Introdução às Equações Diferenciais Ordinárias – Mestrado

Part 1. Classical theory

• Newton’s equations; Classification of differential equations; First order autonomous equations; Finding explicit solutions; Qualitative analysis of first-order equations; Qualitative analysis of first-order periodic equations

• Initial value problems; Fixed point theorems; The basic existence and uniqueness result; Some extensions; Dependence on the initial condition; Regular perturbation theory; Extensibility of solutions; Euler’s method and the Peano theorem

• Linear equations; The matrix exponential; Linear autonomous first-order systems; Linear autonomous equations of order n; General linear first-order systems; Linear equations of order n; Periodic linear systems; Perturbed linear first order systems;

• Differential equations in the complex domain; The basic existence and uniqueness result; The Frobenius method for second-order equations; Linear systems with singularities; The Frobenius method;

• Boundary value problems; Compact symmetric operators; Sturm–Liouville equations; Regular Sturm–Liouville problems; Oscillation theory; Periodic Sturm–Liouville equations.

Part 2. Dynamical systems

• Dynamical systems; The flow of an autonomous equation; Orbits and invariant sets; The Poincaré map; Stability of fixed points; Stability via Liapunov’s method; Newton’s equation in one dimension

• Planar dynamical systems; Examples from ecology; Examples from electrical engineering; The Poincaré–Bendixson theorem; Higher dimensional dynamical systems; Attracting sets; The Lorenz equation.