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[Vladimir_I._Arnol'd]_Ordinary_Differential_Equati.pdf
Cover
Ordinary Differential Equations
Copyright
Preface to the Third Edition
Preface to the First Edition
Frequently used notation
Contents
Chapter 1 . Basic Concepts
1 . Phase Spaces
1 . Examples of Evolutionary Processes
2 . Phase Spaces
3 . The Integral Curves of a Direction Field
4 . A Differential Equation and its Solutions
5 . The Evolutionary Equation with a One-dimensional Phase Space
6 . Example: The Equation of Normal Reproduction
7 . Example: The Explosion Equation
8 . Example: The Logistic Curve
9 . Example: Harvest Quotas
10 . Example: Harvesting with a Relative Quota.
11 . Equations with a Multidimensional Phase Space
12 . Example: The Differential Equation of a Predator-Prey System
13 . Example: A Free Particle on a Line
14 . Example: Free Fall
15 . Example: Small Oscillations
16 . Example: The Mathematical Pendulum
17 . Example: The Inverted Pendulum
18 . Example: Small Oscillations of a Spherical Pendulum
2 . Vector Fields on the Line
1 . Existence and Uniqueness of Solutions
2 . A Counterexample
3 . Proof of Uniqueness
4 . Direct Products
5 . Examples of Direct Products
6 . Equations with Separable Variables
7 . An Example: The Lotka-Volterra Model
3 . Linear Equations
1 . Homogeneous Linear Equations
2 . First-order Homogeneous Linear Equations with Periodic Coefficients
3 . Inhomogeneous Linear Equations
4 . The Influence Function and delta-shaped Inhomogeneities
5 . Inhomogeneous Linear Equations with Periodic Coefficients
4 . Phase Flows
1 . The Action of a Group on a Set
2 . One-parameter Transformation Groups
3 . One-parameter Diffeomorphism Groups
4 . The Phase Velocity Vector Field
5 . The Action of Diffeomorphisms on Vector Fields and Direction Fields
1 . The Action of Smooth Mappings on Vectors
2 . The Action of Diffeomorphisms on Vector Fields
3 . Change of Variables in an Equation
4 . The Action of a Diffeomorphism on a Direction Field
5 . The Action of a Diffeoinorphisin on a Phase Flow
6 . Symmetries
1 . Symmetry Groups
2 . Application of a One-parameter Symmetry Group to Integrate an Equation
3 . Homogeneous Equations
4 . Quasi-homogeneous Equations
5 . Similarity and Dimensional Considerations
6 . Methods of Integrating Differential Equations
Chapter 2 . Basic Theorems
7 . Rectification Theorems
1 . Rectification of a Direction Field
2 . Existence and Uniqueness Theorems
3 . Theorems on Continuous and Differentiable Dependence of the Solutions on the Initial Condition
4 . Transformation over the Time Interval from t_0 zero to t
5 . Theorems on Continuous and Differentiable Dependence on a Parameter
6 . Extension Theorems
7 . Rectification of a Vector Field
8 . Applications to Equations of Higher Order than First
1 . The Equivalence of an Equation of Order n and a System of n First-order Equations
2 . Existence and Uniqueness Theorems
3 . Differentiability and Extension Theorems
4 . Systems of Equations
5 . Remarks on Terminology
9 . The Phase Curves of an Autonomous System
1 . Autonomous Systems
2 . Translation over Time
3 . Closed Phase Curves
10 . The Derivative in the Direction of a Vector Field and First Integrals
1 . The Derivative in the Direction of a Vector
2 . The Derivative in the Direction of a Vector Field
3 . Properties of the Directional Derivative
4 . The Lie Algebra of Vector Fields
5 . First Integrals
6 . Local First Integrals
7 . Time-Dependent First Integrals
11 . First-order Linear and Quasi-linear Partial Differential Equations
1 . The Homogeneous Linear Equation
2 . The Cauchy Problem
3 . The Inhomogeneous Linear Equation
4 . The Quasi-linear Equation
5 . The Characteristics of a Quasi-linear Equation
6 . Integration of a Quasi-linear Equation
7 . The First-order Nonlinear Partial Differential Equation
12 . The Conservative System with one Degree of Freedom
1 . Definitions
2 . The Law of Conservation of Energy
3 . The Level Lines of the Energy
4 . The Level Lines of the Energy Near a Singular Point
5 . Extension of the Solutions of Newton's Equation
6 . Noncritical Level Lines of the Energy
7 . Proof of the Theorem of Sect . 6
8 . Critical Level Lines
9 . An Example
10 . Small Perturbations of a Conservative System
Chapter 3 . Linear Systems
13 . Linear Problems
1 . Example: Linearization
