Dynamical Systems

Credits

Prerequisites

Mathematical Analysis 2, Mathematical Physics.

Scientific-disciplinary sector (SSD)

MAT/07 Mathematical Physics.

Examination method

Written exam (exercises and numerical problems, possibly multiple-choice) and oral exam.

Learning
objectives

To gain mastery of the main analytical techniques for the study of ordinary differential equations. Diffeomorphisms between Euclidean spaces. Study of evolution phenomena in the Applied Sciences through the use of dynamical systems.

Syllabus

Review of linear algebra. Equilibria of vector fields and classification of equilibria. Lyapunov stability (Lyapunov functions and stability theorems). Invariant manifolds, centre manifold theory and normal forms. Notable criteria for autonomous vector fields (La Salle’s theorem, Poincaré-Bendixson theorem, Hartman-Grobman theorem). Elements of bifurcation theory.

Expected learning
outcomes

At the end of the course, students must demonstrate that they

know and understand the general issues related to “dynamical systems”;
are able to apply the knowledge acquired during the course to analyse the long-time asymptotic behaviour of dynamical systems representing evolution problems in the Applied Sciences;
are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
are able to identify the most appropriate methods to analyse and solve a problem related to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Ability to place the question within the topics of the syllabus, to choose the appropriate solution techniques and to interpret the results obtained correctly. In addition to testing the contents, the exam also aims to verify that a sufficient command of the specific language has been achieved.

Dynamical Systems