- #1
- 22,095
- 6,743
- 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:
Code:
[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]
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