(We would like to carry reviews of any of these books in future issues of UK Nonlinear News.)
This book solves a problem that has been open for over 20 years - the complete classification of structurally stable quadratic vector fields modulo limit cycles. The 1950s saw the first real impetus given to the development of the qualitative theory of quadratic vector fields, although prior and ongoing interest in the topic can be shown by the more than 800 papers that have been published on the subject. One of the problems in the qualitative theory of quadratic vector fields is the classification of all structurally stable ones: In this work the authors solve this problem completely modulo limit cycles and give all possible phase portraits for such structurally stable quadratic vector fields.
Memories of the American Mathematical Society, Volume 134, Number 639. July 1998, 108 pages, ISBN 0-8218-0796-X.
The study of dynamical systems usually concentrates on the properties and the structure of invariant sets, since the understanding of these is the first step in describing the long time behaviour of orbits of the entire dynamical system. There are two different sets of problems related to the study of dynamical systems. One, the study of the dynamics in the neighbourhood of the critical elements such as fixed points or periodic orbits, is relatively well understood. This volume tackles the second set of problems, related to global dynamics and global bifurcations. In this volume the author studies dynamics of cyclic feedback systems. Global dynamics are described by a Morse decomposition of the global attractor, defined with the help of a discrete Lyapunov function. The author shows that the dynamics inside individual Morse sets may be very complicated. A three-dimensional system of ODEs with two linear equations is constructed, such that the invariant set is at least as complicated as a suspension of a full shift on two symbols. The questions posed are perhaps as significant as the reported results.
Memories of the American Mathematical Society, Volume 134, Number 637. July 1998, 73 pages, Softcover, ISBN 0-8218-0783-8.
The contemporary study of complex dynamics, which has flourished so much in recent years, is based largely upon work by G. Julia (1918) and P. Fatou (1919-1920). The goal of this book is to analyse this work from an historical perspective and show in detail how it grew out of a corpus regarding the iteration of complex analytic functions. This began with investigations by E. Schroder (1870-71), which he made when he studied Newton's method. In the 1880s, Gabriel Koenigs fashioned this study into a rigorous body of work and thereby strongly influence the subsequent development. But only when Fatou and Julia applied set theory and Paul Montel's theory of normal families was it possible to develop a global approach to the iteration of rational maps. This book shows how this intriguing piece of modern mathematics became a reality.
Vieweg Aspects of Mathematics, Volume 24. Vieweg Verlag. March 1998, 165 pages, Hardcover, ISBN 3-538-06520-6.
The main purpose of this book is to present all known results on the existence of formal, holomorphic and singular solutions of singular nonlinear ordinary and partial differential equations in the complex domain. It contains a new approach to regular singularities for nonlinear PDE, Maillet type theorems for nonlinear PDE, Briot-Bouquet type PDE, higher order nonlinear Fuchsian PDE, Poincare's and Siegel's results for vector fields, and also a general form of the Cauchy-Kowalewski theorem. Readers of the book are assumed to be familiar with only the basics of differential equations and function theory of complex variables.
Vieweg Aspects of Mathematics, Volume 28. Vieweg Verlag.
March 1998, 269 pages, Hardcover, ISBN 3-528-06659-8.
This book serves as an elementary, self-contained, introduction to some important aspects of the theory of global solutions to initial-value problems for nonlinear evolution equations. The presentation is made using the classical method of continuation of local solutions with the help of a prioir estimates obtained for small data.
Vieweg Aspects of Mathematics, Volume 19. Vieweg Verlag. March 1998, 259 pages, Hardcover, ISBN 3-528-06421-8.
In this book the authors investigate the nonlinear dynamics of the self-regulation of social and economic behaviour, and of the closely related interactions among species in ecological communities. Replicator equations describe how successful strategies spread and thereby create new conditions that can alter the basis of their success, i.e. to enable us to understand the strategic and genetic foundations of the endless chronicle of invasions and extinctions that punctuate evolution.
1998 348 pp. 62365-0 Hardback 62570-X Paperback.
Presents the tools for studying local bifurcations of limit cycles in families of planar vector fields.
1998 224 pp. Hardcover ISBN 3-7643-5900-5.
This textbook introduces the core concepts and results of Control and System Theory. Unique in its emphasis on foundational aspects, it takes a ``hybrid'' approach in which basic results are derived for discrete and continuous time scales, and discrete and continuous state variables. Primarily geared towards mathematically advanced undergraduate or graduate students, it may also be suitable for a second engineering course in control which goes beyond the classical frequency domain and state-space material. The choice of topics, together with detailed end-of-chapter links to the bibliography, makes it an excellent research reference as well.
The second edition constitutes a substantial revision and extension of the first edition, mainly adding or expanding upon advanced material, including: Lie-algebraic accessibility theory, feedback linearisation, controllability of neural networks, reachability under input constraints, topics in nonlinear feedback design (such as backstepping, damping, control-Lyapunov functions, and topological obstructions to stabilisation), and introductions to the calculus of variations, the maximum principle, numerical optimal control, and linear time-optimal control. Also covered, as in the first edition, are notions of systems and automata theory, and the algebraic theory of linear systems, including controllability, observability, feedback equivalence, and minimality; stability via Lyapunov, as well as input/output methods; linear-quadratic optimal control; observers and dynamic feedback; Kalman filtering via deterministic optimal observation; parametrisation of stabilising controllers, and facts about frequency domain such as the Nyquist criterion.
Springer-Verlag, New York, 1998, ISBN 0-387-984895.