Second Level Degree in Mathematics
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DYNAMICAL SYSTEMS - 7 CFU
Teacher
Luis Constantino Garcia Naranjo Ortiz De La Huerta
Scheduled Period
I Year - 1 Semester | 04/10/2021 - 15/01/2022
Hours: 56 (24 esercitazione, 32 lezione)
Prerequisites
1. Basic knowledge of the theory of ordinary differential equations (ODEs) and of the qualitative theory of ODEs, at the level of, e.g., the course "Fisica Matematica" which is offered as a a mandatory course at the second year of the Corso di Laurea in Matematica in this University.
2. A basic knowledge of the programming language "Mathematica" (at the level of the tutorials periodically offered by the CCS and available on the YouTube channel of the Department of Mathematics) is useful, as it will be used in the numerical part of the course.
Target skills and knowledge
This course provides an introduction to the theory of Differentiable Dynamical Systems---particularly, continuous Dynamical systems (namely ODEs), but also discrete Dynamical Systems (iterations of maps). The course provides a panoramic of classical results on ODEs, including periodic orbits, Poincare' maps, local classifications, stable and center manifolds, and chaos in the hyperbolic context. The course is completed by a numerical laboratory part, which is devoted to the numerical investigation of ODEs and to the numerical analysis of dynamical systems.
The student will reach an advanced knowledge of the above topics in the theory of differentiable dynamical systems and basic competences and skills on the numerical investigations of dynamical systems.
Examination methods
Oral examination on the topics studied in the course, and with an evaluation and a discussion of the numerical assignments (which will be assigned during the course). Students will prepare the numerical assignments by working either alone or in pairs, at their choice.
This examination format allows to evaluate: 1) The level of the theoretical knowledge and mathematical comprehension of the subject reached by the student. 2) The abilities reached by the student in the analysis and comprehension of the numerical results.
Assessment criteria
Knowledge of the subject, level of the mathematical comprehension, quality of the numerical work, and the ability to analyze and interpret the numerical results within the theoretical framework developed in the course.
Course contents
1. Continuous (ODEs, flows) and discrete (iteration of maps) Dynamical Systems. Linearization, variational equation. Linear dynamical systems; stable, unstable and center subspace.
2. Periodic orbits, Poincare' map; stability; monodromy matrix. Applications. Poincare'-Bendixson theorem.
3. Hyperbolic fixed points: Grobman-Hartman theorem, stable manifold theorem. Persistence and bifurcations of fixed points and periodic orbits.
4. Center manifold and Lyapunov center theorem.
5. Hyperbolic systems and homoclinic phenomena; Smale horseshoe; symbolic dynamics; Melnikov integral; shadowing. Lyapunov exponents.
6. Numerical experiments.
Planned learning activities and teaching methods
Frontal lectures. Lectures in numerical Laboratory. Individual or (recommended) collaborative works on numerical assignments.
Additional notes about suggested reading
For the prerequisites on the qualitative theory of ODEs see e.g.
1. V.I. Arnold, Equazioni Differenziali Ordianrie (MIR, 1979)
2. M.W. Hirsh e S. Smale, Differential equations, dynamical systems, and linear algebra (Academic Press, 1974)
3. F. Fasso`, Primo sguardo ai sistemi dinamici. CLEUP
For the prerequisites on the qualitative theory of ODEs see e.g.
1. V.I. Arnold, Ordinary differential equations (MIR, 1979)
2. M.W. Hirsch e S. Smale, Differential equations, dynamical systems, and linear algebra (Academic Press, 1974)
3. F. Fasso`, Primo sguardo ai sistemi dinamici. CLEUP
The program is covered in lectures notes written by the teacher and distributed during the course and by
4. G. Benettin, "Introduzione ai sistemi dinamici-Cap. 2: Introduzione ai Sistemi Dinamici Iperbolici" (http://www.math.unipd.it/~benettin/)
Reference material includes:
5. E. Zhender, Lectures on Dynamical Systems (EMS, 2010)
6. C. Chicone, Ordinary Differential Equations with Application (II ed), Springer.
The numerical work in the laboratory uses the language Mathematica; a basic knowledge of Mathematica is advised. A tutorial in two parts is downloadable from the Dipartimento di Matematica's YouTube channel:
https://www.youtube.com/watch?v=JfRzv6r0wqM
https://www.youtube.com/watch?v=tUuwgiGipfw
Textbooks (and optional supplementary readings)