常微分方程入门：概念与方法

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"一本关于常微分方程的入门教材，涵盖了标准解法并引入了相平面分析等定性方法，适合一学期或一学期课程使用。书中包含Euler's方法、差分方程、逻辑斯蒂映射动力学和Lorenz方程等主题，强调了该领域的活力，并提供了进一步研究的线索。作者提倡使用图形化方法理解和解决方程，书中有大量插图，并提供MATLAB文件以生成相关图形。书中包含大量例题和练习题，供学生实践和巩固理论知识，教师可获取全套解决方案。" 常微分方程(Ordinary Differential Equations, ODE)是数学的一个重要分支，它研究的是一个或多个变量之间依赖关系的导数。在【标题】"An introduction to ordinary differential equations"中，本书旨在为初学者提供一个轻松进入这一领域的入口。【描述】指出，这本书不仅涵盖了基本的求解技术，还引入了相平面分析这样的定性方法，这是一种理解动态系统行为的有效工具。相平面分析是一种在二维空间中描绘系统动态的方法，通过绘制状态变量的轨迹，可以直观地看出系统的稳定性和周期性行为。这种方法对于理解没有解析解的复杂系统特别有用，因为它可以帮助我们理解系统的行为模式而无需完全求解方程。【部分内容】中提到的主题如Euler's方法，是一种数值解法，用于近似求解常微分方程。这种方法基于微小时间步长的连续逼近，对于无法找到封闭形式解的方程尤其实用。差分方程则是离散版本的微分方程，它们在计算机模拟和数据分析等领域中广泛使用。逻辑斯蒂映射（Logistic Map）是混沌理论中的经典例子，它描述了种群增长的非线性动态，展示了简单的方程如何产生复杂的长期行为。Lorenz方程则是一个著名的三阶常微分方程组，常用来模拟大气对流，其非线性和敏感依赖于初始条件的特性揭示了混沌现象。书中通过丰富的图形展示和MATLAB代码，鼓励读者采用可视化手段来理解这些概念。MATLAB是一种强大的数值计算软件，它的使用可以让学生更好地理解实际应用中如何处理这些方程。此外，书中包含大量的已解示例，用以激发兴趣，阐明关键概念，并演示如何将理论应用于实践中。配套的练习题则旨在测试和扩展学生的理解，教师可以获取全部解答，以辅助教学过程。 "An introduction to ordinary differential equations"是一本全面且深入浅出的教材，它不仅教授基本的微分方程求解技巧，还引导学生进入动态系统分析的世界，结合实例和图形工具，使得理论与实践相结合，对初学者极具价值。

xiv Preface

which covers reduction of order, the method of variation of constants, and series

solutions.

Part IV turns aside from differential equations, motivating the study of dif-

ference equations by discussing Euler’s method of numerical solution. Constant

coefﬁcient linear difference equations are covered, and then there are two chapters

devoted to nonlinear difference equations. One of these goes beyond the conﬁnes

of an introductory course and discusses the dynamics of the logistic map in some

detail.

Part V treats coupled systems of two linear differential equations, starting with

the substitution method that reduces the problem to a second order differential

equation in one variable, the most reliable way to ﬁnd explicit solutions. The re-

mainder of this portion of the book deals with the matrix approach, showing how a

calculation of the eigenvalues and eigenvectors of an appropriate matrix is enough

to draw the phase portrait. This is done by changing to a coordinate system in

which the equation is put into a standard form, providing an illustration of the

Jordan canonical form of a matrix.

Part VI uses the methods from Part V in order to draw the phase plane diagrams

for a variety of nonlinear systems, with examples taken from mathematical ecology

and simple one-dimensional particle systems, including the pendulum. The book

ends with a brief discussion of Dulac’s criterion and the Poincar

e–Bendixson The-

orem, a chapter that investigates the complicated dynamics of the Lorenz Equa-

tions, and suggestions for further reading.

In addition to those already mentioned above I would like to thank various peo-

ple who have contributed to this book. I ﬁrst learned much of the material here

from Tristram Jones-Parry at Westminster School, to whom much belated thanks

for all his ﬁne teaching many years ago. I also owe a debt of gratitude to all those

who taught the course at Warwick before me, shaping its contents and therefore

those of this book; in particular, I had useful guidance from the course notes of

Alan Newell and Claude Baesens. I am most grateful to Andrew Stuart, who, in

encouraging me to emphasise the links with linear algebra, made me fond of a

subject that I still remembered with a shudder from my own undergraduate days.

Thanks too to James Macdonald, whose ‘Swarm of ﬂies’ program for his MMath

project on the Lorenz equations was the inspiration behind Figure 37.8.

