DYNAMICAL SYSTEMS B
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Ordinary differential equations, topology.
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.
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.
Lectures at the blackboard, computer experiments.
Students may contact the instructor by e-mail to set up an appointment.
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Degree course in: MATHEMATICS