Non-Linear Differential Equations

Contents

Preface

Chapter I. General Theorems About Solutions of Differential Systems


1. Integral Curves 1


1. Integral Curves. Extreme Time in the Future t+


2. Conditions for t+—B. The Case N = 1


3. Conditions for t+= 6. General Case


4. Boundedness of the Integral Curves


5. Integral Curves in the Sense of Caratheodory


2. Lipschitzian and Caratheodory Systems


1. Gronwall's Lemma (Generalized)


2. Lipschitzian Systems. Evaluation of x(t) — y(t)\ for Two Arcs of Integral

Curves


3. Uniqueness Theorem. Continuous Dependence on the Initial Point P0 and on f


4. Caratheodory Systems


3. The Solution Φ(t,t0,x°) of the System (1.1.1)


1. The Function ϕ(t,t0,x°). Cases of Uniqueness


2. Continuity of (ϕ(t,t0,x°)


3. Stability


4. The Function Φ(t, t09 x°) for Linear Systems


5. Differentiability of Φ (t, t0, x°)


6. Systems with Parameters


4. Periodic Solutions


1. Periodic Integral Curves. Periodic Orbits


2. Exceptional Periodic Solutions


5. Autonomous Systems


1. Autonomous Systems. Properties of their Integral Curves


2. Trajectories. Phase Space


3. Singular Points. Cycles. Open Trajectories Complements

Bibliography

Chapter II. Particular Plane Autonomous Systems


1. The Linear Case


1. Singular Points


2. Canonical Forms of Isolated Singular Points of Linear Systems


3. Affine Transformations of the Phase Plane


4. Classification of the Types of Singular Points


2. Homogeneous Systems


1. Homogeneous Systems


2. Invariant Rays. Stellar Node


3. The Center and the Focus


4. Isolated Invariant Rays. Normal Angles


5. Behavior of Trajectories In A Normal Angle


6. Examples


3. The Analytic Case


1. Introductory Remarks


2. Examples


3. The Functions Z(x, y), N(x, y)


4. A Lemma


5. Trajectories Tending To 0. Focus


6. The Equation Ν(Θ) = 0. Dicritical Points


7. Study of Z(x, y). Case of the Fixed Sign for Z(x, y)


8. Classification of Z-Sectors


4. The Problem of the Center


1. The Problem of the Center


2. The Problem of the Center for Ν(Θ)≠θ


3. The Case m = 1. Method of Poincare


4. The Case m = 1. Theorem of Poincare for the Center. The Proof of E. Picard-J. Chazy


5. The Case m = 1. Evaluation of the Period


6. Sufficient Condition of Poincare for the Center. Applications To Delaunay's Equations of Lunar Motion


7. Bibliographic Notes on the Problem of the Center


5. Singular Points at Infinity


1. Poincare's Sphere. Singular Points at Infinity


2. Examples


3. Singular Points at Infinity for Homogeneous Systems

Complements

Bibliography

Chapter III. The Singularities of Briot-Bouquet


1. Theorem of Briot-Bouquet for the Analytic Case


1. Introductory Remarks


2. The Equation of Briot—Bouquet in the Case Where P Is Not A Positive Integer. Study of Holomorphic Solutions


3. The Case of A Positive Integer P. Existence of Holomorphic Solutions


4. Solutions of the Equation for the Case p = 0


2. Reduction of Differential Equations with An Isolated Singular Point To A Typical Form in the Analytic Case. The Theorem of I. Bendixson on the Behavior of the Trajectories of the Reduced Equations of the Second Type


1. Reduced Forms of the First and Second Type


2. Results of I. Bendixson on the Behaviour of the Trajectories of the Reduced Equations of the Second Type


3. Equation of Briot-Bouquet in the Nodal Case in the Real Domain. Theorems of A. Wintner


1. Lemma of A. Wintner


2. First Theorem of A. Wintner


3. Second Theorem of A. Wintner

Complements

Bibliogr4phy

Chapter IV. Plane Autonomous Systems


1. Limiting Sets


1. Limiting Sets Α(λ), Ω(λ) of A Trajectory λ. General Properties


2. Classification of Trajectories


3. Regular Points and Trajectories


4. Closed (Plane) Trajectories. Stability Properties of Plane Cycles


5. Regular (Plane) Limiting Trajectories


6. Structure of Bounded Limiting Sets Ω(λ)


7. Limiting Sets Consisting of A Single (Singular) Point


8. Structure of Unbounded Sets Ω(λ)


2. Plane Cycles


1. Limit Cycles


2. Classification of Limit Cycles. Orbital Stability


3. Examples


4. Bendixson's Theorem


5. Systems of Class C1. Characteristic Exponent of A Cycle


6. Cycles of Analytic Systems


7. Limit Cycles of A System with Polynomial Right Sides


8. Regions with No Plane Cycles


9. Periodic Solutions of A Plane Autonomous System. Existence of Limit Cycles


10. Uniqueness of (Limit) Cycles


3. Isolated Singular Points


1. Classification of Isolated Singular Points. Points of the First Kind (Centerfocus) Center


2. The Neighborhood of A Point of the Second Kind


3. The Focus


4. Exceptional Directions


5. Normal Sectors


4. The Index


1. Kronecker's Index


2. Index of A Point


3. Evaluation of the Index for Particular Singular Points


4. Index on the Sphere and on A Surface of Genus p


5. The Cylinder As Phase Space


1. The Cylinder As Phase Space


2. An Example


6. The Torus As Phase Space


1. The Torus As Phase Space


2. Examples


3. Systems with No Singular Points on the Torus


4. Other Results


7. A Short Account on Dynamical Systems

Bibliography

Chapter V. Autonomous Plane Systems with Perturbations


1. Homogeneous Perturbed Systems


1. The General Problem


2. The Case Ν(Θ)≠0


3. Trajectories Tending to 0. Exceptional Directions


4. In Variance of Normal Sectors. Normal Sectors of the First Type


5. Normal Sectors of the Second Type. First Decision Problem


6. Normal Sectors of the Third Type. Second Decision Problem


7. The Case Ν(Θ) Identically Zero


8. Some Remarks


2. Isolated Singular Points of Systems of Class C1. Elementary Points


1. Introductory Remarks


2. Foci and Weak Foci


3. Attractors. Stellar Node


4. Node with One Tangent


5. Node with Two Tangents


6. Saddle Points


7. Remarks


3. An Asymptotic Study of A Node with Two Tangents and A Saddle Point of H. Weyl


1. Statement of the Problem. Notations 235


2. The Nodal Case (0 < I < k). The First Theorem of H. Weyl


3. Bounds for \elty(t) — 6