Over the past two months I have been able to think of little except phase planes

and drawing ﬁgures in M

ATLAB: my wife, Tania Styles, has managed to endure my

many variations on ‘come and see this picture of a washing machine’ with a smile.

Heartfelt thanks to her for this, and, of course, for everything.

Finally, I would particularly like to thank my Ph.D. student, Oliver Tearne, and

my father, John, both of whom read this book extremely carefully and made a num-

ber of very helpful comments. For whatever imperfections remain, my apologies

to them and to my readers.

Introduction

Differential equations date back to the mid-seventeenth century, when calculus

was discovered independently by Newton (c. 1665) and Leibniz (c. 1684). Mod-

ern mathematical physics essentially started with Newton’s Principia (published in

1687) in which he not only developed the calculus but also presented his three fun-

damental laws of motion that have made the mathematical modelling of physical

phenomena possible.

Historically, advances in the theory of differential equations have come from the

insights gained when trying to treat speciﬁc physical models. Despite this some-

what piecemeal development, the subject has become a well-deﬁned and coherent

area of mathematics. This book adopts a theoretical point of view, developing the

theory to the point at which it can no longer be described as ‘basic differential

equations’ and is about to become entangled with more advanced topics from the

theory of dynamical systems. Of course, applications are used throughout to serve

as motivation and illustration, but the emphasis is on a clean presentation of the

mathematics.

You may ﬁnd that some of the problems covered in the ﬁrst few chapters are

already familiar. The methods of solving these problems are well established, and

you may be well practised at applying them. However, we will take care here to

show why these methods work; giving proper justiﬁcation of the methods can take

some time, but as mathematicians we should not be satisﬁed merely with a set of

‘recipes’. Nevertheless, knowing something about the details should not stop you

from applying the methods you know already; rather you should be able to use

them with more conﬁdence.

Some of the chapters, and some sections within other chapters, are marked

with an asterisk (*). These parts of the book contain either material that is more

advanced, or material that expands on points raised elsewhere; while they could be

omitted in the interests of brevity, they are intended to give some indication of the

richness of the subject beyond the conﬁnes of an introductory course.

Various modern editions of this work are available, translated from its original Latin.

2 Introduction

There are three appendices, covering background material that is necessary at

various points in the book. While some of this is elementary and may already

be familiar (Appendix A recalls some notation and various facts about real and

complex numbers that will be used throughout the book) some is a little more ad-

vanced. Problems with timetabling often mean that certain undergraduate courses

have to rely on material that is yet to be taught in others, hence there are appen-

dices on matrices, eigenvalues and eigenvectors (Appendix B) and on derivatives,

partial derivatives and Taylor series (Appendix C). The calculation of eigenvalues

and eigenvectors is treated in detail in the main part of the book.

The use of mathematical computer packages is now a standard part of the under-

graduate curriculum, and an important tool in the armoury of practising mathemati-

cians, scientists and engineers. Although the emphasis in the text is on pencil and

paper analysis, and the book in no way relies on the availability of such software,

some topics, particularly the treatment of coupled nonlinear equations using phase

plane ideas in Chapters 28–37, can beneﬁt greatly from the graphical possibilities

modern computers provide. Almost all of the ﬁgures in this book have been gener-

ated using M

ATLAB, and very occasionally particular MATLAB commands are men-

tioned in the text. Nevertheless, it should be possible to carry out the numerical ex-

ercises suggested here using any of the major commercially available mathematical

packages; and with a little more ingenuity using any programming language with

graphical capabilities. The M

ATLAB ﬁles used to produce some of the ﬁgures, and

mentioned in certain of the exercises, are available for download from the web at

www.cambridge.org/0521533910.

There is no better way to learn this material than by working through a selection

of examples. One set of examples is included in what is, I hope, a natural way in

the text, with the end of each worked solution marked with a box (

). Another set

of examples is given in the exercises that end each chapter, and these should be

considered an integral part of the book. The majority consist of sample problems

that can be treated with the methods of the chapter – in order to give teachers a

reasonable choice of problems, there are intentionally more of these than you could

reasonably be expected to do. Others, labelled with a ‘T’, are more theoretical

and designed to give an indication of some of the mathematical issues raised, but

not treated in detail, in the text. Finally, those exercises labelled with a ‘C’ are

intended to encourage the use of the computer to perform routine calculations and

investigate equations and their solutions graphically. Those involved in teaching

courses based on this book may obtain copies of solutions to these exercises by

applying to the publisher by email (

solutions@cambridge.org).

I would welcome any comments or suggestions, either by post to the Mathe-

matics Institute, University of Warwick, Coventry, CV4 7AL, U.K. or by email

jcr@maths.warwick.ac.uk; any errata that arise will be posted on my own

website

www.maths.warwick.ac.uk/∼jcr/IntroODEs.html.

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