Mathematical Methods and Models for Applications
A.Y. 2024/2025
Learning objectives
The course aims at providing the fundamentals of models, methods and mathematical tools used to study low dimensional dynamical systems, even those presenting a chaotic behaviour, also in connection with applicative problems. This will be pursued with the aid of laboratory sessions, where suitable numerical schemes will be developed.
Expected learning outcomes
At the end of the course, the students should be able to study simple dynamical systems representing mathematical models also arising from applicative problems, and should be able to develop suitable numerical tools supprting the aforementioned study.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1. Review of continuous-time systems (flows), autonomous and non-autonomous, and their relationship with discrete-time systems (maps).
2. One-dimensional maps: fixed points and periodic points; attractors and repellors; stability; bifurcations. The logistic family (a population growth model). Topological dynamics and chaos. Symbolic dynamics. Structural stability. Rotation number for circle maps (towards the standard map).
3. Flows in the plane: review of linear and nonlinear systems (classification of equilibria and stability). Local behavior (close to equilibria): theorems of local stable curves. Poincaré index and section, limit sets. Global asymptotic behavior: complete classification and Poincaré-Bendixson theorem.
4. Flows in higher dimensions; strange attractors. Lorenz model. Lyapunov exponents (also for maps) and variations.
5. Oscillating systems: review of forced linear oscillators and resonance phenomenon. Nonlinear oscillations (anisochronic). Averaging method and applications. Van der Pol model; forced nonlinear oscillator; synchronization phenomenon. Uncoupled oscillators: resonances and linear flow on tori. Coupled oscillators (Hénon-Heiles model).
6. Two-dimensional maps: baker's map, Arnold's cat. Hyperbolic systems; shadowing lemma. Homoclinic point phenomenon and emergence of chaos in continuous conservative systems.
2. One-dimensional maps: fixed points and periodic points; attractors and repellors; stability; bifurcations. The logistic family (a population growth model). Topological dynamics and chaos. Symbolic dynamics. Structural stability. Rotation number for circle maps (towards the standard map).
3. Flows in the plane: review of linear and nonlinear systems (classification of equilibria and stability). Local behavior (close to equilibria): theorems of local stable curves. Poincaré index and section, limit sets. Global asymptotic behavior: complete classification and Poincaré-Bendixson theorem.
4. Flows in higher dimensions; strange attractors. Lorenz model. Lyapunov exponents (also for maps) and variations.
5. Oscillating systems: review of forced linear oscillators and resonance phenomenon. Nonlinear oscillations (anisochronic). Averaging method and applications. Van der Pol model; forced nonlinear oscillator; synchronization phenomenon. Uncoupled oscillators: resonances and linear flow on tori. Coupled oscillators (Hénon-Heiles model).
6. Two-dimensional maps: baker's map, Arnold's cat. Hyperbolic systems; shadowing lemma. Homoclinic point phenomenon and emergence of chaos in continuous conservative systems.
Prerequisites for admission
Basic knowledge of analysis, linear algebra and geometry (introduced in Geometria 1, 2 and 3, and in Analisi 1, 2 and 3). Basic knowledge of dynamical systems as introduced in Fisica Matematica 1 (linearization and classification of equilibria in planar systems, Lyapunov stability, periodically forced linear oscillator).
Teaching methods
Lectures.
Lab classes. These computer-based exercises involve developing simple numerical codes to support the investigation of dynamic systems and the related models discussed in the theory. The activities may involve a combination of small group work and individual tasks. Regular attendance is highly recommended. Course materials available on Ariel will be utilized.
Lab classes. These computer-based exercises involve developing simple numerical codes to support the investigation of dynamic systems and the related models discussed in the theory. The activities may involve a combination of small group work and individual tasks. Regular attendance is highly recommended. Course materials available on Ariel will be utilized.
Teaching Resources
Lecture notes available on the Ariel web page;
additionally: Introduction to Dynamical Systems, Brin&Stuck, Cambridge
additionally: Introduction to Dynamical Systems, Brin&Stuck, Cambridge
Assessment methods and Criteria
The final examination consists of an oral exam and of the evaluation of all the activities performed during the lab sessions.
- In the oral exam, the student will be required to illustrate results presented during the course, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
-The lab exam is based on the activities done in each lab session.
The examination is passed if the oral part is successfully passed and if the lab activities are positively evaluated. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
Further options for the examination could be discussed at the beginning of the course.
- In the oral exam, the student will be required to illustrate results presented during the course, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
-The lab exam is based on the activities done in each lab session.
The examination is passed if the oral part is successfully passed and if the lab activities are positively evaluated. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
Further options for the examination could be discussed at the beginning of the course.
MAT/07 - MATHEMATICAL PHYSICS - University credits: 6
Laboratories: 24 hours
Lessons: 36 hours
Lessons: 36 hours
Professors:
Paleari Simone, Penati Tiziano
Shifts:
Professor(s)
Reception:
Contact me via email
Office 1039, 1st floor, Dipartimento di Matematica, Via Saldini, 50
Reception:
to be fixed by email
office num. 1039, first floor, Dep. Mathematics, via Saldini 50