This is a preprint of: “N-dimensional zero-Hopf Bifurcation of Polynomial Differential Systems via Averaging Theory of Second Order”, Sara Kassa, Jaume Llibre, Ammar Makhlouf, J. Dyn. Control Syst., 2020. DOI: [10.1007/s10883-020-09501-6] N-DIMENSIONAL ZERO-HOPF BIFURCATION OF POLYNOMIAL DIFFERENTIAL SYSTEMS VIA AVERAGING THEORY OF SECOND ORDER SARA KASSA1 , JAUME LLIBRE2 AND AMAR MAKHLOUF1 Abstract. Using the averaging theory of second order we study the limit cycles which bifurcate from a zero-Hopf equilibrium point of polynomial vector fields with cubic nonlinearities in Rn . We prove that there are at least 3n−2 limit cycles bifurcating from such zero-Hopf equilibrium points. Moreover we provide an examples in dimension 6 showing that this number of limit cycles is reached. 1. Introduction and statement of the main result Our goal is to study the limit cycles that bifurcate from a zero-Hopf equilibrium of polynomial differential systems in Rn with cubic nonlinearities by using the averaging theory. In [5] the authors studied the Hopf bifurcation in dimension n > 2, by using the first order averaging method. They proved that at least 2n−3 limit cycles can bifurcate from one singularity with eigenvalues ±bi and n − 2 zeros, i.e. from a zero-Hopf equilibrium of Rn . They proved for the first time that the number of bifurcated limit cycles in a Hopf bifurcation can grow exponentially with the dimension of the system. For a general information about Hopf bifurcations see [7]. In [2] the authors studied the occurrence of the limit cycles bifurcating from the origin of a differential system with cubic homogeneous nonlinearities in R4 . The authors proved that there are at most 9 = 34−2 limit cycles. In this paper we investigate the limit cycles bifurcating in a zero-Hopf bifurcation at the origin of coordinates of the following cubic polynomial differential systems in Rn 2 X X ẋ = (a1 ε + a2 ε2 )x − (b + b1 ε + b2 ε2 )y + εj aj,i1 ,...,in xi1 y i2 z3i3 · · · znin , j=0 i1 +...+in =3 X2 X (1) ẏ = (b + b1 ε + b2 ε2 )x + (a1 ε + a2 ε2 )y2 + εj bj,i1 ,...,in xi1 y i2 z3i3 · · · znin , j=0 i1 +...+in =3 2 X X (k) (k) (k) z˙k = (c1 ε + c2 ε2 )zk + εj cj,i1 ,...,in xi1 y i2 z3i3 . . . znin , j=0 i1 +...+in =3 where k = 3, . . . , n. Our main result is the following. Theorem 1. Consider the differential systems (1) in Rn with n ≥ 2. Applying to these systems the averaging theory of second order they can exhibit at least 3n−2 limit cycles bifurcating from the zero-Hopf equilibrium point localized at the origin of coordinates when ε = 0. In the next corollary we provide a differential system (1) in R6 exhibiting the maximum number of limit cycles stated in Theorem 1. 