### Abstract

© 2018, Springer Science+Business Media B.V., part of Springer Nature. For a Cm+1 differential system on Rn, we study the limit cycles that can bifurcate from a zero–Hopf singularity, i.e., from a singularity with eigenvalues ±bi and n- 2 zeros for n≥ 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 ℓ limit cycles can bifurcate from the origin with ℓ∈ { 0 , 1 , … , 2 n-3} for m= 2 [see Llibre and Zhang (Pac J Math 240:321–341, 2009)], with ℓ∈ { 0 , 1 , … , 3 n-2} for m= 3 , with ℓ≤ 6 n-2 for m= 4 , and with ℓ≤ 4 · 5 n-2 for m= 5. Moreover, ℓ∈ { 0 , 1 , 2 } for m= 4 and n= 3 , and ℓ∈ { 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.

Original language | English |
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Pages (from-to) | 1159-1166 |

Journal | Nonlinear Dynamics |

Volume | 92 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 May 2018 |

### Keywords

- Arbitrary dimension
- Limit cycles
- Zero–Hopf singularity

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## Cite this

Barreira, L., Llibre, J., & Valls, C. (2018). Limit cycles bifurcating from a zero–Hopf singularity in arbitrary dimension.

*Nonlinear Dynamics*,*92*(3), 1159-1166. https://doi.org/10.1007/s11071-018-4115-3