Ordinary Differential Equations and Smooth Dynamical Systems

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1 D.V. Anosov S.Kh. Aranson V.l. Arnold I.U. Bronshtein V.Z. Grines Yu.S. Il'yashenko Ordinary Differential Equations and Smooth Dynamical Systems With 25 Figures Springer

2 I. Ordinary Differential Equations V. I. Arnold, Yu. S. Il'yashenko Translated from the Russian by E. R. Dawson and D. O'Shea Contents Preface 7 Chapter 1. Basic Concepts 8 1. Definitions Direction Fields and Their Integral Curves Vector Fields, Autonomous Differential Equations, Integral Curves and Phase Curves Direction Fields and Differential Equations Diffeomorphisms and Phase Flows Singular Points The Action of a Diffeomorphism on a Vector Field First Integrals Differential Equations with Complex Time Holomorphic Direction Fields in the Complex Domain Higher Order Differential Equations Differential Equations on Manifolds Basic Theorems The Rectifiability Theorem for Vector Fields The Existence and Uniqueness Theorem The Rectifiability Theorem for Direction Fields Methods of Solving Differential Equations The Extension Theorem The Theorem on Differentiable and Analytic Dependence of Solutions on Initial Conditions and Parameters The Variational Equation The Theorem on Continuous Dependence 18

3 2 I. Ordinary Differential Equations 2.9. The Local Phase Flow Theorem The First Integral Theorem Linear Differential Equations The Exponential of a Linear Operator The Theorem on the Relation Between Phase Flows of Linear Vector Fields and Exponentials of Linear Operators Complexification of the Phase Space Saddles, Nodes, Foci, Centers The Liouville-Ostrogradsky Formula Higher Order Linear Equations Stability Lyapunov Stability and Asymptotic Stability Lyapunov's Theorem on Stability by Linearization Lyapunov and Chetaev Functions Generic Singular Points Cycles The Structure of the Phase Curves of Real Differential Equations The Monodromy Transformation of a Closed Phase Curve. Limit Cycles The Multiplicity of a Cycle Multipliers Limit Sets and the Poincare-Bendixson Theorem Systems with Symmetries The Group of Symmetries of a Differential Equation Quotient Systems Homogeneous Equations Use of Symmetries to Reduce the Order Implicit Differential Equations Basic Definitions; the Criminant, Integral Curves Regulär Singular Points Folded Saddles, Nodes, and Foci Normal Forms of Folded Singular Points Whitney Pleats Attractors Definitions An Upper Bound for the Dimension of the Maximal Attractor Applications 38 Chapter 2. Differential Equations on Surfaces Structurally Stable Equations on the Circle and on the Sphere Definitions The One-Dimensional Case Structurally Stable Systems on a Two-Dimensional Sphere 40

4 Contents 3 2. Differential Equations on a Two-Dimensional Torus The Two-Dimensional Torus and Vector Fields on it The Monodromy Mapping The Rotation Number Structurally Stable Differential Equations on the Torus Description of Structurally Stable Equations A Bound on the Number of Cycles Equations on the Torus with Irrational Rotation Numbers The Equivalence of a Diffeomorphism of a Circle to a Rotation of the Circle Diffeomorphisms of a Circle and Vector Fields on S Remarks on the Rotation Number The Rotation Number as a Function of the Parameters Families of Equations on a Torus Endomorphisms of the Circle 46 Chapter 3. Singular Points of Differential Equations in Higher Dimensional Real Phase Space Topological Classification of Hyperbolic Singular Points The Grobman-Hartman Theorem Classification of Linear Systems Lyapunov Stability and the Problem of Topological Classification On the Local Problems of Analysis Algebraic and Analytic Insolubility of the Problem of Lyapunov Stability Algebraic Solubility up to Degeneracies of Finite Codimension Topologically Unstabilizable Jets Formal Classification of Germs of Vector Fields Formal Vector Fields and Their Equivalence Resonances. The Poincare-Dulac Normal Forms and Their Generalizations Applications of the Theory of Formal Normal Forms Polynomial Normal Forms Invariant Manifolds and the Reduction Theorem The Hadamard-Perron Theorem The Center Manifold Theorem The Reduction Principle Criteria for Stability and the Topological Classification of Singular Points in the Case of Degeneracies of Low Codimension Structure of the Criteria Topological Classification of Germs of Smooth Vector Fields up to and Including Degeneracies of Codimension Two Phase Portraits of Normal Forms Criteria for Lyapunov Stability of Degeneracies of Codimension up to and Including Three 62

5 4 I. Ordinary Differential Equations 5.5. The Adjacency Diagram Theorems on Algebraic Solubility Smooth Classification of Germs of Vector Fields The Relation Between Formal Classification and Smooth Classification Germs of Vector Fields with Symmetries Quasi-Hyperbolicity Finitely Smooth Equivalence of Germs of Vector Fields Normal Forms of Vector Fields in which the Linear Part is a Nilpotent Jordan Block Centralized Chains Non-Removable Terms The Standard Representation of the Group SL (2) and of the Algebra sl (2) Extension of a Nilpotent Operator to a Representation of the Lie Algebra sl (2) Conclusion of the Proof of the Theorem 71 Chapter 4. Singular Points of Differential Equations in Higher Dimensional Complex Phase Space Linear Normal Forms Poincare Domains and Siegel Domains. Small Denominators Convergence of the Normalizing Series Analytic Theorems on Divergence of the Normalizing Series Geometrie Theorems on the Divergence of the Normalizing Series The Relation Between Formal and Analytic Classification ConditionA Problems Involving the Smooth and the Analytic Classification Analytic Invariant Manifolds The Invariant Manifold Theorem Analyticity of the Center Manifold Reversible Systems Analytic Center Manifolds of Differential Equations in the Plane Topological Classification of Singular Points in the Complex Domain Linear Vector Fields The Nonlinear Case 79 Chapter 5. Singular Points of Vector Fields in the Real and Complex Planes Resolution of Singularities The Polar Blow-up and the a-process in the Plane 80

