## 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 technical-administrative 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, 2018 | Jan 31, 2019 |

II semestre | Mar 4, 2019 | Jun 14, 2019 |

Session | From | To |
---|---|---|

Sessione invernale d'esame | Feb 1, 2019 | Feb 28, 2019 |

Sessione estiva d'esame | Jun 17, 2019 | Jul 31, 2019 |

Sessione autunnale d'esame | Sep 2, 2019 | Sep 30, 2019 |

Session | From | To |
---|---|---|

Sessione di laurea estiva | Jul 22, 2019 | Jul 22, 2019 |

Sessione di laurea autunnale | Oct 15, 2019 | Oct 15, 2019 |

Sessione di laurea autunnale straordinaria | Nov 21, 2019 | Nov 21, 2019 |

Sessione di laurea invernale | Mar 19, 2020 | Mar 19, 2020 |

Period | From | To |
---|---|---|

Sospensione attività didattica | Nov 2, 2018 | Nov 3, 2018 |

Vacanze di Natale | Dec 24, 2018 | Jan 6, 2019 |

Vacanze di Pasqua | Apr 19, 2019 | Apr 28, 2019 |

Vacanze estive | Aug 5, 2019 | Aug 18, 2019 |

## 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

## Study 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 |
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Modules | Credits | TAF | SSD |
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Modules | Credits | TAF | SSD |
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1° Year

Modules | Credits | TAF | SSD |
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2° Year

Modules | Credits | TAF | SSD |
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3° Year

Modules | Credits | TAF | SSD |
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Modules | Credits | TAF | SSD |
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#### 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 (2019/2020)

Teaching code

4S00244

Credits

6

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/05 - MATHEMATICAL ANALYSIS

The teaching is organized as follows:

##### Teoria

__Dynamical Systems - Parte I teoria__of the course Bachelor's degree in Applied Mathematics

Credits

5

Period

II semestre

Academic staff

Nicola Sansonetto

##### Esercitazioni

__Dynamical Systems - Parte I esercitazioni__of the course Bachelor's degree in Applied Mathematics

Credits

1

Period

II 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 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

Activity | Author | Title | Publishing house | Year | ISBN | Notes |
---|---|---|---|---|---|---|

Teoria | G. Benettin | Appunti per il corso di Fisica Matematica | 2017 | |||

Teoria | G. Benettin | Appunti per il corso di Meccanica Analitica | 2018 | |||

Teoria | F. Fasso` | Primo sguardo ai sistemi dinamici | CLEUP | 2016 | ||

Teoria | G. Benettin | Una passeggiata tra i Sistemi Dinamici | 2012 |

## 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.

## Type D and Type F activities

**Modules not yet included**

## 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.

## 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 |

## Attendance

As stated in point 25 of the Teaching Regulations for the A.Y. 2021/2022, 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 on-site.Please refer to the Crisis Unit's latest updates for the mode of teaching.