(We would like to carry reviews of any of these books in future issues of UK Nonlinear News.)
In this book, the authors study new problems related to the theory of infinite-dimensional dynamical 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 attractos usually have infinite dimension, the research is focussed on the Kolmogorov ε-entropy of attractors. Upper estimates for the ε-entropy of uniform attractors of non-autonomous equations in terms of ε-entropy of time-dependent coefficients are proved.
The book gives systematic treatment to the theory of attractors of autonomous and non-autonomous evolution equations of mathematical physics. It can be used by both specialists and by those who want to get accquainted with this rapidly growing and important area of mathematics.
Colloquium Publications, Volume 49; 2002; 363 pages
Hardcover ISBN 0-8218-2950-5.
This book is a systematic introduction to smooth ergodic theory. The topics discussed include the general (abstract) theory of Lyapunov exponents and its applications to the stability theory of differential equations, stable manifold theory, absolute continuity, and the ergodic theory of dynamical systems with nonzero Lyapunov exponents (including geodesic flows).
The authors consider several nontrivial examples of dynamical systems with nonzero Lyapunov exponents to illustrate some basic methods and ideas of the theory.
This book is self-contained. The reader needs a basic knowledge of real analysis, measure theory, differential equations, and topology. The authors present basic concepts of smooth ergodic theory and provide complete proofs of the main results. They also state some more advanced results to give readers a broader view of smooth ergodic theory. This volume may be used by those nonexperts who wish to become familiar with the field.
University Lecture Series, Volume 23.
Memoirs of the American Mathematical Society,
Volume 155, Number 737.
Softcover. ISBN 0-8218-2739-1.
This comprehensive reference text gives an overview of the current state of nonlinear wave mechanics in both elastic and fluid media. Consisting of self-contained chapters, the book covers new apsects on strong discontinuities (shock waves) and localized self preserving (permenant) shapes (solitary waves and solitons).
2001/276 pp., 12 illustrations.
Hardcover. ISBN 0-8176-4059-2
Offers a broad investigative tool in ergodic theory and measurable dynamics. Measures similarities between two dynamical systems by asking how mucht eh time structure of orbits of one system must be distorted for it to become the other.
Cambridge Tracts in Mathematics 146
Hardback. ISBN 0-521-80795-6
The classic resource for scientific computing, now available in C++. For more information see: http://www.cambridge.org/numericalrecipes
Available from 14th February 2002
Hardback. ISBN 0-521-75033-4
The aim of this book is to present a recently developed approach suitable for investigating a variety of qualitative aspects of order-preserving random dynamical systems and to give the background for further development of the theory. The main objects considered are equilibria and attractors. The effectiveness of this approach is demonstrated by analysing the long-time behaviour of some classes of random and stochastic ordinary differential equations which arise in many applications
Lecture Notes in Mathematics. VOL. 1779
2002. VIII, 230 pp.
Softcover. ISBN 3-540-43246-9
This volume is an offspring of the special semester "Ergodic Theory, Geometric Rigidity and Number Theory" held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK, from January until July, 2000. Some of the major recent developments in rigidity theory, geometric group theory, flows on homogeneous spaces and Teichmüller spaces, quasi-conformal geometry, negatively curved groups and spaces, Diophantine approximation, and bounded cohomology are presented here. The authors have given special consideration to making the papers accessible to graduate students, with most of the contributions starting at an introductory level and building up to presenting topics at the forefront in this active field of research. The volume contains surveys and original unpublished results as well, and is an invaluable source also for the experienced researcher.
2002. XIV, 492 pp.
Hardcover. ISBN 3-540-43243-4
Covering a breadth of topics, ranging from the basic concepts to applications in the physical sciences, the book is highly illustrated and written in a clear and comprehensible style.
Nonlinear Dynamics and Chaos: Second Edition provides an excellent introduction to the subject for students of mathematics, engineering, physics and applied science. It will also appeal to the many researchers who work with computer models of systems that change over time.
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Nonlinear diffusion equations, an important class of parabolic equations, come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which enrich the theory of partial differential equations.
This book provides a comprehensive presentation of the basic problems, main results and typical methods for nonlinear diffusion equations with degeneracy. Some results for equations with singularity are touched upon.