Academic Year 2021-2022

APPLIED DYNAMICAL SYSTEMS

Teachers

Dimitri Breda
Unit Credits
6
Teaching Period
First Period
Course Type
Supplementary
Prerequisites. – arguments of the course Approximation Theory and Practice

– arguments of functional and complex analysis will be recalled and used, this not impeding attendance and learning to those without such requisites.

Teaching Methods. – front theoretical lectures

– laboratory activities on implementation and use of methods for dynamical systems

– possible seminars on specific arguments.

Verification of Learning. Oral examination with discussion on the program and on the laboratory activities of the course, with questions on either theoretical and application aspects, proofs of theorems and possible exercises concerning the study of the dynamics of simple systems.
More Information. – the credits are valid for Percorso Formativo 24 CFU – DM 616/2017 (A-28 “Matematica e Scienze”)

– teaching language: Italian (the course can be taught in English on proposal of the competent didactic structure)

– the course notes, written by the lecturer in English, are complete and self-contained relatively to the program, and include laboratory activities and relevant codes

– several arguments are available for thesis projects, especially in the field of numerical and qualitative analysis of population dynamics.

Objectives
It is an advanced course devoted to the study of dynamical systems, mainly in their numerical and applicative aspects.The student will have to:

Knowledge and comprehension:

– know the basic aspects of the analysis of the dynamics of a system

– have clear the recipe for the local stability analysis based on the principle of linearization

– learn the guidelines to study the changes in the dynamical behaviors due to varying parameters

– understand the fundamentals of the methods of numerical continuation and spectral approximation, also in infinite-dimensional contexts

– become familiar with some essential differences between spaces with finite and infinite dimension

Capacity of applying knowledge and comprehension:

– be able of setting the qualitative and numerical analysis of certain solutions and of their stability

– know how to perform a basic analysis of the dynamics under parameter variation

– know how to apply numerical methods to the study of the dynamics of mathematical models, also realistic ones

Autonomy of judgement:

– know how to individuate the main steps and the suitable methods for the analysis of the dynamics

Communication skills:

– know how to illustrate the analysis of the dynamical behaviors also to a non-specialized audience

– know how to discuss the principal features of certain mathematical models

Learning skills:

– deepen the study autonomously starting from the suggested bibliography

– extend results and methods also to other models.

Contents
The course is focused on the analysis of stability and bifurcation of equilibria and periodic orbits of dynamical systems based on the principle of linearized stability. The systems under consideration are continuous in time and refer either to ordinary differential equations (first part) and to problems on infinite-dimensional Banach spaces (second part), through the study of functional differential and integral equations of retarded type. The program include both basic and advanced arguments of mathematical and functional analysis (e.g., fundamental systems of solutions, matrix exponential, Floquet theory, Lyapunov exponents, semigroup theory and generators and relevant spectral theory) and, in parallel, of numerical analysis, necessary to tackle the problems also from the application and computational standpoint (e.g., bifurcation and continuation methods, methods for boundary value problems, approximation of spectra of operators). The program is integrated with laboratory activities concerning the dynamical analysis of basic models with particular interest in applications. Concerning the latter, the context of reference is that of population dynamics, with emphasis on both classic and advanced models in epidemics and ecology. Due to its advanced character, both contents and exam can be adapted, also in relation to the interest of the students.
Texts
Course notes written by the lecturer in English, including laboratory activities, relevant codes and specific references with respect to the treated arguments, made available on the e-learning webpage of the course subject to registration (https://elearning.uniud.it/moodle/).