This is a preprint of: “4-dimensional zero-Hopf bifurcation for polynomial differentials systems with cubic homogeneous nonlinearities via averaging theory”, Amina Feddaoui, Jaume Llibre, Ammar Makhlouf, Int. J. Dyn. Syst. Differ. Equ., vol. 10(4), 321–328, 2020. 4-DIMENSIONAL ZERO-HOPF BIFURCATION FOR POLYNOMIAL DIFFERENTIALS SYSTEMS WITH CUBIC HOMOGENEOUS NONLINEARITIES VIA AVERAGING THEORY AMINA FEDDAOUI1 , JAUME LLIBRE 2 AND AMAR MAKHLOUF1 Abstract. The averaging theory of second order shows that for polynomial dif- ferential systems in R4 with cubic homogeneous nonlinearities at least nine limit cycles can be born in a zero-Hopf bifurcation. 1. Introduction and statement of the main result Our goal is to study the periodic solutions which can bifurcate at a zero-Hopf bifurcation in a polynomial differential systems in R4 with cubic homogeneous non- linearities by using the averaging theory of the second order. In [7] the authors studied the zero-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. In [5] (resp. [2]) the authors studied the zero-Hopf bifurcation in polynomial dif- ferential systems in R3 (resp. R4 ) with quadratic homogeneous nonlinearities. By applying the averaging theory of the second order to these systems, they show that at most 3 limit cycles can bifurcate from a singular point having eigenvalues of the form ±bi and one zero (resp. two zeros). The zero-Hopf bifurcation in polynomial differential systems in R3 with cubic homogeneous nonlinearities has been studied recently in [3]. In this paper we are interested on the existence of periodic solutions bifurcating from the origin of coordinates of a polynomial differential systems in R4 with cubic homogeneous nonlinearities having eigenvalues ±bi and two zeros, i.e for the differ- ential systems 2 ∑ 2 2 ẋ = (a1 ε + a2 ε )x − (b + b1 ε + b2 ε )y + εj Xj (x, y, z, w), j=0 (1) 2 ∑ ẏ = (b + b1 ε + b2 ε2 )x + (a1 ε + a2 ε2 )y + εj Yj (x, y, z, w), j=0 2010 Mathematics Subject Classification. Primary: 34C07, 34C05, 34C40. Key words and phrases. Hopf bifurcation, averaging theory, cubic polynomial differential systems. 1 http://www.gsd.uab.cat 2 A. FEDDAOUI, J. LLIBRE AND A. MAKHLOUF 2 ∑ ż 2 = (c1 ε + c2 ε )z + εj Zj (x, y, z, w), j=0 2 ∑ ẇ = (d1 ε + d2 ε )w + 2 εj Wj (x, y, z, w), j=0 where Xj (x, y, z, w) = aj0 x3 + aj1 x2 y + aj2 x2 z + aj3 x2 w + aj4 xy 2 + aj5 xyz + aj6 xyw +aj7 xz 2 + aj8 xzw + aj9 xw2 + aj10 y 3 + aj11 y 2 z + aj12 y 2 w + aj13 2 yz + aj14 yzw + aj15 yw2 + aj16 z 3 + aj17 z 2 w + aj18 zw2 + aj19 w3 , Yj (x, y, z, w), Zj (x, y, z, w) and Wj (x, y, z, w) have the same expression as Xj (x, y, z, w) by replacing aji by bji , cji and dji for j = 0, 1, 2 and i = 0, 1, . . . , 19, respectively. The coefficients aij , bij , cij , dij , a1 , a2 , b, b1 , b2 , c1 , c2 , d1 , d2 are real parameters with b ̸= 0. Note that system (1) for ε = 0 at the origin has eigenvalues ±bi, 0, 0. So for ε = 0 the origin is a zero-Hopf equilibrium. Our main result is the following one. Theorem 1. By applying averaging theory of second order system (1) can exhibit at least 9 periodic solutions bifurcating from the origin when ε = 0, and this number of periodic solutions is reached if and only if the following condition is satisfied (3a00 + a04 + b01 + 3b010 )b ̸= 0. Theorem 1 is proved in section 3. In section 2 we recall the averaging theory of first and second order as it was stated in [1]. This will be the main tool for proving Theorem 1. 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, 4, 6]. The following result is Theorem 4.2 of [1]. Theorem 2. We consider the 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. 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 the averaging functions of 4-DIMENSIONAL ZERO-HOPF BIFURCATION VIA AVERAGING THEORY 3 first and second order f1 , f2 : D −→ Rn as ∫ T 1 f1 (z) = F1 (s, z)ds, T 0 (3) ∫ T [ ∫ 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) ̸= 0. Then for |ε| > 0 sufficiently small there exists a T -periodic solution x(t, ε) of the system (2) such that x(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 ))) ̸= 0 in order to have that dB (f1 + εf2 , V, 0) ̸= 0, for more details see [8]. On the other hand if one of the real parts of the eigenvalues of the Jacobian matrix D(f1 +εf2 )(aε ) is positive the periodic solution x(t; ε) is unstable. If all the real parts of the eigenvalues of this matrix are negative the periodic solution is locally stable. For a proof see Theorem 11.6 of [10]. For a general information on the averaging theory see for instance the books [9, 10]. 3. Proof of Theorem 1 First we scale the variables (x, y, z, w) doing the change of variables (x, y, z, w) = (εX, εY, εZ, εW ), second we pass to cylindrical coordinates doing (X, Y, Z, W ) = (ρ cos θ, ρ sin θ, η, ξ), and third we take the angle θ as the new independent variable. Thus in the variables (ρ, η, ξ) system (1) writes dρ = εF11 (θ, ρ, η, ξ) + ε2 F21 (θ, ρ, η, ξ) + O(ε3 ), dθ dη (4) = εF12 (θ, ρ, η, ξ) + ε2 F22 (θ, ρ, η, ξ) + O(ε3 ), dθ dξ = εF13 (θ, ρ, η, ξ) + ε2 F23 (θ, ρ, η, ξ) + O(ε3 ). dθ 4 A. FEDDAOUI, J. LLIBRE AND A. MAKHLOUF where a1 ρ F11 (θ, ρ, η, ξ) = , b c1 η F12 (θ, ρ, η, ξ) = , b d1 ξ F13 (θ, ρ, η, ξ) = , b 1 1 F21 (θ, ρ, η, ξ) = 2 (−b1 cos(θ)2 − b1 sin(θ)2 )(a1 ρ cos(θ)2 + a1 ρ sin(θ)2 ) + ((a016 η 3 + b b a017 η ξ + a018 ηξ + a019 ξ ) cos(θ) + (a2 ρ + a07 η ρ + a08 ηξρ + a09 ξ 2 ρ 2 2 3 2 ) cos(θ)2 + (a02 ηρ2 + a03 ξρ2 ) cos(θ)3 + a00 ρ3 cos(θ)4 + (b016 η 3 + b017 η 2 ξ + b018 ηξ 2 + b019 ξ 3 ) sin(θ) + ((a013 + b07 )η 2 ρ + (a014 + b08 )ηξρ + ( a015 + b09 )ξ 2 ρ) cos(θ) sin(θ) + ((a05 + b02 )ηρ2 + (a06 + b03 )ξρ2 ) cos(θ)2 sin(θ) + (a01 + b00 )ρ3 cos(θ)3 sin(θ) + (a2 ρ + b013 η 2 ρ + b014 ηξρ + b015 ξ 2 ρ) sin(θ)2 + ((a011 + b05 )ηρ2 + (a012 + b06 )ξρ2 ) cos(θ) sin(θ)2 + (a04 +b01 )ρ3 cos(θ)2 sin(θ)2 + (b011 ηρ2 + b012 ξρ2 ) sin(θ)3 + (a010 + b04 )ρ3 cos(θ) sin(θ)3 + b010 ρ3 sin(θ)4 ), 1 F22 (θ, ρ, η, ξ) = (bc2 η + bc016 η 3 + bc017 η 2 ξ + bc018 ηξ 2 + bc019 ξ 3 + b(c07 η 2 ρ + c08 ηξρ b2 +c09 ξ 2 ρ) cos(θ) + (−b1 c1 η + bc02 ηρ2 + bc03 ξρ2 ) cos(θ)2 + b(c013 η 2 ρ + c014 ηξρ + c015 ξ 2 ρ) sin(θ) + bc00 ρ3 cos(θ)3 + bc01 ρ3 cos(θ)2 sin(θ) + b( c05 ηρ2 + c06 ξρ2 ) sin(θ) cos(θ) + (−b1 c1 η + bc011 ηρ2 + bc012 ξρ2 ) sin(θ)2 +bc04 ρ3 cos(θ) sin(θ)2 + bc010 ρ3 sin(θ)3 ), 1 F23 (θ, ρ, η, ξ) = (bd2 η + bd016 η 3 + bd017 η 2 ξ + bd018 ηξ 2 + bd019 ξ 3 + b(d07 η 2 ρ + d08 ηξρ b2 +d09 ξ 2 ρ) cos(θ) + (−b1 d1 η + bd02 ηρ2 + bd03 ξρ2 ) cos(θ)2 + b(d013 η 2 ρ + d014 ηξρ + d015 ξ 2 ρ) sin(θ) + bd00 ρ3 cos(θ)3 + bd01 ρ3 cos(θ)2 sin(θ) + b( d05 ηρ2 + d06 ξρ2 ) sin(θ) cos(θ) + (−b1 d1 η + bd011 ηρ2 + bd012 ξρ2 ) sin(θ)2 +bd04 ρ3 cos(θ) sin(θ)2 + bd010 ρ3 sin(θ)3 ). System (4) is written into the normal form (2) for applying the averaging theory taking x=z = (ρ, η, ξ), t = θ, F1 (t, x) = (F11 (θ, ρ, η, ξ), F12 (θ, ρ, η, ξ), F13 (θ, ρ, η, ξ)), F2 (t, x) = (F21 (θ, ρ, η, ξ), F22 (θ, ρ, η, ξ), F23 (θ, ρ, η, ξ)), T = 2π. . 4-DIMENSIONAL ZERO-HOPF BIFURCATION VIA AVERAGING THEORY 5 From (3) we have that the first averaging function f1 = (f11 , f12 , f13 ) is ∫ 2π 1 f1i (ρ, η, ξ) = F1i (θ, ρ, η, ξ)dθ. 2π 0 Doing these computations we get that a1 ρ c1 η d1 ξ f11 (ρ, η, ξ) = , f12 (ρ, η, ξ) = , f13 (ρ, η, ξ) = . b b b Since we look for solutions (ρ∗ , η ∗ , ξ ∗ ) of f1 (ρ, η, ξ) = 0 with ρ∗ > 0, if a1 ̸= 0 the first averaging function does not provide any information on the periodic solutions of the differential system (3). In order that the second averaging function can give information on the periodic solutions of the differential system (3) the first averaging function must be identically zero. So we take a1 = c1 = d1 = 0, and compute the second averaging function. Then from (3) we have that f2 = (f21 , f22 , f23 ) = (f21 (ρ, η, ξ), f22 (ρ, η, ξ), f23 (ρ, η, ξ) is given by ρ f21 = (4(a07 + b013 )η 2 + 4(a08 + b014 )ηξ + (3a00 + a04 + b01 + 3b010 )ρ2 + 8b 4(a09 + b015 )ξ 2 + 8a2 ), 1 (5) f22 = (2c016 η 3 + 2c2 η + 2ξ(c017 η 2 + c018 ηξ + c019 ξ 2 ) + (c02 + c011 )ηρ2 + 2b (c03 + c012 )ρ2 ξ), 1 f23 = (2d016 η 3 + 2d2 η + 2ξ(d017 η 2 + d018 ηξ + d019 ξ 2 ) + (d02 + d011 )ηρ2 + 2b (d03 + d012 )ρ2 ξ). We isolate ρ2 from the equationf21 (ρ, η, ξ) = 0, and we substitute it in f2i (ρ, η, ξ) = 0 for i = 2, 3. Then we get two polynomials (g22 , g23 ) = (g22 (η, ξ), g23 (η, ξ)) given by 1 g22 = (C1 η + C2 ξ + C3 ηξ 2 + C4 η 2 ξ + C5 η 3 + C6 ξ 3 ) = 0, (3a00 + a04 + b01 + 3b010 )b 1 g23 = (D1 η + D2 ξ + D3 ηξ 2 + D4 η 2 ξ + D5 η 3 + D6 ξ 3 ) = 0, (3a00 + a04 + b01 + 3b010 )b where C1 = −4a2 c02 − 4a2 c011 + 3a00 c2 + a04 c2 + b01 c2 + 3b010 c2 , C2 = −4a2 (c03 + c012 ), C3 = −2(a08 c03 + a08 c012 + a09 c02 + a09 c011 + b014 c03 + b014 c012 + b015 c02 + b015 c011 )+ 3a00 c018 + a04 c018 + b01 c018 + 3b010 c018 , C4 = −2(a07 c03 + a07 c012 + a08 c02 + a08 c011 + b013 c03 + b013 c012 + b014 c02 + b014 c011 )+ 3a00 c017 + a04 c017 + b01 c017 + 3b010 c017 , C5 = −2(c02 + c011 )(a07 + b013 ) + 3a00 c016 + a04 c016 + b01 c016 + 3b010 c016 , 6 A. FEDDAOUI, J. LLIBRE AND A. MAKHLOUF C6 = 3a00 c019 + a04 c019 − 2a09 c03 − 2a09 c012 + b01 c019 + 3b010 c019 − 2b015 c03 − 2b015 c012 , D1 = −4a2 (d02 + d011 ) + 3a00 d2 + a04 d2 + b01 d2 + 3b010 d2 , D2 = −4a2 (d03 + d012 ), D3 = −2(a08 d03 + a08 d012 + a09 d02 + a09 d011 + b014 d03 + b014 d012 + b015 d02 + b015 d011 )+ 3a00 d018 + a04 d018 + b01 d018 + 3b010 d018 , D4 = −2(a07 d03 + a07 d012 + a08 d02 + a08 d011 + b013 d03 + b013 d012 + b014 d02 + b014 d011 )+ 3a00 d017 + a04 d017 + b01 d017 + 3b010 d017 , D5 = −2(d02 + d011 )(a07 + b013 ) + 3a00 d016 + a04 d016 + b01 d016 + 3b010 d016 , D6 = 3a00 d019 +a04 d019 −2a09 d03 −2a09 d012 +b01 d019 +3b010 d019 −2b015 d03 −2b015 d012 . We suppose that 3a00 + a04 + b01 + 3b010 ̸= 0. Looking only at the coefficients of system (1) which appear in Cj and Dj we see that C1 , C2 , C3 , C4 , C5 , C6 , D1 , D2 , D3 , D4 , D5 , D6 are all independent because the rank of the Jacobian matrix of the functions C1 , C2 , C3 , C4 , C5 , C6 , D1 , D2 , D3 , D4 , D5 , D6 with respect to the variables a00, a04 , a07 , a08 , a09 , a2 , b01 , b010 , b013 , b014 , b015 , c02 , c03 , c011 , c012 , c016 , c017 , c018 , c019 , c2 , d02 , d03 , d011 , d012 , d016 , d017 , d018 , d019 , d2 is 12, as it can be easily checked using maple or mathematica. In short, since all coefficients of the polynomials g22 (η, ξ) = 0 and g23 (η, ξ) = 0 are independent they can be chosen arbitrary. By Bezout Theorem, we know that at most system g22 (η, ξ) = 0, g23 (η, ξ) = 0 has 9 solutions. We give an example of polynomial differential system with nine limit cycles xε2 ẋ = 3w2 x + xz 2 + a013 yz 2 + a016 z 3 + − y, 2 yε2 ẏ = 4w2 y + (−4)x2 y + 2yz 2 + + x, (6) 2 21w2 z 3x2 z 3wz 2 ż = −4w3 + wy 2 + − − + 2z 3 + zε2 , 4 2 2 3wz 2 15z 3 zε2 ẇ = −7w2 z + x2 z + y 2 z − − − . 4 4 4 From system (5) we have for system (6) that 1 f21 (ρ, η, ξ) = ρ(1 + 3η 2 + 7ξ 2 − ρ2 ), 2 1 3 (7) f22 (ρ, η, ξ) = (8η − 6η 2 ξ + η(4 + 21ξ 2 − 3ρ2 ) + 2ξ(−8ξ 2 + ρ2 )), 4 1 f23 (ρ, η, ξ) = − η(1 + 15η 2 + 3ηξ + 28ξ 2 − 4ρ2 ). 