Journal of Nonlinear Modeling and Analysis http://jnma.ca; http://jnma.ijournal.cn Volume 1, Number 2, June 2019, 167–177 DOI:10.12150/jnma.2019.167 Periodic Solutions of a Class of Duffing Differential Equations∗ Rebiha Benterki1 and Jaume Llibre2,† Abstract In this work we study the existence of new periodic solutions for the well knwon class of Duffing differential equation of the form x00 + cx0 + a(t)x + b(t)x3 = h(t), where c is a real parameter, a(t), b(t) and h(t) are continuous T –periodic functions. Our results are proved using three different results on the averaging theory of first order. Keywords Periodic solution, averaging method, Duffing differential equa- tion, bifurcation, stability. MSC(2010) 34C15, 34C25. 1. Introduction and statement of the main result Several classes of Duffing differential equations have been investigated by many authors. They are mainly interested in the existence of periodic solutions, in their multiplicity, stability and bifurcation. See the survey of J. Mawhin [12] and for example the articles [2–4, 6, 9, 10, 13, 16, 18, 19]. In this work we shall study the class of Duffing differential equations of the form x00 + cx0 + a(t)x + b(t)x3 = h(t), (1.1) where c > 0 is a constant, and a(t), b(t) and h(t) are continuous T –periodic func- tions. These differential equations were studied by Chen and Li in the papers [2, 3]. These authors studied the periodic solutions of equation (1.1) with the following additional conditions, either b(t) > 0, h(t) > 0 and a(t) satisfies T π2 c2 Z 1 a(t) ≤ 2 + , and a0 = a(t)dt > 0; (1.2) T 4 T 0 or a(t) = a > 0, b(t) = 1 and c > 0, a, c constants. † the corresponding author. Email address:r benterki@yahoo.fr (R. Benterki), jllibre@mat.uab.cat (J. Lli- bre) 1 Département de Mathématiques, Université de Bordj Bou Arréridj, Bordj Bou Arréridj 34265, Elanasser, Algeria 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain ∗ This work is supported by the Ministerio de Economı́a, Industria y Com- petitividad, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and MDM-2014-0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. 168 R. Benterki & J. Llibre In [8] the authors studied the existence and the stability of periodic solutions of the Duffing differential equation (1.1) with c = εC > 0, a(t) = ε2 A(t), A0 b0 > 0 1 T 1 T Z Z where A0 = A(t)dt and b0 = b(t)dt, and ε is sufficiently small. T 0 T 0 Instead of working with the Duffing differential equation (1.1) we shall work with the equivalent differential system x0 = y, (1.3) y 0 = −cy − a(t)x − b(t)x3 + h(t). We define the polynomial ! T t T e−cT Z Z Z −ct cs p(x0 ) = − e e b(s) ds dt − cs e b(s) ds x30 0 0 c 0 ! T t e−cT T cs Z Z Z −ct cs − e e a(s) ds dt − e a(s) ds x0 0 0 c 0 T Z T Z t e−cT Z cs −ct + e h(s) ds + e ecs h(s) ds dt. c 0 0 0 Our first result on the periodic solutions of the differential system (1.3) is the following. Theorem 1.1. For every simple real root of the polynomial p(x0 ) the differential system (1.3) has a periodic solution (x(t), y(t)) such that (x(0), y(0)) = (x0 , 0). Theorem 1.1 will be proved in section 3 using Theorem 4.1 of the averaging theory. Now we define the polynomial ! ! Z T Z T Z T q(x0 ) = − b(s) ds x30 − a(s) ds x0 + h(s) ds. 0 0 0 Theorem 1.2. For every simple real root of the polynomial q(x0 ) the differential system (1.3) has a