2 . Example: One-parameter Groups of Linear Transformations of R^n
3 . The Linear Equation
14 . The Exponential Function
1 . The Norm of an Operator
2 . The Metric Space of Operators
3 . Proof of Completeness
4 . Series
5 . Definition of the Exponential e^A
6 . An Example
7 . The Exponential of a Diagonal Operator
8 . The Exponential of a Nilpotent Operator
9 . Quasi-polynomials
15 . Properties of the Exponential
1 . The Group Property
2 . The Fundamental Theorem of the Theory of Linear Equations with Constant Coefficients
3 . The General Form of One-parameter Groups of Linear Transformations of the Space R^n
4 . A Second Definition of the Exponential
5 . An Example: Euler’s Formula for e^z
6 . Euler’s Broken Lines
16 . The Determinant of an Exponential
1 . The Deteminant of an Operator
2 . The Trace of an Operator
3 . The Connection Between the Determinant and the Trace
4 . The Determinant of the 0perator e^A
17 . Practical Computation of the Matrix of an Exponential - The Case when the Eigenvalues are Real and Distinct
1 . The Diagonalizable Operator
2 . An Example
3 . The Discrete Case
18 . Complexification and Realification
1 . Realification
2 . Complexification
3 . The Complex Conjugate
4 . The Exponential, Determinant, and Trace of a Complex Operator
5 . The Derivative of a Curve with Complex Values
19 . The Linear Equation with a Complex Phase Space
1 . Definitions
2 . The Fundamental Theorem
3 . The Diagonalizable Case
4 . Example: A Linear Equation whose Phase Space is a Complex Line
5 . Corollary
20 . The Complexification of a Real Linear Equation
1 . The Complexified Equation
2 . The Invariant Subspaces of a Real Operator
3 . The Linear Equation on the Plane
4 . The Classification of Singular Points in the Plane
5 . Example: The Pendulum with Friction
6 . The General Solution of a Linear Equation in the Case when the Characteristic Equation Has Only Simple Roots
21 . The Classification of Singular Points of Linear Systems
1 . Example: Singular Points in Three-dimensional Space
2 . Linear, Differentiable, and Topological Equivalence
3 . The Linear Classification
4 . The Differentiable Classification
22 . The Topological Classification of Singular Points
1 . Theorem
2 . Reduction to the Case m_- = 0
3 . The Lyapunov Function
4 . Construction of the Lyapunov Function
5 . An Estimate of the Derivative
6 . Construction of the Homeomorphism h
7 . Proof of Lemma 3
8 . Proof of the Topological Classification Theorem
23 . Stability of Equilibrium Positions
1 . Lyapunov Stability
2 . Asymptotic Stability
3 . A Theorem on Stability in First Approximation
4 . Proof of the Theorem
24 . The Case of Purely Imaginary Eigenvalues
1 . The Topological Classification
2 . An Example
3 . The Phase Curves of Eq . (4) on the Torus
4 . Corollaries
5 . The Multidimensional Case
6 . The Uniform Distribution
25 . The Case of Multiple Eigenvalues
1 . The Computation of e^{At}, where A is a Jordan Block
2 . Applications
3 . Applications to Systems of Equations of Order Higher than the First
4 . The Case of a Single nth-order Equation
5 . On Recursive Sequences
6 . Small Oscillations
26 . Quasi-polynomials
1 . A Linear Function Space
2 . The Vector Space of Solutions of a Linear Equation
3 . Translastion-invariance
4 . Historical Remark
5 . Inhomogeneous Equations
6 . The Method of Complex Amplitudes
7 . Application to the Calculation of Weakly Nonlinear Oscillations
27 . Nonautonomous Linear Equations
1 . Definition
2 . The Existence of Solutions
3 . The Vector Space of Solutions
4 . The Wronskian Determinant
5 . The Case of a Single Equation
6 . Liouville's Theorem
7 . Sturm's Theorems on the Zeros of Solutions of Second-order Equations
28 . Linear Equations with Periodic Coefficients
1 . The Mapping over a Period
2 . Stability Conditions
3 . Strongly Stable Systems
4 . Computations
29 . Variation of Constants
1 . The Simplest Case
2 . The General Case
3 . Computations
Chapter 4 . Proofs of the Main Theorems
30 . Contraction Mappings
1 . Definition
2 . The Contraction Mapping Theorem
3 . Remark
31 . Proof of the Theorems on Existence and Continuous Dependence on the Initial Conditions
1 . The Successive Approximations of Picard
2 . Preliminary Estimates
3 . The Lipschitz Condition
4 . Differentiability and the Lipschitz Condition
5 . The Quantities C , L , a', b'
6 . The Metric Space M
