Nonlinear Dynamics

, Volume 92, Issue 3, pp 1159–1166 | Cite as

Limit cycles bifurcating from a zero–Hopf singularity in arbitrary dimension

  • Luis Barreira
  • Jaume Llibre
  • Claudia Valls
Original Paper


For a \(C^{m+1}\) differential system on \(\mathbb {R}^n\), we study the limit cycles that can bifurcate from a zero–Hopf singularity, i.e., from a singularity with eigenvalues \(\pm ~bi\) and \(n-2\) zeros for \(n\ge 3\). If the singularity is at the origin and the Taylor expansion of the differential system (without taking into account the linear terms) starts with terms of order m, then \(\ell \) limit cycles can bifurcate from the origin with \(\ell \in \{0,1,\ldots , 2^{n-3}\}\) for \(m=2\) [see Llibre and Zhang (Pac J Math 240:321–341, 2009)], with \(\ell \in \{ 0,1,\ldots ,3^{n-2}\}\) for \(m=3\), with \(\ell \le 6^{n-2}\) for \(m=4\), and with \(\ell \le 4\cdot 5^{n-2}\) for \(m=5\). Moreover, \(\ell \in \{0,1,2\}\) for \(m=4\) and \(n=3\), and \(\ell \in \{0,1,2,3,4,5\}\) for \(m=5\) and \(n=3\). In particular, the maximum number of limit cycles bifurcating from the zero–Hopf singularity grows up exponentially with n for \(m=2,3\).


Limit cycles Zero–Hopf singularity Arbitrary dimension 

Mathematics Subject Classification

Primary 37J35 37K10 



The first and third author are partially supported by FCT/Portugal through UID/MAT/04459/2013. The second author is partially supported by a FEDER-MINECO Grant MTM2016-77278-P, a MINECO Grant MTM2013-40998-P, and an AGAUR Grant 2014SGR-568.


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Authors and Affiliations

  1. 1.Departamento de Matemática, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  2. 2.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBarcelonaSpain

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