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Università di Padova - Dipartimento di Matematica "Tullio Levi-Civita" Francesco Fassò

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Dynamical Systems Lauree magistrale in Matematica a.a. 2020-2021

Information on the course

Teachers: Francesco Fassò and Luis Garcia-Naranjo

The course is in the spring term.
The course has 7 CFU.

Students must register to the Moodle page of the course. If a password is requested, and you don't know it (I'll communicate it in class on the first day) you may ask me.

Lectures are in presence and broadcasted on Zoom. The Zoom credentials are posted on the Moodle page of the course

Schedule

Monday 13:30-15:15 Aula 1AD100  + Zoom 
Wednesday 14:30-16:15  Aula 1AD100  + Zoom 
Thursday  08:30-09:15   Aula 1AD100  + Zoom 

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.

Objectives of the course

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 first part of the course provides a panoramic of classical results on ODEs, including periodic orbits, Poincare' maps, local classifications, stable and center manifolds, etc. Then, the course focuses on the difference between integrability and chaoticity (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.

Topics

1. Continuous (ODEs, flows) and discrete (iteration of maps) Dynamical Systems. Linearization, variational equation. Linear dynamical systems; stabel, unstable and center subspace.
2. Periodic orbits: Poincare' map; stability; monodromy matrix. Applications.
3. Hyperbolic fixed points: Grobman-Hartman theorem, stable manifold theorem.
4. Integrability. Invariance of an ODE under a group action, reduction. Dynamical symmetries. Bogoyavlenskij's integrability theorem. Application to Hamiltonian systems.
5. Hyperbolic systems and homoclinic phenomena; Smale horseshoe; symbolic dynamics; Melnikov integral; shadowing.
6. Lyapunov exponents.
7. Numerical esperiments on ODEs.

Examination

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. During the examination, students may also be asked to solve some simple exercises.

Textbooks

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) or chapter 1 of F. Fasso`, Istituzioni di Fisica Matematica (CLEUP)

Most of 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/)

Optional reference material includes:
5. E. Zhender, Lectures on Dynamical Systems (EMS, 2010)
6. C. Chicone, Ordinary Differential Equations with Application (II ed), Springer.

There exist a huge number of books on on the different aspects of the theory of Dynamical Systems. If you need a mathematically elementary introduction but with many applications see e.g.
S. Strogatz,  Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview Press, 1993, 2010). For further suggestions contact me.