This is a preprint of: “Limit cycles bifurcating from a zero-Hopf equilibrium of a 3-dimensional continuous differential system”, Sara Kassa, Jaume Llibre, Ammar Makhlouf, Sao Paulo J. Math. Sci., 8 pp., 2021. DOI: [10.1007/s40863-021-00212-9] Noname manuscript No. (will be inserted by the editor) Limit cycles bifurcating from a zero-Hopf equilibrium of a 3–dimensional continuous differential system Sara Kassa · Jaume Llibre · Amar Makhlouf Received: date / Accepted: date Abstract A zero–Hopf equilibrium of a differential system in R3 is an equi- librium point whose linear part has eigenvalues 0 and ±ωi with ω > 0. We provide necessary and sufficient conditions for the existence of two or one limit cycles bifurcating from a zero–Hopf equilibrium of the following 3–dimensional Lypschizian differential systems ẋ = y, ẏ = z, ż = −a|x| − y + 3y 2 − xz − b, when a = b = 0. Note that due to the existence of an absolute value the vector field associated to this system is only Lypschitz. We shall prove that these limit cycles persist for ab > 0 with a and b sufficiently small, i.e. when the differential system has no equilibria. Additionally we provide an estimation of the size of the bifurcating small limit cycles and also characterize their kind of stability or instability. We remark that there are no works which study the zero-Hopf bifurcation of nonsmooth differential systems as we do in this paper, and that the algorithm that we use here can be applied for studying the zero-Hopf bifurcation of an arbitrary nonsmooth differential system. Keywords continuous differential system · periodic orbit · limit cycle · averaging theory · zero–Hopf bifurcation · zero–Hopf equilibrium 2010 Mathematics Subject Classification. 37G15. S. Kassa and A. Makhlouf Department of Mathematics, University of Annaba, Laboratory LMA, P.O.Box 12, Annaba 23000, Algeria E-mail: kassasara03@outlook.com, makhloufamar@yahoo.fr J. Llibre Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain E-mail: jllibre@mat.uab.cat 2 Sara Kassa et al. 1 Introduction and statements of the main results Jafari, Sprott and Golpayegani in [7] introduced continuous differential sys- tems without equilibria, and studied the existence of chaotic motions in the dynamics of such systems. Azar et al. [1] motivated by the paper of Jafari, Sprott and Golpayegani consider the continuous differential systems ẋ = y, ẏ = z, (1) ż = −a|x| − y + 3y 2 − xz − b, with a > 0 and b > 0. Since the equation |x| = −b/a has no solution if ab > 0, differential systems (1) has no equilibrium points when ab > 0. Azar et al. analyzed the dynamical properties and the synchronization of the differential system (1), and they implemented a physical realization of this differential system in order to illustrate its feasibility. Our objective is to enrich the dynamical properties of the differential sys- tem (1) found in [1], showing that this system can exhibit for convenient values of its parameters a zero–Hopf equilibrium from which can bifurcate two or one limit cycles. A zero-Hopf equilibrium of system (1) is an equilibrium point whose linear part has eigenvalues 0 and ±ωi with ω > 0. A limit cycle of system (1) is a periodic orbit isolated in the set of all periodic orbits of system (1). Define the functions r ! 1 w2 w f1 (r, w) = 2w 1 − 2 − πrw + 2r arcsin , 2π r r and r ! 