Mathematical methods and models for applications

A.Y. 2020/2021
Overall hours
Learning objectives
The course aims at providing the fundamentals of models, methods and mathematical tools used to study low dimensional dynamical systems, even those presenting a chaotic behaviour, also in connection with applicative problems. This will be pursued with the aid of laboratory sessions, where suitable numerical schemes will be developed.
Expected learning outcomes
At the end of the course, the students should be able to study simple dynamical systems representing mathematical models also arising from applicative problems, and should be able to develop suitable numerical tools supprting the aforementioned study.
Course syllabus and organization

Single session

Lesson period
First semester
The subsequent informations concern the lectures; presently it is not possible to publish the details of the lab part. We suggest to check the web page hosted on Ariel to look for updates.

Video lectures will be made available, hosted on Ariel.
If technically feasible, live lectured will be attempted, probably using Zoom.

Changes in the program and in the references are not expected.

Oral examinations, using Zoom, will be essentially identical to the traditional ones.
Course syllabus
1. One dimensional discrete systems: fixed points and periodic orbits; attractors and repulsors; stability; bifurcations; logistic model of population dynamics; chaotic and symbolic dynamics.

2. Two dimensional systems: stationary orbits; linear systems; stability of equilibria; asymptotic orbits; strange attractors; relations with differential equations; Poincare' index and section; bifurcations.

3. Oscillations: forced and dumped linear systems; nonlinear oscillations; averaging method; limit cycles and Van der Pol oscillator; synchronization; resonances; Henon-Heiles model.

4. Chaotic behaviour: hyperbolic maps and shadowing lemma; homoclinic points; symbolic dynamics.
Prerequisites for admission
Basic knowledge of analysis, linear algebra and geometry. Basic knowledge of dynamical systems as introduced in Fisica Matematica 1.
Teaching methods
Lectures and lab classes with the computer.
Teaching Resources
Lecture notes available on the web page:
additionally: Introduction to Dynamical Systems, Brin&Stuck, Cambridge
Assessment methods and Criteria
The final examination consists of an oral exam and of the evaluation of all the activities performed during the lab sessions.

- In the oral exam, the student will be required to illustrate results presented during the course, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
-The lab exam is based on the activities done in each lab session.

The examination is passed if the oral part is successfully passed and if the lab activities are positively evaluated. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.

Further options for the examination could be discussed at the beginning of the course.
MAT/07 - MATHEMATICAL PHYSICS - University credits: 6
Laboratories: 24 hours
Lessons: 36 hours
Professor: Paleari Simone