Studying at the University of Verona

Here you can find information on the organisational aspects of the Programme, lecture timetables, learning activities and useful contact details for your time at the University, from enrolment to graduation.

Study Plan

A.A. 2024/2025 A.A. 2023/2024 A.A. 2022/2023 A.A. 2021/2022 A.A. 2020/2021 A.A. 2019/2020 A.A. 2018/2019 A.A. 2017/2018 A.A. 2016/2017 A.A. 2015/2016 A.A. 2014/2015 A.A. 2013/2014 A.A. 2012/2013 A.A. 2011/2012 A.A. 2010/2011 A.A. 2009/2010

This information is intended exclusively for students already enrolled in this course.
If you are a new student interested in enrolling, you can find information about the course of study on the course page:

Laurea in Matematica applicata - Enrollment from 2025/2026

The Study Plan includes all modules, teaching and learning activities that each student will need to undertake during their time at the University.
Please select your Study Plan based on your enrollment year.

CURRICULUM TIPO:

1° Year 

ModulesCreditsTAFSSD
12
A
MAT/02 ,MAT/03
12
A
MAT/05
6
B
MAT/08
12
A
FIS/01
6
B
MAT/01
12
A
INF/01

2° Year   activated in the A.Y. 2019/2020

ModulesCreditsTAFSSD
6
A
MAT/02
12
B
MAT/05
6
B
MAT/08
6
B
MAT/03
6
C
SECS-P/01
6
C
SECS-P/01
6
B
MAT/06
6
B
MAT/05
English B1
6
E
-

3° Year   activated in the A.Y. 2020/2021

ModulesCreditsTAFSSD
6
C
SECS-P/05
12
C
SECS-S/06
6
B
MAT/08
6
B
MAT/09
6
B
MAT/06
Final exam
6
E
-
activated in the A.Y. 2019/2020
ModulesCreditsTAFSSD
6
A
MAT/02
12
B
MAT/05
6
B
MAT/08
6
B
MAT/03
6
C
SECS-P/01
6
C
SECS-P/01
6
B
MAT/06
6
B
MAT/05
English B1
6
E
-
activated in the A.Y. 2020/2021
ModulesCreditsTAFSSD
6
C
SECS-P/05
12
C
SECS-S/06
6
B
MAT/08
6
B
MAT/09
6
B
MAT/06
Final exam
6
E
-
Modules Credits TAF SSD
Between the years: 1°- 2°- 3°
12
D
-
Between the years: 1°- 2°- 3°
Other activities
6
F
-

Legend | Type of training activity (TTA)

TAF (Type of Educational Activity) All courses and activities are classified into different types of educational activities, indicated by a letter.

A Basic activities
B Characterizing activities
C Related or complementary activities
D Activities to be chosen by the student
E Final examination
F Other training activities
S Placements in companies, public or private institutions and professional associations

Dynamical Systems (2019/2020)

Teaching code

4S00244

Credits

9

Coordinator

Nicola Sansonetto

Language

Italian

Also offered in courses:

Scientific Disciplinary Sector (SSD)

MAT/05 - MATHEMATICAL ANALYSIS

MoodleMoodle Seminars 0

The teaching is organized as follows:

Parte I teoria

Credits

5

Period

II semestre

Academic staff

Nicola Sansonetto

Lessons timetable
Parte II Esercitazioni

Credits

2

Period

II semestre

Academic staff

Alberto Benvegnu'

Lessons timetable
Parte II teoria

Credits

1

Period

II semestre

Academic staff

Nicola Sansonetto

Lessons timetable
Parte I esercitazioni

Credits

1

Period

II semestre

Academic staff

Giacomo Canevari

Lessons timetable

Learning outcomes

The aim of the course is the introduction of the theory and of some applications of continuous and discrete dynamical systems, that describe the time evolution of quantitative variables.
At the end of the course a student will be able to investigate the stability and the character of an equilibrium and to produce and investigate the qualitative analysis of a system of ordinary differential equations and the phase portrait of a dynamical system in dimension 1 and 2.
Moreover a student will be able to study the presence and the nature of limit cycles and to analyse some basic applications of dynamical systems arising from population dynamics, mechanics and traffic flows. Eventually a student will be also able to produce proofs using the typical tools of modern dynamical systems and will be able to read and report specific books and articles on dynamical systems and related applications.

