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.
Academic calendar
The academic calendar shows the deadlines and scheduled events that are relevant to students, teaching and technicaladministrative staff of the University. Public holidays and University closures are also indicated. The academic year normally begins on 1 October each year and ends on 30 September of the following year.
Course calendar
The Academic Calendar sets out the degree programme lecture and exam timetables, as well as the relevant university closure dates..
Period  From  To 

I semestre  Oct 1, 2020  Jan 29, 2021 
II semestre  Mar 1, 2021  Jun 11, 2021 
Session  From  To 

Sessione invernale d'esame  Feb 1, 2021  Feb 26, 2021 
Sessione estiva d'esame  Jun 14, 2021  Jul 30, 2021 
Sessione autunnale d'esame  Sep 1, 2021  Sep 30, 2021 
Session  From  To 

Sessione di laurea estiva  Jul 22, 2021  Jul 22, 2021 
Sessione di laurea autunnale  Oct 14, 2021  Oct 14, 2021 
Sessione di laurea autunnale  Dicembre  Dec 9, 2021  Dec 9, 2021 
Sessione invernale di laurea  Mar 16, 2022  Mar 16, 2022 
Period  From  To 

Festa dell'Immacolata  Dec 8, 2020  Dec 8, 2020 
Vacanze Natalizie  Dec 24, 2020  Jan 3, 2021 
Vacanze di Pasqua  Apr 2, 2021  Apr 6, 2021 
Festa del Santo Patrono  May 21, 2021  May 21, 2021 
Festa della Repubblica  Jun 2, 2021  Jun 2, 2021 
Vacanze Estive  Aug 9, 2021  Aug 15, 2021 
Exam calendar
Exam dates and rounds are managed by the relevant Science and Engineering Teaching and Student Services Unit.
To view all the exam sessions available, please use the Exam dashboard on ESSE3.
If you forgot your login details or have problems logging in, please contact the relevant IT HelpDesk, or check the login details recovery web page.
Academic staff
Gonzato Guido
guido.gonzato@univr.it 045 802 8303Study Plan
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 enrolment year.
Modules  Credits  TAF  SSD 

Modules  Credits  TAF  SSD 

Modules  Credits  TAF  SSD 

1° Year
Modules  Credits  TAF  SSD 

2° Year activated in the A.Y. 2021/2022
Modules  Credits  TAF  SSD 

3° Year activated in the A.Y. 2022/2023
Modules  Credits  TAF  SSD 

Modules  Credits  TAF  SSD 

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.
Dynamical Systems (2021/2022)
Teaching code
4S00244
Credits
6
Coordinatore
Not yet assigned
Language
Italian
Scientific Disciplinary Sector (SSD)
MAT/05  MATHEMATICAL ANALYSIS
The teaching is organized as follows:
Parte I teoria
Credits
5
Period
Secondo semestre
Academic staff
Giacomo Canevari
Parte I esercitazioni
Credits
1
Period
Secondo semestre
Academic staff
Giacomo Canevari
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 study and investigate the stability and the character of an equilibrium and 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 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 I
1. Topics in the theory of ordinary differential equations
Qualitative analysis of ODE: existence and uniqueness of solutions; maximal and global solutions; Gronwall’s Lemma; continuous dependence on the initial data.
2. Vector fields and ordinary differential equations
Vector fields: phase space, integral curves, orbits, equilibria, phase portrait. 1dimensional examples of phase portraits. Secondorder systems of differential equations; phasespace analysis and equilibria.
3. Linear systems
Linearisation of a vector field about an equilibrium. Classification of twodimensional linear systems (over the real numbers) that are diagonalisable over the complex numbers. (If time permits, we will briefly discuss the nilpotent case as well.) ndimensional linear systems: invariant subspace decomposition; the stable, unstable and central subspaces. Comparing a vector field with its linearisation about a hyperbolic equilibrium.
4. Flow of a vector field
Flow of a vector field. Change of coordinates: conjugate vector fields; pullback and pushforward of a vector field by a diffeomorphism. Nonautonomous differential equations: timedependent change of coordinates; scaling of vector fields and time reparametrisations. The local rectification theorem.
5. First integrals
Invariant sets; first integrals; Lie derivative. Invariant foliations; reduction of the order. First integrals and attractive equilibria.
6. Stability theory
Stability 'à la Lyapunov' of an equilibrium; the method of Lyapunov functions; the spectral method. Applications and examples.
7. 1dimensional Newton equation.
Phase portraits of the 1dimensional Newton equation, in the conservative case. Linearisation. Reduction of the order. Systems with friction.
Part II
8. Bifurcations
Bifurcatios from equilibria, with 1dimensional examples; applications.
9. Introduction to the 1dimensional Calculus of Variations
The indirect method for onedimensional integral functionals. Necessary conditions for the existence of minimisers: the EulerLagrange equations. Jacobi integral; conservation laws. Geodesics on a surface.
10. Hamiltonian systems
Hamiltonian vector fields. Legendre transform. Poisson brackets. Canonical transformations. Lie conditions, generating functions. The HamiltonJacobi equations. Integrability. Geometry of the phase space: Liouville's theorem and Poincaré's recurrence theorem.
Examination Methods
The exam consists of two parts: a written part, and an oral one.
The written part consists of exercises  for instance, qualitative analysis of an ordinary differential equation; explicit solution of an ordinary differential equation; phase portrait of a twodimensional, nonlinear system; stability of equilibria; change of coordinates; first integrals; bifurcations; Hamiltonian systems and canonical transformations...
The written part 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.
The oral part consists of 3 questions. The oral part is compulsory and must be completed within the same session as written part of the exam.
The oral exam tests the following learning outcomes:
 To be able to present precise proofs and recognise them.
According to the pandemic situation the structure of the exam may vary.
Type D and Type F activities
Le attività formative in ambito D o F comprendono gli insegnamenti impartiti presso l'Università di Verona o periodi di stage/tirocinio professionale.
Nella scelta delle attività di tipo D, gli studenti dovranno tener presente che in sede di approvazione si terrà conto della coerenza delle loro scelte con il progetto formativo del loro piano di studio e dell'adeguatezza delle motivazioni eventualmente fornite.
years  Modules  TAF  Teacher  

