Outline

1. Introduction to nearly-integrable systems (Arnold-Liouville theorem, action-angle variables, etc.)
2. Rotational dynamics, Andoyer variables, the spin-orbit problem
3. Perturbation theory
4. Application to the precession of the equinoxes
5. KAM theory (Part I)
6. KAM theory (Part II)
7. An application to the spin-orbit problem
8. A dissipative version of KAM theory and an application to the dissipative spin-orbit problem

Notes

• Lectures 1-3
Alessandra Celletti
• Lectures 4-6
Alessandra Celletti
• Lectures 7-9
Alessandra Celletti
• Appendix to the course
Alessandra Celletti

Outline

1. Introduction/review of symbolic dynamics for d=1: shifts of finite type (SFTs), sofic shifts, topological/measure-theoretic entropy, measures of maximal entropy.
2. Computational aspects of SFTs for d>1: aperiodic SFTs, embedding of Turing machines in SFTs, the undecidability problem, computability properties of entropy, periodic points, and subdynamics.
3. Mixing properties for SFTs for d>1: topological mixing, block gluing, strong irreducibility, implications of these properties on factors, periodic points, the undecidability problem, and topological entropy.
4. Measures on SFTs for d>1: Lanford-Ruelle theorem, possible non-uniqueness of measure(s) of maximal entropy, spatial mixing properties and their implications for computability of entropy.
5. "Random SFTs": Probabilistic framework on set of SFTs, ways in which a "typical" SFT is well-behaved.

Notes

• Slides
Ronnie Pavlov

Abstract

The course will provide an introduction to numerical techniques for the computation of invariant manifolds of large-scale dissipative dynamical systems, and the loci of their bifurcations. Although being aimed at high-dimensional systems these techniques are also useful for low-dimensional problems.

We will start with a brief introduction to the iterative numerical linear algebra methods (mainly those based on Krylov subspaces) required to solve linear systems and eigenvalue problems (GMRES, subspace iteration, Arnoldi methods), followed by a description of Newton-Krylov continuation methods. Then the computation of fixed points, and periodic orbits by single and multiple shooting will be described. The parallelization of the latter and how to achieve almost linear speedups will be considered. The continuation of bifurcations of periodic orbits in systems depending on two parameters will be treated next. Finally, methods to compute two-dimensional invariant tori will be described. Examples of applications to reaction-diffusion and fluid mechanics problems will be shown.

Outline

1. Equilibria and periodic orbits of PDEs
2. Newton-Krylov continuation methods
3. Inexact Newton's methods
4. Iterative linear solvers and GMRES
5. Stability
6. Subspace iteration and Arnoldi methods
7. Continuation of codimension-one bifurcation points of equilibria and periodic orbits
8. Continuation of invariant tori
9. Periodic orbits by multiple shooting.

Notes

• Bibliography of the course
Juan Sánchez Umbría
• Note
Juan Sánchez Umbría
• Slides
Juan Sánchez Umbría

1. From non-smooth to smooth friction models, using regularisation and slow-fast theory, Elena Bossolini, Morten Brøns and Kristian Uldall Kristiansen (Technical University of Denmark, Denmark)
2. The reversibility problem for dynamical systems, Isabel Checa, Antonio Algaba, Estanislao Gamero and Cristobal García (Universidad de Huelva, Spain)
3. Dynamics and bifurcations in 3D Filippov Systems at a Degenerate T-singularity, Rony Cristiano and Daniel J. Pagano (Federal University of Santa Catarina, Brazil)
4. Masses and Vortices Dynamics on a Cylinder, Gladston Duarte, Stefanella Boatto and Teresa Stuchi (Universidade Federal do Rio de Janeiro, Brazil)
5. Towards the analysis of a canonical form for PWL systems with hysteresis, Marina Esteban Pérez (Universidad de Sevilla, Spain)
6. Inverse integrating factor for degenerated vector fields, Natalia Fuentes, Antonio Algaba, Cristobal García and Manuel Reyes (Universidad de Huelva, Spain)
7. Arnold diffusion using several combinations of Scattering maps,Rodrigo Gonçalves Schaefer (Universitat Politècnica de Catalunya, Spain)
8. A fractalization route for affine skew-products on the complex plane,Marc Jorba-Cuscó, Núria Fagella, Ángel Jorba and Joan Carles Tatjer (Universitat de Barcelona, Spain)
9. Dynamical compactness and sensitivity, Danylo Khilko, Wen Huang, Sergiy Kolyada, Alfred Peris and Guohua Zhang (Taras Shevchenko National University of Kyiv, Ukraine) 
10. Quasideterminant solutions of NC Painlevé II equation with the Toda solution at n = 1 as a seed solution in its Darboux transformation, Irfan Mahmood (University of the Punjab, Pakistan)
11. A Study on Temperature Distribution, Efficiency and Effectiveness of longitudinal porous fins by using Adomian Decomposition Sumudu Transform Method, Trushit V. Patel (Sardar Vallabhbhai National Institute of Technology, India)
12. The role of isochrons in neural communication, Alberto Pérez Cervera (Universitat Politécnica de Catalunya, Spain)
13. Odd Global Continuation and Linear Stability of the equilibrium solution for Nonlinear Oscillators of Pendulum type, Andrés Rivera and Daniel Núñez (Pontificia Universidad Javeriana Cali, Colombia)

Notes

• Abstract of Posters
Organizing Committee