(We would like to carry reviews of any of these books in future issues of UK Nonlinear News.)
Designed for senior undergraduates and graduate students in applied mathematics, the natural sciences and engineering, this book covers standard material for an introduction to dynamical systems theory using Maple as a tool throughout the text. The book contains over 250 examples and exercises with solutions, has a very hands-on approach, and includes an introductory tutorial guide to Maple. Approximately 40 figures and over 200 Maple plots with simple commands and programs are listed at the end of each chapter. Figures, related Maple commands and programs can be viewed at the web sites http://www.maplesoft.com./apps and http://www.doc.mmu.ac.uk/STAFF/S.Lynch/cover1.html.
In Part I differential equations are used to model examples from various topics such as mechanical systems, chemical kinetics, electric circuits, interacting species, and economics. In Part II both real and complex discrete dynamical systems are considered and examples are taken from economics, population dynamics, nonlinear optics and materials science. Also discussed are bifurcation, bistability, chaos, instability, multistability and periodicity as well as some recently published research articles providing a useful resource for open problems in nonlinear dynamical systems.
January 2001, approximately 352 pages, 175 illustrations
Hardcover ISBN 0-8176-4150-5.
This self-contained book explores the vast field of nonlinear analysis by emphasizing underlying ideas rather than sophisticated refinements of the theory. Two classical examples from physics, elasticity and diffusion, serve to motivate the theoretical parts. Existence, uniqueness, regularity and approximation of solutions for quasilinear and monotone problems are studied.
Birkhäser Advanced Texts, November 2000, approximately 272 pages
Hardcover, ISBN 3-7643-6406-8.
This collection of survey papers by leading researchers covers a wide variety of recent developments in descriptive set theory and the theory of dynamical systems and the interconnections of these two subjects. Researchers and graduate students interested in either of these areas will find this volume to be an excellent introduction to problems and research directions arising from their interconnections.
London Mathematical Society Lecture Note Series 277
2000 304 pages
ISBN 0-521-78644-4 Paperback
This textbook presents a systematic study of the qualitative and geometric theory of nonlinear differential equations and dynamical systems. Although the main topic of the book is the local and global behaviour of nonlinear systems and their bifurcations, a thorough treatment of linear systems is given at the beginning of the text. All the material necessary for a clear understanding of the qualitative behaviour of dynamical systems is contained in this textbook, including an outline of the proof of the Hartman-Grobman theorem, the use of the Poincaré map in the theory of limit cycles, the theory of rotated vector fields and its use in the study of limit cycles and homoclinic loops, and a description of the behaviour and termination of one-parameter families of limit cycles.
In addition to minor corrections and updates throughout, this new edition includes materials on higher order Melnikov theory and the bifurcation of limit cycles for planar systems of differential equations, including new sections on Francoise's algorithm for higher order Melnikov functions and on the finite codimension bifurcations that occur in the class of bounded quadratic systems.
September 2000, approximately 562 pages, 229 illustrations
Hardcover ISBN 0-387-95116-4
Texts in Applied Mathematics, Volume 7.
The theory and applications of infinite dimensional dynamical systems have attracted the attention of scientists for quite some time. Dynamical issues arise in equations which attempt to model phenomena that change with time, and the infinite dimensional aspects occur when forces that describe the motion depend on spatial variables. This book may serve as an entree for scholars beginning their journey into the world of dynamical systems, especially infinite dimensional spaces. The main approach involves the theory of evolutionary equations. It begins with a brief essay on the evolution of evolutionary equations and introduces the origins of the basic elements of dynamical systems, flow and semiflow.
2001, approximately 615 pages, 20 figures
Hardcover ISBN 0-387-98347-3
Discrete dynamical systems are essentially iterated functions. Given the ease with which computers can do iteration, it is now possible for anyone with access to a personal computer to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in publications both mathematical and not. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. The level of the presentation is suitable for advanced undergraduates with a year of calculus behind them. Students in the author's courses using this material have come from numerous disciplines; many have been majors in other disciplines who are taking mathematics courses out of general interest. Concepts from calculus are reviewed as necessary. Mathematica programs that illustrate the dynamics and that will aid the student in doing the exercises are included in an appendix.
XV+223 pp, 56 figures
Softcover ISBN 0-387-94780-9
This is the first book to bring together concepts and methods dealing with hybrid systems from various areas, and to look at these from a unified perspective. The authors have chosen a mode of exposition that is largely based on illustrative examples rather than on abstract theorem-proof format. The examples are taken from many different application areas, ranging from power converters to communication protocols and from chaos to mathematical finance. Subjects covered include: definition of hybrid systems; description formats; existence and uniqueness of solutions; special subclasses; reachability and verification; stability and stabilizability; control design methods.
