Basic Theory of Ordinary Differential Equations

I. Fundamental Theorems of Ordinary Differential Equations.- I-1. Existence and uniqueness with the Lipschitz condition.- I-2. Existence without the Lipschitz condition.- I-3. Some global properties of solutions.- I-4. Analytic differential equations.- Exercises I.- II. Dependence on Data.- II-1. Continuity with respect to initial data and parameters.- II-2. Differentiability.- Exercises II.- III. Nonuniqueness.- III-l. Examples.- III-2. The Kneser theorem.- III-3. Solution curves on the boundary of R(A).- III-4. Maximal and minimal solutions.- III-5. A comparison theorem.- III-6. Sufficient conditions for uniqueness.- Exercises III.- IV. General Theory of Linear Systems.- IV-1. Some basic results concerning matrices.- IV-2. Homogeneous systems of linear differential equations.- IV-3. Homogeneous systems with constant coefficients.- IV-4. Systems with periodic coefficients.- IV-5. Linear Hamiltonian systems with periodic coefficients.- IV-6. Nonhomogeneous equations.- IV-7. Higher-order scalar equations.- Exercises IV.- V. Singularities of the First Kind.- V-1. Formal solutions of an algebraic differential equation.- V-2. Convergence of formal solutions of a system of the first kind.- V-3. TheS-Ndecomposition of a matrix of infinite order.- V-4. TheS-Ndecomposition of a differential operator.- V-5. A normal form of a differential operator.- V-6. Calculation of the normal form of a differential operator.- V-7. Classification of singularities of homogeneous linear systems.- Exercises V.- VI. Boundary-Value Problems of Linear Differential Equations of the Second-Order.- VI- 1. Zeros of solutions.- VI- 2. Sturm-Liouville problems.- VI- 3. Eigenvalue problems.- VI- 4. Eigenfunction expansions.- VI- 5. Jost solutions.- VI- 6. Scattering data.- VI- 7. Reflectionless potentials.- VI- 8. Construction of a potential for given data.- VI- 9. Differential equations satisfied by reflectionless potentials.- VI-10. Periodic potentials.- Exercises VI.- VII. Asymptotic Behavior of Solutions of Linear Systems.- VII-1. Liapounoff’s type numbers.- VII-2. Liapounoff’s type numbers of a homogeneous linear system.- VII-3. Calculation of Liapounoff’s type numbers of solutions.- VII-4. A diagonalization theorem.- VII-5. Systems with asymptotically constant coefficients.- VII-6. An application of the Floquet theorem.- Exercises VII.- VIII. Stability.- VIII- 1. Basic definitions.- VIII- 2. A sufficient condition for asymptotic stability.- VIII- 3. Stable manifolds.- VIII- 4. Analytic structure of stable manifolds.- VIII- 5. Two-dimensional linear systems with constant coefficients.- VIII- 6. Analytic systems in ?n.- VIII- 7. Perturbations of an improper node and a saddle point.- VIII- 8. Perturbations of a proper node.- VIII- 9. Perturbation of a spiral point.- VIII-10. Perturbation of a center.- Exercises VIII.- IX. Autonomous Systems.- IX-1. Limit-invariant sets.- IX-2. Liapounoff’s direct method.- IX-3. Orbital stability.- IX-4. The Poincaré-Bendixson theorem.- IX-5. Indices of Jordan curves.- Exercises IX.- X. The Second-Order Differential Equation
$$\frac{{{d^2}x}}{{d{t^2}}} + h(x)\frac{{dx}}{{dt}} + g(x) = 0
$$.- X-1. Two-point boundary-value problems.- X-2. Applications of the Liapounoff functions.- X-3. Existence and uniqueness of periodic orbits.- X-4. Multipliers of the periodic orbit of the van der Pol equation.- X-5. The van der Pol equation for a small ?> 0.- X-6. The van der Pol equation for a large parameter.- X-7. A theorem due to M. Nagumo.- X-8. A singular perturbation problem.- Exercises X.- XI. Asymptotic Expansions.- XI-1. Asymptotic expansions in the sense of Poincaré.- XI-2. Gevrey asymptotics.- XI-3. Flat functions in the Gevrey asymptotics.- XI-4. Basic properties of Gevrey asymptotic expansions.- XI-5. Proof of Lemma XI-2-6.- Exercises XI.- XII. Asymptotic Expansions in a Parameter.- XII-1. An existence theorem.- XII-2. Basic estimates.- XII-3. Proof of Theorem XII-1-2.- XII-4. A block-diagonalization theorem.- XII-5. Gevrey asymptotic solutions in a parameter.- XII-6. Analytic simplification in a parameter.- Exercises XII.- XIII. Singularities of the Second Kind.- XIII-1. An existence theorem.- XIII-2. Basic estimates.- XIII-3. Proof of Theorem XIII-1-2.- XIII-4. A block-diagonalization theorem.- XIII-5. Cyclic vectors (A lemma of P. Deligne).- XIII-6. The Hukuhara-Turrittin theorem.- XIII-7. An n-th-order linear differential equation at a singular point of the second kind.- XIII-8. Gevrey property of asymptotic solutions at an irregular singular point.- Exercises XIII.- References.