7 . The Contraction Mapping A: M --> M
8 . The Existence and Uniqueness Theorem
9 . Other Applications of Contraction Mappings
32 . The Theorem on Differentiability
1 . The Equation of Variations
2 . The Differentiability Theorem
3 . Higher Derivatives with Respect to x
4 . Derivatives in x and t
5 . The Rectification Theorem
6 . The Last Derivative
Chapter 5 . Differential Equations on Manifolds
33 . Differentiable Manifolds
1 . Examples of Manifolds
2 . Definitions
3 . Examples of Atlases
4 . Compactness
5 . Connectedness and Dimension
6 . Differentiable Mappings
7 . Remark
8 . Submanifolds
9 . An Example
34 . The Tangent Bundle. Vector Fields on a Manifold
1 . The Tangent Space
2 . The Tangent Bundle
3 . A Remark on Parallelizability
4 . The Tangent Mapping
5 . Vector Fields
35 . The Phase Flow Defined by a Vector Field
1 . Theorem
2 . Construction of the Diffeomorphisms g^t for Small t
3 . The Construction of g^t for any t
4 . A Remark
36 . The Indices of the Singular Points of a Vector Field
1 . The Index of a Curve
2 . Properties of the Index
3 . Examples
4 . The Index of a Singular Point of a Vector Field
5 . The Theorem on the Sum of the Indices
6 . The Sum of the Indices of the Singular Points on a Sphere
7 . Justification
8 . The Multidimensional Case
Examination Topics
Sample Examination Problems
Supplementary Problems
Subject Index
Vladimir I. Arnol'd
Ordinary
Differential Equations
Translated from the Russian
by Roger Cooke
With 272 Figures
Springer-Verlag
Berlin Heidelberg NewYork
London Paris Tokyo
Hong Kong Barcelona
_Budapest
Vladimir I.Arnol'd
Steklov Mathematical Institute
ul. Vavilova 42
Moscow 117 966, USSR
Translator:
Roger Cooke
Department of Mathematics
University ofVermont
Burlington, VT 05405, USA
Title of the original Russian edition: Obyknovennye differentsial'nye
uravneniya, 3rd edition, Publisher Nauka, Moscow 1984
Mathematics Subject Classification (1991): 34-01
ISBN 3-540-54813-0 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-54813-0 Springer-Verlag New York Heidelberg Berlin
Library of Congress Cataloging-in-Publication Data
Arnol'd, V. I. (Vladimir Igorevich), 1937-
[Obyknovennye differentsial'nye uravnenua.
English] Ordinary differential equations I Vladimir I. Amol'd; translated from the Russian by
Roger Cooke. - 3rd ed. Translation of: Obyknovennye differentsial'nye uravnenila. Includes
bibliographical references and index.
ISBN 3-540-54813-0 (Berlin).- ISBN 0-387-54813-0 (NewYork)
1. Differential equations. !.Title. QA327.A713 1992 515'.352-dc20 91-44188
This work is subject to copyright. All rights are reserved, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data
banks. Duplication of this publication or parts thereof is permitted only under the provisions
of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee
must always be obtained from Springer-Verlag. Violations are liable for prosecution under
the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1992
Printed in the United States of America
Cover design: Struve & Partner, Heidelberg
Typesetting: camera-ready by author
41/3140-5432
-Printed on acid-free paper
Preface to the Third Edition
The first two chapters of this book have been thoroughly revised and sig
nificantly expanded. Sections have been added on elementary methods of in
tegration (on homogeneous and inhomogeneous first-order linear equations
and o,n homogeneous and quasi-homogeneous equations), on first-order linear
and quasi-linear partial differential equations, on equations not solved for the
derivative, and on Sturm's theorems on the zeros of second-order linear equa
tions. Thus the new edition contains all the questions of the current syllabus
in the theory of ordinary differential equations.
In discussing special devices for integration the author has tried through
out to lay bare the geometric essence of the methods being studied and to
show how these methods work in applications, especially in mechanics. Thus
to solve an inhomogeneous linear equation we introduce the delta-function and
calculate the retarded Green's function; quasi-homogeneous equations lead to
the theory of similarity and the law of universal gravitation, while the theorem
on differentiability of the solution with respect to the initial conditions leads
to the study of the relative motion of celestial bodies in neighboring orbits.
