TY - JOUR

T1 - Limit cycles bifurcating from a zero–Hopf singularity in arbitrary dimension

AU - Barreira, Luis

AU - Llibre, Jaume

AU - Valls, Claudia

PY - 2018/5/1

Y1 - 2018/5/1

N2 - © 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.

AB - © 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.

KW - Arbitrary dimension

KW - Limit cycles

KW - Zero–Hopf singularity

UR - https://ddd.uab.cat/record/199351

U2 - https://doi.org/10.1007/s11071-018-4115-3

DO - https://doi.org/10.1007/s11071-018-4115-3

M3 - Article

VL - 92

SP - 1159

EP - 1166

JO - Nonlinear Dynamics

JF - Nonlinear Dynamics

SN - 0924-090X

IS - 3

ER -