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Author : Anatoliĭ Vladimirovich Babin,M. I. Vishik
Publisher : American Mathematical Soc.
Page : 184 pages
File Size : 54,6 Mb
Release : 1992
Category : Attractors (Mathematics)
ISBN : 0821841092
Author : Anatoliĭ Vladimirovich Babin,M. I. Vishik
Publisher : Unknown
Page : 174 pages
File Size : 53,9 Mb
Release : 2019
Category : Differentiable dynamical systems
ISBN : 7560375456
Author : Vladimir V. Chepyzhov,M. I. Vishik
Publisher : American Mathematical Soc.
Page : 377 pages
File Size : 51,6 Mb
Release : 2002
Category : Mathematics
ISBN : 9780821829509
One of the major problems in the study of evolution equations of mathematical physics is the investigation of the behavior of the solutions to these equations when time is large or tends to infinity. The related important questions concern the stability of solutions or the character of the instability if a solution is unstable. In the last few decades, considerable progress in this area has been achieved in the study of autonomous evolution partial differential equations. For anumber of basic evolution equations of mathematical physics, it was shown that the long time behavior of their solutions can be characterized by a very important notion of a global attractor of the equation. In this book, the authors study new problems related to the theory of infinite-dimensionaldynamical systems that were intensively developed during the last 20 years. They construct the attractors and study their properties for various non-autonomous equations of mathematical physics: the 2D and 3D Navier-Stokes systems, reaction-diffusion systems, dissipative wave equations, the complex Ginzburg-Landau equation, and others. Since, as it is shown, the attractors usually have infinite dimension, the research is focused on the Kolmogorov $\varepsilon$-entropy of attractors. Upperestimates for the $\varepsilon$-entropy of uniform attractors of non-autonomous equations in terms of $\varepsilon$-entropy of time-dependent coefficients are proved. Also, the authors construct attractors for those equations of mathematical physics for which the solution of the corresponding Cauchyproblem is not unique or the uniqueness is not proved. The theory of the trajectory attractors for these equations is developed, which is later used to construct global attractors for equations without uniqueness. The method of trajectory attractors is applied to the study of finite-dimensional approximations of attractors. The perturbation theory for trajectory and global attractors is developed and used in the study of the attractors of equations with terms rapidly oscillating with respect tospatial and time variables. It is shown that the attractors of these equations are contained in a thin neighborhood of the attractor of the averaged equation. The book gives systematic treatment to the theory of attractors of autonomous and non-autonomous evolution equations of mathematical physics.It can be used both by specialists and by those who want to get acquainted with this rapidly growing and important area of mathematics.
Author : James C. Robinson
Publisher : Cambridge University Press
Page : 488 pages
File Size : 40,8 Mb
Release : 2001-04-23
Category : Mathematics
ISBN : 0521632048
This book treats the theory of global attractors, a recent development in the theory of partial differential equations, in a way that also includes much of the traditional elements of the subject. As such it gives a quick but directed introduction to some fundamental concepts, and by the end proceeds to current research problems. Since the subject is relatively new, this is the first book to attempt to treat these various topics in a unified and didactic way. It is intended to be suitable for first year graduate students.
Author : Alexander Komech,Elena Kopylova
Publisher : Cambridge University Press
Page : 229 pages
File Size : 47,7 Mb
Release : 2021-09-30
Category : Mathematics
ISBN : 9781316516911
The first monograph on the theory of global attractors of Hamiltonian partial differential equations.
Author : Cheban David N
Publisher : World Scientific
Page : 616 pages
File Size : 42,7 Mb
Release : 2014-12-15
Category : Mathematics
ISBN : 9789814619844
The study of attractors of dynamical systems occupies an important position in the modern qualitative theory of differential equations. This engaging volume presents an authoritative overview of both autonomous and non-autonomous dynamical systems, including the global compact attractor. From an in-depth introduction to the different types of dissipativity and attraction, the book takes a comprehensive look at the connections between them, and critically discusses applications of general results to different classes of differential equations.The new Chapters 15-17 added to this edition include some results concerning Control Dynamical Systems — the global attractors, asymptotic stability of switched systems, absolute asymptotic stability of differential/difference equations and inclusions — published in the works of author in recent years.
