Teaching code

4S00244

Credits

9

Coordinator

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 course introduces theoretical and practical aspects of Dynamical Systems, describing the evolution in time of quantitative variables. Even though classical examples come from physics, for dynamics of point particles, dynamical systems now model several different phenomena, such as population dynamic, models in computer science, road traffic…
The two main goals for this course are: first, the knowledge of basic results about Dynamical Systems (When is a system well-defined? When is it stable? How can one check its main properties?); second, the ability to study and build models based on dynamical systems (see examples in the syllabus).
After the course, the student will be able to define and study a given dynamical system (write the equations, study its trajectories and its stability); he will know both main theorems about Dynamical Systems and classical models.

Program

Part 1
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. 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.

Part 2
4. Stability
Definition. Lyapunov theory. Alpha and Omega-limits. Poincaré-Bendixson Theorem.

Examination Methods

Part 1
A written exam (1h30) with 2 exercises: phase portrait in 2D for a non-linear dynamical system; computation of trajectories and stability for a discrete-time system.
The maximal grade for this exam is 30. The minimal grade for the oral exam is 18/30.
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; if it is not passed, the student is required to repeat the written exam too. If it is passed, the grade from the written exam is increased or decreased of 6 grades maximum.
The oral exam tests the following learning outcomes:
- To be able to present precise proofs and recognize them.

Part 2
A written exam (2h) with 3 exercises: 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 oral exam and the learning outcomes are described in Part 1.