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    Full Reviews

    UK Nonlinear News Book Review
    Chaotic Dynamics: An Introduction Based on Classical Mechanics

    "Recalling my undergraduate mechanics lectures I half-expected a book full of complicated trigonometry, lengthy equations, peculiar coordinate systems and weighty analysis. What I read, and enjoyed, was a well-organised, beautifully illustrated volume covering the main points of chaotic dynamical systems, using intuitive mechanical examples to motivate clear and lucid discussions.
    The target audience of the book is undergraduate students of science, engineering and computational science, and one of its stated aims is to ''contribute to clarifying some misconceptions arising from everyday usage of the term 'chaos'.'' This it fulfills handsomely. It is certainly a book intended to teach or learn from, rather than as a research reference text. Indeed its opening chapter even contains a list of instructions on how to examine chaotic motion. Despite this, the book does far more than simply ``go through the motions''. Diagrams are plentiful and detailed, which is crucial for the geometric viewpoint presented. Many well-known mechanical systems (for example, the magnetic pendulum, the water-wheel, the spinning top) introduce concepts which are first abstracted and then used to discuss, albeit briefly, a wider range of applications, including environmental issues, chaotic scattering and the three-body problem. Throughout the work self-contained text capsules appear, expanding on issues raised in the discussions, and containing excellent summaries, histories, explanations and more potential applications.
    One of the book's main strengths is its excellent layout and ordering. It is unusual but pleasing to see the book open with a chapter describing the temporal phenomenon of chaotic motion, followed by a new and entirely separate chapter discussing the geometric phenomenon of fractal structures. Too often are the terms ''chaotic attractor'' and ''strange attractor'' mistakenly taken to be synonyms. The student reading this book is left with a thorough understanding of the connections and differences between the two concepts. At this stage the reader has been armed with the basic ideas of chaotic motion and fractal geometry, but is not yet permitted to attack problems displaying these traits. First, the authors introduce the necessary ideas from regular and periodically driven motion that will be needed, stressing the importance of studying unstable behaviour. All of this background material is presented at a level suitable for undergraduates.
    Inevitably there are omissions. Despite the chapter on driven motion there is no mention, for example, of synchronization, which is perhaps surprising given in the mechanical setting. The emphasis is on firmly presenting tools used to study chaotic systems, rather than giving a rigorous mathematical treatment. For example, while stable and unstable manifolds are discussed in terms of their use in dividing phase space, the reader must look elsewhere for results on the existence and smoothness of such objects.
    I would happily lend this book to any student wishing to begin learning the subject of chaotic dynamics, but I would certainly also demand its return."

    Rob Sturman
    Department of Applied Mathematics
    University of Leeds.
    Source link


    Journal of Statistical Physics, Vol. 127, No. 3, May 2007
    Book Review: Chaotic Dynamics
    Published Online: February 22, 2007

    "For the more than three hundred years since the time of Newton the evolution and stability of mechanical systems, and especially that of the solar system, have been central in physics and mathematics. During the eighteen century Euler, Lagrange and Laplace made substantial developments in that subject area by predicting changes in the planetary orbits due to small perturbations, in thatway establishing a framework for the study of global stability. All of this culminated in the nineteenth centurywork by Hamilton and Jacobi who reformulated the Lagrangian formalism of classical mechanics in terms of phase space, a step which proved to be most fruitful for developments in both statistical and quantum mechanics. These successes consolidated the idea of classical determinism. However, at the end of nineteenth century, the works of Poincar´e not only closed the door to an age but generated the first serious fracture in the philosophical conception of determinism. These works showed the impossibility of proving the convergence of perturbation series and, therefore relevant questions such as: is the solar system stable? remained unanswered. Poincar´e was thus the first to study what much later has been called (apparently by J. Yorke in 1975) chaos.
    In their book Tél and Gruiz give a plain, albeit not simplistic, definition of chaos: "Chaos is the complicated temporal behavior of simple systems." Thus, and contrary to common thinking, chaos is not spatial and static disorder but a characteristic of certain motions and is essentially a dynamical concept.Chaos may appear in a variety of fields: physics, chemical reactions, the spread of illnesses, economics and, of course, in planetary motions. For, contrary to intuition, chaotic behavior is not an exclusive property of dissipative systems but of conservative systems as well.
    Among the many books on the subject published in recent years and that basically focus on interdisciplinary examples and applications, the emphasis of the present one is on classical mechanics which, in turn, enhances a hidden and quite forgotten issue: classical indeterminism. All in all, the book is very readable, only requiring an elementary knowledge of physics and mathematics rendering it suitable for science or engineering undergraduate students and for those in other branches of science wanting an introduction to the subject.
    Tél and Gruiz's book consists of three parts having a total of ten chapters. These are devoted to topics like regular and chaotic motions, fractal objects, chaos in conservative and in dissipative systems, chaotic scattering and applications of chaos, among these, as most striking, I would single out chaos in the solar system and in fluid flows. The book has been carefully edited by Cambridge University Press and contains a complete and useful bibliography with many useful graphs and color plates and is otherwise highly recommended as an introductory text."