periodic solution (x(t), y(t)) such that (x(0), y(0)) = (0, 0). Theorem 1.2 will be proved in section 4 using Theorem 4.2 of the averaging theory. As we shall see Theorem 4.3 of the averaging theory will provide results on the periodic solutions of system (1.3) which are already contained in Theorems 1.1 and 1.2. In order to apply the three theorems of the averaging theory of first order for studying the periodic solutions of the differential system (1.3) in section 2 we rescale the variables, the parameters and the functions of system (1.3). The results of averaging theory that we use in this paper are described in section 4. Periodic solutions of Duffing equations 169 2. Preliminary results We start doing a rescaling of the variables (x, y), of the functions a(t), b(t) and h(t) and of the parameter c as follows: x = εm1 X, y = εm2 Y, c = εm3 C, a(t) = εn1 A(t), (2.1) b(t) = εn2 B(t), h(t) = εn3 H(t). In such a way that the differential equation (1.3) becomes X 0 = εm2 −m1 Y, Y 0 = −εm3 CY − εn1 +m1 −m2 A(t)X − εn2 +3m1 −m2 B(t)X 3 + εn3 −m2 H(t), (2.2) where 0 ≤ m3 , 0 ≤ m1 ≤ m2 ≤ n3 , m2 ≤ n1 + m1 , m2 ≤ n2 + 3m1 , and {m2 − m1 , m3 , n1 + m1 − m2 , n2 + 3m1 − m2 , n3 − m2 } ∩ {1} = 6 ∅. We distinguish the following seven cases with their corresponding subcases, recall that we want to apply the averaging theory of first order for studying the periodic solutions of the differential system (1.2), see a summary on this theory at the appendix. Case I: m2 − m1 = 0 and m3 = 0. Then we have the following subcases (I.1) n1 + m1 − m2 = 0, n2 + 3m1 − m2 = 0, n3 − m2 = 1; (I.2) n1 + m1 − m2 = 0, n2 + 3m1 − m2 = 1, n3 − m2 = 0; (I.3) n1 + m1 − m2 = 1, n2 + 3m1 − m2 = 0, n3 − m2 = 0; (I.4) n1 + m1 − m2 = 0, n2 + 3m1 − m2 = 1, n3 − m2 = 1; (I.5) n1 + m1 − m2 = 1, n2 + 3m1 − m2 = 0, n3 − m2 = 1; (I.6) n1 + m1 − m2 = 1, n2 + 3m1 − m2 = 1, n3 − m2 = 0; (I.7) n1 + m1 − m2 = 1, n2 + 3m1 − m2 = 1, n3 − m2 = 1. Case II: m2 − m1 = 0 and m3 ≥ 1. Case III: m2 − m1 = 1 and m3 = 0. Case IV: m2 − m1 = 1 and m3 = 1. Case V: m2 − m1 > 1 and m3 = 1. Case VI: m2 − m1 = 1 and m3 > 1. Case VII: m2 − m1 > 1 and m3 > 1. Every case α from II to VII can be split into the following eight subcases: (α.1) n1 + m1 − m2 = 0, n2 + 3m1 − m2 = 0, n3 − m2 = 0, (α.2) n1 + m1 − m2 = 0, n2 + 3m1 − m2 = 0, n3 − m2 = 1, (α.3) n1 + m1 − m2 = 0, n2 + 3m1 − m2 = 1, n3 − m2 = 0, (α.4) n1 + m1 − m2 = 1, n2 + 3m1 − m2 = 0, n3 − m2 = 0, (α.5) n1 + m1 − m2 = 0, n2 + 3m1 − m2 = 1, n3 − m2 = 1, 170 R. Benterki & J. Llibre (α.6) n1 + m1 − m2 = 1, n2 + 3m1 − m2 = 0, n3 − m2 = 1, (α.7) n1 + m1 − m2 = 1, n2 + 3m1 − m2 = 1, n3 − m2 = 0, (α.8) n1 + m1 − m2 = 1, n2 + 3m1 − m2 = 1, n3 − m2 = 1. We have applied the three theorems of averaging (see section 4) for studying the existence of periodic solutions of the 55 previous subcases of differential systems (2.2). Theorem 1.1 comes from the subcase (III.1), and Theorem 1.2 follows from the subcase (IV.1). All the subcases, different from (III.1) or (IV.1), either do not satisfy the hy- potheses of one of the three theorems of averaging, or provide partial results of the ones stated in Theorems 1.1 and 1.2. So in what follows we shall consider only the subcases (III.1) or (IV.1). Theorem 4.3 has been applied for studying the subcases (α, 8) for α = IV, . . . , V II, and either do not provide periodic solutions, or provide particular cases of the results given in Theorems 1.1 and 1.2. In short, in what follows we only provide the details of the positive results, i.e. we shall give the proofs of Theorems 1.1 and 1.2. 