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Ordinary Differential Equations
Introduction and Qualitative Theory, Third Edition
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Book Description
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 twodimensional 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
What This Chapter Is About
Existence Theorem by Successive Approximations
Differentiability Theorem
Existence Theorem for Equation with a Parameter
Existence Theorem Proved by Using a Contraction Mapping
Existence Theorem without Uniqueness
Extension Theorems
Examples
Linear Systems
Existence Theorems for Linear Systems
Homogeneous Linear Equations: General Theory
Homogeneous Linear Equations with Constant Coefficients
Homogeneous Linear Equations with Periodic Coefficients: Floquet Theory
Inhomogeneous Linear Equations
Periodic Solutions of Linear Systems with Periodic Coefficients
Sturm–Liouville Theory
Autonomous Systems
Introduction
General Properties of Solutions of Autonomous Systems
Orbits near an Equilibrium Point: The TwoDimensional Case
Stability of an Equilibrium Point
Orbits near an Equilibrium Point of a Nonlinear System
The Poincaré–Bendixson Theorem
Application of the Poincaré–Bendixson Theorem
Stability
Introduction
Definition of Stability
Examples
Stability of Solutions of Linear Systems
Stability of Solutions of Nonlinear Systems
Some Stability Theory for Autonomous Nonlinear Systems
Some Further Remarks Concerning Stability
The Lyapunov Second Method
Definition of Lyapunov Function
Theorems of the Lyapunov Second Method
Applications of the Second Method
Periodic Solutions
Periodic Solutions for Autonomous Systems
Stability of the Periodic Solutions
Sell’s Theorem
Periodic Solutions for Nonautonomous Systems
Perturbation Theory: The Poincaré Method
Introduction
The Case in which the Unperturbed Equation Is Nonautonomous and Has an Isolated Periodic Solution
The Case in which the Unperturbed Equation Has a Family of Periodic Solutions: The Malkin–Roseau Theory
The Case in which the Unperturbed Equation Is Autonomous
Perturbation Theory: Autonomous Systems and Bifurcation Problems
Introduction
Using the Averaging Method: An Introduction
Introduction
Periodic Solutions
Almost Periodic Solutions
Appendix
Ascoli’s Theorem
Principle of Contraction Mappings
The Weierstrass Preparation Theorem
Topological Degree
References
Index
Exercises appear at the end of each chapter.
Author(s)
Biography
Cronin, Jane
Reviews
… 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 graduatelevel introduction to ordinary differential equations, especially for applications. …
—MAA Reviews, August 2010
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