Author : A.V. Babin,M.I. Vishik
Publisher : Elsevier
Page : 543 pages
File Size : 41,8 Mb
Release : 1992-03-09
Category : Mathematics
ISBN : 9780080875460
Problems, ideas and notions from the theory of finite-dimensional dynamical systems have penetrated deeply into the theory of infinite-dimensional systems and partial differential equations. From the standpoint of the theory of the dynamical systems, many scientists have investigated the evolutionary equations of mathematical physics. Such equations include the Navier-Stokes system, magneto-hydrodynamics equations, reaction-diffusion equations, and damped semilinear wave equations. Due to the recent efforts of many mathematicians, it has been established that the attractor of the Navier-Stokes system, which attracts (in an appropriate functional space) as t - ∞ all trajectories of this system, is a compact finite-dimensional (in the sense of Hausdorff) set. Upper and lower bounds (in terms of the Reynolds number) for the dimension of the attractor were found. These results for the Navier-Stokes system have stimulated investigations of attractors of other equations of mathematical physics. For certain problems, in particular for reaction-diffusion systems and nonlinear damped wave equations, mathematicians have established the existence of the attractors and their basic properties; furthermore, they proved that, as t - +∞, an infinite-dimensional dynamics described by these equations and systems uniformly approaches a finite-dimensional dynamics on the attractor U, which, in the case being considered, is the union of smooth manifolds. This book is devoted to these and several other topics related to the behaviour as t - ∞ of solutions for evolutionary equations.
Author : Jan W. Cholewa,Tomasz Dlotko
Publisher : Cambridge University Press
Page : 252 pages
File Size : 40,6 Mb
Release : 2000-08-31
Category : Mathematics
ISBN : 9780521794244
This book investigates the asymptotic behaviour of dynamical systems corresponding to parabolic equations.
Author : David N. Cheban
Publisher : World Scientific
Page : 524 pages
File Size : 53,5 Mb
Release : 2004
Category : Mathematics
ISBN : 9789812563088
The study of attractors of dynamical systems occupies an important position in the modern qualitative theory of differential equations. This engaging volume presents an authoritative overview of both autonomous and non-autonomous dynamical systems, including the global compact attractor.
Author : Messoud Efendiev
Publisher : American Mathematical Soc.
Page : 233 pages
File Size : 55,7 Mb
Release : 2013-09-26
Category : Mathematics
ISBN : 9781470409852
This book deals with the long-time behavior of solutions of degenerate parabolic dissipative equations arising in the study of biological, ecological, and physical problems. Examples include porous media equations, -Laplacian and doubly nonlinear equations, as well as degenerate diffusion equations with chemotaxis and ODE-PDE coupling systems. For the first time, the long-time dynamics of various classes of degenerate parabolic equations, both semilinear and quasilinear, are systematically studied in terms of their global and exponential attractors. The long-time behavior of many dissipative systems generated by evolution equations of mathematical physics can be described in terms of global attractors. In the case of dissipative PDEs in bounded domains, this attractor usually has finite Hausdorff and fractal dimension. Hence, if the global attractor exists, its defining property guarantees that the dynamical system reduced to the attractor contains all of the nontrivial dynamics of the original system. Moreover, the reduced phase space is really "thinner" than the initial phase space. However, in contrast to nondegenerate parabolic type equations, for a quite large class of degenerate parabolic type equations, their global attractors can have infinite fractal dimension. The main goal of the present book is to give a detailed and systematic study of the well-posedness and the dynamics of the semigroup associated to important degenerate parabolic equations in terms of their global and exponential attractors. Fundamental topics include existence of attractors, convergence of the dynamics and the rate of convergence, as well as the determination of the fractal dimension and the Kolmogorov entropy of corresponding attractors. The analysis and results in this book show that there are new effects related to the attractor of such degenerate equations that cannot be observed in the case of nondegenerate equations in bounded domains. This book is published in cooperation with Real Sociedad Matemática Española (RSME).
