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Other titles in the Monographs and Textbooks in Pure and Applied Mathematics series:
Ordinary Differential Equations: Introduction and Qualitative Theory, Third Edition
Synopses & Reviews
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 Poincar 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 Poincar method.
Book News Annotation:
Cronin (mathematics, Rutgers U.) ensures students build their understanding of core material by offering background information in the existence and properties of solutions, linear equations, autonomous equations and stability as well as more advanced work in periodic solutions of nonlinear equations. She begins with existence theorems, providing examples including the Volterra equations for predator-prey systems, the Hodgkin-Huxley equations, the Field-Noyes model for the Belousov-Zhabotinsky reaction and the Goodwin equations for a chemical reaction system. He proceeds to linear systems, including homogeneous linear equations, Flouquet theory and inhomogeneous linear equations, autonomous systems, including the Poincaré-Bendixson theorem and applications, stability, the Lyapunov second methods, periodic solutions, perturbation by the Poincaré methods and by autonomous systems and bifurcation problems, and the averaging method. Cronin has updated this text to provide detailed methods and examples. She recommends students complete the first semester of advanced calculus and a semester of linear algebra as prerequisites. Annotation Â©2008 Book News, Inc., Portland, OR (booknews.com)
Ordinary Differential Equations: Introduction and Qualitative Theory, Third Edition provides a detailed account, based on the work of Lefschetz, of the Bendixson theory of solutions of two-dimensional autonomous systems in a neighborhood of a singular point. Updated and revised, this third edition presents a unified treatment of the perturbation problem for periodic solutions, covering nonautonomous equations and bifurcation problems. The text includes a self-contained account of topological degree and its use as well as a detailed description of how the averaging method can be used to examine the problem of periodic solutions. It also contains additional exercises, applications, and solutions.
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