3. Proof of Theorem 1.1 For the case (III.1), i.e. for m2 = m1 + 1, m3 = 0, m2 = n1 = n2 = n3 = 1 and m1 = m3 = 0; (3.1) system 2.2 becomes Ẋ = εY, (3.2) Ẏ = −CY − A(t)X − B(t)X 3 + H(t). We shall apply the averaging Theorem 4.1 to system (3.2) and we shall obtain Theorem 1.1. In what follows we shall use the notation of Theorem 4.1, see the appendix. Thus x = (X, Y )T and   0 F0 (t, x) =  , −CY − A(t)X − B(t)X 3 + H(t)   Y F1 (t, x) =   , 0   0 F2 (t, x) =   . 0 The unperturbed differential system (4.5) has the solution x(t, z, 0) = (X(t), Y (t))T such that x(0, z, 0) = z = (X0 , Y0 )T , where X(t) = X0 ,  Z t  Y (t) = e−Ct Y0 + eCs −B(s)X03 − A(s)X0 + H(s) ds .  0 Periodic solutions of Duffing equations 171 In order that x(t, z, 0) be a periodic solution we must choose Z T 1 eCs −B(s)X03 − A(s)X0 + H(s) ds.  Y0 = CT e −1 0 So we get ! Z T 1 Cs −B(s)X03  zα = (α, β0 (α)) = X0 , CT e − A(s)X0 + H(s) ds . e −1 0 Therefore, following the notation of Theorem 4.1, we have n = 2 and k = 1. Now we compute the fundamental matrix Mzα (t) associated to the first varia- tional system (4.6) such that Mzα (0) = Id of R2 , and we obtain   1 0   Mzα (t) =  Z t  . Cs  −Ct −Ct  −e e (3B(s)X0 + A(s)) ds e 0 Its inverse matrix is   1 0 Mz−1 (t) =  Z t .   α Cs 3B(s)X02  Ct e + A(s) ds e 0 Since the matrix   0 0 Mz−1 (0) − Mz−1 (T ) =  t  Z  α α  eCs 3B(s)X02 + A(s) ds 1 − eCT  − 0 has a zero 1 × 1 matrix in the upper right corner and a non–zero 1 × 1 matrix in its lower right corner equal to 1 − eCT , because T 6= 0. We can apply the averaging theory described in Theorem 4.1 for studying the periodic solutions which can be prolonged from the unperturbed differential system to the perturbed one. Therefore, since for our differential system we have ξ(X, Y ) = X, then we must compute the function F(α) = F(X0 ) given in (4.7), i.e. ! Z T F(X0 ) = ξ Mz−1 α (t)F1 (t, x(t, zα , 0))dt 0 " Z T Z T −Ct 1 eCs −B(s)X03 − X0 A(s) + H(s) ds  = e 0 −1 + eCT 0 Z t  eCs −B(s)X03 − X0 A(s) + H(s) ds dt  + 0 Z T = Y (t) dt. 0 Theorem 4.1 says that for every simple real root X0 of the polynomial F(X0 ) the differential system (3.2) with ε 6= 0 sufficiently small has a periodic solution (X(t), Y (t)) such that (X(0), Y (0)) tends to (X0 , β0 (X0 )) when ε → 0. 172 R. Benterki & J. Llibre Now it is to check that the function F(X0 ) after the change of variables (2.1) satisfying (3.1), i.e. y h(t) b(s) a(s) X = x, Y = , H(t) = , B(s) = , A(s) = , ε ε ε ε becomes the polynomial p(x0 ) defined in section 1 just before the statement of Theorem 1.1. Hence Theorem 1.1 is proved. 4. Proof of Theorem 1.2 For the case (IV.1), i.e. m2 = m1 + 1, m3 = 1, m1 = 1, m2 = 2, n1 = −n2 = 1 and n3 = 2; (4.1) system (2.2) becomes Ẋ = εY, (4.2) Ẏ = −εCY − A(t)X − B(t)X 3 + H(t). We shall apply the averaging Theorem 4.2 to system (4.2) and we shall obtain Theorem 1.2. In what follows we shall use the notation of Theorem 4.2. Thus x = (X, Y )T and   0 F0 (t, x) =  , −A(t)X − B(t)X 3 + H(t)   Y F1 (t, x) =  , −CY   0 F2 (t, x) =   . 0 The unperturbed differential system (4.5) has the solution x(t, z, 0) = (X(t), Y (t))T such that x(0, z, 0) = z = (X0 , Y0 )T , where X(t) = X0 , Z t −B(s)X03 − A(s)X0 + H(s) ds.  