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.
Problem sheet 1
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
Exam 2011 , Exam 2013 , Exam 2014