2010 Mathematics Subject Classification. Primary: 34C07, 34C05, 34C40. Key words and phrases. Hopf bifurcation, averaging theory, cubic polynomial differential systems. 1 2 S. KASSA, J. LLIBRE AND A. MAKHLOUF Corollary 2. Consider the polynomial differential system ẋ = 21 ε2 x − y − 21 x3 , ẏ = x + 21 ε2 y + 32 x2 y − y 3 , z˙3 = 23 ε2 (z + x2 y) − 21 z 3 , z˙4 = −ε2 u + 31 u3 , z˙5 = 32 ε2 v − 13 v 3 , z˙6 = 41 ε2 w − w3 + ε(−v 3 + x3 + y 3 ). It has 81 limit cycles bifurcating from the zero–Hopf equilibrium localized at the origin of coordinates when ε = 0. 2. The averaging theory of first and second order The aim of this section is to present the averaging theory of first and second order as it was developed in [1, 3, 4]. The following result is Theorem 4.2 of [1]. Theorem 3. We consider the following differential system (2) ẋ(t) = εF1 (t, x) + ε2 F2 (t, x) + ε3 R(t, x, ε), where F1 , F2 : R × D → Rn , R : R × D × (−εf , εf ) → Rn are continuous functions, T -periodic in the first variable, and D is an open subset of Rn . Assume that the following hypotheses (i) and (ii) hold. We assume: (i) F1 , F2 , R are locally Lipschitz with respect to x, F1 (t, .) ∈ C 1 (D) for all t ∈ R, and R is differentiable with respect to ε. We define f1 , f2 : D −→ Rn as Z T 1 f1 (z) = F1 (s, z)ds, T 0 (3) Z T  Z s  1 f2 (z) = Dz F1 (s, z) F1 (t, z)dt + F2 (s, z) ds. T 0 0 (ii) For V ⊂ D an open and bounded set and for each ε ∈ (−εf , εf ) \ {0}, there exists a ∈ V such that f1 (a) + εf2 (a) = 0 and dB (f1 + εf2 , V, a) 6= 0. Then for |ε| > 0 sufficiently small there exists a T -periodic solution ϕ(·, ε) of the system (4) such that ϕ(0, ε) → a when ε → 0. Where dB (f1 + εf2 , V, 0) denotes the Brouwer degree of the function f1 + εf2 in the neighborhood V of zero. It is known that if the function f1 + εf2 is C 1 then it is sufficient to check that det(D(f1 + εf2 (aε ))) 6= 0 in order to have that dB (f1 + εf2 , V, 0) 6= 0, for more details see [6]. For additional information on the averaging theory see the books [8, 10]. N-DIMENSIONAL ZERO-HOPF BIFURCATION OF POLYNOMIAL DIFFERENTIAL SYSTEMS 3 3. Proof of Theorem 1 We consider the polynomial differential system (1) with cubic nonlinearities in Rn . By doing the scaling (x, y, z3 . . . , zn ) = (εX, εY, εZ3 . . . , εZn ), system (1) becomes (4) 2 1X j X Ẋ = (a1 ε + a2 ε2 )X − (b + b1 ε + b2 ε2 )Y + ε aj,i1 ,...,in εi1 +...