Author : A. Katok,B. Hasselblatt
Publisher : Elsevier
Page : 1235 pages
File Size : 41,9 Mb
Release : 2005-12-17
Category : Mathematics
ISBN : 9780080478227
This second half of Volume 1 of this Handbook follows Volume 1A, which was published in 2002. The contents of these two tightly integrated parts taken together come close to a realization of the program formulated in the introductory survey “Principal Structures of Volume 1A.The present volume contains surveys on subjects in four areas of dynamical systems: Hyperbolic dynamics, parabolic dynamics, ergodic theory and infinite-dimensional dynamical systems (partial differential equations). . Written by experts in the field.. The coverage of ergodic theory in these two parts of Volume 1 is considerably more broad and thorough than that provided in other existing sources. . The final cluster of chapters discusses partial differential equations from the point of view of dynamical systems.
Author : Andrew Comech,Alexander Komech,Mikhail Vishik
Publisher : Springer Nature
Page : 334 pages
File Size : 47,8 Mb
Release : 2023-11-15
Category : Mathematics
ISBN : 9783031336812
Mark Vishik was one of the prominent figures in the theory of partial differential equations. His ground-breaking contributions were instrumental in integrating the methods of functional analysis into this theory. The book is based on the memoirs of his friends and students, as well as on the recollections of Mark Vishik himself, and contains a detailed description of his biography: childhood in Lwów, his connections with the famous Lwów school of Stefan Banach, a difficult several year long journey from Lwów to Tbilisi after the Nazi assault in June 1941, going to Moscow and forming his own school of differential equations, whose central role was played by the famous Vishik Seminar at the Department of Mechanics and Mathematics at Moscow State University. The reader is introduced to a number of remarkable scientists whose lives intersected with Vishik’s, including S. Banach, J. Schauder, I. N. Vekua, N. I. Muskhelishvili, L. A. Lyusternik, I. G. Petrovskii, S. L. Sobolev, I. M. Gelfand, M. G. Krein, A. N. Kolmogorov, N. I. Akhiezer, J. Leray, J.-L. Lions, L. Schwartz, L. Nirenberg, and many others. The book also provides a detailed description of the main research directions of Mark Vishik written by his students and colleagues, as well as several reviews of the recent development in these directions.
Author : T.L. Gill,Woodford W. Zachary
Publisher : Springer
Page : 194 pages
File Size : 47,5 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540477914
The original idea of the organizers of the Washington Symposium was to span a fairly narrow range of topics on some recent techniques developed for the investigation of nonlinear partial differential equations and discuss these in a forum of experts. It soon became clear, however, that the dynamical systems approach interfaced significantly with many important branches of applied mathematics. As a consequence, the scope of this resulting proceedings volume is an enlarged one with coverage of a wider range of research topics.
Author : Jack K. Hale
Publisher : Unknown
Page : 210 pages
File Size : 55,5 Mb
Release : 2014-06-29
Category : Differentiable dynamical systems
ISBN : 1470412527
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. (SURV/25.S)
Author : B. Fiedler
Publisher : Gulf Professional Publishing
Page : 1099 pages
File Size : 40,8 Mb
Release : 2002-02-21
Category : Science
ISBN : 9780080532844
This handbook is volume II in a series collecting mathematical state-of-the-art surveys in the field of dynamical systems. Much of this field has developed from interactions with other areas of science, and this volume shows how concepts of dynamical systems further the understanding of mathematical issues that arise in applications. Although modeling issues are addressed, the central theme is the mathematically rigorous investigation of the resulting differential equations and their dynamic behavior. However, the authors and editors have made an effort to ensure readability on a non-technical level for mathematicians from other fields and for other scientists and engineers. The eighteen surveys collected here do not aspire to encyclopedic completeness, but present selected paradigms. The surveys are grouped into those emphasizing finite-dimensional methods, numerics, topological methods, and partial differential equations. Application areas include the dynamics of neural networks, fluid flows, nonlinear optics, and many others. While the survey articles can be read independently, they deeply share recurrent themes from dynamical systems. Attractors, bifurcations, center manifolds, dimension reduction, ergodicity, homoclinicity, hyperbolicity, invariant and inertial manifolds, normal forms, recurrence, shift dynamics, stability, to namejust a few, are ubiquitous dynamical concepts throughout the articles.
