Ordinary Differential Equations with Applications

Sze-Bi Hsu

• Publisher: WSPC
• Year: 2013
• ISBN:
  9789814452922
• Edition: 2
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

During the past three decades, the development of nonlinear analysis, dynamical systems and their applications to science and engineering has stimulated renewed enthusiasm for the theory of Ordinary Differential Equations (ODE).

This useful book, which is based on the lecture notes of a well-received graduate course, emphasizes both theory and applications, taking numerous examples from physics and biology to illustrate the application of ODE theory and techniques.

Written in a straightforward and easily accessible style, this volume presents dynamical systems in the spirit of nonlinear analysis to readers at a graduate level and serves both as a textbook and as a valuable resource for researchers.

This new edition contains corrections and suggestions from the various readers and users. A new chapter on Monotone Dynamical Systems is added to take into account the new developments in ordinary differential equations and dynamical systems.

Contents:

• Introduction
• Fundamental Theory
• Linear Systems
• Stability of Nonlinear Systems
• Method of Lyapunov Functions
• Two-Dimensional Systems
• Second Order Linear Equations
• The Index Theory and Brouwer Degree
• Perturbation Methods
• Introduction to Monotone Dynamical Systems

Readership: Graduate students in mathematics, applied mathematics, and engineering.

Chapter 1

INTRODUCTION

1.1Where do ODEs arise

The theory of ordinary differential equations deals with the large time behavior of the solution x(t,x₀) of the initial value problem (I.V.P.) of the first order system of differential equations:

or in vector form

where

D is open in

.

If the right-hand side of (1.1) is independent of time t, i.e.,

then we say that (1.2) is an autonomous system. In this case, we call f a vector field on its domain Ω. If the right-hand side depends on time t, then we say that (1.1) is a nonautonomous system. The most important nonautonomous system is the periodic system, i.e., f(t,x) satisfies

for some w > 0 (w is called the period). If f(t,x) = A(t)x where A(t) ∈

then we say that

is a linear system of differential equations. It is easy to verify that if φ(t),ψ(t) are solutions of (1.3), then αφ(t) + βψ(t) is also a solution of the linear system (1.3) for

The system

is called a linear system with nonhomogeneous part g(t). If A(t) ≡ A, then

is a linear system with constant coefficients. We say that system (1.1) is nonlinear if it is not linear. It is usually much harder to analyze nonlinear systems than the linear ones. The main difference between linear systems and nonlinear systems is the superposition principle. The superposition principle states that the linear combination of solutions is also a solution. For linear systems (1.3), (1.4), (1.5) as we can see in Chapter 3 that we have nice solution structures. However, nonlinear systems arise in many areas of science and engineering. It is still a great challenge to understand the nonlinear phenomena. In the following we present some important examples of differential equations from physics, chemistry and biology.

Example 1.1.1

The equation describes the motion of a spring with damping and restoring forces. Applying Newton's law, F = ma, we have

Let

Then we covert the equation into a first order system of two equations

Example 1.1.2

...

Table of contents

• Front Cover
• Half Title
• Series Title
• Title Page
• Copyright
• Preface to the First Edition
• Preface to the Second Edition
• Contents
• 1. INTRODUCTION
• 2. FUNDAMENTAL THEORY
• 3. LINEAR SYSTEMS
• 4. STABILITY OF NONLINEAR SYSTEMS
• 5. METHOD OF LYAPUNOV FUNCTIONS
• 6. TWO-DIMENSIONAL SYSTEMS
• 7. SECOND ORDER LINEAR EQUATIONS
• 8. THE INDEX THEORY AND BROUWER DEGREE
• 9. PERTURBATION METHODS
• 10. INTRODUCTION TO MONOTONE DYNAMICAL SYSTEMS
• Bibliography
• Index