Basic theory of ordinary differential equations

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