LxDS Spring School 2023


24-26 May 2023, ISEG-Lisbon School of Economics & Management, Universidade de Lisboa


The LxDS-Lisbon Dynamical Systems group, the Mathematics Department of ISEG, CEMAPRE, REM and CMAFcIO are organizing a 3-day spring school (24th to 26th May 2023) 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, 2023. 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 21, 2023).

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


Josef Hofbauer (University Vienna), Nicolas Chevallier (Université de Haute Alsace), Silvius Klein (PUC-Rio).


Pedro Duarte (Universidade de Lisboa), José Pedro Gaivão (Universidade de Lisboa), João Lopes Dias (Universidade de Lisboa) and Telmo Peixe (Universidade de Lisboa)


Diophantine approximations and the space of lattices (Nicolas Chevallier)

The objective of the course is to show the fruitful relations between  Diophantine approximations,  lattices and the action of the diagonal flow in the space of  unimodular lattices. We will start with the classics: continued fractions in one dimension, Dirichlet`s theorem in one dimension  and Dirichlet`s theorem in simultaneous approximation. Then we will introduce lattices, the space of unimodular lattices, the shortest vector function and the diagonal flow. We will reinterpret the notions of badly approximable and singular vectors (Dani correspondence). We will also reinterpret the notion of best Diophantine approximation in terms of minimal vectors in lattices. Finally we will give applications  of the ergodicity of the diagonal flow to Diophantine approximations.

Statistical properties for certain dynamical systems (Silvius Klein)

The main objective of this mini-course is the study of some recent topics in ergodic theory about limit laws (i.e. the large deviations principle and the central limit theorem) for certain types of dynamical systems. The main tool in this study is the existence of the spectral gap of the Markov transition operator or the Ruelle transfer operator in an appropriate space of observables.

Game dynamics (Josef Hofbauer)

Evolutionary Game Theory provides a natural way to associate dynamical systems to games, such as the replicator dynamics and the best response dynamics (fictitious play). Depending on the underlying game, this can lead to Hamiltonian systems, volume preserving systems, gradient systems, or monotone flows. We wish to understand the long run behaviour of these dynamics, in particular which Nash equilibria are stable.


Time24th May25th May26th May
10h00 - 11h30J. Hofbauer
(Part 1)
J. Hofbauer
(Part 2)
J. Hofbauer
(Part 3)
11h30 - 11h45Coffee breakCoffee breakCoffee break
11h45 - 13h15S. Klein
(Part 1)
S. Klein
(Part 2)
S. Klein
(Part 3)
13h15 - 15h00LunchLunchLunch
15h00 - 16h30N. Chevallier
(Part 1)
N. Chevallier
(Part 2)
N. Chevallier
(Part 3)
16h30 - 17h00Coffee breakCoffee breakClosure
17h00 - 17h30Talk 1 - A. Rodrigues
(University of Lisboa)
Heteroclinic Bifurcations:
Large Strange Attractors
Talk 3 - A. Ferreira
(University of Porto)
The Jungle Game dynamics
17h30 - 18h00Talk 2 - O. Etubi
(University of Porto)
Hölder Continuity of Density and
Entropy for Piecewise Expanding Maps
Talk 4 - L. Garrido da Silva
(University of Porto)
Heteroclinic attractors in
biological dynamical systems

20h00Social Dinner

Target audience

PhD students, post-docs and researchers.


ISEG, Building Quelhas 6, Room Novo Banco.


For more information please contact us.


Talk 1 - A. Rodrigues (University of Lisboa)
Title: Heteroclinic Bifurcations: Large Strange Attractors
Abstract: In this talk, we deal with bifurcations associated to (small) heteroclinic networks. We are interested in large strange attractors which are not confined to a small portion of the phase space. We focus on two different configurations: the first one involves the destruction of a torus; the other is connected to heteroclinic tangles. The proof of the existence of strange attractors relies on the analysis of the corresponding 1D reduction (the singular limit).

Talk 2 - O. Etubi (University of Porto)
Title: Holder Continuity of Density and Entropy for Piecewise Expanding Maps
Abstract: In this work, we show the Holder continuity of invariant densities and entropy for piecewise expanding maps, thereby extending the results in [1, 2]. Furthermore, we built a family of circle maps with neutral fixed points possessing an absolutely continuous invariant measure. Our technique is to show that the Hölder continuity of the density and entropy for this family of maps possessing a weak Gibbs Markov inducing scheme is inherited by the density and entropy for the original map. References [1] Alves, J., Pumariño, A., and Vigil, E. (2017). Statistical stability for multidimensional piecewise expanding maps. Proceedings of the American Mathematical Society, 145(7), 3057-3068. [2] Alves, J. F., and Pumarino, A. (2021). Entropy formula and continuity of entropy for piecewise expanding maps. In Annales de l Institut Henri Poincaré C, Analyse non linéaire, 38(1), 91-108.

Talk 3 - A. Ferreira (University of Porto)
Title: The Jungle Game dynamics
Abstract: The Jungle Game is used in population dynamics illustrating how the cyclic competition among species and the reproduction of each species lead to the coexistence of all or only a subset of species. In the Jungle Game the interaction among species is achieved via a food chain, where a hierarchically superior species preys on all the species below. The last species in the chain wins when competing with the top species. The dynamics is usually described by mean-field equations in $R^n_+$ , or equivalently, Lotka-Volterra dynamics, with the strength of the interaction among species expressed in the coefficients of the equations. The dynamics of the Jungle Game supports a heteroclinic network whose cycles represent species that may coexist. We study the stability of the 2-dimensional short connection in a $\\\\\\\\Delta-$clique that allow us to calculate the stability indices of all the cycles of the network. A stable cycle indicates that the species involved are likely to coexist.


CEMAPRE - Centre for Applied Mathematics and Economics

Rua do Quelhas, n.º 6
1200-781 Lisboa

Email: cemapre@iseg.ulisboa.pt
Tel: (+351) 213 925 876