(We would like to carry reviews of any of these books in future issues of UK Nonlinear News.)
This book is a collection of papers on the dynamical and statistical theory of nonlinear wave propagation in dispersive conservative media. The emphasis is on waves on the surface of ideal fluids and on Rossby waves in the atmosphere. Although the book deals mainly with weakly nonlinear waves, it is more than simply a description of standard perturbation techniques. The goal is to show that the theory of weakly interacting waves is naturally related to such areas of mathematics a Diophantine equations, differential geometry of waves, Poincaré normal forms and the inverse scattering method.
American Mathematical Society Translations - Series 2,
Advances in the Mathematical Sciences, Volume 182.
November 1997, 197 pages, Hardcover ISBN 0-8218-4113-0.
This book is a very readable exposition of the modern theory of topological dynamics and presents diverse applications to such areas as ergodic theory, combinatorial number theory and differential equations. There are three parts: 1) The abstract theory of topological dynamics is discussed, including a comprehensive survey by Furstenberg and Glasner on the work and influence of R. Ellis. Presented in book form for the first time are new topics in the theory of dynamical systems, such as almost-periodicity, hidden eigenvalues, a natural family of factors and topological analogues of ergodic decomposition. 2) The power of abstract techniques is demonstrated by giving a very wide range of applications to areas of ergodic theory, combinatorial number theory, random walks on groups and others. 3) Applications to non-autonomous linear differential equations are shown. Exposition on recent results about Floquet theory, bifurcation theory and Lyapanov exponents is given.
This text will also be of interest to those working in analysis.
Contemporary Mathematics, Volume 215
December 1997, 336 pages, Softcover, ISBN 0-8218-0608-4
This book describes the progress that has been made towards the development of a comprehensive understanding of the formation of complex, disorderly patterns under conditions far from equilibrium. The application of fractal geometry and scaling concepts to the quantitative description and understanding of structure formed under non-equilibrium conditions is described. Self-similar fractals, multi-fractals and scaling methods are discussed, with examples, to facilitate applications in the physical sciences. Computer simulations and experimental studies are emphasised, but the author also includes discussion of theoretical advances in the subject. Much of the book deals with diffusion-limited growth processes and the evolution of rough surfaces, although a broad range of other applications is also included. The techniques and topics will be relevant to graduate students and researchers in physics, chemistry, materials science, engineering and the earth sciences, interested in applying the ideas of fractals and scaling.
Contents:
Preface; 1. Pattern formation far from equilibrium; 2. Fractals
and scaling; 3. The basic models; 4. Experimental studies; 5. The growth of
surfaces and interfaces; Appendix I: Instabilities; Appendix II:
Multifractals; Bibliography.
Cambridge Nonlinear Science Series 1997 247 x 174 mm 688 pp.
151 line diagrams 26 half-tones 7 colour plates
ISBN 0 521 45253 8
Deterministic chaos provides a novel framework for the analysis of irregular time series. Traditionally, nonperiodic signals are modelled by linear stochastic processes. But even very simple chaotic dynamical systems can exhibit strongly irregular time evolution without random inputs. Chaos theory offers completely new concepts and algorithms for time series analysis which can lead to a thorough understanding of the signal. This book introduces a broad choice of such concepts and methods, including phase space embeddings, nonlinear prediction and noise reduction, Lyapunov exponents, dimensions and entropies, as well as statistical tests for nonlinearity. Related topics such as chaos control, wavelet analysis and pattern dynamics are also discussed. Applications range from high quality, strictly deterministic laboratory data to short, noisy sequences which typically occur in medicine, biology, geophysics or the social sciences. All material is discussed and illustrated using real experimental data. Sample program listings are provided in FORTRAN and C. This book will be of value to any graduate student and researcher who needs to be able to analyse time series data, especially in the fields of physics, chemistry, biology, geophysics, medicine, economics and the social sciences.
Contents: PART ONE: BASIC CONCEPTS:
1. Introduction: why nonlinear
methods?; 2. Linear tools and general considerations; 3. Phase space
methods; 4. Determinism and predictability; 5. Instability: Lyapunov
exponents; 6. Self-similarity: dimensions; 7. Using nonlinear methods when
determinism is weak; 8. Selected nonlinear phenomena; PART TWO: ADVANCED
TOPICS: 9. Advanced embedding methods; 10. Chaotic data and noise; 11. More
about invariant quantities; 12. Modeling and forecasting; 13. Chaos
control; 14. Other selected topics; Appendix 1. Efficient neighbour
searching; Appendix 2. Program listings; Appendix 3. Description of the
experimental data sets.
Cambridge Nonlinear Science Series 7 1997247 x 174 mm 320 pp.
ISBN 0 521 55144 7