LEFSCHETZ, SOLOMON — Differential Equations: Geometric Theory

New York : Dover Publications, 1977 First Edition ADDED papers 608 p. tables, formulas, index SOFTCOVER: (Orig. paperback. Illustrated wrappers. Name end paper. VERY GOOD) This Dover edition, first published in 1977, is an unabridged republication of the second (1963) edition of the work originally published by Interscience Publishers in 1957 as Volume VI of the Interscience Pure and Applied Mathematics Series. CONTENTS: Preface; I Preliminary Questions; II Existence Theorems. General Properties of the Solutions; III Linear Systems; IV Stability; V The Differential Equation dx/dt = Px + q(x;t); VI The Differential Equation dx/dt = Px + q(x;y); VII The Differential Equation dx/dt = P(t)z + q(x;t); VIII Periodic Systems and Their Stability; IX Two Dimensional Systems. Simple Critical Points. The Index. Behavior at Infinity; X Two Dimensional Systems (continued); XI Differential Equations of the Second Order; XII Oscillations in Systems of the Second Order. Methods of Approximation; Appendix I Complement on Matrices; Appendix II Some Topological Complements Problems; Bibliography; List of Principal Symbols; Index. SYNOPSIS: This book investigates the area of nonlinear differential equations of the second order. Professor Lefschetz is interested in the important group of problems stirred up over a generation ago by Van der Pol, problems that still concern students, professional mathematicians, physicists, and engineers. The first four chapters on preliminary questions, existence theorems, linear systems, and stability provide a thorough and comprehensive coverage of the classical literature. The next three chapters deal with point stability, and considerable emphasis is given to the fundamental work of Liapunov, particularly to his so-called direct method, and to his Soviet disciples, noted authorities in this field. In Chapter 8 the discussion of periodic solutions--the n dimensional part is concluded. The remaining chapters cover two-dimensional systems. Chapters 9 and 10 are devoted to the results of Poincare and Bendixson: critical points, the index, behavior at infinity, some special systems. Also included here are the important notion of structural stability and the contributions of Andronov-Pontrjagin and DeBaggis. The last two chapters concern equations of the second order: notably, the work of Cartwright-Littleword, Levinson-Smith and Levinson, and the application of the perturbation method. There are two appendices, one on vectors and matrices, the other on topology. The book concludes with a list of problems that function as logical extensions of topics discussed in the body of the tekst. The late Solomon Lefschetz, a prominent authority in the field of differential equations, was Chairman of the Mathematics Department at Princeton University from 1945-53, and its Henry Burchardt Fine Professor of Mathematics, Emeritus. Former President of the American Mathematical Society and editor of Annals of Mathematics, he directed the Center for Differential Equations, RIAS, Baltimore.