Ordinary differential equations introduction and qualitative theory 3rd Edition by Jane Cronin ISBN 0824723378 978-0824723378

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Product details: 

ISBN 10: 0824723378 

ISBN 13: 978-0824723378

Author: Jane Cronin 

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: 

1. Existence Theorems
    1.1 What This Chapter Is About
    1.2 Existence Theorem by Successive Approximations
    1.3 Differentiability Theorem
    1.4 Existence Theorem for Equation with a Parameter
    1.5 Existence Theorem Proved by Using a Contraction Mapping
    1.6 Existence Theorem without Uniqueness
    1.7 Extension Theorems
    1.8 Examples

2. Linear Systems
    2.1 Existence Theorems for Linear Systems
    2.2 Homogeneous Linear Equations: General Theory
    2.3 Homogeneous Linear Equations with Constant Coefficients
    2.4 Homogeneous Linear Equations with Periodic Coefficients: Floquet Theory
    2.5 Inhomogeneous Linear Equations
    2.6 Periodic Solutions of Linear Systems with Periodic Coefficients
    2.7 Sturm–Liouville Theory

3. Autonomous Systems
    3.1 Introduction
    3.2 General Properties of Solutions of Autonomous Systems
    3.3 Orbits near an Equilibrium Point: The Two-Dimensional Case
    3.4 Stability of an Equilibrium Point
    3.5 Orbits near an Equilibrium Point of a Nonlinear System
    3.6 The Poincaré–Bendixson Theorem
    3.7 Application of the Poincaré–Bendixson Theorem

4. Stability
    4.1 Introduction
    4.2 Definition of Stability
    4.3 Examples
    4.4 Stability of Solutions of Linear Systems
    4.5 Stability of Solutions of Nonlinear Systems
    4.6 Some Stability Theory for Autonomous Nonlinear Systems
    4.7 Some Further Remarks Concerning Stability

5. The Lyapunov Second Method
    5.1 Definition of Lyapunov Function
    5.2 Theorems of the Lyapunov Second Method
    5.3 Applications of the Second Method

6. Periodic Solutions
    6.1 Periodic Solutions for Autonomous Systems
    6.2 Stability of the Periodic Solutions
    6.3 Sell’s Theorem
    6.4 Periodic Solutions for Nonautonomous Systems

7. Perturbation Theory: The Poincaré Method
    7.1 Introduction
    7.2 The Case in which the Unperturbed Equation Is Nonautonomous and Has an Isolated Periodic Solution
    7.3 The Case in which the Unperturbed Equation Has a Family of Periodic Solutions: The Malkin–Roseau Theory
    7.4 The Case in which the Unperturbed Equation Is Autonomous

8. Perturbation Theory: Autonomous Systems and Bifurcation Problems
    8.1 Introduction
    8.2 Using the Averaging Method: An Introduction
    8.3 Introduction
    8.4 Periodic Solutions
    8.5 Almost Periodic Solutions

9. Appendix
    9.1 Ascoli’s Theorem
    9.2 Principle of Contraction Mappings
    9.3 The Weierstrass Preparation Theorem
    9.4 Topological Degree

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Tags: Jane Cronin, Ordinary differential, equations introduction, qualitative theory