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The communication of knowledge on nonlinear dynamical systems, between the mathematicians working on the analytic approach and the scientists working mostly on the applications and numerical simulations has been less than ideal. This volume hopes to bridge the gap between books written on the subject by mathematicians and those written by scientists. The second objective of this volume is to draw attention to the need for cross-fertilization of knowledge between the physical and biological scientists. The third aim is to provide the reader with a personal guide on the study of global nonlinear dynamical systems.
This is a graduate text in differentiable dynamical systems. It focuses on structural stability and hyperbolicity, a topic that is central to the field. Starting with the basic concepts of dynamical systems, analyzing the historic systems of the Smale horseshoe, Anosov toral automorphisms, and the solenoid attractor, the book develops the hyperbolic theory first for hyperbolic fixed points and then for general hyperbolic sets. The problems of stable manifolds, structural stability, and shadowing property are investigated, which lead to a highlight of the book, the Ω-stability theorem of Smale. While the content is rather standard, a key objective of the book is to present a thorough treatment for some tough material that has remained an obstacle to teaching and learning the subject matter. The treatment is straightforward and hence could be particularly suitable for self-study. Selected solutions are available electronically for instructors only. Please send email to [email protected] for more information.
This is a graduate text in differentiable dynamical systems. It focuses on structural stability and hyperbolicity, a topic that is central to the field. Starting with the basic concepts of dynamical systems, analyzing the historic systems of the Smale horseshoe, Anosov toral automorphisms, and the solenoid attractor, the book develops the hyperbolic theory first for hyperbolic fixed points and then for general hyperbolic sets. The problems of stable manifolds, structural stability, and shadowing property are investigated, which lead to a highlight of the book, the \Omega-stability theorem of S.
This monograph considers structural stability on open 2-manifolds in the [italic]C[superscript italic]r-Whitney topology. The statements of the theorems in this monograph are analogous to the statements of Peixoto's theorem for compact 2-manifolds. However, to obtain the proofs of these results for the noncompact case the authors provide a large measure of original mathematics.
The main purpose of developing stability theory is to examine dynamic responses of a system to disturbances as the time approaches infinity. It has been and still is the object of intense investigations due to its intrinsic interest and its relevance to all practical systems in engineering, finance, natural science and social science. This monograph provides some state-of-the-art expositions of major advances in fundamental stability theories and methods for dynamic systems of ODE and DDE types and in limit cycle, normal form and Hopf bifurcation control of nonlinear dynamic systems. Presents comprehensive theory and methodology of stability analysis Can be used as textbook for graduate students in applied mathematics, mechanics, control theory, theoretical physics, mathematical biology, information theory, scientific computation Serves as a comprehensive handbook of stability theory for practicing aerospace, control, mechanical, structural, naval and civil engineers
Since the first edition of this book, geometrical methods in the theory of ordinary differential equations have become very popular and some progress has been made partly with the help of computers. Much of this progress is represented in this revised, expanded edition, including such topics as the Feigenbaum universality of period doubling, the Zoladec solution, the Iljashenko proof, the Ecalle and Voronin theory, the Varchenko and Hovanski theorems, and the Neistadt theory. In the selection of material for this book, the author explains basic ideas and methods applicable to the study of differential equations. Special efforts were made to keep the basic ideas free from excessive technicalities. Thus the most fundamental questions are considered in great detail, while of the more special and difficult parts of the theory have the character of a survey. Consequently, the reader needs only a general mathematical knowledge to easily follow this text. It is directed to mathematicians, as well as all users of the theory of differential equations.
Dynamic Stability of Structures covers the proceedings of an International Conference on Dynamic Stability of Structures, held in Northwestern University, Evanston, Illinois on October 18-20, 1965, jointly sponsored by the Air Force of Scientific Research and Northwestern University. The conference aims to delineate the various categories of dynamic stability phenomena. This book is organized into six sections encompassing 20 chapters that tackle general topics such as mathematical methods of analysis, physical phenomena, design applications in engineering, and reports of field research. The first two sections deal with the fundamentals, principles, and concept of dynamic stability, as well as an introduction to the use of computing machines as an aid in studying the motions of complicated dynamical systems. The succeeding two sections highlight the statistical aspects in the structural stability theory and certain problems of structural dynamic. These sections also look into the dynamic buckling of elastic structures and the buckling of long slender ships due to wave-induced whipping. The last two sections explore the stability and vibration problems of mechanical systems under harmonic excitation and the dynamic buckling under step loading. These sections also include discussions on the nonlinear dynamic response of shell-type structures and of a column under random loading, as well as Italian research in the field. Structural and mechanical engineers will find this book invaluable.