(We would like to carry reviews of any of these books in future issues of UK Nonlinear News.)
This book presents the study of ergodic properties of so-called chaotic dynamical systems. One of the central topics is the interplay between deterministic and quasi-stochastic behaviour in chaotic dynamics and between properties of continuous dynamical systems and those of their discrete approximations. Using simple examples, the author describes the main phenomena known in chaotic dynamical systems, studying topics such as the operator approach in chaotic dynamics, stochastic stability, and the so-called coupled systems. The last two chapters are devoted to problems of numerical modelling of chaotic dynamics.
Contents: Introduction; Operator approach in chaotic dynamics; Random perturbations of dynamical systems; Weakly coupled dynamical systems; Phase space discretisation in dynamical systems; Ergodic properties of some methods for numerical modelling of chaotic dynamics; Bibliography; Index.
Translations of Mathematical Monographs, Volume 161
March 1997, 161 pages, Hardcover,
ISBN 0-8218-0370-0.
This volume is a collection of papers dealing with harmonic analysis and nonlinear differential equations and stems from a conference on these two areas and their interface held in November 1995 at the University of California, Riverside, in honour of V.L. Shapiro. There are four papers dealing directly with the use of harmonic analysis techniques to solve challenging problems in nonlinear partial differential equations. There are also several survey articles on recent developments in multiple trigonometric series, dyadic harmonic analysis, special functions, analysis on fractals, and shock waves, as well as papers with new results in nonlinear differential equations. These survey articles, along with several of the research articles, cover a wide variety of applications such as turbulence, general relativity and black holes, neural networks, and diffusion and wave propagation in porous media. A number of the papers contain open problems in their respective areas.
This text will also be of interest to those working in differential equations.
Contemporary Mathematics, Volume 208.
August 1997, 350 [ages, Softcover, ISBN 0-8218-0565-7.
The perturbation theory for the operator div is of particular interest in the study of boundary-value problems for the general nonlinear equation F(\dot{y},y,x)=0. Taking as the linearisation the first order operator Lu = C_{ij}u^{i}_{xj} +C_{i}u^{i}, one can under certain conditions regard the operator L as a compact perturbation of the operator div.
This book presents results on boundary-value problems for L and the theory of nonlinear perturbations of L. Specifically, necessary and sufficient solvability conditions in explicit form are found for various boundary-value problems for the operator L. An analog of the Weyl decomposition is proved.
Translations of Mathematical Monographs, Volume 160;
1997;
104 pages; Hardcover; ISBN 0-8218-0586-X.
This annual directory provides a handy reference to various organisations in the mathematical sciences community. Listed in the directory are the following: officers and committee members of over thirty professional mathematical organisations (terms of office and other pertinent information are also provided in some cases); key mathematical sciences personnel of selected government agencies; academic departments in the mathematical sciences; mathematical units in nonacademic organisations; and alphabetical listings of colleges and universities. Current addresses, telephone numbers, and electronic addresses for individuals are listed in the directory when provided.
1997; approximately 200 pages;This book aims at providing a handy explanation of the notions behind the self-similar sets called "fractals" and "chaotic dynamical systems". The authors emphasise the beautiful relationship between fractal functions (such as Weierstrass's) and chaotic dynamical systems; these nowhere-differentiable functions are generating functions of chaotic dynamical systems. These functions are shown to be in a sense unique solutions of certain boundary problems. The last chapter of the book treats harmonic functions on fractal sets.
Contents: The fundamentals of fractals; Self-similar sets; An alternative computation for differentiation; In quest of fractal analysis; Recommended reading; Index.
Translations of Mathematical Monographs, Volume 107.
August 1997, approximately 96 pages, Hardcover.
ISBN 0-8218-0537-1.
The lectures, given at the DMV Seminar "Classical Nonintegrability, Quantum Chaos", provide an introduction to the ideas and mathematical techniques of classical and quantum nonlinear dynamics. The chapters on irregular scattering and expanding maps illustrate techniques in nonlinear dynamics using the simplest nontrivial examples. The chapters on quantum chaos and Liouville surfaces stress a phase space geometry approach to semi-classical quantum theory. These lectures can serve as a basis for graduate seminars in mathematics and physics.
1997 approx 104pp,
ISBN 3-7643-5708-8
DMV Seminar Series, Volume 27.
This book deals with the probabilistic concepts used in the analysis of one-dimensional chaotic systems. The precise description of chaotic behaviour requires the notion of a measure on the phase space. Since this measure describes asymptotic behaviour, it is time invariant, and hence is called invariant. The authors examine those invariant measures which occur in practice, namely the ones which are absolutely continuous with respect to Lebesgue measure, and thus have a probability density function. Important applications are given, ranging from biology to the design of drills.
