Description

The LxDS-Lisbon Dynamical Systems group, the Mathematics Departments of ISEG and FCUL, and the research centers CEMAPRE and CEMS.UL, are organizing a 3-day spring school (28th-30th May 2025) on dynamical systems to be held at ISEG/ULisboa.

The school will consist of three mini-courses in specific areas of dynamical systems lectured by specialists of recognized international merit.

REGISTRATION is required to confirm participation by April 30, 2025. Please fill the form on this web page.

Those wishing to give a 20-minute talk should mention it in the comments section of the registration form, indicating the corresponding title and abstract.

The spring school will have a limited number of participants.

PhD students can apply for financial support (deadline - April 30, 2025).

The organizers are grateful for the sponsorship of CIM - Centro Internacional de Matemática.

This event is supported by national funds through FCT – Foundation for Science and Technology, projects

CEMAPRE/REM - UIDB /05069/2020 - https://doi.org/10.54499/UIDB/05069/2020 and

Center For Mathematical Studies - CEMS.UL - UID/04561/2025

 Speakers:

Daniel Peralta-Salas

(Institute of Mathematical Sciences - ICMAT)


Joel Moreira

(University of Warwick)

Santiago Ibáñez

(Universidad de Oviedo)

Outline:

An introduction to 3D Euler flows (Daniel Peralta-Salas)

This short course is focused on the stationary solutions to the 3D Euler equations from the dynamical systems viewpoint. I will review the celebrated structure theorem by Arnold (1965) and some properties of the Beltrami flows, a particularly flexible class of stationary solutions.

Ergodic Ramsey Theory (Joel Moreira)

In 1977 Furstenberg established a correspondence between combinatorial properties of sets of integers and dynamical properties in ergodic theory. He used this correspondence to give an ergodic theoretic proof to a theorem of Szemerédi on the existence of arbitrarily long arithmetic progressions in any set of integers with positive density. Since then, Furstenberg's correspondence, together with many ideas and techniques from ergodic theory have been used to successfully address many problems in additive combinatorics and Ramsey theory, creating a new area now called Ergodic Ramsey Theory. In the minicourse I will present the Furstenberg correspondence, and explore several combinatorial results that can be proved with it. No previous knowledge of additive combinatorics, Ramsey theory or ergodic theory will be required..

Unfolding chaos: Singularities of vector fields (Santiago Ibañez)

Detecting chaotic behavior in a dynamical system is far from a simple task. In most cases, numerical methods are employed, and, in fact, there are now computational techniques available that allow us to handle chaotic scenarios with relative ease. On the other hand, analytical tools to prove the existence of chaos are rare; they require arduous demonstrations and are difficult to implement in practice. Certain configurations are known to exist that can explain the genesis of chaotic behavior (strange attractors). However, these configurations are not easy to identify in a given model. It is at the end of this journey that one encounters singularities. Specifically, we refer to singularities of vector fields. It is possible to prove the existence of singularities that exhibit configurations from which strange attractors emerge. The good news is that singularities are more manageable objects and easier to locate, provided that they exist in the system. The aim of this course is to introduce these singularities and explain how they contribute to the genesis of chaotic dynamics. Specifically, Hopf-Zero singularities and 3-dimensional nilpotent singularities will be the central focus, with the global configurations that accompany them—primarily homoclinic orbits of Shilnikov type—playing key roles. The course is designed to be engaging and accessible to an audience that may not be experts in local bifurcation theory or chaotic systems analysis.

Organizers:

Pedro Duarte

(Universidade de Lisboa)


João Lopes Dias

(Universidade de Lisboa)

José Pedro Gaivão

(Universidade de Lisboa)


Telmo Peixe

(Universidade de Lisboa)

Alexandre Rodrigues

(Universidade de Lisboa)

Pedro Soares

(Universidade de Lisboa)

Target Audience

PhD students, post-docs and researchers.

Accommodation

For more information please contact us.

Supporting institutions

Contacts

CEMAPRE - Centro de Matemática Aplicada à Previsão e Decisão Económica

Rua do Quelhas, n.º 6

1200-781 Lisboa

Portugal

Email: cemapre@iseg.ulisboa.pt

Tel: (+351) 213 925 876