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
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.
1° Year
| Modules | Credits | TAF | SSD |
|---|
2° Year It will be activated in the A.Y. 2026/2027
| Modules | Credits | TAF | SSD |
|---|
3° Year It will be activated in the A.Y. 2027/2028
| Modules | Credits | TAF | SSD |
|---|
| Modules | Credits | TAF | SSD |
|---|
| Modules | Credits | TAF | SSD |
|---|
| 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 (2026/2027)
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
Courses Single
Authorized
The teaching is organized as follows:
Teoria 1
Teoria 2
Learning objectives
The aim of the course is to introduce the theory and some applications of dynamical systems, which describe the time evolution of quantitative variables. By the end of the course, the students will be able to investigate the stability and the character of an equilibrium, the qualitative analysis of a system of ordinary differential equations, the phase portrait of a (parametric) dynamical system in dimension 1 and 2, and to analyse finite-dimensional Hamiltonian systems. Moreover, the students will be able to study some basic applications of dynamical systems arising from population dynamics, mechanics and traffic flows. Finally, students 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.
Prerequisites and basic notions
The material covered in first-year and second-year, first-semester courses - in particular, Mathematical Analysis 1 and 2, Linear Algebra.
Program
Part I (for everybody)
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. 1-dimensional examples of phase portraits. Second-order systems of differential equations; phase-space analysis and equilibria.
3. Linear systems
Linearisation of a vector field about an equilibrium. Classification of two-dimensional 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.) n-dimensional 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; pull-back and push-forward of a vector field by a diffeomorphism. Non-autonomous differential equations: time-dependent 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. 1-dimensional Newton equation.
Phase portraits of the 1-dimensional Newton equation, in the conservative case. Linearisation. Reduction of the order. Systems with friction.
Part II (for those who take the 9 CFU course)
8. Bifurcations: Notion of bifurcation; examples of bifurcations from equilibria in dimension one; applications.
9. Some properties of volume-preserving vector fields: Transport Theorem and Recurrence Theorem.
10. Introduction to 1-dimensional calculus of variations The indirect method for integral functionals in dimension one. Necessary conditions for minimality: the Euler-Lagrange equations. Jacobi first integral and conservation laws.
11. The mechanical case and Lagrangian mechanics: invariance of the Euler-Lagrange equations under uplifts. Geodesics on a surface. Gauge equivalent Lagrangians. Stability theory for mechanical Lagrangian systems.
Bibliography
Didactic methods
Lectures, group work, homework, weekly group recaps.
Learning assessment procedures
The exam consists of a written part, with exercises and/or questions on the course content, and an oral part.
Evaluation criteria
Knowledge of the content of the course. Ability to solve simple problems. Logical rigour. Clarity of presentation.
Criteria for the composition of the final grade
The written exam is given a grade out of thirtieths. The grade of the written exam serves as the basis for the final evaluation, out of thirtieths, which also takes into account the oral exam (and may be higher, equal to or lower than the grade of the written exam alone).
Exam language
Italiano
Sustainable Development Goals - SDGs
This initiative contributes to the achievement of the Sustainable Development Goals of the UN Agenda 2030. More information on sustainability
