UK Nonlinear News, May 2000
(We would like to carry reviews of any of these books in future issues of UK Nonlinear News.)
This book presents the expanded notes from ten lecturers given by the author at the NSF/CBMS conference held at California State University (Bakersfield). The author describes what he calls single orbit dynamics, which is an approach to the analysis of dynamical systems via the study of single orbits, rather than the study of a system as a whole. He presents single orbit interpretations of several areas of topological dynamics and ergodic theory and some new applications of dynamics to graph theory.
In the concluding lectures, single orbit approaches to generalizations of the Shannon-Breiman-McMillan theorem and related problems of compressions and universal coding are presented. Complete proofs and illuminating discussions are included and references for further study are given. Some of the material appears here for the first time in print.
This item will also be of interest to those working in geometry and topology.
CBMS Regional Conference Series in Mathematics,
113 pages, Softcover, ISBN 0-8218-0414-6
Mechanics and Dynamical Systems with Mathematica provides a systematic and unified treatment of mechanics and dynamical systems, addressing modeling, qualitative analysis, and simulations of physical systems using ordinary differential equations. The scientific computational components are presented using the software program Mathematica, both in worked examples and in the end-of-chapter problems. Special attention is given to classical mechanics models in light of new computational methods and concepts from dynamical systems. The book's nine chapters are organized into three unified parts: mathematical methods for differential equations; methods of classical mechanics; and dynamics, stochastic models and discretization of continuous models.
1999. 432 pages. Hardcover.
The Mandelbrot set is a fractal shape that classifies the dynamics of quadratic polynomials. It has a remarkably rich geometric and combinatorial structure. This volume provides a systematic exposition of current knowledge about the Mandelbrot set and presents the latest research in complex dynamics. Topics discussed include the universality and the local connectivity of the Mandelbrot set, parabolic bifurcations, critical circle homeomorphisms, absolutely continuous invariant measures and matings of polynomials, along with the geometry, dimension and local connectivity of Julia sets. In addition to presenting new work, this collection documents important results hitherto unpublished or difficult to find in the literature. This book will be of interest to graduate students in mathematics, physics and mathematical biology, as well as researchers in dynamical systems and Kleinian groups.
April 2000. 286 pages, 69 line diagrams. Paperback.
Explores diffusion population dynamics, autonomous differential equations and the stability of ecosystems, biogeography, pharmacokinetics, biofluid mechnics, cardiac mechanics, and the spectral analysis of heart sounds using FFT techniques. A key feature is a new chapter on epidemiology, including modeling the spread of AIDS through a population. Exercises are included.
Cambridge Studies in Mathematical Biology
Algorithmic, or automatic, differentiation (AD) is concerned with the accurate and efficient evaluation of derivatives for functions defined by computer programs. No truncation errors are incurred, and the resulting numerical derivative values can be used for all scientific computations that are based on linear, quadratic, or even higher order approximations to nonlinear scalar or vector functions. In particular, AD has been applied to optimization, parameter identification, equation solving, the numerical integration of differential equations, and combinations thereof. Apart from quantifying sensitivities numerically, AD techniques can also provide structural information, e.g., sparsity pattern and generic rank of Jacobian matrices. This first comprehensive treatment of AD describes all chainrule-based techniques for evaluating derivatives of composite functions with particular emphasis on the reverse, or adjoint, mode. The corresponding complexity analysis shows that gradients are always relatively cheap, while the cost of evaluating Jacobian and Hessian matrices is found to be strongly dependent on problem structure and its efficient exploitation. Attempts to minimize operations count and/or memory requirement lead to hard combinatorial optimization problems in the case of Jacobians and a well-defined trade-off curve between spatial and temporal complexity for gradient evaluations. The book is divided into three parts: a stand-alone introduction to the fundamentals of AD and its software, a thorough treatment of methods for sparse problems, and final chapters on higher derivatives, nonsmooth problems, and program reversal schedules. Each of the chapters concludes with examples and exercises suitable for students with a basic understanding of differential calculus, procedural programming, and numerical linear algebra. Audience This volume will be valuable for designers and users of algorithms and software for nonlinear computational problems. It opens up an exciting opportunity to develop new algorithms that reflect the availability of accurate derivatives and their true cost to achieve improvements in speed and reliability. Some familiarity with modern approaches to the seemingly straightforward task of evaluating derivatives will benefit any mathematician, scientist or engineer.