The author has permitted himself to include some historical digressions
in this preface. Differential equations were invented by Newton ( 1642-1 727).
Newton considered this invention of his so important that he encoded it as an
anagram whose meaning in modern terms can be freely translated as follows:
"The laws of nature are expressed by differential equations."
One of Newton's fundamental analytic achievements was the expansion
of all functions in power series (the meaning of a second, long anagram of
Newton's to the effect that to solve any equation one should substitute the
series into the equation and equate coefficients of like powers). Of particular
importance here was the discovery of Newton's binomial formula (not with
integer exponents, of course, for which the formula was known, for example,
to Viete (1540-1603), but -what is particularly important -with fractional
and negative exponents). Newton expanded all the elementary functions in
"Taylor series" (rational functions, radicals, trigonometric, exponential, and
logarithmic functions). This, together with a table of primitives compiled by
Newton (which entered the modern textbooks of analysis almost unaltered),
enabled him, in his words, to compare the areas of any figures "in half of a
quarter of an hour."
2
Preface to the Third Edition
Newton pointed out that the coefficients of his series were proportional
to the successive derivatives of the function, but did not dwell on this, since
he correctly considered that it was more convenient to carry out all the com
putations in analysis not by repeated differentiation, but by computing the
first terms of a series. For Newton the connection between the coefficients of
a series and the derivatives was more a means of computing derivatives than
a means of constructing the series.
On of Newton's most important achievements is his theory of the solar
system expounded in the Mathematical Principles of Natural Philosophy (the
Principia) without using· mathematical analysis. It is usually assumed that
Newton discovered the law of universal gravitation using his analysis. In fact
Newton deserves the credit only for proving that the orbits are ellipses (1680)
in a gravitational field subject to the inverse-square law; the actual law of
gravitation was shown to Newton by Hooke (1635-1703) ( cf. § 8) and seems
to have been guessed by several other scholars.
Modern physics begins with Newton's Principia. The completion of the
formation of analysis as an independent scientific discipline is connected with
the name of Leibniz (1646-1716). Another of Leibniz' grand achievements is
the broad publicizing of analysis (his first publication is an article in 1684)
and the development of its algorithms1 to complete automatization: he thus
discovered a method of teaching how to use analysis (and teaching analysis
itself) to people who do not understand it at all - a development that has to
be resisted even today.
Among the enormous number of eighteenth-century works on differential
equations the works of Euler (1707-1783) and Lagrange (1736-1813) stand
out. In these works the theory of small oscillations is first developed, and
consequently also the theory of linear systems of differential equations; along
the way the fundamental concepts of linear algebra arose (eigenvalues and
eigenvectors in the n-dimensional case). The characteristic equation of a lin
ear operator was long called the secular equation, since it was from just such
an equation that the secular perturbations (long-term, i.e., slow in comparison
with the annual motion) of planetary orbits were determined in accordance
with Lagrange's theory of small oscillations. After Newton, Laplace and La
grange and later Gauss (1777-1855) develop also the methods of perturbation
theory.
When the unsolvability of algebraic equations in radicals was proved, Li
ouville (1809-1882) constructed an analogous theory for differential equations,
establishing the impossibility of solving a variety of equations (including such
classical ones as second-order linear equations) in elementary functions and
quadratures. Later S. Lie (1842-1899), analyzing the problem of integration
equations in quadratures, discovered the need for a detailed investigation of
groups of diffeomorphisms (afterwards known as Lie groups) - thus from the
1 Incidentally the concept of a matrix, the notation aij, the beginnings of the theory
of determinants and systems of linear equations, and one of the first computing
machines, are due to Leibniz.
Preface to the Third Edition
3
theory of differential equations arose one of the most fruitful areas of mod
ern mathematics, whose subsequent development was closely connected with
completely different questions (Lie algebras had been studied even earlier by
Poisson (1781-1840), and especially by Jacobi (1804-1851)).
A new epoch in the development of the theory of differential equations
begins with the works of Poincare (1854-1912), the "qualitative theory of
differential equations," created by him, taken together with the theory of
functions of a complex variable, lead to the foundation of modern topology.