1 w2 w 2 f2 (r, w) = − πβ + 2r 1 − 2 − 2πr + 2w arcsin . π r r Then our main result is the following one. Theorem 1 Consider the Lipchitz differential system (1) with a = ε, b = ε2 β and |ε| sufficiently small. Then for each zero (r0 , w0 ) with r0 > 0 of the func- tion (f1 , f2 ) such that the determinant of the Jacobian matrix ∂(f1 , f2 )/∂(r, w) evaluated at (r0 , w0 ) is non–zero, there exist a limit cycle (x(t, ε), y(t, ε), z(t, ε)) of system (1) such that (x(0, ε), y(0, ε), z(0, ε)) = ε(r0 + w0 , 0, −r0 ) + O(ε2 ). If the two eigenvalues of the matrix ∂(f1 , f2 ) |(r,w)=(r0 ,w0 ) (2) ∂(r, w) have negative real parts then the periodic solution (x(t, ε), y(t, ε), z(t, ε)) is stable. If some of the eigenvalues has positive real part the periodic solution (x(t, ε), y(t, ε), z(t, ε)) is unstable. Zero-Hopf bifurcation 3 Theorem 1 is proved in section 3 using the averaging theory for computing periodic solutions developed in [2] and [3] for Lipschitz differential systems. We remark that the classical averaging theory for computing periodic solutions needs that the differential system be of class C 2 , see for instance Theorems 11.5 and 11.6 of the book of Verhulst [9]. For another study of the limit cycles of a Lipschitz system see for instance [5]. Corollary 1 Consider the Lipchitz differential system (1) with a = ε, b = ε2 β and |ε| sufficiently small. (a) System (1) has two periodic solutions (x± (t, ε), y± (t, ε), z± (t, ε)) such that p 2 1 ± 1 + 2π 2 β (x±(0, ε), y± (0, ε), z± (0, ε)) = ε(r± , 0, −r± )+O(ε ), r± = , 2π p p if 1 + 2π 2 β > 0, 1 − 1 + 2π 2 β > 0 and 1 + 2π 2 β − 7 1 + 2π 2 β 6= 0. (b) System (1) has one periodic solutions (x(t, ε), y(t, ε), z(t, ε)) such that p 2 1 + 1 + 2π 2 β (x(0, ε), y(0, ε), z(0, ε)) = ε(r+ , 0, −r+ ) + O(ε ), r+ = , 2π p p if 1 + 2π 2 β > 0, 1 − 1 + 2π 2 β < 0 and 1 + 2π 2 β − 7 1 + 2π 2 β 6= 0. (c) If additional topthe assumptions of statements (a) and (b) we have that −1 − 2π 2 β + 7 1 + 2π 2 β > 0, then the periodic solution associated to the zero (r+ , 0) of the function (f1 , f2 ) is unstable. Corollary 1 is also proved in section 3 using Theorem 2. 2 The averaging theory for Lipschitz differential systems In this section we present the basic results on the averaging theory for the Lipschitz differential systems that we need for proving our main result. The averaging theory reduces the problem of finding isolated periodic solu- tions to the problem of finding zeros of a function, called the averaged function. We need the averaging theory of first order for differential systems developed in [2] and [3] whose associated vector field is Lipschiz but not C 1 . The sufficient conditions for the existence of simple isolated zeros of the averaged function are given in terms of the Brouwer degree, see [6] for precise definitions. The classical averaging theory for finding isolated periodic solutions of a differential system needs that the vector field associated to the system be C 2 . See for general results on averaging theory the book of Sanders, Verhulst and Murdock [8]. Theorem 2 We consider the following differential system ẋ(t) = εF (t, x) + ε2 R(t, x, ε), (3) 4 Sara Kassa et al. where F : R × D → Rn , R : R × D × (−εf , εf ) → Rn are continuous functions, T –periodic in t, and D is an open subset of Rn . We define