1997-Spring approx. 440pp 50 Illustrations
ISBN 0-8176-4003-7
This new advanced text/reference contains an introduction to the theory and applications of nonlinear partial differential equations which appear in many areas of modern applied mathematics, physics, geometry, and engineering. It provides the reader the basic ideas, principles, results, and various modern methods of solutions of nonlinear partial differential equations, and their applications to a wide variety of problems as encountered in numerous interdisciplinary areas. The book contains about 450 examples and exercises.
Approximately 532pp, hardcover
ISBN 0-8716-3902-0
This authoritative volume shows how modern dynamical systems theory can help us in understanding the evolution of cosmological models. It leads to an understanding of how special (high symmetry) models determine the evolution of more general families of models; and how these families relate to real cosmological observations.
1997 357 pp Hardback
ISBN 55457-8
The authors concentrate on the techniques used to set up mathematical models and describe many systems in full detail, covering both differential and difference equations in depth. Among the broad spectrum of topics studied in this book are: mechanics, genetics, thermal physics, economics, and population studies.
Australian Mathematical Society Lecture Series 10.
1997 c. 400pp
44069-6 Hardback.
44618-X Paperback.
Concerns the theory of nonlinear boundary value problems in function spaces. Focus is on the analysis of semilinear elliptic boundary value problems in Sobolev spaces of fractional order. Here methods from Fourier analysis and the theory of function spaces are used to study existence and multiplicity results in the general framework of Besov-Triebel-Lizorkin spaces.
de Gruyter Series in Nonlinear Analysis Volume 3.
1996. x + 547 pages. Hardcover.
ISBN 3-11-015113-8.
In recent years, there has been a great deal of interest in variational problems for nonlinear elliptic equations characterised by lack of compactness. This book provides a detailed and extensive review of the methods, constrained minimisation, min-max theory of critical points and concentration-compactness principles, that have been developed to deal with these problems. The techniques discussed go far beyond elliptic equations and can be applied to Hamiltonian systems, nonlinear wave equations, and problems related to surfaces of prescribed mean curvature.
One of the most effective theorems applied in many variational problems is the Ambrosetti-Rabinowitz mountain pass theorem, the original proof of which is based on a deformation lemma. The author is concerned with generalisations of this theorem and also presents a different proof based on Ekeland's variational principle combined with techniques from the Clarke theory of generalised gradient. Moreover, on the basis of P.L. Lion's first and second concentration-compactness principle, he presents a novel and simplified approach to concentration-compactness principles, giving rise to the concentration-compactness principle at infinity in both the subcritical and critical case.
The material covered by this book requires some knowledge of functional analysis, nonlinear partial differential equations, and the theory of Sobolev spaces. For the reader's convenience, most of the prerequisites are given in an appendix.
1997. ix + 290 pages. Hardcover.
ISBN 3-11-015269-X.
Clearly and systematically develops the analysis of time series of data, both regular and chaotic, for nonlinear systems. The book emphases modern mathematical tools and leads readers from the measurements of one or more variables through building models of the source as a dynamical system, classifying the source by its dynamical characteristics, and predicting and controlling the system. The text not only examines methods for separating the signal from unwanted noise, and for investigating the phase space of the chaotic signal and its properties, but also develops a tool kit for analysing signals from nonlinear sources.
1996, 272 pp, 140 illustrationsThe aim of this book is to introduce students in mathematics and the sciences to the exciting world of nonlinear dynamical systems and chaos, including: discrete dynamical systems (maps), fractals, and systems of nonlinear differential equations. Computer experiments, designed to be used with many standard software packages, are included throughout.
1997, 603pp, 224 illustrations, softcover
ISBN 0-387-94677-2
Textbooks in Mathematical Sciences
There is now a new third edition of the highly acclaimed book Economic Dynamics by Giancarlo Gandolfo. Long out-of-print, but still in high demand, this completely rewritten and updated edition treats all of the mathematical methods used in economic dynamics, from elementary linear difference and differential equations and simultaneous systems to the qualitative analysis of non-linear dynamical systems. The reader is guided through a step-by-step analysis of each topic, from mathematical methods to economic models. This user-friendly feature is also present in the exercises.
1997, approximately 610pp, 65 illustrations, softcover
ISBN 3-540-62760-X
The nonlinear Schrödinger (NLS) equation is a fundamental nonlinear partial differential equation (PDE) that arises in many areas and engineering, e.g. in plasma physics, nonlinear waves, and nonlinear optics. It is an example of a completely integrable PDE where phase space structure is known in some detail. In this monograph the authors present detailed and pedagogic proofs of persistence theorems for normally hyperbolic invariant manifolds and their stable and unstable manifolds for classes of perturbations of the NLS equations. The authors' techniques are based on an infinite dimensional generalisation of the graph transform and can be viewed as an infinite dimensional generalisation of Feinchel's results. This book also shows that the authors' techniques are quite general and can be applied to a broad class of infinite dimensional dynamical systems.
1997, approx 180pp, hardcover
ISBN 0-387-94925-9
Applied Mathematical Sciences, Volume 128