Academic Year 2021-2022

QUALITATIVE THEORY OF DYNAMICAL SYSTEMS

Teachers

Fabio Zanolin
Unit Credits
6
Teaching Period
Second Period
Course Type
Supplementary
Prerequisites. Courses of “Mathematical Analysis I-II” and “Differential Equations”. A knowledge of some basic concepts of Topology and Linear Algebra is also required.
Teaching Methods. In addition to the activities related to the theoretical lessons, there are also activities of exercises / applications in the classroom, with an illustration on how to deal with various problems and exercises related to the theory presented and also presentation of some numerical simulations related to the non-linear systems studied.

Depending on the conditions connected to possible health emergencies, the lessons could be held online.

Verification of Learning. The exam is oral and can be done both in a traditional form and in the form of a seminar on a topic related to the course and provided by the teacher (for example a research article to study and comment) that the student will have to develop and present after having autonomously reworked on it.
More Information. Additional notes (lecture notes, articles, research problems, in-depth material) available on the “materiale didattico” website.
Objectives
To acquire some fundamental concepts in the area of the qualitative theory of continuous and discrete dynamical systems, as well as in the theory of fixed points and also to acquire mastery of some mathematical methods for the study of systems of nonlinear differential equations.
Contents
The course of Qualitative Theory of Dynamical Systems aims to present several fundamental aspects related to the theory of continuous and discrete dynamical systems and their applications to the qualitative theory of differential equations, in particular, to the use of topological methods for the study of nonlinear systems.

The first part of the course concerns autonomous and non-autonomous ordinary differential equations, especially those with periodic coefficients, and the associated dynamic systems. Furthermore, abstract dynamic systems (in metric spaces) and their topological properties (limit sets, orbits, recurrence properties) will be introduced. Some fundamental arguments will also be presented from the application point of view, such as the theory of stability and invariant sets. The second part of the course concerns the introduction of some topological methods in nonlinear analysis, such as the theory of fixed points in finite dimension and infinite dimension and on manifolds, with different applications to non-linear problems. Finally, in the final part of the course some applications of the developed methods will be proposed and some current research fields will be indicated where these methods are used, including the issues related to “deterministic chaos” and complex systems.

At the request of the students, the course can be held in English.

Texts
Besides the classical texts:

N.P. Bhatia, G.P. Szegö: Stability Theory of Dynamical Systems, Springer, 1970

K. Deimling: Nonlinear Functional Analysis, Springer, 1985

J.K. Hale, Ordinary Differential Equations, Krieger, 1980

students will be provided with more recent articles as well as classroom notes by the teacher.