This is a preprint of: “Zero-Hopf Periodic Orbit of a Quadratic System of Differential Equations Obtained from a Third-Order differential equation”, Jaume Llibre, Ammar Makhlouf, Differ. Equ. Dyn. Syst., vol. 27, 75–82, 2019. DOI: [10.1007/s12591-017-0375-5] ZERO–HOPF PERIODIC ORBIT OF A QUADRATIC SYSTEM OF DIFFERENTIAL EQUATIONS OBTAINED FROM A THIRD–ORDER DIFFERENTIAL EQUATION JAUME LLIBRE1 AND AMMAR MAKHLOUF 2 Abstract. We study the zero-Hopf bifurcation of the third–order differential equations x′′′ + (a1 x + a0 )x′′ + (b1 x + b0 )x′ + x2 = 0, where a0 , a1 , b0 and b1 are real parameters. The prime denotes derivative with respect to an independent variable t. We also provide an estimate of the zero–Hopf periodic solution and their kind of stability. The Hopf bifurcations of these differential systems were studied in [5], here we complete these studies adding their zero-Hopf bifurcations. 1. Introduction and statement of the main result Quadratic systems of differential equations in R3 are some of the simplest systems after the linear ones and have been extensively studied in the last years. There are many examples of such systems, see for instance Lorenz system [19], the Chen system [4], the Liu system [14], the Rössler system [21], the Rikitake system [20], the Lu system [13], among several others. One of the most interesting problems related with quadratic systems of differ- ential equations is the study of their limit cycles, i.e. of their isolated periodic orbits in the set of all periodic orbits. It is known that every quadratic system of differential equations in R2 has finitely many limit cycles, see for instance [1], [6], [11]. For quadratic systems of differential equations in Rn with n > 2 the scenario is very different. There are quadratic systems of differential equations for n > 2 with infinitely many limit cycles, see for instance [7]. In [5] the authors study the nonlinear dynamics including the Hopf periodic orbits but not the zero–Hopf periodic orbits of a quadratic system of differential equations in R3 which comes from a third–order differential equation. More precisely, they analyze the third–order differential equation x′′′ + (a1 x + a0 )x′′ + (b1 x + b0 )x′ + c2 x2 + c1 x + c0 = 0, or equivalently the first–order system of differential equations x′ = y, (1) y ′ = z, z ′ = −(a0 + a1 x)z − (b0 + b1 x)y − (c2 x2 + c1 x + c0 ), 2010 Mathematics Subject Classification. 37G15, 37C80, 37C30. Key words and phrases. periodic orbit, third–order differential equation, quadratic system, averaging theory. 1 2 J. LLIBRE AND A. MAKHLOUF where the prime denotes derivative with respect to the independent variable t. The study of the different Hopf bifurcation of this system of differential equations in [5] is done in two steps, in the first the authors reduces the polynomial c2 x2 + c1 x + c0 to x2 , and in the second to x(x + 1). In both cases the authors characterize the Hopf bifurcations, i.e. when from an equilibrium point of system (1) having eigenvalues α(ε) ± β(ε)i, λ(ε) it bifurcates a periodic orbit at ε = 0 with α(0) = 0, α′ (0) ̸= 0 and λ(0) ̸= 0. The goal of this paper is to improve the study of the zero–Hopf bifurcation which exhibits in the case x2 the system (1), i.e. from an equilibrium point of system (1) having eigenvalues α(ε) ± β(ε)i, λ(ε) it bifurcates a periodic orbit at ε = 0 with α(0) = 0, α′ (0) ̸= 0 and λ(0) = 0. Such bifurcation was studied in [5] under the name fold–Hopf bifurcation, our two main contributions are to provided for ε ̸= 0 sufficiently small, first an approximation of the periodic orbit which bifurcate in it, and second its kind of