This is a preprint of: “The zero-Hopf bifurcations of a four-dimensional hyperchaotic system”, Jaume Llibre, Yuzhou Tian, J. Math. Phys., vol. 62(052703), 1–9, 2021. DOI: [10.1063/5.0023155] ∗ THE ZERO-HOPF BIFURCATIONS OF A FOUR-DIMENSIONAL HYPERCHAOTIC SYSTEM JAUME LLIBRE1 AND YUZHOU TIAN2,∗ Abstract. We consider the four-dimensional hyperchaotic system ẋ = a(y − x), ẏ = bx + u − y − xz, ż = xy − cz, u̇ = −du − jx + exz, where a, b, c, d, j, e are real parameters. This system extend the famous Lorenz system to dimension four and was introduced in the paper of the Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750021. We characterize the values of the parameters for which its equilibrium points are zero-Hopf points. Using the averaging theory we obtain sufficient conditions for the existence of periodic orbits bifurcating from these zero-Hopf equilibria, and give some examples to illustrate the conclusions. Moreover the stability conditions of these periodic orbits are given using the Routh-Hurwitz criterion. 1. Introduction Chaos phenomenon is a complex dynamic behavior in nonlinear dynamical sys- tem, which appears in nature widely. In 1963, the meteorologist Edward Lorenz [24] was the first to introduce the mathematical and physical chaotic model in R3 , which is known as the Lorenz system. The Lorenz system planted the seed in the chaos science. This system plays an important effect in other areas as in the mod- eling of lasers [11] and dynamos [12]. As one of the simplest models presenting chaos, the Lorenz system exhibits a rich range of dynamical properties, and it has been researched from different points of view, such as positively invariant [17], in- tegrability [22, 16, 14], global dynamics [34, 4, 26] and bifurcation [3, 32]. After that Lorenz system, mathematicians and physicists from physical or purely abstract mathematical point of view proposed various polynomial differential systems in R3 , whose trajectories exhibit chaotic dynamics of the Lorenz system type. As exam- ples, one can refer to Rikitake system [20], Sprott A system [1], Shimizu-Morioka system [13], etc. Nowadays three-dimensional nonlinear systems cannot provide adequate descrip- tion of many phenomena in neural networks, social sciences and engineering, etc. To better describe the real world, we often necessitate to introduce high-dimensional (dimension at least four) nonlinear systems. Recently the hyperchaotic system has become a focus research subject, see [6, 9, 30, 31, 5, 35, 27, 7] and the references therein. The concept of hyperchaos was given by Rössler in [29]. The precise defini- tion of hyperchaotic system is: (i) at least four-dimensional autonomous differential system, (ii) a dissipative structure, and (iii) at least two unstable directions, of ∗ Corresponding author: Yuzhou Tian. 2020 Mathematics Subject Classification. Primary 34C23, Secondary 34C25, 34C29. Key words and phrases. Hyperchaotic system, zero-Hopf bifurcation, periodic orbits, averaging theory. 