Program

Part 1
Module 1. Complements of ordinary differential equations.
First and second order differential equations. Methods of the variations of the constants. Existence and uniqueness theorem. Qualitative analysis of ODE: maximal solutions, Gronwall’s Lemma. Esplicit solutions of particular equations: separations of variable, Riccati and total equations. Linear systems.

Module 2. Vector fields and ODE.
Orbits and phase space. Equilibria, phase portrait in 1 dimension. ODE of the second order and their equilibria. LInearisation about an equilibrium and periodic solutions of an ODE.

Module 3. Linear systems.
Linear systems in in R2, real and complex eigenvalues. Elements of Jordan theory. Diagram of biforcation in R2.
Linear systems in Rn, stable, unstable and central subspeces. Linearization about an equilibrium.

Module 4. Flows and flows conjugations.
Flow of a vector field. Dependance on the parameters. time dependent vector fields.
Change of coordinates, conjugations of flows, pull-back and push-forward of functions and vector fields. Time dependent change of coordinates.riscalamenti di campi vettoriali e riparametrizzazioni del tempo.
Rectification theorem.

Module 5. First integrals.
Invariant sets, first integrals and Lie derivative. Invariant foliations, reduction of order. First integrals and attractive equilibria.

Module 6. 1-dimensional Newton equation. Phase portrait in the conservative case. Linearisation. Reduction of order. Systems with friction.

Module 7. Stability theory.
Lyapunov Stability, Lyapunov functions and spectral method.

Part 2.

Module 8. Bifurcations and applications.
Definition of bifurcation, bifurcation at equilibria. Applications and numerical simulations.

Module 9. Calculus of variations.

Module 10. Hamiltonian dynamics.
Hamiltonian systems, basic properties, Poisson bracket and canonical transformations. Lie conditions, generating functions, action-angle variables, integrability and Hamilton-Jacobi equation.

Bibliography

Reference texts
Activity Author Title Publishing house Year ISBN Notes
Parte I teoria G. Benettin Appunti per il corso di Fisica Matematica 2017
Parte I teoria G. Benettin Appunti per il corso di Meccanica Analitica 2018
Parte I teoria F. Fasso` Primo sguardo ai sistemi dinamici CLEUP 2016
Parte I teoria G. Benettin Una passeggiata tra i Sistemi Dinamici 2012
Parte II Esercitazioni M.W. Hirsch e S. Smale Differential equations, dynamical systems, and linear algebra Academic Press 1974
Parte II Esercitazioni S. Strogatz Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering Westview Press 2010
Parte II Esercitazioni F. Fasso` Primo sguardo ai sistemi dinamici CLEUP 2016
Parte II teoria M.W. Hirsch e S. Smale Differential equations, dynamical systems, and linear algebra Academic Press 1974
Parte II teoria S. Strogatz Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering Westview Press 2010
Parte II teoria F. Fasso` Primo sguardo ai sistemi dinamici CLEUP 2016

Examination Methods

A written exam with exercises: phase portrait in 2D for a non-linear dynamical system; computation of trajectories and stability for a discrete-time system, phase portrait in 2D for a non-linear dynamical system; computation of trajectories and stability for a discrete-time system; stability analysis for a system.
The written exam tests the following learning outcomes:
- To have adequate analytical skills;
- To have adequate computational skills;
- To be able to translate problems from natural language to mathematical formulations;
- To be able to define and develop mathematical models for physics and natural sciences.

An oral exam with 2-3 theoretical questions. The oral exam is compulsory and must be completed within the session
in which the written part has been done.
The oral exam tests the following learning outcomes:
- To be able to present precise proofs and recognise them.

Students with disabilities or specific learning disorders (SLD), who intend to request the adaptation of the exam, must follow the instructions given HERE