Ordinary Differential Equations and Stability Theory

An Introduction

David A. Sanchez

• Publisher: Dover Publications
• Year: 2019
• Print ISBN: 9780486837598
• eBook ISBN: 9780486843865
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This brief modern introduction to the subject of ordinary differential equations emphasizes stability theory. Concisely and lucidly expressed, it is intended as a supplementary text for advanced undergraduates or beginning graduate students who have completed a first course in ordinary differential equations.
The author begins by developing the notions of a fundamental system of solutions, the Wronskian, and the corresponding fundamental matrix. Subsequent chapters explore the linear equation with constant coefficients, stability theory for autonomous and nonautonomous systems, and the problems of the existence and uniqueness of solutions and related topics. Problems at the end of each chapter and two Appendixes on special topics enrich the text.

CHAPTER
1

Introduction

1.1 Preliminary Notation

For convenience, we will employ vector notation throughout the text, and n-dimensional vector-valued functions of n- or (n + 1)-dimensional vectors will be most frequently used. Thus if the positive integer n is unspecified, then any results stated will be applicable to one- or many-dimensional problems.

In general, suppose we are given the function f mapping a subset of R^m, Euclidean m-dimensional space, into Rⁿ, Euclidean n-dimensional space. If x = (x₁, . . . , x_m) is in the domain of f, and we denote its image under f by y = (y₁, . . . , y_n), then we may write

where we define

We will say that f is continuous in x if each f_i is continuous in x. Furthermore, we define the vector of partial derivatives as

where 1 ≤ j ≤ m.

Given the n-dimensional vector x(t) = (x₁(t), . . . , x_n(t)), where t is a real variable and each x_i(t) is real-valued, we say x(t) is continuous at t = t₀ if each x_i(t) is continuous at t = t₀, and it is differentiable if each x_i(t) is differentiable. We then may express the derivative vector as

and successive derivatives will be denoted by

If x(t) is given as above, we denote the norm of x(t) by

and for each t this is a mapping of x(t) into the nonnegative real numbers. It has the properties (i) ||x(t)|| = 0 if and only if x(t) = 0, that is, each x_i(t) is zero; (ii) ||kx(t)|| = ||k|||x(t)|| for any real or complex scalar k; and (iii) ||x(t) + y(t)|| ≤ ||x(t)|| + ||y(t)||. The above norm has certain computational advantages over the usual Euclidean norm,

which also satisfies the characteristic properties of the norm listed above. For geometrical convenience (for example, when using polar coordinates) we will occasionally use the latter norm; any results given will not, however, depend on the norm chosen.

Frequently in the text we will be considering a given function f mapping a subset of Rⁿ⁺¹ into Rⁿ. If we denote a point in Rⁿ⁺¹ by (t, x), where t is real and x = (x₁, . . . , x_n), then its image, wherever defined, may be denoted by

In particular, if x = x(t) = (x₁(t), . . . , x_n(t)), then y = y(t) = f(t...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• Contents
• Chapter 1 Introduction
• Chapter 2 The Linear Equation: General Discussion
• Chapter 3 The Linear Equation with Constant Coefficients
• Chapter 4 Autonomous Systems and Phase Space
• Chapter 5 Stability for Nonautonomous Equations
• Chapter 6 Existence, Uniqueness, and Related Topics
• Appendix A Series Solutions of Second-Order Linear Equations
• Appendix B Linear Systems with Periodic Coefficients
• References
• Index