1 2 JAUME LLIBRE AND YUZHOU TIAN which at least one direction is nonlinear [7]. The hyperchaotic systems are very useful in secure communication due to the fact that the dynamic information of such systems are difficult to characterize and predict, see [37]. In this work we use the classical averaging theory to investigate the zero-Hopf bifurcation of a hyperchaotic system. A zero-Hopf equilibrium is an equilibrium point of a four-dimensional autonomous differential system which has a double zero eigenvalue and a pair of purely imaginary eigenvalues. There are rich works on three-dimensional zero-Hopf bifurcation, see for example [8, 23, 21, 19, 15], etc. The zero-Hopf bifurcation of hyperchaotic Lorenz system (i.e. four-dimensional) can be found in [7, 6, 18]. Actually there are few results on the n-dimensional zero-Hopf bifurcation with n > 3. In [38] Zhou et al. present the following four-dimensional hyperchaotic system ẋ = a(y − x), ẏ = bx + u − y − xz, (1) ż = xy − cz, u̇ = −du − jx + exz, where a, b, c, d, j, e are real parameters. The hyperchaotic system (1) extend the Lorenz system to dimension four, and is invariant under the symmetry with respect to the z-axis, i.e. under the symmetry τ (x, y, z, u) = (−x, −y, z, −u). For the zero- Hopf bifurcation of system (1) at the origin, partial results are given by Yang et al. in [36]. The objective of this paper is to study all the zero-Hopf bifurcations of system (1). The equilibria and zero-Hopf equilibria of system (1) are described in the next two results. Proposition 1. Let ∆ = c (bd − d − j) / (d − e) with d 6= e. The hyperchaotic system (1) has the following equilibria. (i) If c = 0, system (1) has a straight line of equilibria (0, 0, z, 0). (ii) If ∆ ≤ 0 and c 6= 0, system (1) has an unique equilibrium point E0 = (0, 0, 0, 0). (iii) If ∆ > 0 and c 6= 0, system (1) has three equilibria E0 = (0, 0, 0, 0), √ ! √ √ bd − d − j (e + j − be) ∆ E1 = ∆, ∆, ,− and d−e d−e √ ! √ √ bd − d − j (e + j − be) ∆ E2 = − ∆, − ∆, , . d−e d−e Proposition 1 follows easily by direct computations. Theorem 2. For the hyperchaotic system (1) the following statements hold. (i) There is a two-parameter family of systems (1) for which the origin of coordinates is a zero-Hopf equilibrium point. Namely, c = 0, d = −a − 1, b = −(1 + a + a2 + ω 2 )/a, j = ((1 + a)3 + (1 + a)ω 2 )/a. ZERO-HOPF BIFURCATIONS OF THE HYPERCHAOTIC SYSTEM 3 (ii) There is a three-parameter family of systems (1) for which the equilibria E1,2 are zero-Hopf equilibrium  points. Namely, a = 0, j = bd, c = −d − 1 and d2 e + de + e − d3 (d − e) > 0. (iii) When c = 0 there is a three-parameter family of system (1) for which the equilibria (0, 0, z0 , 0) is a zero-Hopf equilibrium points. Namely a = −1 − d, j = (b − 1)d + z0 (e − d) and (b − d − z0 ) (d + 1) > 1. Theorem 2 is proved in section 3. In the following theorem we characterize the periodic orbits bifurcating from the zero-Hopf equilibrium E0 of system (1). Theorem 3. Let a2 + a + 1 + ω 2 b=− + εb1 , a c = εc1 , d = −a − 1 + εd1 , (a + 1)3 + (a + 1)ω 2 j= + εj1 , a with ω > 0 and ε > 0 a sufficiently small parameter. If η = (a + 1)2 d1 + a(a + 1)b1 + aj1 , d1 6= 0, c1 ηa (a + e + 1) > 0 and c1 ω 2 d1 + η a (a + e + 1) > 0, then for ε > 0 sufficiently small the hyperchaotic system (1) has a zero-Hopf bifurcation at the equilibrium point located at E0 and at most four periodic orbits can bifurcate from this equilibrium when ε = 0, and two of them are stable if either c1 < 0, d1 < η 0, η < 0, or η > 0, c1 < 0, − 2 < d1 < 0. Moreover there are systems (1) for ω which this zero-Hopf bifurcation exhibits the four periodic orbits, see example 1. The proofs of Theorem 3 and of the Example 1 are given in section 4, and use the averaging theory