经典ODE教程.pdf

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Ordinary Differential Equations with Applications Carmen Chicone Springer

To Jenny, for giving me the gift of time.

Preface This book is based on a two-semester course in ordinary diﬀerential equa- tions that I have taught to graduate students for two decades at the Uni- versity of Missouri. The scope of the narrative evolved over time from an embryonic collection of supplementary notes, through many classroom tested revisions, to a treatment of the subject that is suitable for a year (or more) of graduate study. If it is true that students of diﬀerential equations give away their point of view by the way they denote the derivative with respect to the independent variable, then the initiated reader can turn to Chapter 1, note that I write , and thus correctly deduce that this book is written with an eye ˙x, not x toward dynamical systems. Indeed, this book contains a thorough intro- duction to the basic properties of diﬀerential equations that are needed to approach the modern theory of (nonlinear) dynamical systems. However, this is not the whole story. The book is also a product of my desire to demonstrate to my students that diﬀerential equations is the least insular of mathematical subjects, that it is strongly connected to almost all areas of mathematics, and it is an essential element of applied mathematics. When I teach this course, I use the ﬁrst part of the ﬁrst semester to pro- vide a rapid, student-friendly survey of the standard topics encountered in an introductory course of ordinary diﬀerential equations (ODE): existence theory, ﬂows, invariant manifolds, linearization, omega limit sets, phase plane analysis, and stability. These topics, covered in Sections 1.1–1.8 of Chapter 1 of this book, are introduced, together with some of their im- portant and interesting applications, so that the power and beauty of the subject is immediately apparent. This is followed by a discussion of linear

viii Preface systems theory and the proofs of the basic theorems on linearized stabil- ity in Chapter 2. Then, I conclude the ﬁrst semester by presenting one or two realistic applications from Chapter 3. These applications provide a capstone for the course as well as an excellent opportunity to teach the mathematics graduate students some physics, while giving the engineering and physics students some exposure to applications from a mathematical perspective. In the second semester, I introduce some advanced concepts related to existence theory, invariant manifolds, continuation of periodic orbits, forced oscillators, separatrix splitting, averaging, and bifurcation theory. However, since there is not enough time in one semester to cover all of this material in depth, I usually choose just one or two of these topics for presentation in class. The material in the remaining chapters is assigned for private study according to the interests of my students. My course is designed to be accessible to students who have only stud- ied diﬀerential equations during one undergraduate semester. While I do assume some knowledge of linear algebra, advanced calculus, and analysis, only the most basic material from these subjects is required: eigenvalues and eigenvectors, compact sets, uniform convergence, the derivative of a func- tion of several variables, and the deﬁnition of metric and Banach spaces. With regard to the last prerequisite, I ﬁnd that some students are afraid to take the course because they are not comfortable with Banach space theory. However, I put them at ease by mentioning that no deep properties of inﬁnite dimensional spaces are used, only the basic deﬁnitions. Exercises are an integral part of this book. As such, many of them are placed strategically within the text, rather than at the end of a section. These interruptions of the ﬂow of the narrative are meant to provide an opportunity for the reader to absorb the preceding material and as a guide to further study. Some of the exercises are routine, while others are sections of the text written in “exercise form.” For example, there are extended ex- ercises on structural stability, Hamiltonian and gradient systems on man- ifolds, singular perturbations, and Lie groups. My students are strongly encouraged to work through the exercises. How is it possible to gain an un- derstanding of a mathematical subject without doing some mathematics? Perhaps a mathematics book is like a musical score: by sight reading you can pick out the notes, but practice is required to hear the melody. The placement of exercises is just one indication that this book is not written in axiomatic style. Many results are used before their proofs are pro- vided, some ideas are discussed without formal proofs, and some advanced topics are introduced without being fully developed. The pure axiomatic approach forbids the use of such devices in favor of logical order. The other extreme would be a treatment that is intended to convey the ideas of the subject with no attempt to provide detailed proofs of basic results. While the narrative of an axiomatic approach can be as dry as dust, the excite- ment of an idea-oriented approach must be weighed against the fact that

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