First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SCP3051008, A.A. 2019/20

Information concerning the students who enrolled in A.Y. 2019/20

Information on the course unit
Degree course Second cycle degree in
SC1172, Degree course structure A.Y. 2011/12, A.Y. 2019/20
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Degree course track GENERALE [010PD]
Number of ECTS credits allocated 7.0
Type of assessment Mark
Course unit English denomination DYNAMIC SISTEMS
Website of the academic structure
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction Italian
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Teacher in charge FRANCESCO FASSO' MAT/07
Other lecturers OLGA BERNARDI MAT/07

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Core courses MAT/07 Mathematical Physics 7.0

Course unit organization
Period Second semester
Year 1st Year
Teaching method frontal

Type of hours Credits Teaching
Hours of
Individual study
Practice 3.0 24 51.0 No turn
Lecture 4.0 32 68.0 No turn

Start of activities 02/03/2020
End of activities 12/06/2020
Show course schedule 2019/20 Reg.2011 course timetable

Examination board
Board From To Members of the board
8 Sistemi Dinamici - a.a. 2019/2020 01/10/2019 30/09/2020 FASSO' FRANCESCO (Presidente)
BERNARDI OLGA (Membro Effettivo)
7 Sistemi Dinamici - a.a. 2018/2019 01/10/2018 25/11/2019 FASSO' FRANCESCO (Presidente)
CARDIN FRANCO (Membro Effettivo)

Prerequisites: 1. Basic knowledge of the theory of ordinary differential equations (ODEs) and of the qualitative theory of ODEs, at the level of, e.g., the course "Fisica Matematica" which is offered as a a mandatory course at the second year of the Corso di Laurea in Matematica in this University.
2. A basic knowledge of the programming language "Mathematica" (at the level of the tutorials periodically offered by the CCS and available on the YouTube channel of the Department of Mathematics) is useful, as it will be used in the mumerical part of the course.
Target skills and knowledge: This course provides an introduction to the theory of Differentiable Dynamical Systems---particularly, continuous Dynamical systems (namely ODEs), but also discrete Dynamical Systems (iterations of maps). The first part of the course provides a panoramic of classical results on ODEs, including periodic orbits, Poincare' maps, local classifications, stable and center manifolds, etc. Subsequently,the course will focus on the difference between integrability and chaoticity (in the hyperbolic context). The course is completed by a numerical laboratory part, which is devoted to the numerical investigation of ODEs and to the numerical analysis of dynamical systems.
The student will reach an advanced knowledge of the above topics in the theory of differentiable dynamical systems and basic competences and skills on the numerical investigations of dynamical systems.
Examination methods: Oral examination on the topics studied in the course, and with an evaluation and a discussion of the numerical assignments (which will be assigned during the course). Students will prepare the numerical assignments by working either alone or (recommended) in pairs, at their choice. During the examination, students may also be asked to solve some simple exercies.
This examination format allows to evaluate: 1) The level of the theoretical knowledge of the matter reached by the student. 2) The level of the mathematical comprehension of the matter reached by the student. 3) The abilities reached by the student in the numerical investigation of dynamical systems, and in particular in the analysis and comprehension of the numerical results.
Assessment criteria: Knowledge of the subject, level of the mathematical comprehension, quality of the numerical work, and the abaility to analyze and interpret the numerical results within the theoretical framework developed in the course.
Course unit contents: 1. Continuous (ODEs, flows) and discrete (iteration of maps) Dynamical Systems. Linearization, variational equation. Linear dynamical systems; stabel, unstable and center subspace.
2. Periodic orbits: Poincare' map; stability; monodromy matrix. Applications.
3. Hyperbolic fixed points: Grobman-Hartman theorem, stable manifold theorem.
4. Integrability. Invariance of an ODE under a group action, reduction. Dynamical symmetries. Bogoyavlenskij's integrability theorem. Application to Hamiltonian systems.
5. Hyperbolic systems and homoclinic phenomena; Smale horseshoe; symbolic dynamics; Melnikov integral; shadowing.
6. Lyapunov exponents.
7. Numerical esperiments on ODEs.
Planned learning activities and teaching methods: Frontal lectures. Lectures in numerical Laboratory. Individual or (recommended) collaborative works on numerical assignments.
Additional notes about suggested reading: For the prerequisites on the qualitative theory of ODEs see e.g.
1. V.I. Arnold, Equazioni Differenziali Ordianrie (MIR, 1979)
2. M.W. Hirsh e S. Smale, Differential equations, dynamical systems, and linear algebra (Academic Press, 1974)
3. F. Fasso`, Primo sguardo ai sistemi dinamici. CLEUP

The program is covered in lectures notes written by the teacher and distributed during the course and by
4. G. Benettin, "Introduzione ai sistemi dinamici-Cap. 2: Introduzione ai Sistemi Dinamici Iperbolici" (

Reference material includes:
5. E. Zhender, Lectures on Dynamical Systems (EMS, 2010)
6. C. Chicone, Ordinary Differential Equations with Application (II ed), Springer.

The numerical work in the laboratory uses the language Mathematica; a basic knowledge of Mathematica is advised. A tutorial in two parts is downloadable from the Dipartimento di Matematica's YouTube channel:
Textbooks (and optional supplementary readings)

Innovative teaching methods: Teaching and learning strategies
  • Lecturing
  • Laboratory
  • Working in group
  • Use of online videos
  • Loading of files and pages (web pages, Moodle, ...)

Innovative teaching methods: Software or applications used
  • Moodle (files, quizzes, workshops, ...)
  • Mathematica