Cover
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 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 Rn
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 eA
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 Rn
4 . A Second Definition of the Exponential
5 . An Example: Eulerโs Formula for ez
6 . Eulerโs 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 0n 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 gt for Small t
3 . The Construction of gt 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 Singu1a.r Poiiits on a Sphere
7 . Justification
8 . The Multidimensional Case
Examination Topics
Sample Examination Problems
Supplementary Problems
Subject Index