Apr 2000. 369 pages. Softcover.
Dynamical systems arise in all fields of applied mathematics. The author focuses on the description of numerical methods for the detection, computation, and continuation of equilibria and bifurcation points of equilibria of dynamical systems. This subfield has the particular attraction of having links with the geometric theory of differential equations, numerical analysis, and linear algebra. Several features make this book unique. The first is the systematic use of bordered matrix methods in the numerical computation and continuation of various bifurcations. The second is a detailed treatment of bialternate matrix products and their Jordan structure. Govaerts discusses their use in the numerical methods for Hopf and related bifurcations. A third feature is a unified treatment of singularity theory, with and without a distinguished bifurcation parameter, from a numerical point of view. Finally, numerical methods for symmetry-breaking bifurcations are discussed in detail, up to the fundamental cases covered by the equivariant branching lemma. Audience Anyone interested in computational methods for ODEs, PDEs, and bifurcation theory will find this volume a great addition to their library. This volume can be used as a text for graduate courses on numerical computation of bifurcations or numerical singularity theory. A basic knowledge of linear algebra, numerical linear algebra, calculus, and differential equations is required for full understanding of the material.
Jan 2000. 362 pages. Softcover.
Presents an outstanding selection of executable programs, including explanatory introductory texts on chaos theory and its simulation. Many numerical experiments and suggestions for further studies help the reaeder to become familiar with this fascinating topic. The second edition includes a CD-ROM with executable programs, Windows 95 compatible. Programs span cases such as billiards, the double pendulum, scattering, Fermi accleration, the Duffing oscillator, electronic circuits, and Mandelbrot and Julia sets.
1999/311pp, 250 illustations, CD-ROM
The Institute for Mathematics and its Applications (IMA) devoted its 1997-1998 program to Emerging Applications of Dynamical Systems. Dynamical systems theory and related numerical algorithms provide powerful tools for studying the solution behavior of differential equations and mappings. In the past 25 years computational methods have been developed for calculating fixed points, limit cycles, and bifurcation points. A remaining challenge is to develop robust methods for calculating more complicated objects, such as higher- codimension bifurcations of fixed points, periodic orbits, and connecting orbits, as well as the calcuation of invariant manifolds. Another challenge is to extend the applicability of algorithms to the very large systems that result from discretizing partial differential equations. Even the calculation of steady states and their linear stability can be prohibitively expensive for large systems (e.g. 103-106 equations) if attempted by simple direct methods. Several of the papers in this volume treat computational methods for low and high dimensional systems and, in some cases, their incorporation into software packages. A few papers treat fundamental theoretical problems, including smooth factorization of matrices, self-organized criticality, and unfolding of singular heteroclinic cycles. Other papers treat applications of dynamical systems computations in various scientific fields, such as biology, chemical engineering, fluid mechanics, and mechanical engineering.
Contents: Numerical bifurcation techniques for chemical reactor problems
Series: The IMA Volumes in Mathematics and its
Applications VOL 119
2000, approx 495pp, 134 figures.
Resonances are widely studied in most areas of engineering and physics, but the approach remains mostly computational or experimental because even reduced models of resonant interactions are typically higher dimensionally and exhibit great complexity; therefore, they are inaccessible to textbook techniques from dynamical systems theory. Chaos Near Resonance offers the first systematic exposition of recent analytic results that can be used to understand and predict the global effect of resonances in phase space. The geometric methods discussed here enable one to identify complicated multi-time-scale solution sets and slow-fast chaos in physical problems. This self-contained monograph should be useful to mathematicians interested in the geometric theory of multi-and infinite-dimensional dynamical systems, as well as to the applied scientist who wishes to analyze resonances in physical problems.
1999. 448 pages, 155 illustrations. Hardcover
This monograph aims to fill the gap between the mathematical literature which has significantly contributed to the understanding of the collapse phenomenon, and applications to domains like plasma physics and nonlinear optics, where this process provides a fundamental mechanism for small scale formation and wave dissipation.
