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
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 2025/2026The 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.
1° Year
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2° Year activated in the A.Y. 2018/2019
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3° Year activated in the A.Y. 2019/2020
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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 (2018/2019)
Teaching code
4S00244
Credits
9
Language
Italian
Also offered in courses:
- Dynamical Systems of the course Bachelor's degree in Applied Mathematics
Scientific Disciplinary Sector (SSD)
MAT/05 - MATHEMATICAL ANALYSIS
The teaching is organized as follows:
Parte I
Credits
6
Period
II semestre
Academic staff
Marta Zoppello
Esercitazioni 2 parte II
Esercitazioni parte II
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
Program
1. Basic principles.
The Cauchy Problem. Completeness. Flows and orbits. Re-parametrization. Local rectifiability Theorem. First examples: exponential growth, the logistic equation, the Lotka–Volterra equation, the SIS and SIR models, car-following…
2. Models and examples
One-dimensional systems. Conservative systems with one degree of freedom, Newtonian systems. Linear systems: dimension 1, 2, n. Non-linear systems in R^2.
3. Discrete-time systems.
Definitions. Examples: bacterial growth, Fibonacci, structured populations, AIMD…
Linear systems and z-transform. Stability.
4. Stability
Definition. Lyapunov theory. Alpha and Omega-limits. Poincaré-Bendixson Theorem.
Bibliography
| Activity | Author | Title | Publishing house | Year | ISBN | Notes |
|---|---|---|---|---|---|---|
| Parte I | M.W. Hirsch e S. Smale | Differential equations, dynamical systems, and linear algebra | Academic Press | 1974 | ||
| Parte I | S. Strogatz | Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering | Westview Press | 2010 | ||
| Parte I | F. Fasso` | Primo sguardo ai sistemi dinamici | CLEUP | 2016 | ||
| Esercitazioni 2 parte II | M.W. Hirsch e S. Smale | Differential equations, dynamical systems, and linear algebra | Academic Press | 1974 | ||
| Esercitazioni 2 parte II | S. Strogatz | Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering | Westview Press | 2010 | ||
| Esercitazioni 2 parte II | F. Fasso` | Primo sguardo ai sistemi dinamici | CLEUP | 2016 | ||
| Esercitazioni parte II | M.W. Hirsch e S. Smale | Differential equations, dynamical systems, and linear algebra | Academic Press | 1974 | ||
| Esercitazioni parte II | S. Strogatz | Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering | Westview Press | 2010 | ||
| Esercitazioni parte II | 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.