    Jaume Masoliver
    Departament de Fisica Fonamental
    Universitat de Barcelona
    Spain


    Physik Journal:
    Tél, T., M. Gruiz: Chaotic Dynamics

    "Systems exhibiting chaotic behaviour play an important role in several sciences. The book in hand, building only on classical mechanics, provides a nice introduction to chaotic dynamical systems.
    The first, introductory part illustrates important aspects of chaotic behaviour via several examples, and presents the concept of fractal objects.
    In the second part the authors introduce basic concepts such as space representation, instability, hyperbolic points and invariant manifolds.
    The most expansive third part is devoted to a detailed discussion of chaotic attractors of dissipative systems, of transient chaos, of conservative systems, and of chaotic scattering. The book concludes with an overview of applications and a short outlook on turbulence and spatiotemporal chaos.
    A special advantage of the book is the large number of problems embedded into the text, along with, in selected cases, their detailed solution in the appendix. More solutions for tutors are available on the website of the book, upon request. A detailed, thematically ordered bibliography completes the book.
    Each chapter contains several examples that, along with a large number of high quality illustrations, help to gain intuitive understanding of the mechanisms in question.
    Handling the topics, a plausible, qualitative form is choosen by the authors, which in my opinion, unfortunately, leaves the mathematical aspects somewhat suppressed.
    The colour plates should better have been integrated into the text, rather than being all presented at the end of the book, in particular since the captions appear, in a strange way, on the first pages of the book.
    As preliminary knowledge only the elementary knowledge of e.g., ordinary differential equations, linear algebra, and Newtonian equations is assumed, the book is thus very well suited to be an introductory text, and can also serve individual learning It contains especially useful material also for completing a course on theoretical mechanics with regards to chaotic systems."

    Dr. Arnd Bäcker
    Institute for Theoretical Physics
    TU Dresden
    Physik Journal, October, 2007


    Choice Reviews Online

    "A basic understanding of chaos theory is becoming increasingly important in a wide number of fields. Tél and Gruiz (both, Eötvös Univ., Budapest) introduce this subject at a basic level accessible to undergraduates. The text is well written with lots of diagrams; color plates; boxes with interesting asides on the history, specific applications, and explanations of the relevance; and concrete examples that that will aid the student in grasping difficult concepts. Part 1 is a phenomenological introduction to chaos, phase space, and fractals, with many examples and few equations. Part 2 builds on that foundation by explaining the basic concepts through the introduction of various canonical example problems. Part 3 develops the joy of exploring chaos, concluding with discussions of various applications such as classic mechanics (the spinning top), celestial mechanics, climate change, and pollutant dispersion. The authors do not shy away from the mathematics, but neither do they become so bogged down as to lose the excitement of discovery. This book would make a very nice book for a first course in chaos theory--it includes appendixes to explain some advanced mathematical concepts and the solutions to problem sets. Summing Up: Highly recommended. Upper-division undergraduates through professionals."

    S. E. Haupt
    Pennsylvania State University
    Choice Reviews Online


    Clippings from Issue 2008b
    Mathematical Reviews
    Established in 1940


    Bernard Lani-Wayda
    D-GSSN
    Giessen


    Zentralblatt MATH Database 1931 - 2008
    c 2008 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag
    Chaotic dynamics. An introduction based on classical mechanics

    "Even though there exists a lot of rather recent books on this subject, this manual is advisable for its clear presentation of the topics and for the simple and rigorous use of mathematical tools. A merit of this book is the cornerstone starting point: classical mechanics, which is the fundamental model for any other dynamical science. The book is planned mainly for undergraduate students of science, engineering and computational mathematics, but in the reviewer's opinion it could be useful to anybody interested in complex dynamics. The bibliography should have been more exhaustive, and cite many other authors of different schools.
    Contents:
    Introduction; Part I. The phenomenon: complex motion, unusual geometry: 1. Chaotic motions; 2. Fractal objects; Part II. Introductory concepts: 3. Regular motion; 4. Driven motion; Part III. Investigation of chaotic motion: 5. Chaos in dissipative systems; 6. Transient chaos in dissipative systems; 7. Chaos in conservative systems; 8. Chaotic scattering; 9. Applications of chaos; 10. Epilogue: Outlook; Appendix; Solutions to the problems; Bibliography; Index.
    Keywords: dissipative systems; conservative systems; scattering
    Classification:
    - Textbooks (mechanics of particles and systems)
    - Transition to stochasticity (chaotic behavior)
    - Dynamical systems in classical and celestial mechanics"

    Franco Cardin
    Department of Pure and Applied Mathematics
    University of Padova


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