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

For a C^m+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.