# DYNAMICAL SYSTEMS B

Degree course:
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree:
2017/2018
Year:
2
Academic year in which the course will be held:
2018/2019
Course type:
Compulsory subjects, characteristic of the class
Credits:
8
Period:
First Semester
Standard lectures hours:
64
Detail of lecture’s hours:
Lesson (64 hours)
Requirements:

Basic calculus, including ordinary differential equations. Basic notions in tolopogy.

Take home exam, and oral discussion. The discussion is intended to verify the acquired knowledge of the basis tenets of the theory, while the take-home exam requires to show the capability of applying them to a problem suggested by the instructor. Both parts contribute equally to the final evaluation of the exam.

Assessment:
Voto Finale

Scope of the course is to introduce the basic concepts of the theory of dynamical systems. Students should learn the foundations of theory, and develop the capability to apply them to abstract situations, as well as to problems arising in other fields of mathematics and physics.

Homeomorphisms, diffeomorphisms generated by differential equations. Phase space. Periodic points, Lyapunov stability. Lyapunov function Method. Van der Pol oscillator, Lorenz equations. Conjugation and equivalence of dynamical systems, flow tube theorem. Hyperbolic points of nonlinear systems. Circle maps. Rotation number. Denjoy theorem. Asymptotic behavior. Limit sets, non-wandering set. Planar flows. Lotka–Volterra Model, gradient flow. Index theory and examples. Poincaré–Bendixson theorem. Bendixson criterion. Elliptic points. Local analysis: local stable and unstable manifolds. Grossman Hartman's theorem. Calculation of stable and unstable manifolds. Examples, dynamical systems of biological interest. Smale's horseshoe. Symbolic dynamics. Shift and subshifts of finite type. Homoclinic intersections. Smale theorem on homoclinic points and examples. The notion of hyperbolic set. Anosov systems. Topological transitivity and minimality. Characterization theorems. Examples of topologically transitive systems. Weyl's theorem. Birkhoff's theorem and applications. Topological mixing. Bowen's shadow theorem. Examples. Cardinality of the set of periodic trajectories. Markov partitions. Markov partitions for the Arnol'd cat, transfer matrix, trace and powers. Topological entropy. Definition and properties.

Online notes by the teacher and by professor Benettin of the University of Padova.