of first order, see subsection 2.1. Example 1. The hyperchaotic system (2) ẋ = x − y, ẏ = 2x + u − y − xz, ż = xy, u̇ = −xz, has four small periodic orbits bifurcating from the equilibrium point (0, 0, 0, 0), and two of them are stable. In order to study the zero-Hopf bifurcation at the equilibria E1 and E2 it is sufficient to study it for the equilibrium point E1 due to the symmetry exhibited by system (1). After translating the equilibrium E1 at the origin of coordinates and do the convenient changes of variables, similar to the ones of the proof of Theorem 3 we see that we cannot write system (1) in the normal form (3) and consequently we cannot apply to it the averaging theory described in Theorem 4. On the other hand, doing the changes of variables similar to the ones of the proof of Theorem 3, we can write system (1) in the normal form (6) for applying the averaging theory stated in Theorem 5, but unfortunately this system does not satisfy the assumption (ii) of Theorem 5. Therefore the averaging theory does not give any information on the possible periodic orbits of the zero-Hopf bifurcation at the equilibrium E1 . We can apply the averaging theory for studying the zero-Hopf bifurcation at the equilibria (0, 0, z0 , 0) for all z0 ∈ R, after writing it in the normal form (3) 4 JAUME LLIBRE AND YUZHOU TIAN doing similar changes of variables to the ones of the proof of Theorem 3. But the determinant (5) evaluated at the zeros of the averaged function becomes zero, so the averaging theory of Theorem 4 does not provide any information on the periodic orbits which could exist in the zero-Hopf bifurcation at the equilibria (0, 0, z0 , 0). In section 2 we present some basic results that we shall need for proving our theorems. 2. Preliminaries 2.1. Averaging theory. In this subsection we present the results on averaging theory that we need for proving our results. Consider the following differential equation (3) ẋ = εF (t, x) + ε2 G (t, x, ε) , (t, x, ε) ∈ [0, ∞) × Ω × (0, ε0 ], where Ω is an open subset of R , F (t, x) and G (t, x, ε) are T -periodic in t. We n introduce the averaged function Z 1 T (4) F (x) = F (t, x) dt. T 0 Theorem 4. Assume that F, its Jacobian ∂F/∂x and its Hessian ∂ 2 F/∂x2 ; G, its Jacobian ∂G/∂x are defined, continuous and bounded by a constant independent of ε in [0, ∞) × Ω × (0, ε0 ], and that the period T is a constant independent of ε. Then the following statements hold. (i) If p is the zero of the averaged function F (x) such that the Jacobian   ∂F (5) det 6= 0, ∂x x=p then there exists a T -periodic solution x (t, ε) of equation (3) such that x (0, ε) → p as ε → 0. (ii) The stability of the periodic solution x (t, ε) is determined by the eigenvalues of the Jacobian matrix (∂F/∂x) |x=p . For more details about a proof of Theorem 4 see [33]. We consider the problem of the bifurcation of T –periodic solutions from differ- ential systems of the form (6) ẋ = F0 (t, x) + εF1 (t, x) + ε2 F2 (t, x, ε), with ε = 0 to ε 6= 0 sufficiently small. Here the functions F0 , F1 : R × Ω → Rn and F2 : R × Ω × (−ε0 , ε0 ) → Rn are C 2 functions, T –periodic in the first variable, and Ω is an open subset of Rn . The main assumption is that the unperturbed system (7) ẋ = F0 (t, x), has a submanifold of periodic solutions. A solution of this problem is given using the averaging theory. Let x(t, z, ε) be the solution