Ordinary Differential Equations by Vladimir I. Arnold

[Astronuc]

• Author: Vladimir I. Arnold, R. Cooke (Translator)
• Title: Ordinary Differential Equations
• Amazon Link: https://www.amazon.com/dp/3540345639/?tag=pfamazon01-20
• Prerequisities: Introductory Calculus
• Level: Adanced Undergraduate or Graduate

Table of Contents:

[LIST]
[*] Basic Concepts
[LIST]
[*] Phase Spaces
[LIST]
[*] Examples of Evolutionary Processes
[*] Phase Spaces
[*] The Integral Curves of a Direction Field
[*] A Differential Equation and its Solutions
[*] The Evolutionary Equation with a One-dimensional Phase Space
[*] Example: The Equation of Normal Reproduction
[*] Example: The Explosion Equation
[*] Example: The Logistic Curve
[*] Example: Harvest Quotas
[*] Example: Harvesting with a Relative Quota
[*] Equations with a Multidimensional Phase Space
[*] Example: The Differential Equation of a Predator-Prey system
[*] Example: A Free Particle on a Line
[*] Example: Free Fall
[*] Example: Small Oscillations
[*] Example: The Mathematical Pendulum
[*] Example: The Inverted Pendulum
[*] Example: Small Oscillations of a Spherical Pendulum
[/LIST]
[*] Vector Fields on the Line
[LIST]
[*] Existence and Uniqueness of Solutions
[*] A Counterexample
[*] Proof of Uniqueness
[*] Direct Products
[*] Examples of Direct Products
[*] Equations with Separable Variables
[*] An Example: The Lotka-Volterra Model
[/LIST]
[*] Linear Equations
[LIST]
[*] Homogeneous Linear Equations
[*] First-order Homogeneous Linear Equations with Periodic Coefficients
[*] Inhomogeneous Linear Equations
[*] The Influence Function and \delta-shaped Inhomogeneities
[*] Inhomogeneous Linear Equations with Periodic Coefficients
[/LIST]
[*] Phase Flows
[LIST]
[*] The Action of a Group on a Set
[*] One-parameter Transformation Groups
[*] One-parameter Diffeomorphism Groups
[*] The Phase Velocity Vector Field
[/LIST]
[*] The Action of DifFeomorphisms on Vector Fields and Direction Fields
[LIST]
[*] The Action of Smooth Mappings on Vectors
[*] The Action of Diffeomorphisms on Vector Fields
[*] Change of Variables in an Equation
[*] The Action of a Diffeomorphism on a Direction Field
[*] The Action of a Diffeomorphism on a Phase Flow
[/LIST]
[*] Symmetries
[LIST]
[*] Symmetry Groups
[*] Application of a One-parameter Symmetry Group to Integrate an Equation
[*] Homogeneous Equations
[*] Quasi-homogeneous Equations
[*] Similarity and Dimensional Considerations
[*] Methods of Integrating Differential Equations
[/LIST]
[/LIST]
[*] Basic Theorems
[LIST]
[*] Rectification Theorems
[LIST]
[*] Rectification of a Direction Field
[*] Existence and Uniqueness Theorems
[*] Theorems on Continuous and Differentiable Dependence of the Solutions on the Initial Condition
[*] Transformation over the Time Interval from t_0 to t
[*] Theorems on Continuous and Differentiable Dependence on a Parameter
[*] Extension Theorems
[*] Rectification of a Vector Field
[/LIST]
[*] Applications to Equations of Higher Order than First
[LIST]
[*] The Equivalence of an Equation of Order n and a System of n First-order Equations
[*] Existence and Uniqueness Theorems
[*] Differentiability and Extension Theorems
[*] Systems of Equations
[*] Remarks on Terminology
[/LIST]
[*] The Phase Curves of an Autonomous System
[LIST]
[*] Autonomous Systems
[*] Translation over Time
[*] Closed Phase Curves
[/LIST]
[*] The Derivative in the Direction of a Vector Field and First Integrals
[LIST]
[*] The Derivative in the Direction of a Vector
[*] The Derivative in the Direction of a Vector Field
[*] Properties of the Directional Derivative
[*] The Lie Algebra of Vector Fields
[*] First Integrals
[*] Local First Integral
[*] Time-Dependent First Integrals
[/LIST]
[*] First-order Linear and Quasi-linear Partial Differential Equations
[LIST]
[*] The Homogeneous Linear Equation
[*] The Cauchy Problem
[*] The Inhomogeneous Linear Equation
[*] The Quasi-linear Equation
[*] The Characteristics of a Quasi-linear Equation
[*] Integration of a Quasi-linear Equation 
[*] The First-order Nonlinear Partial Differential Equation
[/LIST]
[*] The Conservative System with one Degree of Freedom
[LIST]
[*] Definitions
[*] The Law of Conservation of Energy
[*] The Level Lines of the Energy
[*] The Level Lines of the Energy Near a Singular Point
[*] Extension of the Solutions of Newton's Equation
[*] Noncritical Level Lines of the Energy
[*] Proof of the Theorem of Sect. 6
[*] Critical Level Lines
[*] An Example
[*] Small Perturbations of a Conservative System
[/LIST]
[/LIST]
[*] Linear Systems
[LIST]
[*] Linear Problems
[LIST]
[*] Example: Linearization
[*] Example: One-parameter Groups of Linear Transformations of R^n
[*] The Linear Equation
[/LIST]
[*] The Exponential Function
[LIST]
[*] The Norm of an Operator
[*] The Metric Space of Operators
[*] Proof of Completeness
[*] Series
[*] Definition of the Exponential e^A
[*] An Example
[*] The Exponential of a Diagonal Operator
[*] The Exponential of a Nilpotent Operator
[*] Quasi-polynomials
[/LIST]
[*] Properties of the Exponential
[LIST]
[*] The Group Property
[*] The Fundamental Theorem of the Theory of Linear Equations with Constant Coefficients
[*] The General Form of One-parameter Groups of Linear Transformations of the Space R^n
[*] A Second Definition of the Exponential
[*] An Example: Euler's Formula, for e^z
[*] Euler's Broken Lines