1999. 366 pages. Hardcover
This textbook offers graduate students a rapid introduction to the language of the subject of ordinary differential equations followed by a careful treatment of the central topics of the qualitative theory: existence, uniqueness, extensiblity, phase plane analysis, Poincar‰-Bendixson theory, linearization, linear systems theory, Floquet theory, stability theory, invariant manifolds, continuation theory, averaging, bifurcation theory, nonlinear dynamics, and chaos. In addition, special attention is given to the origins and applications of differential equations in physical science and engineering. Of special interest to mathematics students is a multifaceted approach to existence theory, for solutions and for invariant manifolds, including an integrated new treatment of smoothness of solutions and invariant manifolds based on the fiber contraction principle. Mastery of the material in this book will provide a solid background for research in the subject of ordinary differential equations and applications of the theory to real world problems.
1999. 584 pages, 68 illustrations. Hardcover.
Computing all the zeros of an analytic function and their respective multiplicities, locating clusters of zeros of analytic functions, computing zeros and poles of meromorphic functions, and solving systems of analytic equations are problems in computational complex analysis that lead to a rich blend of mathematics and numerical analysis. This book treats these four problems in a unified way. It contains not only theoretical results (based on formal orthogonal polynomials or rational interpolation) but also numerical analysis and algorithmic aspects, implementation heuristics, and polished software (the package ZEAL) that is available via the CPC Program Library. Graduate students and researchers in numerical mathematics will find this book very readable.
This EMS volume, the first edition of which was published as Dynamical Systems II, EMS 2, sets out to familiarize the reader to the fundamental ideas and results of modern ergodic theory and its applications to dynamical systems and statistical mechanics. The exposition starts from the basic of the subject, introducing ergodicity, mixing and entropy. The ergodic theory of smooth dynamical systems is treated. Numerous examples are presented carefully along with the ideas underlying the most important results. Moreover, the book deals with the dynamical systems of statistical mechanics, and with various kinetic equations. For this second enlarged and revised edition, published as Mathematical Physics I, EMS 100, two new contributions on ergodic theory of flows on homogeneous manifolds and on methods of algebraic geometry in the theory of interval exchange transformations were added.
Due 5/2000. 2nd enlarged and corr. ed. 2000. Approx.
400 pp. Hardcover.
The summer school on Mathematics inspired by Biology was held at Martina Franca, Apulia, Italy in 1997. This volume presents five series of six lectures each. The common theme is the role of structure in shaping transient and ultimate dynamics. But the type of structure ranges from spatial (hadeler and maini in the deterministic setting, Durrett in the stochastic setting) to physiological (Diekmann) and order (Smith). Each contribution sketches the present state of affairs while, by including some wishful thinking, pointing at open problems that deserve attention.
2000. 268 pages. Softcover
"A good book for a nice price!"
Monatshefte f’r Mathematik
"... for lecture courses that cover the classical
nonlinear differential equations associated with Poincar‰ and
Lyapunov and introduce the student to the ideas of bifurcation
theory and chaos this is an ideal text ..."
"The pedagogical style is excellent, consisting
typically of an
insightful overview followed by theorems, illustrative examples and
Corrected 2nd printing, 2000. 2nd rev. and expanded ed.
1996. 303 pages, 127 figures.
This invaluable book presents a comprehensive introduction to bifurcation theory in the presence of symmetry, an applied mathematical topic which has developed considerably over the past twenty years and has been very successful in analysing and predicting pattern formation and other critical phenomena in most areas of science where nonlinear models are involved, like fluid flow instabilities, chemical waves, elasticity and population dynamics.
The book has two aims. One is to expound the mathematical methods of equivariant bifurcation theory. Beyond the classical bifurcation tools, such as center manifold and normal form reductions, the presence of symmetry requires the introduction of the algebraic and geometric formalism of Lie group theory and transformation group methods. For the first time, all these methods in equivariant bifurcations are presented in a coherent and self-consistent way in a book.
The other aim is to present the most recent ideas and results in this theory, in relation to applications. This includes bifurcations of relative equilibria and relative periodic orbits for compact and noncompact group actions, heteroclinic cycles and forced symmetry-breaking perturbations. Although not all recent contributions could be included and a choice had to be made, a rather complete description of these new developments is provided. At the end of every chapter, exercises are offered to the reader.
March 2000. 420 pages.
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