1999, 174 pages, 28 illustrations, softcover
Lecture notes in control and information sciences, vol 251
The formation of patterns in developing biological systems involves the spatio-temporal coordination of growth, cell-cell signaling, tissue movement, gene expression and cell differentiation. The interactions of these complex processes are generally nonlinear, and this mathematical modeling and analysis are needed to provide the framework in which to compute the outcome of different hypotheses on modes of interaction and to make experimentally testable predictions.
This collection contains papers exploring several aspects of the hierarchy of processes occurring during pattern formation. A number of papers address the modeling of cell movement and deformation, with application to pattern formation within a collection of cells in response to external signaling cues. The results are considered in the context of pattern generation in Dictyostelium discoideum and bacterial colonies.
A number of models at the macroscopic level explore the possible mechanisms underlying spatio-temporal pattern generation in early development, focusing on primitive streak, somitogenesis, vertebrate limb development and pigmentation patterning. The latter two applications consider in detail the effects of growth on patterning.
The potential of models to generate more complex patterns are considered and models involving different modes of cell-cell signaling are investigated. Pattern selection is analysed in the context of chemical Turing patterns, which serve as a paradigm for morphogenesis, and a model for vegetation patterns is presented.
September 2000, approximately 320 pages, 111 illustrations
Hardcover ISBN 0-387-95103-2
The IMA Volumes in Mathematics and its Applications, Volume 121
Systems with sub-processes evolving on many different time scales are ubiquitous in applications: chemical reactions, electro-optical and neuro-biological systems, to name just a few. This volume contains papers that expose the state of the art in mathematical techniques for analyzing such systems. Recently developed geometric ideas are highlighted in this work that includes a theory of relaxation-oscillation phenomena in higher dimensional phase spaces. Subtle exponentially small effects result from singular perturbations implicit in certain multiple time scale systems. Their role in the slow motion of fronts, bifurcations, and jumping between invariant tori are all explored here. Neurobiology has played a particularly stimulating role in the development of these techniques and one paper is directed specifically at applying geometric singular perturbation theory to reveal the synchrony in networks of neural oscillators.
The IMA Volumes in Mathematics and its Applications Volume 122
2000. Approx. 290 pp. 76 figs.
The book collects many techniques that are helpful in obtaining regularity results for solutions of nonlinear systems of partial differential equations. They are then applied in various cases to provide useful examples and relevant results, particularly in fields like fluid mechanics, solid mechanics, semiconductor theory, or game theory.
In general, these techniques are scattered in the journal literature and developed in the strict context of a given model. In the book, they are presented independently of specific models, so that the main ideas are explained, while remaining applicable to various situations. Such a presentation will facilitate application and implementation by researchers, as well as teaching to students.
Keywords: Regularity Analysis, Control Theory, Variational Methods MSC : 35-XX ; 49-XX ; 74-XX ; 91A15 ; 76-XX
Contents: I Introduction.- II General Technical Results.- III General Regularity Results.- IV Nonlinear Elliptic Systems Arising from Stochastic Games.- V Nonlinear Elliptic Systems Arising from Ergodic Control.- VI Harmonic Mappings.- VII Nonlinear Elliptic Systems Arising from the Theory of Semiconductors.- VIII Stationary Navier Stokes Equations.- IX Strongly Coupled Elliptic Systems.- X Dual Approach to Nonlinear Elliptic Systems.- XI Nonlinear Elliptic Systems Arising from Plasticity Theory.
2001. Approx. 345 pp. 3-540-67756-9
The book provides a self-contained introduction to the mathematical theory of non-smooth dynamical problems, as they frequently arise from mechanical systems with friction and/or impacts. It is aimed at applied mathematicians, engineers, and applied scientists in general who wish to learn the subject.
Keywords: dynamical systems, friction, impacts, differential inclusions, qualitative methods . Mathematics Subject Classification : 34Cxx, 34Dxx, 37-xx, 70Exx, 70Kxx
Contents: Introduction. Some general theory of differential inclusion. Bounded, unbounded, periodic, and almost periodic solutions. Lyapunov exponents for non-smooth dynamical systems. On the application of Conley index theory to non-smooth dynamical systems. On the application of KAM theory to non-smooth dynamical systems. Planar non-smooth dynamical systems. Melnikov's method for non-smooth dynamical systems. Further topics and notes.
2000. X+228 pp.