Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
Academic year in which the course will be held: 
Course type: 
Compulsory subjects, characteristic of the class
First Semester
Standard lectures hours: 
Detail of lecture’s hours: 
Lesson (64 hours)

Ordinary differential equations, topology.

Final Examination: 

Take home project on a topic of the class chosen by the instructor. Final oral examination at the blackboard, including a simple exercise. There is only a final mark not parted between project and oral examination.

Voto Finale

Aim of this class is to guide the student into the theory of topological dynamical systems. He/she will be taught to understand the basic tenets of the theory and to apply it to other fields of mathematics and physics.

This course is a primer on dynamical systems, from a topological point of view.

1. Homeomorphisms and diffeomorphisms generated by differential equations. 2. Phase space. 3. Periodic points. 4. Lyapunov stability. Denjoy theorem. 4. Examples: Van der Pol oscillator, Lorenz equazions. 5. Local analysis, the flow tube theorem. 6.Hyperbolic points of non-linear systems. 7. Maps of the circle. Winding number. 8. Asymptotic properties. Limit sets, non-wandering set. 9. Planar flows. Lotka-Volterra models, gradient flows. 10. Index theory and examples. 11. Poincare'-Bendixson theorem. Bendixson criterion. 12. Elliptic points. 13. Local stable and unstable varieties theorem. 14. Hartman Grossman theorem. 15. Computation of stable and unstable varieties. 16. Smale horseshoe. Symbolic dynamics. 17. Smale homoclinic point theorem. 18. Anosov systems. 19. Topological transitivity and minimality. Weyl’s theorem. 20. Birkhoff theorem and applications. 21. Topological mixing. 22. Bowen shadow theorem. 23. Markov partitions. 24.Topological entropy.

Intructor's notes, on-line material. All published in the web page of the course.

Lectures at the blackboard, computer experiments.

Students may contact the instructor by e-mail to set up an appointment.