of the system (7) such that x(0, z, ε) = z. We write the linearization of the unperturbed system along a periodic solution x(t, z, 0) as (8) ẏ = Dx F0 (t, x(t, z, 0))y. ZERO-HOPF BIFURCATIONS OF THE HYPERCHAOTIC SYSTEM 5 In what follows we denote by Mz (t) some fundamental matrix of the linear differ- ential system (8), and by ξ : Rk × Rn−k → Rk the projection of Rn onto its first k coordinates; i.e. ξ(x1 , . . . , xn ) = (x1 , . . . , xk ). We assume that there exists a k–dimensional submanifold Z of Ω filled with T –periodic solutions of (7). Then an answer to the problem of bifurcation of T – periodic solutions from the periodic solutions contained in Z for system (6) is given in the following result. Theorem 5. Let W be an open and bounded subset of Rk , and let β : Cl(W ) → Rn−k be a C 2 function. We assume that (i) Z = {zα = (α, β(α)) , α ∈ Cl(W )} ⊂ Ω and that for each zα ∈ Z the solution x(t, zα ) of (7) is T –periodic; (ii) for each zα ∈ Z there is a fundamental matrix Mzα (t) of (8) such that the matrix Mz−1α (0) − Mz−1α (T ) has in the upper right corner the k × (n − k) zero matrix, and in the lower right corner a (n − k) × (n − k) matrix ∆α with det(∆α ) 6= 0. We consider the function F : Cl(W ) → Rk Z ! 1 T −1 (9) F(α) = ξ Mzα (t)F1 (t, x(t, zα ))dt . T 0 If there exists a ∈ W with F(a) = 0 and det ((dF/dα) (a)) 6= 0, then there is a T –periodic solution ϕ(t, ε) of system (6) such that ϕ(0, ε) → za as ε → 0. Theorem 5 goes back to Malkin [25] and Roseau [28], for a shorter proof see [2]. 2.2. Roots of cubic equation. The Routh-Hurwitz Criterion gives necessary and sufficient conditions in order that all the roots of a polynomial p (x) ∈ R[x] have negative real parts, for more details see page 231 of [10]. Theorem 6 (Routh-Hurwitz Criterion). All roots of the real polynomial p (x) = b0 xn + b1 xn−1 + · · · + bn−1 x + bn (b0 > 0) have negative real parts if and only if ∆1 > 0, ∆2 > 0, . . . , ∆n > 0, where   b1 b3 b5 ···  b0 b2 b4 ···     0 b1 b3 ···    ∆i = det  0 b0 b2 b4  (bk = 0 if k > n)    .. .. .. .. ..   . . . . .  bi is the Hurwitz determinant of order i (i = 1, 2, · · · , n). Corollary 7. All roots of the real polynomial b0 x3 + b1 x2 + b2 x + b3 (b0 > 0) have negative real parts if and only if ∆1 = b1 > 0, ∆2 = b1 b2 − b3 b0 > 0, b3 > 0. 6 JAUME LLIBRE AND YUZHOU TIAN 3. Proof of Theorem 2 (i) The characteristic polynomial p (λ) of the linearization of system (1) at the origin is p (λ) =λ4 + (a + c + d + 1) λ3 + (a(1 − b + c + d) + cd + c + d) λ2 + (a (c(1 − b + d) − bd + d + j) + cd) λ + ac(j − bd + d). Since the origin of the hyperchaotic system  (1) is a zero-Hopf equilibrium, p (λ) must be of the form p (λ) = λ2 λ2 + ω 2 with ω ∈ R+ . Then we obtain a2 + a + 1 + ω 2 (a + 1)3 + (a + 1)ω 2 c = 0, d = −a − 1, b = − , j= . a a (ii) Let ∆ = c (bd − d − j) / (d − e). Then c = ∆ (d − e) / (bd − d − j). The char- acteristic polynomial of the linear part of the system (1) at E1 is given by   ∆(d − e) ∆(d(a + b + d) − e(a + d) − e − j) 2 p (λ) =λ4 + a − + d + 1 λ3 + λ j − bd + d bd − d − j a(b − 1)e + (a + 1)d2 − (a + 1)de − aj 2 + λ d−e   ∆ a d(2b − e − 2) + (b − 1)e + d2 − 3j + (d − e)(bd − j) + λ + 2a∆(d − e). bd − d − j If the equilibrium E  1 is a zero-Hopf equilibrium, then p (λ) must be of the form p (λ) = λ2 λ2 + ω 2 with ω ∈ R+ . So we get that the parameters must satisfy  2 2 d d2 + ω 2 a = 0, j = bd, ∆ = d + d + ω + 1, e = 2 . d + d + ω2 + 1 Clearly ∆ > 0, otherwise the equilibrium E1 does not exist. (iii) The characteristic polynomial at equilibrium point (0, 0, z0 , 0) is p (λ) = λ4 + (a + d + 1)λ3 + (a(d + 1 − b) + az0 + d) λ2 + a (z0 (d − e) + d + j − bd) λ. Since (0, 0, z0 , 0) is a zero-Hopf equilibrium, the parameters must be satisfied  d2 + d + ω 2 + 1 d d2 + ω 2 a = −1 − d, b = + z0 , j = + ez0 , d+1 d+1 where ω ∈ R+ . This completes the proof of Theorem 2. 