Aviso: Se está a ler esta mensagem, provavelmente, o browser que utiliza não é compatível com os "standards" recomendados pela W3C. Sugerimos vivamente que actualize o seu browser para ter uma melhor experiência de utilização deste "website". Mais informações em

Warning: If you are reading this message, probably, your browser is not compliant with the standards recommended by the W3C. We suggest that you upgrade your browser to enjoy a better user experience of this website. More informations on



    Research IDs

    ORCID: 0000-0002-0344-5959

    ResearcherID: F-8725-2018

    Scopus Author ID: 55308107200

    Google Scholar


    Centro de investigação

    Member of CEMAPRE-Centro de Matemática Aplicada à Previsão e Decisão Económica, based at ISEG-Lisbon School of Economics & Management, Universidade de Lisboa, and of the LxDS-Lisbon Dynamical Systems Group.

    Interesses científicos em Matemática

    1. Dynamical Systems, particularly Hamiltonian dynamics, Lotka-Volterra systems, replicator equation, game theory, and evolutionary game dynamics.
    2. Mathematical psychology.

    Publicações em revistas internacionais com arbitragem

    1. Recorrências, Progressões Aritméticas e Teoria Ergódica: Teoremas de van der Warden e de Green-Tao, J. Buescu and T. Peixe, Revista Matemática Universitária, 48/49, 39-51 (2010).

    2. Rank of Stably Dissipative Graphs, P. Duarte and T. Peixe, Linear Algebra and Its Applications, 437, 2537-2586 (2012).

    3. Conservative and Dissipative Polymatrix Replicators, H. Alishah, P. Duarte and T. Peixe, Journal of Dynamics and Games, 2, 157-185 (2015).

    4. Asymptotic Poincaré Maps along the edges of Polytopes, H. Alishah, P. Duarte and T. Peixe, Nonlinearity, 33, 469-510, (2020).

    5. Permanence in polymatrix replicators, T. Peixe, Journal of Dynamics and Gamespublished online, November, (2020).

    6. Periodic attractor in the discrete time best-response dynamics of the Rock-Paper-Scissors game, J. P. Gaivão and T. Peixe,  Dynamic Games and Applications, published online, November, (2020).
    7. Asymptotic Dynamics of Hamiltonian Polymatrix Replicators, H. Alishah, P. Duarte and T. Peixe, preprint, February, (2021).
    8. Persistent strange attractors in 3D Polymatrix Replicators, T. Peixe and A. A. Rodrigues, submitted, March, 2021.
    9. On the Roots of Underdevelopment: "Wrong Equilibrium" or "Miscoordination"?, José P. Pontes and T. Peixe, submitted, July, (2021).

    Flows on Polytopes - Mathematica Code

    We provide here the Mathematica code we had developed to explore the dynamics of polymatrix replicators for lower dimensional polytopes. Download the "Mathematica_code" file and load it at the beginning of the examples files. For any information please contact us (Pedro Duarte and Telmo Peixe).

    Artigos em revistas nacionais sem arbitragem

    1. Observar Golfinhos... Com Trigonometria, P. Duarte, T, Peixe and T. Caissotti, Gazeta de Matemática , 169, 16-22 (2013).

    Comunicações em Conferências

    1.  Evaluating the effect of measurement error in pairs of 3D bearings in point transect sampling estimates of density,  Tiago A Marques, Pedro Duarte, Telmo Peixe, David Moretti, Len Thomas, ISEC International Statistical Ecology Conference, 2018.


    1. Sistemas Lotka-Volterra Dissipativos, T. Peixe, Master's Thesis, Universidade de Lisboa, Faculdade de Ciências, Portugal, (2010).
    2. Lotka-Volterra Systems and Polymatrix Replicators, T. Peixe, PhD Thesis, Universidade de Lisboa, Faculdade de Ciências, (2015).

    Projetos de investigação

    Dynamics on polytopes and polymatrix replicators

    Evolutionary game theory provides many examples of ODEs and flows where the phase space is a 
    polytope. A computational technique to analyze the asymptotic dynamics along the polytope's edges has been 
    developed by Duarte (2011), and Alishah, Duarte and Peixe (2014, arXiv:1411.6227v2). 
    A particular class of such ODEs defined on polytopes are the polymatrix replicators (see Alishah and 
    Duarte(2015) and Alishah, Duarte and Peixe (2015)), which includes well known classes of 
    evolutionary game dynamics, such as the symmetric and asymmetric games associated to replicator 
    equations. We aim to expand and make these techniques user friendlier in order to disseminate their 


    Piecewise Linear Dynamics meets Economics and Biology

    This project studies piecewise-smooth dynamical systems arising in real-world applications.
    Piecewise-smooth dynamical systems are found in applications to problems in engineering, biology,
    computer science, economics and management. We pretend to develop new methods to analyse the dynamics of a certain class of piecewise linear dynamical systems defined on polytopes. Using the novel methods, we pretend also to answer several standing open questions raised in the 90's concerning the dynamics of two basic models in manufacturing and logistics. Additionally, we want to study the dynamics of evolutionary game theory models from a piecewise-smooth dynamical systems perspective, and develop software for simulation and analysis of these models. From a pure mathematical side, this project also contributes to the development of the theory of piecewise-smooth dynamical systems which still today presents many challenging open problems.


    New Trens in Lyapunov Exponents - FCT PTDC/MAT-PUR/29126/2017

    This project deals primarily with Lyapunov exponents (LE), which measure the sensitivity of a dynamical system to initial conditions. The Lyapunov exponents will be studied in the framework of Ergodic Theory (ET), Mathematical Physics (MP) and the Theory of Dynamical Systems (DS). The applied strand of the project will focus on ranking algorithms for large sparse graphs, in the context of bibliometric analysis.