Short syllabus: Existence, uniqueness, smooth dependence on parameters; global existence, Lyapunov functions, Hamiltonian systems, gradient systems; Topological equivalence; Limit sets, attractors; Systems on a plane; Stable, unstable and saddle equilibrium states and periodic orbits; Smale horseshoe, zero-dimensional hyperbolic sets; Transverse homoclinic, Melnikov function.

Dynamical systems. Lectures 1-3

Lectures 4 and 5 (local behaviour near hyperbolic equilibria and periodic orbits) see in My book (part 1) Chapters 2 and 3.

Lecture 6 (systems on a plane) see here and (with a general overview) there .

Exercises with phase portraits on a plain see here and here

See also My book (part 2) and Lecture notes on hyperbolic sets (Lecture 7) with some problems and solutions