stability. More precisely we first study the zero–Hopf equilibria of system x′ = y, (2) y′ = z, z′ = −(a1 x + a0 )z − (b1 x + b0 )y − x2 . That is, the equilibrium points of a 3-dimensional autonomous system of differential equations which have a zero eigenvalue and a pair of purely imaginary eigenvalues. In our system this zero-Hopf equilibrium is the point (0, 0, 0) when a0 = 0. System (2) possesses the unique singular point (x, y, z) = (0, 0, 0). In order to simplify the computations we change the parameter b0 by a new parameter β > 0 (which will be the imaginary part of the eigenvalues of the linear part of system (5) 1 at the origin), as follows b0 = (ε2 a2 + β 2 ). 4 Our main result on the periodic solutions of the third–order differential equation (2) is the following. Theorem 1. Assume that in system (2) (a1 β 2 − 4)(a1 β 2 − 8)2 > 0, β > 0, b0 = (a20 + β 2 )/4 > 0 and a0 = εa where ε is a small parameter, then the differential equation (3) x′′′ + (a1 x + a0 )x′′ + (b1 x + (ε2 a2 + β 2 )/4)x′ + x2 = 0, has the periodic solution x(t, ε) such that 4(ω ∗ − r∗ ) (4) x(0, ε) = ε + O(ε2 ), a20 + β 2 bifurcating from a zero–Hopf equilibrium point when ε = 0, where ( ) ∗ ∗ aβ 4 aβ 4 (r , ω ) = √ ,− . 2(a1 β 2 − 4)(a1 β 2 − 8)2 4(a1 β 2 − 8) Moreover the following statements hold. (a) The periodic solution x(t, ε) has a stable and an unstable manifolds each one formed by two cylinders if a1 β 2 − 8 < 0. (b) The periodic solution x(t, ε) is locally unstable if a1 β 2 − 8 > 0 and a > 0. (c) The periodic solution x(t, ε) is locally stable if a1 β 2 − 8 > 0 and a < 0. ZERO–HOPF PERIODIC ORBIT 3 We note that Dias and Mello in [5] they proved the existence of such zero–Hopf bifurcation when the parameter a1 ∈/ {2/b0 , 1/b0 , 9/10/b0 , 0}. The parameter β is taken in such a way that at the zero–Hopf bifurcation (i.e. when ε = 0) the eigenvalues are ±β and 0, in such a way the computations become easier. For the meaning of the other parameters see the paper [5]. Theorem 1 is proved in section 2. Its proof is based in the averaging theory for computing periodic orbits, see section 3. For others applications of the averaging theory to the study of limit cycles, see for instance [3, 15, 16] and mainly the books [18, 22] where there are more than three hundred references on the averaging theory. We shall study the system of differential equations of first order x′ = y, (5) y ′ = z, z ′ = −(εa + a1 x)z − ((ε2 a2 + β 2 )/4 + b1 x)y − x2 , associated to the differential equation (3). The periodic solution u(t, ε) satisfying (4) of Theorem 1 written as solution of system (5) is a periodic solution which tends to the equilibrium point localized at the origin of system (5). So this periodic solution bifurcates from the origin at ε = 0. Consequently it is a Hopf periodic solution. Moreover, since at ε = 0 the√eigenvalues of the linear part at the equilib- rium (0, 0, 0) are ±βi/2, 0 with β = 2 b0 , we have that such periodic solution is a zero–Hopf periodic solution. 2. Proof of Theorem 1 First we shall see that the system of differential equations x′ = y, (6) y ′ = z, z ′ = −(a0 + a1 x)z − (b0 + b1 x)y − x(x + 1), has no zero–Hopf equilibria. Indeed, this system has two equilibria: (0, 0, 0) and (−1, 0, 0). The characteristic polynomials in these equilibria are −λ3 +(a1 −a0 )λ2 + (b1 − b0 )λ + 1 and −λ3 − a0 λ2 − b0 λ − 1, respectively. Since the product of the three eigenvalues of the linear part of system (6) in these two equilibria are 1 and −1, these equilibria never have a zero eigenvalue. So they never can be a zero–Hopf equilibrium. The Jacobian matrix of system (5) possesses the eigenvalues 0, (−εa ± iβ)/2. Hence when a0 = εa = 0, we have that (0, 0, 0) is a zero-Hopf equilibrium. We shall study if from this zero–Hopf equilibrium it bifurcates or not a periodic orbit. We shall write system (5) in such a way that the linear part at the origin will be in its real Jordan normal form. Then, doing the change of variables (x, y, z) → (u, v, w) given by  a  0 ε 1    2   u   x  v = β   0 0  y ,  2  w   z 1 2 2 (ε a + β 2 ) εa 1 4 4 J. LLIBRE AND A. MAKHLOUF which is equivalent to  4 4εa 4  − −    ε2 a2+ β2 (ε2 a2 + β 2 )β ε2 a2 2 + β   x   u  2   y = 0 0  v ,  β  z   w  εa  1 − 0 β the system of differential equations (5) becomes (7) εa β 4(a1 (ε2 a2 + β 2 ) − 4) 2 8(εa(εab1 − 4) + b1 β 2 ) u′ = − u − v + u + uv 2 2 (ε2 a2 + β 2 )2 β(ε2 a2 + β 2 )2 4(a1 (ε2 a2 + β 2 ) − 8) − uw− (ε2 a2 + β 2 )2 4εa(εa(4 + ε2 a2 a1 − 2εab1 ) + (εaa1 − 2b1 )β 2 ) 2 − v β 2 (ε2 a2 + β 2 )2 4εa(8 + ε2 a2 a1 − 2εab1 ) + 4(εaa1 − 2b1 )β 2 16 + vw − 2 2 w2 , β(ε2 a2 + β 2 )2 (ε a + β 2 )2 β εa v′ = u − v, 2 2 4(a1 (ε a + β 2 ) − 4) 2 8(εa(−4 + εab1 ) + b1 β 2 ) 2 2 w′ = u + uv (ε2 a2 + β 2 )2 β(ε2 a2 + β 2 )2 4(a1 (ε2 a2 + β 2 ) − 8) 4εa(εa(4 + ε2 a2 a1 − 2εab1 ) + (εaa1 − 2b1 )β 2 ) 2 − uw − v (ε2 a2 + β 2 )2 β 2 (ε2 a2 + β 2 )2 4εa(8 + ε2 a2 a1 − 2εab1 ) + 4(εaa1 − 2b1 )β 2 16 + 2 2 2 2 vw − 2 2 w2 . β(ε a + β ) (ε a + β 2 )2 Note that the linear part of the system of differential equations (7) at the origin is in its real normal form of Jordan. Doing the rescaling (u, v, w) = (εU, εV, εW ), and system (7) becomes β a 1 ( U ′ = − V − ε U − ε 4 − 8a1 U 2 β 2 + 8a1 U W β 2 − 16b1 U V β+ 2 2 2β ) 16b1 V W β + 32U + 32W 2 − 64U W + O(ε2 ), 2 (8) β a V′ = U − ε V, 2 2 ( ) ′ 4(U − V ) (a1 β 2 − 4 U + 2βb1 V + 4W ) W = ε + O(ε2 ). β4 ZERO–HOPF PERIODIC ORBIT 5 Consider the polar coordinates U = r cos θ, V = r sin θ and W = ω. The system of differential equations (8) in these coordinates writes dr 1 ( = ε 4 8r2 (a1 β 2 − 4) cos3 θ − arβ 4 sin2 θ dt 2β −16ω cos θ(2ω + b1 rβ sin θ) ) +r cos2 θ(64ω − 8a1 β 2 ω − aβ 4 + 16b1 βr sin θ) + O(ε2 ), (9) dθ β 4 = + ε 4 (ω − r cos θ) sin θ(4ω + r(a1 β 2 − 4) cos θ + 2b1 βr sin θ) dt 2 β r +O(ε2 ), dω 4 = −ε (ω − r cos θ)(4ω + r(a1 β 2 − 4) cos θ + 2b1 βr sin θ) + O(ε2 ). dt β4 Therefore the solutions of system (9) in the region θ̇ > 0 can be studied analyzing the solution of the system of differential equations (10) dr 1 ( = ε 5 8r2 (a1 β 2 − 4) cos3 θ − arβ 4 sin2 θ − 16ω cos θ(2ω + b1 rβ sin θ) dθ β ) +r cos2 θ(64ω − 8a1 β 2 ω − aβ 4 + 16b1 βr sin θ) + O(ε2 ), dω 8 = −ε 5 (ω − r cos θ)(4ω + r(a1 β 2 − 4) cos θ + 2b1 βr sin θ) + O(ε2 ). dθ β For applying the averaging theorem we compute the averaged system (15). We obtain dr 1 ( ) = −ε 5 r aβ 4 + 4ω(a1 β 2 − 8) = εg1 (r, ω), dθ β (11) dω 4 ( ) = ε 5 −8ω 2 + r2 (a1 β 2 − 4) = εg2 (r, ω). dθ β The unique equilibrium point of system (11) with r > 0 is ( ) ∗ ∗ aβ 4 aβ 4 (r , ω ) = √ ,− , 2(a1 β 2 − 8)2 (a1 β 2 − 4) 4(a1 β 2 − 8) recall that by assumptions (a1 β 2 − 8)2 (a1 β 2 − 4) > 0. The zero (r∗ , ω ∗ ) of system ( ) ( ) g1 (r, ω) 0 = , g2 (r, ω) 0 with respect to the variables r and ω provide a periodic solution of system (10) with ε ̸= 0 sufficiently small if it is simple , i.e. if ( ) ∂(g1 , g2 ) 16a2 (12) det = 2 ̸= 0, ∂(r, ω) (r,ω)=(r∗ ,ω∗ ) β (a1 β 2 − 8) which is the case by assumptions. By Theorem 2, system (10) has a periodic solution (r(θ, ε), ω(θ, ε)) such that (r(0, ε), ω(0, ε)) = (r∗ , ω ∗ ) + O(ε). This periodic solution in the coordinates (r, θ, ω) of system (9) writes (r(t, ε), θ(t, ε), ω(t, ε)) and satisfies (r(0, ε), θ(0, ε), ω(0, ε)) = (r∗ , 0, ω ∗ ) + O(ε). 