1° 2°  History and Didactics of Geology  D 
Guido Gonzato
(Coordinatore)


1° 2° 3°  Algorithms  D 
Roberto Segala
(Coordinatore)


1° 2° 3°  Scientific knowledge and active learning strategies  F 
Francesca Monti
(Coordinatore)


1° 2° 3°  Genetics  D 
Massimo Delledonne
(Coordinatore)

years  Modules  TAF  Teacher 

1° 2° 3°  Algorithms  D 
Roberto Segala
(Coordinatore)

1° 2° 3°  Python programming language  D 
Vittoria Cozza
(Coordinatore)

1° 2° 3°  Organization Studies  D 
Giuseppe Favretto
(Coordinatore)

years  Modules  TAF  Teacher  

1°  Subject requirements: mathematics  D 
Rossana Capuani


1° 2° 3°  ECMI modelling week  F  Not yet assigned  
1° 2° 3°  ESA Summer of code in space (SOCIS)  F  Not yet assigned  
1° 2° 3°  Google summer of code (GSOC)  F  Not yet assigned  
1° 2° 3°  Introduzione all'analisi non standard  F 
Sisto Baldo


1° 2° 3°  C Programming Language  D 
Pietro Sala
(Coordinatore)


1° 2° 3°  LaTeX Language  D 
Enrico Gregorio
(Coordinatore)

Career prospects
Module/Programme news
News for students
There you will find information, resources and services useful during your time at the University (Student’s exam record, your study plan on ESSE3, Distance Learning courses, university email account, office forms, administrative procedures, etc.). You can log into MyUnivr with your GIA login details: only in this way will you be able to receive notification of all the notices from your teachers and your secretariat via email and soon also via the Univr app.
Further services
I servizi e le attività di orientamento sono pensati per fornire alle future matricole gli strumenti e le informazioni che consentano loro di compiere una scelta consapevole del corso di studi universitario.
Graduation
Attachments
Title  Info File 

1. Come scrivere una tesi  31 KB, 29/07/21 
2. How to write a thesis  31 KB, 29/07/21 
5. Regolamento tesi (valido da luglio 2022)  171 KB, 17/02/22 
List of theses and work experience proposals
theses proposals  Research area 

Formule di rappresentazione per gradienti generalizzati  Mathematics  Analysis 
Formule di rappresentazione per gradienti generalizzati  Mathematics  Mathematics 
Proposte Tesi A. Gnoatto  Various topics 
Mathematics Bachelor and Master thesis titles  Various topics 
Stage  Research area 

Internship proposals for students in mathematics  Various topics 
Erasmus+ and other experiences abroad
Attendance
As stated in the Teaching Regulations for the A.Y. 2022/2023, except for specific practical or lab activities, attendance is not mandatory. Regarding these activities, please see the web page of each module for information on the number of hours that must be attended onsite.
Please refer to the Crisis Unit's latest updates for the mode of teaching.