Existence and uniqueness theorem. Differentiable dependence on initial conditions. Linear equations. Exponential matrices. Classification of linear fields. Jordan canonical form. Non-autonomous linear equations: fundamental solution and Liouville’s theorem. Non-homogeneous linear equations. Equations with periodic coefficients, Floquet’s theorem. Stability and asymptotic instability of a singular point of an autonomous equation. Lyapounov functions. Hyperbolic fixed points. Statement of the Grobman-Hartman linearization theorem. Flow associated with an autonomous equation. Limit sets. Gradient fields. Hamiltonian fields. Fields in the plane: periodic orbits and Poincaré-Bendixon theorem. Hyperbolic periodic orbits. Van der Pol equation.

References:
ARNOLD, V. – Ordinary Differential Equations. Moscow, Ed. Mir, 1974.
HIRSCH, M. and SMALE, S. – Differential Equations, Dynamical Systems and Linear Algebra. New York, Academic Press, 1974.
PONTRYAGIN, L. S. – Ordinary Differential Equations. Reading, Mass., Addison-Wesley, 1969.
SOTOMAYOR, J. – Lessons in Ordinary Differential Equations. Rio de Janeiro, IMPA, Projeto Euclides, 1979.

* Basic syllabus. The teacher has the autonomy to make any changes.