Ordinary Differential Equations

Exploring the qualitative aspects of periodic solutions of ODEs, this edition extends the treatment of two-dimensional systems as well as periodic solutions in small parameter problems. It illustrates existence theorems with various examples, provides a detailed account of the Bendixson theory of solutions of two-dimensional autonomous systems, and presents a unified treatment of the perturbation problem for periodic solutions. The text also shows how topological degree is used to obtain significant extensions of perturbation theory and describes how the averaging method is used to study periodic solutions. It contains numerous exercises.

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.

Autorentext

Cronin, Jane

Zusammenfassung

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.

Inhalt

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.