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Introduction to the theory of infinite-dimensional dissipative systems

  • Author(s): Igor Chueshov (Kharkiv National University, Kharkiv, Ukraine)
  • Publisher: ACTA Scientific Publishing House
  • Year: 2002

[I. Chueshov, Introduction to the theory of infinite-dimensional dissipative systems, ACTA, Kharkiv, 2002]

This textbook gives a systematic introduction to the main ideas, methods and problems of the mathematical theory of infinite-dimensional dissipative dynamical systems. The main attention is paid to systems generated by nonlinear partial differential equations arising in the modern mechanics. An inspection of the table of contents will give an impression about the scope of the material covered in the book.

The book begins with the first chapter on basic concepts and definitions of the mathematical theory of infinite-dimensional dissipative dynamical systems. It deals with the notion of dynamical systems, trajectories and invariant sets, attractors, dissipativity and asymptotic compactness, theorems on the existence of global attractors, structure of global attractors, stability properties of attractors and reduction principle, invariant sets, existence and properties of attractors of a class of infinite-dimensional dissipative systems.

Chapter 2. Long-time behaviour of solutions to semilinear parabolic equations. It examines positive operators with discrete spectrum, semilinear parabolic equations in Hilbert space, existence conditions and properties of global attractors, systems with Lyapunov function, models of nonlinear diffusion, the model of turbulence in fluid, retarded semilinear parabolic equations.

Chapter 3 deals with basic equation and concept of inertial manifolds. It describes the integral equation for determination of inertial manifolds, existence and properties of inertial manifolds, dependence of inertial manifolds on parameters, approximate inertial manifolds for semilinear parabolic equations, inertial manifolds for in time second-order equations, approximate inertial manifolds for in time second-order equations, the nonlinear Galerkin method.

Chapter 4 investigates the problem on nonlinear oscillations of a plate in a supersonic gas flow. It considers the auxiliary linear problem, the theorem on the existence and uniqueness of solutions, smoothness of solutions, dissipativity and asymptotic compactness, global attractors and inertial sets, conditions of regularity of attractors, singular limit in the problem of oscillations of a plate, inertial and approximate inertial manifolds.

Chapter 5 presents basic notions and properties of the theory of functionals that uniquely determine long-time dynamics. It displays the concept of a set of determining functionals, completeness defect, estimates on completeness defect in Sobolev spaces, functionals for abstract semilinear parabolic equations, functionals for reaction-diffusion systems, functionals in the problem of nerve impulse transmission, functionals for the in time second-order equations, boundary determining functionals.

Chapter 6 deals with the homoclinic chaos in infinite-dimensional systems. It describes Bernoulli shift as a model of chaos, exponential dichotomy and difference equations, hyperbolicity of invariant sets for differentiable mappings, Anosov’s lemma on ε-trajectories, the Birkhoff-Smale theorem, the possibility of chaos in the problem of nonlinear oscillations of a plate, existence of transversal homoclinic trajectories.

Every chapter includes a separate list of references consisting of the publications referred to in this book and offers additional works recommended for further reading. There is a lot of exercises in the book. Some exercises contain additional information on the object under consideration.

The core of the book is composed of the courses given by the author at the Department of Mechanics and Mathematics at Kharkov University during several years.

The well chosen material given in an appropriate form and style makes the book very useful for the university students as well as for mathematicians and specialists in mechanics which are interested in the mathematical theory of dynamical systems. The book will benefit both teacher and student.