Y (t) = Y0 + 0 In order that x(t, z, 0) be a periodic solution X0 must satisfy Z T −B(s)X03 − A(s)X0 + H(s) ds = 0,  (4.3) 0 and Y0 is arbitrary. Therefore we get  zα = (α, β0 (α)) = Y0 , X̄0 , Periodic solutions of Duffing equations 173 where X̄0 is a real root of the cubic polynomial (4.3). In short the unperturbed system (i.e. system (4.3) with ε = 0) has at most three families of periodic solutions because Y0 is arbitrary and X̄0 is a real root of the cubic polynomial (4.3). Therefore, using the notation of Theorem 4.2, we have n = 2 and k = 1 for each one of these possible families of periodic solutions. We compute the fundamental matrix Mzα (t) associated to the first variational system (4.6) associated to the vector field (Ẏ , Ẋ) given by (4.2) with ε = 0, and such that Mzα (0) = Id of R2 , and we obtain t  Z  3B(s)X02 + A(s) ds   1 − Mzα (t) =  0 . 0 1 Its inverse matrix is  Z t  2  1 3B(s)X 0 + A(s) ds Mz−1 (t) =  .   α 0 0 1 The matrix T  Z  3B(s)X02  0 − + A(s) ds  Mz−1 α (0) − Mz−1 α (T ) =  0  0 0 has a non–zero 1 × 1 matrix in the upper right corner if the real root X̄0 of the cubic polynomial (4.3) is simple, and a zero 1×1 matrix in its lower right corner. Therefore the assumptions of Theorem 4.2 hold, then by applying this theorem we study the periodic solutions which can be prolonged from the unperturbed differential system to the perturbed one. Since for our differential system we have ξ ⊥ (Y, X) = X, then we must compute the function G(α) = G(Y0 ) given in (4.7), i.e. ! Z T Z T ⊥ G(Y0 ) = ξ Mz−1 α (t)F1 (t, x(t, zα , 0))dt =− CY0 = −CT Y0 . 0 0 Theorem 4.2 says that for every simple real root Y0 = 0 of the polynomial G(Y0 ) the differential system (4.2) with ε 6= 0 sufficiently small has a periodic solution (X(t), Y (t)) such that (X(0), Y (0)) tends to (X̄0 , 0) when ε → 0, being X̄0 a simple real root of the cubic polynomial (4.3). Now it is easy to check that the cubic polynomial (4.3) after the change of variables (2.1) satisfying (4.1), i.e. x y h(t) b(s) a(s) X= , Y = , H(t) = , B(s) = , A(s) = , ε ε2 ε2 ε ε becomes the polynomial q(x0 ) defined in section 1 just before the statement of Theorem 1.2. Hence Theorem 1.2 is proved. 174 R. Benterki & J. Llibre The appendix: Periodic solutions via the averaging theory In this section we present the basic results on the averaging theory of first order that we need for proving our results. We consider the problem of bifurcation of T –periodic solutions from the differ- ential systems of the form ẋ = F0 (t, x) + εF1 (t, x) + ε2 F2 (t, x, ε), (4.4) with ε = 0 to ε 6= 0 sufficiently small. The functions F0 , F1 : R × Ω → Rn and F2 : R × Ω × (−ε0 , ε0 ) → Rn are C 2 , T –periodic in the first variable and Ω is an open subset of Rn . The main assumption is that the unperturbed system ẋ = F0 (t, x), (4.5) 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 system (4.5) such that x(0, z, ε) = z. We write the linearization of the unperturbed system along a periodic solution x(t, z, 0) as ẏ = Dx F0 (t, x(t, z, 0))y. (4.6) In what follows we denote by Mz (t) a fundamental matrix of the linear differ- ential system (4.6), by ξ : Rm × Rn−m → Rm and ξ ⊥ : Rm × Rn−m → Rn−m the projections of Rn onto its first m and n − m coordinates respectively; i.e. ξ(x1 , . . . , xn ) = (x1 , . . . , xm ), and ξ ⊥ (x1 , . . . , xn ) = (xm+1 , . . . , xn ). Theorem 4.1. Let V ⊂ Rm be open and bounded, let β0 : Cl(V ) → Rn−m be a C k function and Z = {zα = (α, β0 (α)) | α ∈ Cl(V )} ⊂ Ω its graphic in Rn . Assume that for each zα ∈ Z the solution x(t, zα , 0) of (4.5) is T -periodic and that there exists a fundamental matrix Mzα (t) of (4.6) such that the matrix Mz−1α (0)−Mz−1α (T ) (a) has in the lower right corner the (n−m)×(n−m) matrix ∆α with det(∆α ) 6= 0, and (b) has in the upper right corner the m × (n − m) zero matrix. Consider the function F : Cl(V ) → Rm defined by ! Z T F(α) = ξ Mz−1 α (t)F1 (t, x(t, zα , 0))dt . (4.7) 0 Suppose that there is α0 ∈ V with F(α0 ) = 0, then the following statements hold for ε 6= 0 sufficiently small. (i) If det((∂F/∂α)(α0 )) 6= 0, then there is a unique T -periodic solution ϕ1 (t, ε) of system (4.4) such that ϕ1 (t, ε) → x(t, zα0 , 0) as ε → 0. (ii) If m = 1 and F 0 (α0 ) = · · · = F (s−1) (α0 ) = 0 and F (s) (α0 ) 6= 0 with s ≤ k, then there are at most s T -periodic solutions ϕ1 (t, ε), . . . , ϕs (t, ε) of system (4.4) such that ϕi (t, ε) → x(t, zα0 , 0) as ε → 0 for i = 1, . . . , s. Periodic solutions of Duffing equations 175 Theorem 4.1 is a classical result due to Malkin [11] and Roseau [14]. For a shorter proof of Theorem 4.1(a), see [1]. As we shall see in this paper we have cases where Theorem 4.1 cannot be applied for studying the existence of periodic solutions, because its assumptions are not satisfied. Then in [7] the following result on averaging has been proved. Theorem 4.2. Let V ⊂ Rm be open and bounded, let β0 : Cl(V ) → Rm be a C k function and Z = {zα = (α, β0 (α)) | α ∈ Cl(V )} ⊂ Ω its graphic in R2m . Assume that for each zα ∈ Z the solution x(t, zα , 0) of (4.5) is T -periodic and that there exists a fundamental matrix Mzα (t) of (4.6) such that the matrix Mz−1α (0)−Mz−1α (T ) (a) has in the upper right corner the m × m matrix ∆α with det(∆α ) 6= 0, and (b) has in the lower right corner the m × m zero matrix. Consider the function G : Cl(V ) → Rm defined by Z T ! ⊥ −1 G(α) = ξ Mzα (t)F1 (t, x(t, zα , 0))dt . (4.8) 0 Suppose that there is α0 ∈ V with G(α0 ) = 0, then the following statements hold for ε 6= 0 sufficiently small. (i) If det((∂G/∂α)(α0 )) 6= 0, then there is a unique T -periodic solution ϕ1 (t, ε) of system (4.4) such that ϕ1 (t, ε) → x(t, zα0 , 0) as ε → 0. (ii) If m = 1 and G 0 (α0 ) = · · · = G (s−1) (α0 ) = 0 and G (s) (α0 ) 6= 0 with s ≤ k, then there are at most s T -periodic solutions ϕ1 (t, ε), . . . , ϕs (t, ε) of system (4.4) such that ϕi (t, ε) → x(t, zα0 , 0) as ε → 0 for i = 1, . . . , s. In any case now we shall recall the more classical result on averaging theory for studying periodic solutions. We consider the initial value problems ẋ = εF1 (t, x) + ε2 F2 (t, x, ε), x(0) = x0 , (4.9) and ẏ = εg(y), y(0) = x0 , (4.10) n with x , y and x0 in some open Ω of R , t ∈ [0, ∞), ε ∈ (0, ε0 ]. We assume that F1 and F2 are periodic of period T in the variable t, and we set 1 T Z g(y) = F1 (t, y)dt. T 0 Theorem 4.3. Assume that F1 , Dx F1 ,Dxx F1 and Dx F2 are continuous and bounded by a constant independent of ε in [0, ∞) × Ω × (0, ε0 ], and that y(t) ∈ Ω for t ∈ [0, 1/ε]. Then the following statements holds: 1. For t ∈ [0, 1/ε] we have x(t) − y(t) = O(ε) as ε → 0. 2. 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