Geometrical Methods in the Theory of Ordinary Differential Equations

1 Special Equations.-
1. Differential Equations Invariant under Groups of Symmetries.-
2. Resolution of Singularities of Differential Equations.-
3. Implicit Equations.-
4. Normal Form of an Implicit Differential Equation in the Neighborhood of a Regular Singular Point.-
5. The Stationary Schrödinger Equation.-
6. Geometry of a Second-Order Differential Equation and Geometry of a Pair of Direction Fields in Three-Dimensional Space.- 2 First-Order Partial Differential Equations.-
7. Linear and Quasilinear First-Order Partial Differential Equations.-
8. The Nonlinear First-Order Partial Differential Equation.-
9. A Theorem of Frobenius.- 3 Structural Stability.-
10. The Notion of Structural Stability.-
11. Differential Equations on the Torus.-
12. Analytic Reduction of Analytic Circle Diffeomorphisms to a Rotation.-
13. Introduction to the Hyperbolic Theory.-
14. Anosov Systems.-
15. Structurally Stable Systems Are Not Everywhere Dense.- 4 Perturbation Theory.- 16. The Averaging Method.-
17. Averaging in Single-Frequency Systems.-
18. Averaging in Systems with Several Frequencies.-
19. Averaging in Hamiltonian Systems.-
20. Adiabatic Invariants.-
21. Averaging in Seifert's Foliation.- 5 Normal Forms.-
22. Formal Reduction to Linear Normal Forms.-
23. The Case of Resonance.-
24. Poincaré and Siegel Domains.-
25. Normal Form of a Mapping in the Neighborhood of a Fixed Point.-
26. Normal Form of an Equation with Periodic Coefficients.-
27. Normal Form of the Neighborhood of an Elliptic Curve.-
28. Proof of Siegel's Theorem.- 6 Local Bifurcation Theory.-
29. Families and Deformations.-
30. Matrices Depending on Parameters and Singularities of the Decrement Diagram.-
31. Bifurcations of Singular Points of a Vector Field.-
32. Versal Deformations of Phase Portraits.-
33. Loss of Stability of an Equilibrium Position.-
34. Loss of Stability of Self-Sustained Oscillations.-
35. Versal Deformations of Equivariant Vector Fields on the Plane.-
36. Metamorphoses of the Topology at Resonances.-
37. Classification of Singular Points.- Samples of Examination Problems.