Teaching code

4S00244

Credits

9

Coordinator

Nicola Sansonetto

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

Moodle Seminars 0

The teaching is organized as follows:

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

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.