3rd Edition

Ordinary Differential Equations Introduction and Qualitative Theory, Third Edition

By Jane Cronin Copyright 2008
    408 Pages 48 B/W Illustrations
    by CRC Press

    408 Pages 48 B/W Illustrations
    by CRC Press

    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.

    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 Two-Dimensional 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.

    Biography

    Cronin, Jane

    … 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