[/LIST]
[*] The Determinant of an Exponential
[LIST]
[*] The Determinant of an Operator
[*] The Trace of an Operator
[*] The Connection Between the Determinant and the Trace
[*] The Determinant of the Operator e^A
[/LIST]
[*] Practical Computation of the Matrix of an Exponential - The Case when the Eigenvalues are Real and Distinct
[LIST]
[*] The Diagonalizable Operator
[*] An Example
[*] The Discrete Case
[/LIST]
[*] Complexification and Realification
[LIST]
[*] Realification
[*] Complexification
[*] The Complex Conjugate
[*] The Exponential, Determinant, and Trace of a Complex Operator
[*] The Derivative of a Curve with Complex Values
[/LIST]
[*] The Linear Equation with a Complex Phase Space
[LIST]
[*] Definitions
[*] The Fundamental Theorem
[*] The Diagonalizable Case
[*] Example: A Linear Equation whose Phase Space is a Complex Line
[*] Corollary
[/LIST]
[*] The Complexification of a Real Linear Equation
[LIST]
[*] The Complexified Equation
[*] The Invariant Subspaces of a Real Operator
[*] The Linear Equation on the Plane
[*] The Classification of Singular Points in the Plane
[*] Example: The Pendulum with Friction 
[*] The General Solution of a Linear Equation in the Case when the Characteristic Equation Has Only Simple Roots
[/LIST]
[*] The Classification of Singular Points of Linear Systems
[LIST]
[*] Example: Singular Points in Three-dimensional Space
[*] Linear, Differentiable, and Topological Equivalence
[*] The Linear Classification 
[*] The Differentiable Classification
[/LIST]
[*] The Topological Classification of Singular Points
[LIST]
[*] Theorem
[*] Reduction to the Case m_ = 0
[*] The Lyapunov Function
[*] Construction of the Lyapunov Function
[*] An Estimate of the Derivative
[*] Construction of the Homeomorphism h
[*] Proof of Lemma 3
[*] Proof of the Topological Classification Theorem
[/LIST]
[*] Stability of Equilibrium Positions
[LIST]
[*] Lyapunov Stability
[*] Asymptotic Stability
[*] A Theorem on Stability in First Approximation
[*] Proof of the Theorem
[/LIST]
[*] The Case of Purely Imaginary Eigenvalues
[LIST]
[*] The Topological Classification
[*] An Example
[*] The Phase Curves of Eq. (4) on the Torus
[*] Corollaries
[*] The Multidimensional Case
[*] The Uniform Distribution
[/LIST]
[*] The Case of Multiple Eigenvalues
[LIST]
[*] The Computation of e^A t, where A is a Jordan Block
[*] Applications
[*] Applications to Systems of Equations of Order Higher than the First
[*] The Case of a Single nth-order Equation
[*] On Recursive Sequences
[*] Small Oscillations
[/LIST]
[*] Quasi-polynomials
[LIST]
[*] A Linear Function Space
[*] The Vector Space of Solutions of a Linear Equation
[*] Translation-invariance
[*] Historical Remark
[*] Inhomogeneous Equations
[*] The Method of Complex Amplitudes
[*] Application to the Calculation of Weakly Nonlinear Oscillations
[/LIST]
[*] Nonautonomous Linear Equations
[LIST]
[*] Definition
[*] The Existence of Solutions
[*] The Vector Space of Solutions
[*] The Wronskian Determinant
[*] The Case of a Single Equation
[*] Liouville's Theorem
[*] Sturm's Theorems on the Zeros of Solutions of Second-order Equations
[/LIST]
[*] Linear Equations with Periodic Coefficients
[LIST]
[*] The Mapping over a Period
[*] Stability Conditions
[*] Strongly Stable Systems
[*] Computations
[/LIST]
[*] Variation of Constants
[LIST]
[*] The Simplest Case
[*] The General Case
[*] Computations
[/LIST]
[/LIST]
[*] Proofs of the Main Theorems
[LIST]
[*] Contraction Mappings
[LIST]
[*] Definition
[*] The Contraction Mapping Theorem
[*] Remark
[/LIST]
[*] Proof of the Theorems on Existence and Continuous Dependence on the Initial Conditions
[LIST]
[*] The Successive Approximations of Picard
[*] Preliminary Estimates
[*] The Lipschitz Condition
[*] Differentiability and the Lipschitz Condition
[*] The Quantities C,L,a',b'
[*] The Metric Space M
[*] The Contraction Mapping A : M -> M
[*] The Existence and Uniqueness Theorem
[*] Other Applications of Contraction Mappings
[/LIST]
[*] The Theorem on Differentiability
[LIST]
[*] The Equation of Variations
[*] The Differentiability Theorem
[*] Higher Derivatives with Respect to x
[*] Derivatives in x and t
[*] The Rectification Theorem
[*] The Last Derivative
[/LIST]
[/LIST]
[*] Differential Equations on Manifolds
[LIST]
[*] Differentiate Manifolds
[LIST]
[*] Examples of Manifolds
[*] Definitions
[*] Examples of Atlases
[*] Compactness
[*] Connectedness and Dimension
[*] Differentiable Mappings
[*] Remark
[*] Submanifolds
[*] An Example
[/LIST]
[*] The Tangent Bundle. Vector Fields on a Manifold
[LIST]
[*] The Tangent Space
[*] The Tangent Bundle
[*] A Remark on Parallelizability
[*] The Tangent Mapping
[*] Vector Fields
[/LIST]
[*] The Phase Flow Denned by a Vector Field
[LIST]
[*] Theorem 
[*] Construction of the Diffeomorphisms g^t for Small t
[*] The Construction of g^t for any t
[*] A Remark
[/LIST]
[*] The Indices of the Singular Points of a Vector Field
[LIST]
[*] The Index of a Curve
[*] Properties of the Index
[*] Examples
[*] The Index of a Singular Point of a Vector Field
[*] The Theorem on the Sum of the Indices
[*] The Sum of the Indices of the Singular Points on a Sphere 
[*] Justification
[*] The Multidimensional Case
[/LIST]
[/LIST]
[*] Examination Topics
[*] Sample Examination Problems
[LIST]
[*] Supplementary Problems
[/LIST]
[*] Subject Index 
[/LIST]