6 Contents Elementary Singular Points Good Blow-ups Smooth Orbital Classification of Elementary Singular Points in the Plane Table of Normal Forms; the Analytic Case Normal Forms in the Smooth Case Topological Classification of Compound Singular Points with a Characteristic Trajectory The Fundamental Alternative Topological Classification of Differential Equations on the Plane in a Neighbourhood of a Singular Point Topological Finite Determinacy. Newton Diagrams of Vector Fields Investigation of Vector Fields by Their Principal Part The Problem of Distinguishing Between a Center and a Focus Statement of the Problem Algebraic Insolubility Distinguishing a Center by Linear Terms Nilpotent Jordan Block Singular Points Without Exclusive Directions A Programme for Investigating the General Case The Generalized First Focus Number Polynomial Vector Fields Analytic Classification of Elementary Singular Points in the Complex Domain Germs ofconformal Mappings withtheldentityas Linear Part Classification of Resonant Mappings and Vector Fields with Generic Nonlinearities Degenerate Elementary Singular Points Geometrie Theory of Analytic Normal Forms Embedding in the Flow, Extraction of Roots, Divergence of Normalizing Series, and Holomorphic Center Manifolds Taylor Description of the Analytic Equivalence Classes Orbital Topological Classification of Elementary Singular Points in the Complex Plane The Non-Resonant Case Saddle-Resonant Vector Fields Degenerate Elementary Singular Points 96 Chapter 6. Cycles The Monodromy Mapping Definitions Realization Local Theory of Diffeomorphisms Linear Normal Forms 99

7 6 I. Ordinary Differential Equations 2.2. The Resonant Case Invariant Manifolds of Diffeomorphism Germs Invariant Manifolds of a Cycle Blow-ups Equations with Periodic Right Hand Side Normal Form of a Linear Equation with Periodic Coefficients Linear Normal Forms Resonant Normal Forms Limit Cycles of Polynomial Vector Fields in the Plane The Finiteness Problem and Compound Cycles The Monodromy Map of a Compound Cycle Finiteness Theorems for Limit Cycles Two Particular Finiteness Theorems Method of Proving Dulac's Theorem and its Generalization Quadratic Vector Fields Limit Cycles of Systems Close to Hamiltonian Systems Generation of Real Limit Cycles Generation of Complex Cycles Investigation of the Variation The Weak Hubert Conjecture Abelian Integrals Appearing in Bifurcation Theory Polynomial Differential Equations in the Complex Plane Admissible Fields Polynomial Fields 113 Chapter 7. Analytic Theory of Differential Equations Equations Without Movable Critical Points Definitions Movable Critical Points of a First Order Equation The Riccati Equation Implicit Equations Painleve Equations Local Theory of Linear Equations with Complex Time Regulär and Irregulär Singular Points Formal, Holomorphic, and Meromorphic Equivalence Monodromy Formal Theory of Linear Systems with a Fuchsian Singular Point Formal Theory of Linear Systems with a Non-Fuchsian Singular Point Asymptotic Series and the Stokes Phenomenon Analytic Classification of Nonresonant Systems in a Neighbourhood of an Irregulär Singular Point Theory of Linear Equations in the Large Equations and Systems of the Fuchsian Class 125

8 Preface Extensibility and Monodromy The Riemann-Fuchs Theorem Analytic Functions of Matrices Connection with the Theory of Kleinian Groups Integrability in Quadratures Special Equations of Mathematical Physics The Monodromy Group of the Gauss Equation The Riemann-Hilbert Problem Formulation of the Problem The Riemann-Hilbert Problem for a Disk The Riemann-Hilbert Problem for a Sphere The Riemann-Hilbert Problem for Fuchsian Systems Generalizations for Non-Standard Time Vector Bundles on the Sphere Application to the Riemann-Hilbert Problem Isomonodromic Deformations and the Painleve Equations. 135 Bibliography 135 Index 142 Preface This survey is devoted mainly to the local theory of ordinary differential equations. It does not include bifurcation theory, which will be dealt with in a separate article. The averaging method is dealt with in the survey " Mathematical aspects of classical and celestial mechanics" by V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt. We do not touch on the spectral theory of differential Operators with a Single independent variable (see, for example, [28]); as regards objectives and methods, this is more closely related to functional analysis. Our survey also does not include the theory of integral transforms and their application to linear differential equations. The asymptotic theory of differential equations is dealt with in M. V. Fedoryuk's survey, "Asymptotic methods in analysis"; however, some general theorems of this theory are presented in Chap. 7. The question of actually integrating particular equations is not touched on at all; the Standard book on this subject is E. Kamke's "Differentialgleichungen: Lösungen und Lösungsmethoden", Chelsea (1948). In recent years there has been a sharp increase in research activity involving classical problems of the theory of differential equations. This is due to the penetration into the theory of other disciplines: algebra (the theory of formal normal forms), algebraic geometry (resolution of singularities), and complex analysis. We have tried, as far as possible, to reflect this research in the present article.

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