+in X i1 Y i2 Z3i3 . . . Znin , ε j=0 i +...+i =3 1 n 2 1X j X Ẏ = (b + b1 ε + b2 ε2 )X + (a1 ε + a2 ε2 )Y + ε εi1 +...+in bj,i1 ,...,in X i1 Y i2 Z3i3 . . . Znin , ε j=0 i 1 +...+in =3 2 (k) (k) 1X j X (k) Z˙k = (c1 ε + c2 ε2 )Zk + ε εi1 +...+in cj,i1 ,...,in X i1 Y i2 Z3i3 . . . Znin , ε j=0 i 1 +...+in =3 for k = 3, . . . , n. Since we have i1 + . . . + in = 3, then εi1 +...+in = ε3 . We write system (4) as 2 X X Ẋ = (a1 ε + a2 ε2 )X − (b + b1 ε + b2 ε2 )Y + ε2 εj aj,i1 ,...,in X i1 Y i2 Z3i3 . . . Znin , j=0 i1 +...+in =3 2 X X (5) Ẏ = (b + b1 ε + b2 ε2 )X + (a1 ε + a2 ε2 )Y + ε2 εj bj,i1 ,...,in X i1 Y i2 Z3i3 . . . Znin , j=0 i1 +...+in =3 2 X X (k) (k) (k) Z˙k = (c1 ε + c2 ε2 )Zk + ε2 ε j cj,i1 ,...,in X i1 Y i2 Z3i3 . . . Znin , j=0 i1 +...+in =3 for k = 3, . . . , n. We pass now to the cylindric coordinates (X, Y, Z3 , . . . , Zn ) = (ρ cos θ, ρ sin θ, η3 , . . . , ηn ), system (5) becomes  X 2 X ρ̇ = a1 ρε + ε2 cos θ εj aj,i1 ,...,in (ρ cos θ)i1 (ρ sin θ)i2 η3i3 . . . ηnin j=0 i1 +...+in =3 2 X X  + sin θ εj bj,i1 ,...,in (ρ cos θ)i1 (ρ sin θ)i2 η3i3 . . . ηnin + a2 ρ , j=0 i1 +...+in =3  2 X X (6) θ̇ = 1 ρ bρ + b 1 ρε + ε 2 cos θ εj bj,i1 ,...,in (ρ cos θ)i1 (ρ sin θ)i2 η3i3 . . . ηnin j=0 i1 +...+in =3 2 X X  − sin θ ε j aj,i1 ,...,in (ρ cos θ) i1 (ρ sin θ)i2 η3i3 . . . ηnin + b2 , j=0 i1 +...+in =3 X 2 X  (k) (k) (k) η˙k = c1 εηk + ε 2 εj cj,i1 ,...,in (ρ cos θ)i1 (ρ sin θ)i2 η3i3 . . . ηnin + c2 ηk , j=0 i1 +...+in =3 for k = 3, . . . , n. We take θ as the new independent variable in the neighborhood of (ρ, z3 , ..., zn ) = (0, 0, ..., 0), and system (6) writes  X dρ εa1 ε2 = ρ+ cos θ a0,i1 ,...,in (ρ cos θ)i1 (ρ sin θ)i2 η3i3 . . . ηnin dθ b b X i1 +...+in =3  i1 i2 i3 in a1 b1 + sin θ b0,i1 ,...,in (ρ cos θ) (ρ sin θ) η3 . . . ηn + a2 ρ − + O(ε3 ), i1 +...+in =3 b (k)  X (k)  dηk εc1 ε2 (k) (k) c b1 = ηk + cj,i1 ,...,in (ρ cos θ)i1 (ρ sin θ)i2 η3i3 . . . ηnin + c2 ηk − 1 + O(ε3 ), dθ b b i +...+i =3 b 1 n where k = 3, . . . , n. 4 S. KASSA, J. LLIBRE AND A. MAKHLOUF By using the notation of Theorem 3, i.e. x=z = (θ, ρ, η3 , . . . , ηn ), t = θ, F1 (t, x) = (F11 (θ, ρ, η3 , . . . , ηn ), F12 (θ, ρ, η3 , . . . , ηn ), F13 (θ, ρ, η3 , . . . , ηn ), . . . , F1n (θ, ρ, η3 , . . . , ηn )), F2 (t, x) = (F21 (θ, ρ, η3 , . . . , ηn ), F22 (θ, ρ, η3 , . . . , ηn ), F23 (θ, ρ, η3 , . . . , ηn ), . . . , F2n (θ, ρ, η3 , . . . , ηn )), T = 2π, where  (3) (n)  a1 c1 c F1 = ρ, η3 , . . . , 1 ηn b b b  X 1 F2 = (cos θ a0,i1 ,...,in (ρ cos θ)i1 (ρ sin θ)i2 η3i3 . . . ηnin b i +...