4. Proof of Theorem 3 Let   a2 + a + 1 + ω 2 (a + 1)3 + (a + 1)ω 2 (b, c, d, j) = − + εb1 , εc1 , −a − 1 + εd1 , + εj1 a a ZERO-HOPF BIFURCATIONS OF THE HYPERCHAOTIC SYSTEM 7 where ω > 0 and ε > 0 is a sufficiently small parameter. Then the hyperchaotic system (1) becomes ẋ = a(y − x),   a2 + a + 1 + ω 2 ẏ = u − xz − y + − + b1 ε x, a (10) ż = xy − εc1 z,   (a + 1)ω 2 + (a + 1)3 u̇ = exz − u (−a − 1 + εd1 ) − + εj1 x. a Doing the rescaling of variables (x, y, z, u) 7→ (εx, εy, εz, εu) system (10) writes ẋ = a(y − x), a2 + a + 1 + ω 2 ẏ = u − y − x + εx (b1 − z) , (11) a ż = ε (xy − c1 z) , (a + 1)3 + (a + 1)ω 2 u̇ = u + au − x + ε (exz − d1 u − j1 x) . a After the linear change of variables (x, y, z, u) 7→ (X, Y, Z, U ) aωY + Z aωY − ω 2 X + Z x= 2 , y= , ω ω2 (12)  ωa(a + 1)(aY − ωX + Y ) + (a + 1)2 + ω 2 Z z = U, u = , aω 2 the linear part at the origin of system (11) for ε = 0 can be transformed into its real Jordan normal form   0 ω 0 0  −ω 0 0 0   .  0 0 0 0  0 0 0 0 Under the change of variable (12), system (11) can be written as ε (u − b1 ) (aωy + z) ẋ = ωy + , ω2 ε (aωy + z) A ẏ = −ωx + , aω 2 (13) ε (aωy + z) A ż = d1 ε (a(a + 1)x − z) − ,  ω  (aωy + z) (ω(ay − ωx) + z) u̇ = ε − c1 u , ω4 where we have written (x, y, z, u) instead of (X, Y, Z, U ) and ab1 (a + 1) + d1 (a + 1)2 + (j1 − (a + e + 1)u) a (14) A= . ω 8 JAUME LLIBRE AND YUZHOU TIAN Performing the cylindrical change of variables (x, y, z, u) 7→ (r cos θ, r sin θ, z, u), system (13) becomes   dr sin θ a2 (b1 − u) r cos θ + d1 (a(a + 1)r cos θ − z) = ε dθ aω 2  (b1 − u) z cos θ A (aωr sin θ + z) sin θ  + − + O ε2 ω3 aω 3  = εF1 (θ, r, z, u) + O ε2 ,   (15) dz d1 (z − a(a + 1)r cos θ) A (aωr sin θ + z)  = ε + 2 + O ε2 dθ ω ω 2  = εF2 (θ, r, z, u) + O ε ,  du ε c1 ω 4 u − (aωr sin θ + z) (ω(a sin θ − ω cos θ)r + z)  = 5 + O ε2 dθ ω  = εF3 (θ, r, z, u) + O ε2 . System (15) is written in the normal form (3) for applying the averaging theory, and satisfies all the assumptions of Theorem 4. Then using the notations of the averaging theory described in Theorem 4, we have t = θ, T = 2π, x = (r, z, u),     F1 (θ, r, z, u) F1 (r, z, u) F (θ, r, z, u) =  F2 (θ, r, z, u)  and F (r, z, u) =  F2 (r, z, u)  , F3 (θ, r, z, u) F3 (r, z, u) where Z 2π 1 rA F1 (r, z, u) = F1 (θ, r, z, u) dθ = − , 2π 0 2ω 2 Z 2π 1 (ωd1 + A) z F2 (r, z, u) = F2 (θ, r, z, u) dθ = , 2π 0 ω2 Z 2π 1 a2 ω 2 r2 − 2c1 ω 4 u + 2z 2 F3 (r, z, u) = F3 (θ, r, z, u) dθ = − . 2π 0 2ω 5 The system F1 (r, z, u) = F2 (r, z, u) = F3 (r, z, u) = 0 has the following five solutions s0 = (0, 0, 0) , s ! ω 2c1 η η s1,2 = ∓ , 0, , a a(a + e + 1) a(a + e + 1) s ! 