New lower bounds for the Hilbert numbers using reversible centers

R. Prohens, J. Torregrosa

Research output: Contribution to journalArticleResearchpeer-review

12 Citations (Scopus)


© 2018 IOP Publishing Ltd & London Mathematical Society. In this paper we provide the best lower bounds, that are known up to now, for the Hilbert numbers of polynomial vector fields of degree N, , for small values of N. These limit cycles appear bifurcating from symmetric Darboux reversible centers with very high simultaneous cyclicity. The considered systems have, at least, three centers, one on the reversibility straight line and two symmetric outside it. More concretely, the limit cycles are in a three nests configuration and the total number of limit cycles is at least 2n + m, for some values of n and m. The new lower bounds are obtained using simultaneous degenerate Hopf bifurcations. In particular, H(4) ≥ 28, H(5) ≥ 37, H(6) ≥ 53, H(7) ≥ 74, H(8) ≥ 96, H(9) ≥ 120 and H(10) ≥ 142.
Original languageEnglish
Pages (from-to)331-355
Publication statusPublished - 1 Jan 2019


  • 16th Hilbert number
  • bifurcation and number of periodic orbits
  • limit cycles
  • polynomial differential equation


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