4 4-DIMENSIONAL ZERO-HOPF BIFURCATION VIA AVERAGING THEORY 7 Solving system (7) there are nine solutions zi = (ρ∗i , ηi∗∗ , ξi∗ ) with ρ∗i > 0 for i = 1, . . . , 9 given by z1 = (2, −1, 0), √ z2 = (2 2, 0, −1), z3 = (1, 0, 0), √ z4 = (2 2, 0, 1), z5 = (2, 1, 0), (√ ( √ )) 1 √ √ √ 1 √ √ 3 z6 = (55 − 13 7), − 3 − 7, −4 3 − 7 + (3 − 7) 2 , 2 2 (√ ( √ )) 1 √ √ √ 1 √ √ 3 z7 = (55 − 13 7), 3 − 7, 4 3 − 7 − (3 − 7) 2 , 2 2 (√ ( √ )) 1 √ √ √ 1 √ √ 3 z8 = (55 + 13 7), − 3 + 7, −4 3 + 7 + (3 + 7) 2 , 2 2 (√ ( √ )) 1 √ √ √ 1 √ √ 3 z9 = (55 + 13 7), 3 + 7, 4 3 + 7 − (3 + 7) 2 . 2 2 Since the determinant ( ) ∂(f21 , f22 , f23 ) (8) det ∂(ρ, η, ξ) (ρ,η,ξ)=(ρ∗ ,η ∗ ,ξ ∗ ) 9 3 9 9 √ for these nine solutions zi = (ρ∗i , ηi∗ , ξi∗ ) are − , −6, , −6, − , − (−189 + 67 7), 2 8 2 8 9 √ 9 √ 9 √ − , (−189 + 67 7), (189 + 67 7), (189 + 67 7) respectively, we obtain, us- 8 8 8 ing the averaging theory of second order (see Theorem 2), 9 periodic solutions (ρi (θ, ε), ηi (θ, ε), ξi (θ, ε)) of system (4) such that (ρi (0, ε), ηi (0, ε), ξi (0, ε)) = zi . In fact, these periodic soutions are limit cycles because the solutions zi are isolated solutions of system (7) because the determinants (8) are non-zero. Going back through the changes of variables these 9 limit cycles provide 9 limit cycles (xi (t, ε), yi (t, ε), zi (t, ε), wi (t, ε) of the differential system (1) such that (xi (0, ε), yi (0, ε), zi (0, ε), wi (0, ε) = ε(ρ∗i cos(bt), ρ∗i sin(bt), ηi∗ , ξi∗ ) + O(ε2 ). Since these initial conditions then to the origin of R4 when ε → 0, the corresponding 9 limit cycles tend to the zero-Hopf singular point localized at the origin of R4 . In short, the differential system (1) can exhibit a zero-Hopf bifurcation at the origin of system (6) with at least 9 limit cycles. This completes the proof of Theorem 1. 8 A. FEDDAOUI, J. LLIBRE AND A. MAKHLOUF Acknowledgements This work is supported by the Ministerio de Economı̀a, Industria y competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajusts Universitaris i de Recerca grant 2017SGR1617, and the European project Dynamics-H2020-MSCA-RISE-2017-777911. References [1] A. Buicǎ and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. math. 128 (2004), 7–22. [2] D. Djeddid, J. Llibre and A. Makhlouf, 4-dimensional zero-Hopf bifurcation via averaging theory, prepint, 2018. [3] S. Kassa, J. Llibre and A. Makhlouf, 3-dimensional zero-Hopf ifurcation via averaging theory, preprint, 2018. [4] J. Llibre, C.A. Buzzi and P.R. da Silva, 3-dimensional Hopf bifurcation via averaging theory, Discrete Contin. Dynam. Syst. 17 (2007), 529–540. [5] J. Llibre, A. Makhlouf and S. Badi, 3-dimensional Hopf bifurcation via averaging theory of the second order. Discrete Contin. Dynam. Syst. 25(2009), 1287–1295. 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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 E-mail address: feddaouiamina90@gmail.com, makhloufamar@yahoo.fr 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat