Ordinary Differential Equations
Introduction and Qualitative Theory, Third Edition
Designed for a rigorous first course in ordinary differential equations, Ordinary Differential Equations: Introduction and Qualitative Theory, Third Edition includes basic material such as the existence and properties of solutions, linear equations, autonomous equations, and stability as well as more advanced topics in periodic solutions of nonlinear equations. Requiring only a background in advanced calculus and linear algebra, the text is appropriate for advanced undergraduate and graduate students in mathematics, engineering, physics, chemistry, or biology.
This third edition of a highly acclaimed textbook provides a detailed account of the Bendixson theory of solutions of two-dimensional nonlinear autonomous equations, which is a classical subject that has become more prominent in recent biological applications. By using the Poincaré method, it gives a unified treatment of the periodic solutions of perturbed equations. This includes the existence and stability of periodic solutions of perturbed nonautonomous and autonomous equations (bifurcation theory). The text shows how topological degree can be applied to extend the results. It also explains that using the averaging method to seek such periodic solutions is a special case of the use of the Poincaré method.
Table of Contents
Prefaces. Introduction. Existence Theorems. Linear Systems. Autonomous Systems. Stability. The Lyapunov Second Method. Periodic Solutions. Perturbation Theory: The Poincaré Method. Perturbation Theory: Autonomous Systems and Bifurcation Problems. Using the Averaging Method: An Introduction. Appendix. References. Index.
… a classic treatment of many of the topics an instructor would want in such a course, with particular emphasis on those aspects of the qualitative theory that are important for applications to mathematical biology. … A nice feature of this edition is an extended and unified treatment of the perturbation problem for periodic solutions. … a solid graduate-level introduction to ordinary differential equations, especially for applications. …
—MAA Reviews, August 2010