2 c1 (ω 2 d1 + η) ω 2 d1 + η s3,4 = 0, ∓ω , , a(a + e + 1) a(a + e + 1) where η = (a + 1)2 d1 + a(a + 1)b1 + aj1 . The first solution s0 corresponds to the equilibrium at the origin, so it is not a good solution. For other four solutions, we ZERO-HOPF BIFURCATIONS OF THE HYPERCHAOTIC SYSTEM 9 get     ∂F ∂F c1 d 1 η det (s1 ) = det (s2 ) = ,  ∂x   ∂x  ω5  ∂F ∂F c1 d1 ω 2 d1 + η det (s3 ) = det (s4 ) = − . ∂x ∂x ω5  Since by assumptions d1 6= 0, c1 ηa (a + e + 1) > 0 and c1 ω 2 d1 + η a (a + e + 1) > 0, the solutions si exist and det (∂F (si ) /∂x) 6= 0 for i = 1, 2, 3, 4. From Theorem 4 it follows that system (15) for ε > 0 sufficiently small has four 2π-periodic or- bits γi = (ri (θ, ε) , zi (θ, ε) , ui (θ, ε)) such that (ri (0, ε) , zi (0, ε) , ui (0, ε)) → si as ε → 0 with i = 1, 2, 3, 4. The Jacobian matrices ∂F (s1 ) /∂x and ∂F (s2 ) /∂x have the same characteristic equation  c1 + d1 2 c1 η + ω 2 d1 c1 d1 η (16) λ3 − λ + λ− = 0. ω ω4 ω5 By Corollary 7 all the roots of equation (16) have negative real parts if  c1 + d1 c1 c1 η + d1 (c1 + d1 )ω 2 c1 d 1 η − > 0, − 5 > 0, − 5 > 0, ω ω ω or equivalently if c1 < 0, d1 < 0, η < 0. Thus, the periodic orbits γ1 and γ2 are stable if c1 < 0, d1 < 0, η < 0. The Jacobian matrices ∂F (s3 ) /∂x and ∂F (s4 ) /∂x have the same characteristic equation   2c1 + d1 2 c1 4η + 3ω 2 d1 c1 d1 η + ω 2 d1 (17) λ3 − λ − λ + = 0. 2ω 2ω 4 ω5 Using Corollary 7 all the roots of equation (17) have negative real parts if   2c1 + d1 c1 8c1 (η + d1 ω 2 ) − (2c1 + d1 )d1 ω 2 c1 d1 η + ω 2 d1 − > 0, > 0, > 0, 2ω 4ω 5 ω5 η or equivalently η > 0, c1 < 0, − 2 < d1 < 0. This implies that the periodic orbits ω γ3 and γ4 are stable if one of the three previous conditions hold. This completes the proof of Theorem 3. Proof of Example 1. Taking a = e = −1, b = 2 and c = d = j = 0, system (1) becomes system (2). Since the origin of system (2) has a double zero eigenvalue and a pair of purely imaginary eigenvalues ±i, the origin is a zero-Hopf equilibrium point. Let c1 = d1 = j1 = −1 and ω = 1. Consider the perturbation of Theorem 3, that is, b = 2 + εb1 , j = ε, c = d = −ε in system (2) with ε > 0 a sufficiently small parameter. By the steps of averaging theory, we have the following functions r(u + 1) r2 (18) F1 (r, z, u) = , F2 (r, z, u) = −(u + 2)z, F3 (r, z, u) = − − u − z 2 . 2 2 10 JAUME LLIBRE AND YUZHOU TIAN √  The system (18) has five solutions s0 = (0, 0, 0), s1,2 = 0, ± 2, −2 and s3,4 = √  ± 2, 0, −1 . Since the determinant     ∂ (F 1 , F 2 , F 3 ) ∂ (F 1 , F 2 , F3 ) det   = 2 and det   = −1, ∂ (r, z, u) ∂ (r, z, u) s1,2 s3,4 four periodic orbits can bifurcate from √ the zero-Hopf equilibrium at the origin. The eigenvalues of s3,4 are −1 and −1 ± i 3 /2. For the solutions s1,2 the associated √  eigenvalues are −1/2 and −1 ± 17 /2. Therefore two of four periodic orbits are stable.  Acknowledgements The first author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants M TM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The sec- ond author is partially supported by the National Natural Science Foundation of China (No. 11971495 and No. 11801582), China Scholarship Council (No. 201906380022) and Guangdong Basic and Applied Basic Research Foundation (No. 2019A1515011239). References [1] L. Barreira, J. Llibre, and C. Valls, Integrability and zero-Hopf bifurcation in the Sprott A system, Bull. Sci. Math., 162 (2020), 102874, pp 16. [2] A. Buică, J. P. Françoise and J. Llibre, Periodic solutions of nonlinear periodic differ- ential systems with a small parameter, Comm. on Pure and Appl. Anal. 6 (2007), 103–111. [3] M. R. Cândido, J. 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