f : D → Rn as Z 1 T f (z) = F (s, z)ds, (4) T 0 and assume that (i) F and R are locally Lipschitz in x; (ii) for a ∈ D with f (a) = 0, there exists a neighborhood V of a such that f (z) 6= 0 for all z ∈ V \ {a} and dB (f, V, a) 6= 0 (where dB (f, V, a) denotes the Brouwer degree of f in the neighborhood V of a). Then for |ε| > 0 sufficiently small, there exists an isolated T –periodic solution x(t, ε) of system (3) such that x(0, ε) → a as ε → 0. If the averaged function f (z) is C 1 in some neighborhood of a fixed isolated zero a of f (z), then we can use the following remark in order to verify the hypothesis (ii) of Theorem 2. For more details see again [6]. Remark 1 Let f : D → Rn be a C 1 function, with f (a) = 0, where D is an open subset of Rn and a ∈ D. If a is a simple zero of f , i.e the determinant (det(Df (a)) 6= 0) of the Jacobian matrix of the function f at a is not zero, then there exists a neighborhood V of a such that f (z) 6= 0 for all z ∈ V \ {a}. Then dB (f, a, V, 0) ∈ {−1, 1}. In [3] Theorem 2 is improved as follows. Theorem 3 Under the assumptions of Theorem 2, for small ε the condition det(Df (a)) 6= 0 ensures the existence and uniqueness of a T −periodic solution x(t, ε) of system (3) such that x(0, ε) → a as ε → 0, and if all eigenvalues of the matrix Dh(a) have negative real parts, then the periodic solution x(t, ε) is stable. If some of the eigenvalue has positive real part the periodic solution x(t, ε) is unstable. The averaging theory for studying periodic solutions is very useful see for instance [4]. 3 Proofs of the results Proof (Proof of Theorem 1) Consider a = ε, b = ε2 β and ε ≥ 0 in system (1). We shall write the differential system (1) in the normal form for applying the averaging theory, i.e. in the form (3). Firstly we rescale the variables (x, y, z) = (εX, εY, εZ), we get Ẋ = Y, Ẏ = Z, (5) Ż = −Y + ε(3Y 2 − XZ − |X| − β). Zero-Hopf bifurcation 5 (X, 0, 0) are the singular points of the differential system (5) when ε = 0. The linear part of system (5) with ε = 0 at the origin is written as      Ẋ 0 1 0 X  Ẏ  =  0 0 1   Y  . Ż 0 −1 0 Z Then we do the change of variables      u 0 0 −1 X  v  =  0 −1 0   Y  , w 1 0 1 Z which writes the linear part of system (5) with ε = 0 in its real Jordan normal form. We get the next differential system u̇ = −v + ε(−3v 2 − u(u + w) + |u + w| + β), v̇ = u, (6) 2 ẇ = ε(3v + u(u + w) − |u + w| − β). Now we pass system (6) to the cylindrical coordinates (u, v, w) = (r cos θ, r sin θ, w), and we obtain the differential system ṙ = ε cos θ(r2 cos 2θ − rw cos θ + |w + r cos θ| − 2r2 + β), ε sin θ 2 θ̇ = 1 − (r cos 2θ − rw cos θ + |w + r cos θ| − 2r2 + β), (7) r ẇ = ε(3r2 sin2 θ + r cos θ(w + r cos θ) − |w + r cos θ| − β). Taking θ as a new independent variable in the differential system (7), it be- comes dr = ε cos θ(r2 cos 2θ − rw cos θ + |w + r cos θ| − 2r2 + β) + O(ε2 ), dθ (8) dw = ε(r2 (2 − cos 2θ) − |w + r cos θ| + rw cos θ − β) + O(ε2 ). dθ Now we calculate the averaged function using the formula (4) with the nota- tions x = z = (r, w), t = θ, F (t, x) = F (θ, r, w) T = 2π, where F (θ, r, w) = (F1 (θ, r, w), F2 (θ, r, w)) is given by     F1 (θ, r, w) cos θ(r2 cos 2θ − rw cos θ + |w + r cos θ| − 2r2 + β) = . F2 (θ, r, w) r2 (2 − cos 2θ) − |w + r cos θ| + rw cos θ − β 6 Sara Kassa et al. Then the averaged function f (r, w) = (f11 (r, w), f12 (r, w)) is Z arccos( −w r ) Z 2π−arccos −w 1 r f1 (r, w) = F1 (θ, r, w)dθ + F1 (θ, r, w)dθ 2π 0 arccos( −w r ) Z ! 