+i =3 1 n X  a1 b1 + sin θ b0,i1 ,...,in (ρ cos θ) i1 (ρ sin θ)i2 η3i3 . . . ηnin + a2 ρ − , i1 +...+in =3 b  X (3)  1 (3) (3) c b1 c0,i1 ,...,in (ρ cos θ)i1 (ρ sin θ)i2 η3i3 . . . ηnin + c2 − 1 ,..., b i1 +...+in =3 b  X (n)  1 (n) (n) c1 b1 c0,i1 ,...,in (ρ cos θ)i1 (ρ sin θ)i2 η3i3 . . . ηnin + c2 − , b i1 +...+in =3 b where k = 3, . . . , n. We calculate the averaged function of the first order Z 2π 1 f1 (ρ, η3 , . . . , ηn ) = F1 (θ, ρ, η3 , . . . , ηn )dθ, 2π 0 and we get  a1  b ρ  (3) c1   b η3     .  f1 (ρ, η3 , . . . , ηn ) =  ,  .     .  (n) c1 b ηn The unique solution of f1 (ρ, η3 , . . . , ηn ) = (0, 0, . . . , 0) with respect to ρ, η3 , . . . , ηn is (ρ, η3 , . . . , ηn ) = (0, 0, . . . , 0). Then the averaging theory of the first order can not provide information about the existence of the periodic solutions. To pass to the second order, we make the first averaged function (k) identically null, i.e. we take a1 = c1 = 0 for k = 3, . . . , n. We calculate the averaged function of the second order using the formula (3). We get Z 2π 1 f2 (ρ, η3 , . . . , ηn ) = F2 (θ, ρ, η3 , . . . , ηn )dθ, 2π 0 N-DIMENSIONAL ZERO-HOPF BIFURCATION OF POLYNOMIAL DIFFERENTIAL SYSTEMS 5 because F1 (θ, ρ, η3 , . . . , ηn ) = (0, 0, . . . , 0), where Z 2π  X 1 f21 (ρ, η3 , . . . , ηn ) = cos θ a0,i1 ,...,in (ρ cos θ)i1 (ρ sin θ)i2 η3i3 . . . ηnin 2πb 0 i1 +...+in =3 X  + sin θ b0,i1 ,...,in (ρ cos θ) i1 (ρ sin θ)i2 η3i3 . . . ηnin + a2 ρ dθ i1 +...+in =3 1 = I1 , 2πb Z 2π  X  1 (k) (k) f2k (ρ, η3 , . . . , ηn ) = c0,i1 ,...,in (ρ cos θ)i1 (ρ sin θ)i2 η3i3 . . . ηnin + c2 ηk dθ 2πb 0 i1 +...+in =3 1 = I2 , 2πb where k = 3, . . . , n, and  Z 2π I1 = ρ3 ((a0,1,2,0,..,0 + b0,2,1,0,..,0 ) cos2 θ sin2 θ + a0,3,0,..,0 cos4 θ + b0,0,3,0,..,0 sin4 θ) 0  X X  +ρ a0,1,0,i3 ..in cos2 θη3i3 ..ηnin + b0,0,1,i3 ..in sin2 θη3i3 ..ηnin + a2 dθ i1 +...+in =2 i1 +...+in =2 1 = π(a0,1,2,0,..,0 + b0,2,1,0,..,0 + 3(a0,3,0,..,0 + b0,0,3,0,..,0 ))ρ3 4  X X + π 2a2 + a0,1,0,i3 ..in η3i3 ..ηnin + b0,0,1,i3 ..in η3i3 ..ηnin ρ, i1 +...+in =2 i1 +...+in =2 Z 2π   X X  (k) (k) I2 = ρ2 c0,2,0,i3 ..,in cos2 θη3i3 ..ηnin + c0,0,2,i3 ,..,in η3i3 ..ηnin sin2 θ 0 i3 +..+in =1 i3 +..+in =1 X  (k) (k) + c0,0,0,i3 ..,in η3i3 ..ηnin + c2 ηk dθ i3 +..+in =3   X X  (k) (k) = π ρ2 c0,2,0,i3 ..,in η3i3 ..ηnin + c0,0,2,i3 ,..,in η3i3 ..ηnin i3 +..+in =1 i3 +..+in =1 X  (k) (k) +2 c0,0,0,i3 ..,in η3i3 ..ηnin + c2 ηk , i3 +..+in =3 where k = 3, . . . , n. Then the averaged function of the second order is  ρ f21 = 8a2 + (a0,1,2,0,..,0 + b0,2,1,0,..,0 + 3(a0,3,0,..,0 + b0,0,3,0,..,0 ))ρ2 8b X X  + a0,1,0,i3 ..in η3i3 ..ηnin + b0,0,1,i3 ..in η3i3 ..ηnin , i1 +...+in =2 i1 +...+in =2   X X  1 (k) (k) f2k = ρ2 c0,2,0,i3 ..,in η3i3 . . . ηnin + c0,0,2,i3 ,..,in η3i3 ..ηnin 2b i 3 +..+in =1 i3 +..+in =1 X  (k) (k) +2 c0,0,0,i3 ..,in η3i3 ..ηnin + c2 η k , i3 +..+in =3 where k = 3, . . . , n. 6 S. KASSA, J. LLIBRE AND A. MAKHLOUF Now we solve the system of the averaged functions of the second order with respect to ρ, η3 , . . . , ηn . First we isolate the expression of ρ2 from f21 = 0, and we obtain X X 8a2 + a0,1,0,i3 ..in η3i3 ..ηnin + b0,0,1,i3 ..in η3i3 ..ηnin i1 +...+in =2 i1 +...+in =2 ρ2 = − . a0,1,2,0,..,0 + b0,2,1,0,..,0 + 3(a0,3,0,..,0 + b0,0,3,0,..,0 ) After we substitute the expression of ρ2 in f2k = 0 for k = 3, . . . , n. By using the Bezout Theorem (see [9]) we obtain that these functions admit at most 3n−2 real zeros (ρ∗ , ηk∗ ) for k = 1, . . . , 3n−2 . Since the coefficients of the system f2k = 0 are independent, we can take these 3n−2 real zeros with the coordinate ρ positive. Therefore, going back through the changes of coordinates, these zeros provide at least 3n−2 periodic solutions bifurcating from the zero-Hopf equilibrium at the origin of coordinates. Note that since the number of zeros are the maximum number provided by the Bezout Theorem the determinants ! ∂(f21 , f2k ) det ∂(ρ, ηk ) ∗) (ρ,ηk )=(ρ∗ ,ηk n−2 are non-zero for k = 1, . . . , 3 . This completes the proof of Theorem 1. 4. Proof of the Corollary 2 We consider the cubic polynomial differential system (2). By doing the scaling, passing to the cylindrical coordinates (ρ cos θ, ρ sin θ, η3 , η4 , η5 , η6 ) and taking θ as the new independent variable, we get that the functions F2j (θ, ρ, η3 , η4 , η5 , η6 ) for j = 1, . . . , 5 are 1 F21 (θ, ρ, η3 , η4 , η5 , η6 ) = − ρ(ρ2 (6 cos θ4 − 7 cos θ2 + 2) − 1), 2 1 F22 (θ, ρ, η3 , η4 , η5 , η6 ) = − η3 (2η32 − 9), 3 1 F23 (θ, ρ, η3 , η4 , η5 , η6 ) = η4 (η42 − 3), 3 1 F24 (θ, ρ, η3 , η4 , η5 , η6 ) = − η5 (2η52 − 9), 6 1 F25 (θ, ρ, η3 , η4 , η5 , η6 ) = −η63 + η6 . 4 We integrate these last functions from 0 to 2π, and we get the averaged functions of the second order f2 (ρ, η3 , η4 , η5 , η6 ) = f2j for j = 0, . . . , 5 1 f21 = − ρ(3ρ2 − 4), 8 1 f22 = − η3 (2η32 − 9), 6 1 f23 = η4 (η42 − 3), 3 1 f24 = − η5 (2η52 − 9), 6 1 f25 = − η6 (4η62 − 1). 