6 J. LLIBRE AND A. MAKHLOUF Now the periodic solution written in the coordinates (U, V, W ) of system (8) be- comes (U (t, ε), V (t, ε), W (t, ε)) and satisfies (U (0, ε), V (0, ε), W (0, ε)) = (r∗ , 0, ω ∗ )+ O(ε). Again this last periodic solution written in the variables (u, v, w) of system (7) is (u(t, ε), v(t, ε), w(t, ε)) satisfying (u(0, ε), v(0, ε), w(0, ε)) = ε (r∗ , 0, ω ∗ ) + O(ε2 ). Finally this periodic solution in the variable (x, y, z) of system (5) becomes (x(t, ε), y(t, ε), z(t, ε)) satisfying ( ) 4(ω ∗ − r∗ ) (x(0, ε), y(0, ε), z(0, ε)) = ε , 0, r∗ + O(ε2 ). ε2 a2 + β 2 Hence, this periodic solution in the differential equation (3) is x(t, ε) satisfying 4 x(0, ε) = ε 2 2 (ω ∗ − r∗ ) + O(ε2 ). ε a + β2 In order to complete the proof of Theorem 1, we must prove the statements (a)–(d). The eigenvalues of the matrix ( ) ∂(g1 , g2 ) (13) ∂(r, ω) (r,ω)=(r ∗ ,ω ∗ ) are √ √ 8a − 4a 12 − a1 β 2 8a + 4a 12 − a1 β 2 and . β (a1 β 2 − 8) β (a1 β 2 − 8) So the trace of the matrix (13) is 16a . β(a1 β 2 − 8) From the expression of the determinant of the matrix (13) given in (12) it follows that the equilibrium point (r∗ , ω ∗ ) of the averaged system (11) is a saddle when a1 β 2 − 8 < 0. Therefore by statement (c) of Theorem 2 the corresponding periodic orbit of system (2) has two stable and two unstable manifolds formed by cylinders, and statement (a) of Theorem 1 is proved. If a1 β 2 − 8 > 0 and a > 0, then the trace of the matrix (13) is positive. So some of the two eigenvalues of this matrix has a positive real part. Hence the equilibrium point (r∗ , ω ∗ ) of the averaged system (11) is unstable, and consequently by statement (c) of Theorem 2 the corresponding periodic orbit of system (2) is unstable. This proves statement (b) of Theorem 1. If a1 β 2 − 8 > 0 and a < 0, then the trace of the matrix (13) is negative, and both eigenvalues have negative real part. Therefore the equilibrium point (r∗ , ω ∗ ) of the averaged system (11) is stable, and consequently by statement (c) of Theorem 2 the corresponding periodic orbit of system (2) is stable. This proves statement (c) of Theorem 1. This completes the proof of Theorem 1. ZERO–HOPF PERIODIC ORBIT 7 3. Averaging theory of first order We work with the two initial value problems (14) ẋ = εF1 (t, x) + ε2 F2 (t, x, ε), x(0) = x0 , and (15) ẏ = εg(y), y(0) = x0 , with x , y and x0 in some open Ω of R , t ∈ [0, ∞), ε ∈ (0, ε0 ]. Let F1 and F2 be n periodic functions of period T in the variable t, and we define ∫ 1 T g(y) = F1 (t, y)dt. T 0 Theorem 2. By hypotheses the functions F1 , Dx F1 ,Dxx F1 and Dx F2 are con- tinuous, bounded by a constant independent of ε in [0, ∞) × Ω × (0, ε0 ], and that y(t) ∈ Ω for t ∈ [0, 1/ε]. Then the next statements are satisfied. (a) For t ∈ [0, 1/ε] it follows that x(t) − y(t) = O(ε) as ε → 0. ̸ 0 is an equilibrium point of system (15) and detDy g(p) ̸= 0, then (b) If p = there is a periodic solution ϕ(t, ε) of period T for system (14) such that ϕ(0, ε) − p = O(ε) as ε → 0. (c) The kind of stability of the periodic solution ϕ(t, ε) is the same than the stability of the equilibrium point. The notation Dx g means the Jacobian matrix of g with respect to the compo- nents of x, and Dxx g means the Hessian matrix of g. For a proof of Theorem 2 see [23]. The averaging theory at any order in dimension one is developed in [8], and in arbitrary dimension in [17]. Moreover it has been proved the equivalence of the averaging method at first order in dimension two with the Abelian integral method see [9], and recently the equivalence between the averaging method and the Melnikov function method at any order in [2, 10]. 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