José Pedro Gaivão

CEMAPRE, ISEG - Universidade de Lisboa
Rua do Quelhas, 6
1200-781 Lisboa

Email: jpgaivao at
Phone: (+351) 213 925 874 (Ext. 3874)
Office: 511, Quelhas 6

I am a postdoc fellow at CEMAPRE based at ISEG, Portugal. I am also a member of the Lisbon Dynamical Systems Group - LxDS.

Research interests:

Dynamical Systems, particularly Hamiltonian dynamics, billiards and exponentially small phenomena.


Licenciatura in Matemática Aplicada à Tecnologia, Universidade do Porto (2005)
PhD in Mathematics (supervisor: Vassili Gelfreich), University of Warwick (2010)


  1. Exponentially small splitting of invariant manifolds near a Hamiltonian-Hopf bifurcation
    PhD thesis, University of Warwick, 2010.
  2. Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift-Hohenberg equation as an example
    (with V. Gelfreich)
    Nonlinearity, 24, no.3, p. 677-698, 2011.
  3. Chaos in the square billiard with a modified reflection law
    (with G. Del Magno, J. L. Dias, P. Duarte and D. Pinheiro)
    Chaos, 22, 026106, 2012.
  4. Analytic invariants for the 1:-1 resonance.
    Annales de l'Institut Fourier, vol. 63, no.4, p. 1267-1426, 2013.
  5. SRB measures for polygonal billiards with contracting reflection laws
    (with G. Del Magno, J. L. Dias, P. Duarte and D. Pinheiro)
    Communications in Mathematical Physics, vol. 329, 2, p. 687-723, 2014.
  6. Ergodicity of polygonal slap maps
    (with G. Del Magno, J. L. Dias and P. Duarte)
    Nonlinearity, 27, no.8, p. 1969-1983, 2014.
  7. Dissipative outer billiards: a case study
    (with G. Del Magno and E. Gutkin)
    Dynamical Systems: An International Journal, vol. 30, no.1, p. 45-69, 2015.
  8. Hyperbolic polygonal billiards with finitely many ergodic SRB measures
    (with G. Del Magno, J. L. Dias, P. Duarte)
    Ergodic Theory and Dynamical Systems, 1-24, doi:10.1017/etds.2016.119, 2017.
  9. Hyperbolic billiards on polytopes with contracting reflection laws
    (with P. Duarte and M. Soufi)
    Discrete and Continuous Dynamical Systems - Series A, vol. 37, no. 6, p. 3079-3109, 2017.
  10. Asymptotic periodicity in outer billiards with contraction
    Ergodic Theory and Dynamical Systems, 1-16, doi:10.1017/etds.2018.36, 2018.
  11. $\lambda$-stability of periodic bililard orbits
    (with S. Troubetzkoy)
    Nonlinearity, 31, no.9, p. 4326, 2018.
  12. Ergodic properties of polygonal billiards with strongly contracting reflection laws
    (with G. Del Magno, J. L. Dias, P. Duarte)
    Preprint, 2015.
  13. Renormalization of Gevrey vector fields with a Brjuno type aithmetical condition
    (with J. L. Dias)
    Preprint, 2017.
  14. On the attractor of piecewise expanding maps of the interval
    (with G. Del Magno, J. L. Dias, P. Duarte)
    Preprint, 2017.

Conference proceedings:

  1. Practical implementation issues for switching between stabilizing controllers
    (with E. Santos, T. Mendonça and P. Rocha).
    Proc. of the 7th Portuguese Conference on Automatic Control, M. Ayala-Botto, IST, 6 p., Lisbon, Portugal, September 11-13, 2006.
  2. Observer design in switching control of neuromuscular blockade: clinical cases
    (with H. Magalhães, J. Lemos, T. Mendonça, P. Rocha and S. Esteves).
    28th Annual International Conference IEEE Engineering in Medicine and Biology Proceedings, p. 5436-5439, 2006.
  3. Exponentially small splitting of separatrices.
    Bol. Soc. Port. Mat. Special Issue , p. 181-184, 2012.
  4. Polygonal billiards with a contractive reflection law: a review of some hyperbolic properties
    (with G. Del Magno, J. L. Dias, P. Duarte and D. Pinheiro)
    in Difference Equations, Discrete Dynamical Systems and Applications, Eds. L. Alseda i Soler et al., Springer Proceedings in Mathematics and Statistics 180, 2016.