Differential Equations Geometric Theory

Geared toward upper-level undergraduates and graduate students, this text investigates nonlinear differential equations of the second order and includes an extensive overview of the classical literature. 1957 edition.

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.

This book gathers together papers from a workshop held in Cortona, Italy. The contributions come from a group of outstanding mathematicians and together they cover the most recent advances in the geometric theory of singular phenomena of partial differential equations occurring in real and complex differential geometry. This volume will be of great interest to all those whose research interests lie in real and complex differential geometry, partial differential equations, and gauge theory.

This book collects together lectures by some of the leaders in the field of partial differential equations and geometric measure theory. It features a wide variety of research topics in which a crucial role is played by the interaction of fine analytic techniques and deep geometric observations, combining the intuitive and geometric aspects of mathematics with analytical ideas and variational methods. The problems addressed are challenging and complex, and often require the use of several refined techniques to overcome the major difficulties encountered. The lectures, given during the course "Partial Differential Equations and Geometric Measure Theory'' in Cetraro, June 2–7, 2014, should help to encourage further research in the area. The enthusiasm of the speakers and the participants of this CIME course is reflected in the text.

Written by a celebrated mathematician and teacher, this text investigates nonlinear differential equations of the second order. Geared toward upper-level undergraduates and graduate students, it addresses problems that also concern professional mathematicians, physicists, and engineers. The first four chapters on preliminary questions, existence theorems, linear systems, and stability provide an extensive overview of the classical literature. The next three chapters deal with point stability, placing considerable emphasis on the fundamental work of Liapunov. Chapter 8 discusses periodic solutions, and subsequent chapters cover two-dimensional systems. Chapters 9 and 10 are devoted to the results of Poincar� and Bendixson: critical points, the index, behavior at infinity, and special systems. They also examine 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-Littlewood, Levinson-Smith, and Levinson--and the application of the perturbation method. Two appendices--one on vectors and matrices, and the other on topology--conclude the text, along with a supplemental list of problems.