2π + F1 (θ, r, w)dθ , 2π−arccos( −w r ) Z arccos( −w r ) Z 2π−arccos −w 1 r f2 (r, w) = F2 (θ, r, w)dθ + F2 (θ, r, w)dθ 2π 0 arccos( −w r ) Z ! 2π + F2 (θ, r, w)dθ , 2π−arccos( −w r ) and we obtain q 2 w 1− w r arcsin( w r ) f1 (r, w) = − rw 2 + π r2 + ,  qπ (9) w2 f2 (r, w) = − π1 πβ − 2πr + 2r 2 1− r2 + 2w arcsin( wr ) . Then going back through the change of variables from Theorem 2 and 3 it fol- lows Theorem 1. We solve the system (f1 (r, w), f2 (r, w)) = (0, 0) with respect to the variables r and w which is equivalent to solve the system r w2 w 2w 1 − 2 − πrw + 2r arcsin = 0, r r r w2 w πβ + 2r 1 − 2 − 2πr2 + 2w arcsin = 0, r r By Theorem 2 each zero (r0 , w0 ) of the averaged function (f1 (r, w), f2 (r, w)) given in (9) whose determinant ∂(f1 (r, w), f2 (r, w)) det( )|(ri ,wi ) 6= 0, (10) ∂(r, w) provides a limit cycle (r(θ, ε), w(θ, ε)) such that (r(0, ε), w(0, ε)) = (r0 , w0 ) + O(ε), of system (8), or equivalently a limit cycle (r(t, ε), θ(t, ε), w(t, ε)) such that (r(0, ε), θ(0, ε), w(0, ε)) = (r0 , 0, w0 ) + O(ε), of the differential system (7), and consequently a limit cycle (u(t, ε), v(t, ε), w(t, ε)) such that (u(0, ε), v(0, ε), w(0, ε)) = (r0 , 0, w0 ) + O(ε), of system (6). So we have a limit cycle (X(t, ε), Y (t, ε), Z(t, ε)) such that (X(0, ε), Y (0, ε), Z(0, ε)) = (r0 + w0 , 0, −r0 ) + O(ε), Zero-Hopf bifurcation 7 of system (5). Finally we obtain a limit cycle (x(t, ε), y(t, ε), z(t, ε)) such that (x(0, ε), y(0, ε), z(0, ε)) = ε(r0 + w0 , 0, −r0 ) + O(ε), of system (1). This limit cycle tend to the singular point localized at the origin of coordinates when ε → 0. Now the rest of the proof of the theorem follows directly from Theorem 3. Proof (Proof of Corollary 1) The averaged functionp f1 ((r, w), f2 (r, w)) given 1 ± 1 + 2π 2 β in (9) has the two zeros (r± , 0) with r± = . The determinant 2π (10) at these two zeros takes the values p 1 + 2π 2 β ∓ 7 1 + 2π 2 β d± = . 2π 2 So by Theorem (1) statements (a) and (b) of Corollary 1 follow. Since the eigenvalues of the matrix (2) evaluated at (r+ , 0) are p −1 − 2π 2 β + 7 1 + 2π 2 β ± , 2π 2 by Theorem (1) it follows that the periodic solution defined by (r+ , 0) is un- stable. On the other hand since the eigenvalues of the matrix (2) evaluated at (r− , 0) are purely imaginary we cannot say anything about its stability or instability. Acknowledgements The author is partially supported by the Ministerio de Ciencia, Inno- vación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and PID2019-104658GB-I00 (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Re- cerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE- 2017-777911. Conflict of interest The authors declare that they have no conflict of interest. References 1. A.T. Azar, C. Volos, N.A. Gerodimos, G.S. Tombras, V.-T. Pham, A.G. Radwan, S. Vaidyanathan, A. Ouannas and J.M. Munoz-Pacheco, A novel chaotic system without equilibrium: Dynamics, synchronization and circuit realization, Hindawi Complexity, Article ID 7871467, 11 pages (2017) 2. A. Buică and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math. 128, 7–22 (2004) 3. A. Buică, J. Llibre and O.Yu. 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