractor
- 4. Dissipativity and Asymptotic Compactness
- 5. Theorems on Existence of Global Attractor
- 6. On the Structure of Global Attractor
- 7. Stability Properties of Attractor and Reduction Principle
- 8. Finite Dimensionality of Invariant Sets
- 9. Existence and Properties of Attractors of a Class of Infinite-Dimensional Dissipative Systems
- x. References
- Chapter 2. Long-Time Behaviour of Solutions to a Class of Semilinear Parabolic Equations
- 1. Positive Operators with Discrete Spectrum
- 2. Semilinear Parabolic Equations in Hilbert Space
- 3. Examples
- 4. Existence Conditions and Properties of Global Attractor
- 5. Systems with Lyapunov Function
- 6. Explicitly Solvable Model of Nonlinear Diffusion
- 7. Simplified Model of Appearance of Turbulence in Fluid
- 8. On Retarded Semilinear Parabolic Equations
- x. References
- Chapter 3. Inertial Manifolds
- 1. Basic Equation and Concept of Inertial Manifold
- 2. Integral Equation for Determination of Inertial Manifold
- 3. Existence and Properties of Inertial Manifolds
- 4. Continuous Dependence of Inertial Manifold onProblem Parameters
- 5. Examples and Discussion
- 6. Approximate Inertial Manifolds for Semilinear Parabolic Equations
- 7. Inertial Manifold for Second Order in Time Equations
- 8. Approximate Inertial Manifolds for Second Order in Time Equations
- 9. Idea of Nonlinear Galerkin Method
- Chapter 4. The Problem of Nonlinear Oscillations of a Plate in a Supersonic Gas Flow
- 1. Spaces
- 2. Auxiliary Linear Problem
- 3. Theorem on the Existence and Uniqueness of Solutions
- 4. Smoothness of Solutions
- 5. Dissipativity and Asymptotic Compactness
- 6. Global Attractor and Inertial Sets
- 7. Conditions of Regularity of Attractor
- 8. On Singular Limit in the Problem of Oscillations of a Plate
- 9. On Inertial and Approximate Inertial Manifolds
- x. References
- Chapter 5. Theory of Functionals that Uniquely Determine Long-Time Dynamics
- 1. Concept of a Set of Determining Functionals
- 2. Completeness Defect
- 3. Estimates of Completeness Defect in Sobolev Spaces
- 4. Determining Functionals for Abstract Semilinear Parabolic Equations
- 5. Determining Functionals for Reaction-Diffusion Systems
- 6. Determining Functionals in the Problem of Nerve Impulse Transmission
- 7. Determining Functionals for Second Order in Time Equations
- 8. On Boundary Determining Functionals
- x. References
- Chapter 6. Homoclinic Chaos in Infinite-Dimensional Systems
- 1. Bernoulli Shift as a Model of Chaos
- 2. Exponential Dichotomy and Difference Equations
- 3. Hyperbolicity of Invariant Sets for Differentiable Mappings
- 4. Anosov’s Lemma on e-trajectories
- 5. Birkhoff-Smale Theorem
- 6. Possibility of Chaos in the Problem of Nonlinear Oscillations of a Plate
- 7. On the Existence of Transversal Homoclinic Trajectories
- x. References
- Index
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Introduction to the Theory of Infinite-Dimensional Dissipative Systems
