Methods of Qualitative Theory in Nonlinear Dynamics

by Shilnikov, Leonid P.; Shilnikov, Andrey L.; Turaev, Dmitry V.; Chua, Leon O.

• ISBN13: 9789810233822
• ISBN10: 9810233825
• Format: Hardcover
• Copyright: 1998-06-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

Bifurcation and Chaos has dominated research in nonlinear dynamics for over two decades and numerous introductory and advanced books have been published on this subject. There remains, however, a dire need for a textbook which provides a pedagogically appealing yet rigorous mathematical bridge between these two disparate levels of exposition. This book is written to serve the above unfulfilled need.

Following the footsteps of Poincare, and the renowned Andronov school of nonlinear oscillations, this book focuses on the qualitative study of high-dimensional nonlinear dynamical systems. Many of the qualitative methods and tools presented in this book were developed only recently and have not yet appeared in a textbook form.

In keeping with the self-contained nature of this book, all topics are developed with an introductory background and complete mathematical rigor. Generously illustrated and written with a high level of exposition, this book will appeal to both beginners and advanced students of nonlinear dynamics interested in learning a rigorous mathemati

Table of Contents

Preface	ix
Basic Concepts	1	(20)
Necessary background from the theory of ordinary differential equations	1	(5)
Dynamical systems Basic notions	6	(6)
Qualitative integration of dynamical systems	12	(9)
Structurally Stable Equilibrium States Of Dynamical Systems	21	(90)
Notion of an equilibrium state. A linearized system	21	(3)
Qualitative investigation of 2- and 3-dimensional linear systems	24	(13)
High-dimensional linear systems. Invariant subspaces	37	(10)
Behavior of trajectories of a linear system near saddle equilibrium states	47	(9)
Topological classification of structurally stable equilibrium states	56	(9)
Stable equilibrium states. Leading and non-leading manifolds	65	(13)
Saddle equilibrium states. Invariant manifolds	78	(7)
Solution near a saddle. The boundary-value problem	85	(10)
Problem of smooth linearization Resonances	95	(16)
Structurally Stable Periodic Trajectories of Dynamical Systems	111	(124)
A Poincare map. A fixed point. Multipliers	112	(3)
Non-degenerate linear one- and two-dimensional maps	115	(10)
Fixed points of high-dimensional linear maps	125	(3)
Topological classification of fixed points	128	(7)
Properties of nonlinear maps near a stable fixed point	135	(6)
Saddle fixed points. Invariant manifolds	141	(13)
The boundary-value problem near a saddle fixed point	154	(14)
Behavior of linear maps near saddle fixed points. Examples	168	(13)
Geometrical properties of nonlinear saddle maps	181	(5)
Normal coordinates in a neighborhood of a periodic trajectory	186	(8)
The variational equations	194	(7)
Stability of periodic trajectories. Saddle periodic trajectories	201	(8)
Smooth equivalence and resonances	209	(9)
Autonomous normal forms	218	(5)
The principle of contraction mappings. Saddle maps	223	(12)
Invariant Tori	235	(34)
Non-autonomous systems	236	(6)
Theorem on the existence of an invariant torus. The annulus principle	242	(16)
Theorem on persistence of an invariant torus	258	(6)
Basics of the theory of circle diffeomorphisms. Synchronization problems	264	(5)
Center Manifold. Local Case	269	(56)
Reduction to the center manifold	273	(13)
A boundary-value problem	286	(16)
Theorem on invariant foliation	302	(12)
Proof of theorems on center manifolds	314	(11)
Center Manifold. Non-Local Case	325	(32)
Center manifold theorem for a homoclinic loop	326	(8)
The Poincare map near a homoclinic loop	334	(11)
Proof of the center manifold theorem near a homoclinic loop	345	(3)
Center manifold theorem for heteroclinic cycles	348	(9)
Appendix A. Special Form of Systems Near a Saddle Equilibrium State	357	(14)
Appendix B. First Order Asymptotic for the Trajectories Near a Saddle Fixed Point	371	(10)
Bibliography	381	(8)
Index	389