The qualitative theory of differential equations, or, as it is more frequently
known nowadays, the theory of dynamical systems, is now the most actively
developing area of the theory of differential equations, having the most im
portant applications in physical science. Beginning with the classical works
of A. M. Lyapunov (1857-1918) on the theory of stability of motion, Russian
mathematicians have taken a large part in the development of this area (we
mention the works of A. A. Andronov (1901-1952) on bifurcation theory, A.
A. Andronov and L. S. Pontryagin on structural stability, N. M .Krylov (1879-
1955) and N. N. Bogolyubov on the theory of averaging, A. N. Kolmogorov
on the theory of perturbations of conditionally-periodic motions). A study
of the modern achievements, of course, goes beyond the scope of the present
book (one can become acquainted with some of them, for example, from the
author's books, Geometrical Methods in the Theory of Ordinary Differential
Equations, Springer-Verlag, New York, 1983; Mathematical Methods of Clas
sical Mechanics, Springer-Verlag, New York, 1978; and Catastrophe Theory,
Springer-Verlag, New York, 1984).
The au'thor is grateful to all the readers of earlier editions, who sent their
comments, which the author has tried to take account of in revising the book,
and also to D. V. Anosov, whose numerous comments promoted the improve
ment of the present edition.
V. I. Arnol'd
From the Preface to the First Edition
In selecting the material for this book the author attempted to limit the
material to the bare minimum. The heart of the course is occupied by two
circles of ideas: the theorem on the rectification of a vector field (equivalent
to the usual existence, uniqueness, and differentiability theorems) and the
theory of one-parameter groups of linear transformations (i.e., the theory of
autonomous linear systems).
The applications of ordinary differential equations in mechanics are stud
ied in more detail than usual. The equation of the pendulum makes its ap
pearance at an early stage,; thereafter efficiency of the concepts introduced is
always verified through this example. Thus the law of conservation of energy
appears in the section on first integrals, the "small parameter method" is de
rived from the theorem on differentiation with respect to a parameter, and
the theory of linear equations with periodic coefficients leads naturally to the
study of the swing ("parametric resonance").
The exposition of many topics in the course differs widely from the tradi
tional exposition. The author has striven throughout to make plain the geo
metric, qualitative side of the phenomena being studied. In accordance with
this principle there are many figures in the book, but not a single complicated
formula. On the other hand a whole series of fundamental concepts appears,
concepts that remain in the shadows in the traditional coordinate presenta
tion (the phase space and phase flows, smooth manifolds and bundles, vector
fields and one-parameter diffeomorphism groups). The course could have been
significantly shortened if these concepts had been assumed to be known. Un
fortunately at present these topics are not included in courses of analysis or
geometry. For that reason the author was obliged to expound them in some
detail, assuming no previous knowledge on the part of the reader beyond the
standard elementary courses of analysis and linear algebra.
The present book is based on a year-long course of lectures that the author
gave to second-year mathematics majors at Moscow University during the
years 1968-1970.
In preparing these lectures for publication the author received a great deal
of help from R.I. Bogdanov. The author is grateful to him and all the students
and colleagues who communicated their comments on the mimeographed text
of the lectures (MGU, 1969). The author is also grateful to the reviewers D.
V. Anosov and S. G. Krein for their attentive review of the manuscript.
1911
V. Arnol'd
Frequently used notation
R- the set (group, field) of real numbers.
C- the set (group, field) of complex numbers.
Z - the set (group, ring) of integers.
x E X C Y- x is an element of the subset X of the set Y.
X n Y, X U Y - the intersection and union of the sets X and Y.
f : X ---+ Y - f is a mapping of the set X into the set Y.
x ~----+ y -
fog - the composite of the mappings (g being applied first).
:3, \/ - there exists, for all.
*-a problem or theorem that is not obligatory (more difficult).
Rn - a vector space of dimension n over the field R.
Other structures may be considered in ths set Rn (for example, an affine struc
the mapping takes the point x to the point y.
ture, a Euclidean structure, or the direct product of n lines). Usually this will be
noted specifically ("the affine space Rn", "the Euclidean space Rn", "the coordinate
space Rn", and so forth).
Elements of a vector space are called vectors. Vectors are usually denoted by bold
face letters (v, e, and so forth). Vectors of the coordinate space Rn are identified
with n-tuples of numbers. We shall write, for example, V = (VI, •.• , Vn) = VI ei +
· · · + Vn en; the set of n vectors ei is called a coordinate basis in Rn.
We shall often encounter functions a real variable t called time. The derivative
with respect to t is called velocity and is usually denoted by a dot over the letter:
x = dxjdt.
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