 

[mathwonk]

People with geometric orientation will probably like this book a lot as I do. He begins by introducing the concept of phase space, a tool for visualizing the states of an evolving situation, by an example. Suppose two towns A,B are connected by two disjoint roads. One day two cars traverse the two separate roads between the towns in the same direction, from A to B. The next day two cars traverse the two separate roads between the towns but in opposite directions. He asks if there is a moment of the second day on which the two oppositely traveling cars both occupy the same positions as did the two similarly traveling cars at some moment of the first day?

Actually he is more picturesque: he says the two cars on the first day are connected by a rope of length less than 20 feet, and asks whether on the second day the other two vehicles, which are circular wagons of radius 10 feet, must collide at some time.

To visualize this, he describes a rectangle, with two perpendicular edges parametrizing the two roads, and shows that the two trips are described by two curves in the rectangle, one joining two opposite corners, and the other joining the other two opposite corners. hence these curves meet, so indeed there is some pair of coordinates, corresponding to a pair of points on the two roads, which is occupied on one day by two of the cars, and on the other day by the two other cars.

This shows the power of the method and the clarity of Arnol'd's exposition. He convinces us totally, but without pausing to discuss what is actually a subtle mathematical point, namely why do such curves have to meet? Since this is intuitively obvious he does not mention it, but the reader may enjoy as I did making a proof for that obvious fact, using continuity. The rest of the book seems to just get better.