Non-Linear Differential Equations

ContentsPreface 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 F(t,t0,x°) of the System (1.1.1) 1. The Function ¿(t,t0,x°). Cases of Uniqueness 2. Continuity of (¿(t,t0,x°) 3. Stability 4. The Function F(t, t09 x°) for Linear Systems 5. Differentiability of F (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 BibliographyChapter 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 ¿(T) = 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 ¿(T)¿¿ 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.