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Differential Equations Geometric Theory

ISBN 0486634639 / 9780486634630 / 0-486-63463-9

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Book summary

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 Poincaré 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 ordernotably, the work of Cartwright-Littlewood, Levinson-Smith, and Levinsonand the application of the perturbation method. Two appendicesone on vectors and matrices, and the other on topologyconclude the text, along with a supplemental list of problems.