4 We solve the system of the averaged functions of the second order (f21 , f22 , f23 , f24 , f25 ) = (0, 0, 0, 0, 0) with respect to ρ, η3 , η4 , η5 and η6 , we get 81 solutions zi = N-DIMENSIONAL ZERO-HOPF BIFURCATION OF POLYNOMIAL DIFFERENTIAL SYSTEMS 7 (ρ∗i , ηij∗ ) with ρ∗i > 0 for i = 1, . . . , 81 and j = 3, . . . , 6 √ √ √ z1 = ( 32 √3, 0, 0, 0, 0), √ z2 = ( 23 √3, 0, 0, √ 3 2 2, 0), z3 = ( 32 √3, 0, 0,√− 32 2, 0), z4 = ( 32 √3, 0, √3, 0,√ 0), 2 z5 = ( 3 √3, 0, −√3, 0,√ 0), z6 = ( 32 √3, 0, √3, 32 √ 2, 0), z7 = ( 23 √3, 0, −√3, 32 √ 2, 0), z8 = ( 32 √3, 0, √3, − 32 2, 0), z9 = ( 32 √3, 0, − √3, − 32 2, 0), z10 = ( 32 √3, 32 √2, 0, 0,√ 0), 2 3 z11 = ( 3 √3, − 2 √2, 0, 0,√ 0), z12 = ( 32 √3, 32 √2, 0, 23 √ 2, 0), z13 = ( 23 √3, − 32 √2, 0, 32 √ 2, 0), z14 = ( 32 √3, 32 √2, 0, − 3 √ 2 2, 0), z15 = ( 32 √3, − 32 √2, 0,√ 2− 3 2, 0), z 16 = ( 2 3√ 3, 3 2√ 2, √3, 0, 0), z17 = ( 32 √3, − 32 √2, √ 3, 0, 0), z18 = ( 32 √3, 32 √2, − √ 3,3 √ 0, 0), z19 = ( 32 √3, − 32 √2, − √ 3√ 3, 0, 0), z 20 = ( 2 3√ 3, 3 2√ 2, 3, √ 2 3√ 2, 0), z21 = ( 32 √3, −√ 3 2 2, √ 3, 2√ 2, 0), z 22 = ( 2 3√ 3, 3 2 √ 2, − 3, √ 2 3√ 2, 0), z23 = ( 32 √3, 32 √ 2, √ 3, − 32 √ 2, 0), z24 = ( 32 √3, −√ 3 2 2, − √ 3, 2 √2, 0), 2 3 3 z25 = ( 3 √3, − 2 √2, √ 3, − 2 √ 2, 0), z26 = ( 3 √3, 2 2, − 3, − 32 2, 0), 2 3 2 3 3 z27 = ( 3 √3, − 2 2, − 3, − 2 2, 0), z28 = ( 23 √3, 0, 0, 0,√12 ), 2 1 z29 = ( 3 √3, 0, 0, 0, −√2 ), z30 = ( 32 √3, 0, 0, 32 √2, 21 ), 2 3 1 z31 = ( 3 √3, 0, 0, − 2 √2, 2 ), z32 = ( 32 √3, 0, 0, 3 √ 2 2,1 − 2 ), 1 2 3 1 2 z33 = ( 3 √3, 0, 0,√ − 2 2, − 2 ), z34 = ( 3 √3, 0, √3, 0, 2 ), z35 = ( 23 √3, 0, −√3, 0, 12 ), z36 = ( 32 √3, 0, √3, 0,√ − 21 ), 2 1 2 3 z37 = ( 3 √3, 0, −√3, 0,√ − 2 ), z38 = ( 3 √3, 0, √3, 2 √ 2, 12 ), 2 3 1 2 3 1 z39 = ( 3 √3, 0, −√3, 2 √ 2, 2 ), z40 = ( 3 √3, 0, √3, −√ 2 2, 2 ), 2 3 1 2 3 z41 = ( 3 √3, 0, −√3, −√ 2 2, 2 ), z42 = ( 3 √3, 0, √3, 2 √ 2, − 12 ), 2 3 1 z43 = ( 3 √3, 0, −√3, 2 √ 2, − 2 ), z44 = ( 3 √3, 0,√ 3, − 2 2, − 12 ), 2 3 2 3 1 z45 = ( 3 √3, 0, −√ 3, − 2 2, − 2 ), z46 = ( 32 √3, 32 √2, 0, 0, 21 ), 2 3 1 z47 = ( 3 √3, − 2 √2, 0, 0, 2 ), z48 = ( 32 √3, 32 √2, 0, 0,√ − 21 ), 2 3 1 2 3 3 z49 = ( 3 √3, − 2 √2, 0, 0,√ − 2 ), z50 = ( 3 √3, 2 √2, 0, 2 √ 2, 21 ), 2 3 3 1 2 3 3 1 z51 = ( 3 √3, − 2 √2, 0, 2 √ 2, 2 ), z52 = ( 3 √3, 2 √2, 0, −√ 2 2, 2 ), z53 = ( 32 √3, − 32 √2, 0, −√ 3 2 2, 2 ), 1 z54 = ( 32 √3, 32 √2, 0, 23 √ 2, − 12 ), 2 3 3 1 2 3 3 1 z55 = ( 3 √3, − 2 √2, 0, 2 √ 2, − 2 ), z56 = ( 3 √3, 2 √2, 0, √ − 2 12, − 2 ), 2 3 3 1 2 3 z57 = ( 3 √3, − 2 √2, 0, √ − 2 12, − 2 ), z58 = ( 3 √3, 2 √2, √ 3, 0, 2 ), z59 = ( 23 √3, − 32 √2, √ 3, 0, 2 ), z60 = ( 32 √3, 32 √2, − √ 3, 0, 21 ), 1 2 3 1 2 3 z61 = ( 3 √3, − 2 √2, − √ 3, 0, 21 ), z62 = ( 3 √3, 2 √2, √ 3, 0, − 2 ), z63 = ( 23 √3, − 32 √2, √ 3, 0, − 2 ), z64 = ( 23 √3, 32 √2, − √ 3,3 √ 0, − 12 ), 2 3 1 2 3 z65 = ( 3 √3, − 2 √2, − √ 3,3 √ 0, − 2 ), z66 = ( 3 √3, 2 √2, √ 3, 2 √ 2, 21 ), 2 3 1 2 3 3 z67 = ( 3 √3, −√ 2 2, √ 3, 23 √2, 21 ), z68 = ( 3 √3, 2 √ 2, − √ 3, 2 √ 2, 21 ), 2 3 2 3 3 1 z69 = ( 3 √3, 2 √ 2, √ 3, − 2 √ 2, 2 ), z70 = ( 3 √3, −√ 2 2, − √ 3, 23 √2, 21 ), 2 3 3 1 2 3 z71 = ( 3 √3, − 2 √2, √ 3, − 2 √ 2, 2 ), z72 = ( 3 √3, 2 √2, − √ 3,3 √ − 2 2, 2 ), z73 = ( 23 √3, − 32 √2, − 3, √ 3√ 2 1 2− 3 2, 1 ), z 74 = ( 2 3√ 3, 3 2√ 2, 3, √ 2 3√ 2, − 21 ), 2 3 2 3 z75 = ( 3 √3, −√ 2 2, √ 3, 23 √2, − 21 ), z76 = ( 3 √3, 2 √ 2, − √ 3, 2 √ 2, − 21 ), 2 3 2 3 3 1 z77 = ( 3 √3, 2 √ 2, √ 3, − 2 √ 2, − 2 ), z78 = ( 3 √3, −√ 2 2, − √ 3, 23 √2, − 21 ), 2 3 3 1 2 3 z79 = ( 3 √3, − 2 √2, √ 3, − 2 √ 2, − 2 ), z80 = ( 3 3, 2 2, − 3, − 2 2, − 2 ), 2 3 3 1 z81 = ( 3 3, − 2 2, − 3, − 2 2, − 2 ). The determinants ! ∂(f21 , f22 , f23 , f24 ) det ∂(ρ, η j ) (%,η j )=(ρ∗ j∗ i ,ηi ) 9 9 9 9 9 evaluated at the zeros are given by 16 , − 8 , 4 , − 8 , − 2 and 9. All of these determinants are non-zero. So there are 81 limit cycles bifurcating from the zero Hopf-equilibrium localized at the origin of coordinates. 8 S. KASSA, J. LLIBRE AND A. MAKHLOUF 5. Conclusion Using the averaging theory of the second order we show that the number of the limit cycles bifurcating from a zero-Hopf equilibrium point of a polynomial differential systems with cubic non- linearities increases at least exponentially as 3n−2 if n is the dimension of the differential system. References [1] A. Buicǎ and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. math. 128 (2004), 7–22. [2] A. Feddaoui, J. Llibre and A. Makhlouf, 4-dimensional zero-Hopf bifurcation for polynomial differentials systems with cubic homogeneous nonlinearities via averaging theory. 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[9] I.R.Shafarevich, Basic algebraic geometry, Springer, 1974. [10] F. Verhulst, Nonlinear differential equations and dynamical systems, Universitex, Berlin, Springer-Verlag, 1990. 1 Department of Mathematics, University of Annaba, Laboratory LMA, P.O.Box 12, Annaba 23000, Algeria Email address: kassasara03@outlook.com, makhloufamar@yahoo.fr 2 Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat