This is a preprint of: “Many periodic solutions for a second order cubic periodic differential equa- tion”, Adriana Buica, Armengol Gasull, Monatsh. Math., 2020. DOI: [10.1007/s00605-020-01433-4] MANY PERIODIC SOLUTIONS FOR A SECOND ORDER CUBIC PERIODIC DIFFERENTIAL EQUATION ADRIANA BUICĂ1 AND ARMENGOL GASULL2 Abstract. The aim of this work is to provide results that assure the existence of many isolated T -periodic solutions for T -periodic second-order differential equations of the form x00 = a(t)x + b(t)x2 + c(t)x3 . We use bifurcation meth- ods, including Malkin functions and results of Fonda, Sabatini and Zanolin. In addition, we give a general result that assures the existence of a T -periodic per- turbation of a non-isochronous center with an arbitrary number of T -periodic solutions. 1. Introduction The aim of this work is to provide results that assure the existence of many T -periodic solutions for the class of T -periodic second-order differential equations (1) x00 = a(t)x + b(t)x2 + c(t)x3 . We assume that a, b, c ∈ C(R) are T-periodic functions, and T > 0 is a fixed real number. Recall that, when in the above equation we replace x00 by x0 , we obtain the classical Abel equation. Hence equation (1) can be seen as a generalized Abel equation, and our goal is similar to the celebrated result of Lins-Neto ([15]) where the author proved that there is no upper bound for the number of isolated T - periodic solutions for the classical T -periodic Abel differential equations. Another motivation to look for such results came after reading the papers [12] and [1] where sufficient conditions are given in order to assure the existence of 1 and, respectively, 2 non-null T -periodic solutions. We mention that the authors of [1, 12] were motivated to study equations of this form by Austin [2] who proposed a similar equation as a biomathematical model of an aneurysm, and by Cronin [6] who was the first to study the existence of periodic solutions of it. In the class of equations of the form (1) we distinguish the ones with constant coefficients a, b, c ∈ R. Here T > 0 is not related with the coefficients. Note that, whenever a 6= 0 and b2 −4ac > 0, equation (1) has exactly 2 non-null constant (thus T -periodic for any T ) isolated solutions. The equation x00 = 0 has any constant function as T -periodic solution (for arbitrary T > 0), while the solutions of the equation x00 = −x are all 2π-periodic. Another interesting example of autonomous equation of the form (1) is x00 = −x3 whose set of T -periodic orbits (for arbitrary T > 0) is a countably infinite set. This example will be discussed in detail in Section 2. In addition, the linear Hill equation x00 = a(t)x is in the class (1) when a ∈ C(R) is T -periodic. An interesting aspect to discuss is how to find a(t) such that all the 1 2 A. BUICĂ AND A. GASULL solutions of this equation are T -periodic. An answer will be given in Section 3.4.1. For other results in this direction one can see [13, 17]. For equation (1) with time-dependent coefficients we obtain different sufficient conditions that guarantee the existence of 2, 3, 8 and finally 2N (with arbitrary N ∈ N∗ ) isolated T -periodic solutions. Our examples of such equations are small perturbations of one of the equations x00 = 0, x00 = −x, or x00 = −x3 . To obtain the main result in [1], Araujo-Pedroso used the Mawhin continuation theorem, a result within the topological coincidence degree theory for infinite di- mensional operators. The results in [12], which conclude the existence of a nonnull T -periodic solution, where obtained by Grossinho-Sanchez using variational meth- ods, more precisely, a variant of Mountain-Pass Theorem due to Rabinowitz. Here we use bifurcation methods, more precisely, the Malkin bifurcation theorem and another result due to Fonda-Sabatini-Zanolin [9]. These results will be stated in Section 2. For a short history of the Malkin bifurcation theorem and for a modern proof one can see [3, 4]. The main results on the generalized Abel equation (1) are stated and proved in Section 3. In Section 4, the last one, we revisit the Malkin theorem in the planar case and present new results on non-necessarily Hamilton- ian systems. One of these results assures the existence of 2N different T -periodic solutions, for any arbitrary integer N ≥ 2. Note that, in Section 3, we will ap- ply the Fonda-Sabatini-Zanolin theorem in order to obtain 2N different T -periodic solutions for a planar system with a Hamiltonian structure. 2. Main tools 2.1. The Malkin bifurcation theorem. Let n ≥ 1 be an integer and T > 0 be a real. We consider in this subsection the n-dimensional differential system (2) u0 = f (u) + εg(t, u) where ε ∈ R, f : Rn → Rn and g : R × Rn → Rn are sufficiently smooth and T -periodic with respect to the variable 0 t0 . Let k ∈ Z be such that 1 ≤ k ≤ n. We assume that the unperturbed system (3) u0 = f (u) has a k-dimensional T -period manifold Z ⊂ Rn . More precisely, we assume that there exists a C 2 function ξ : U → Rn , where U is a nonempty, open subset of Rk , such that the Jacobian matrix Dξ(θ) has full rank for any θ ∈ U , and Z = ξ(U ). Moreover, for any θ ∈ U , the solution of (3) with u(0) = ξ(θ), denoted u(t, θ), is T -periodic. Note that T > 0 is not necessarily the minimal period. In addition, we assume that Z = ξ(U ) is normally nondegenerate, that is, for each θ ∈ U , the first variational (T -periodic) system (4) v 0 = Df (u(t, θ)) v has exactly k linearly independent T -periodic solutions. MANY PERIODIC SOLUTIONS 3 These hypotheses assure the existence (see [14]), for each θ ∈ U , of exactly k linearly independent T -periodic solutions of the adjoint system w0 = − [Df (u(t, θ))]tr w, denoted wi (t, θ), i = 1, k. The Malkin bifurcation function M : U → Rk is defined componentwise by Z T θ 7→ h wi (t, θ), g(t, u(t, θ))idt, i = 1, k. 0 Here [·]tr denotes the transpose of a matrix, while h·, ·i denotes the scalar product in Rn . We remark that in the literature there are other representations of the bifurcation function, that eventually differ by a change of variable and/or a factor without zeros. In the hypotheses and notations of this section we have the following result known as the Malkin theorem [16] (for a proof see also [3, 4]). Theorem 1. Assume that there exists θ∗ ∈ U such that M (θ∗ ) = 0 and the Jacobian determinant det DM (θ∗ ) 6= 0. Then there exists ε > 0 such that for each ε ∈ (−ε, ε), system (2) has a T -periodic solution uε with lim uε (0) = ξ(θ∗ ), and ε→0 with the property that it is isolated in the set of T -periodic solutions of (2). We say that a T -periodic solution u(t) of a non-autonomous differential equation is isolated in the set of T -periodic solutions when there exists a neighborhood of u(0) ∈ Rn where u(0) is the only initial value of a T -periodic solution. Note that this definition is meaningful only for non-autonomous differential equations since the only isolated T -periodic solutions of an autonomous differential equations are its isolated critical points. In the end of this subsection we discuss, for further use, different characteri- zations of the condition that the k-dimensional T -period manifold Z is normally nondegenerate. For the case when the unperturbed system (3) is linear, we have the following result. Proposition 2. Assume that the unperturbed system (3) is linear and has exactly k linearly independent T -periodic solutions. Let Z be the linear space spanned by the initial values of the k linearly independent T -periodic solutions. Then Z is a k–dimensional normally nondegenerate T -period manifold. For the case when the unperturbed system (3) is nonlinear, the hypothesis that Z is a k-dimensional T -period manifold assures that the Jacobian matrix Dθ u(t, θ) is an n × k matrix whose columns are k linearly independent T -periodic solutions of the variational system (4). Other solutions of the variational system are deriva- tives of the general solution of the unperturbed system (3) with respect to other 4 A. BUICĂ AND A. GASULL directions in Rn . Thus, one has to decide whether one of these other solutions is T -periodic. 2.2. The Fonda-Sabatini-Zanolin theorem. We consider in this subsection the planar Hamiltonian system (5) u0 = J∇H(u) + εJ∇H(t, u)   0 −1 where J = denotes the symplectic matrix, H : R2 → R and H : 1 0 R × R2 → R are sufficiently smooth and T -periodic with respect to the variable 0 t0 and in ∇H(t, u) the gradient is only with respect the variable u. We assume that the unperturbed autonomous planar Hamiltonian system (6) u0 = J∇H(u) has a period annulus A ⊂ R2 such that the inner and outer components of its boundary are Jordan curves. Assume, in addition, that A is not isochronous, that is, the period of the periodic orbits in A covers an interval [T1 , T2 ], with T1 < T2 . Without loosing the generality, assume that the origin is a singularity of (6) and the periodic orbits in A encircle the origin. Under these assumptions, the authors in [9] proved the following result. Theorem 3. Given a positive integer j satisfying T1 < T /j < T2 there is an ε > 0 such that, if ε ∈ (−ε, ε), then system (5) has at least two T - periodic solutions, whose orbits are contained in A, which make exactly j rotations around the origin in the period time T . Note that this result does not provide any information whether the T -periodic solutions are isolated or not. 3. Results on the generalized Abel equation 3.1. Examples with two non-null periodic solutions. Theorem 4. Let T > 0 and a, b, c ∈ C(R) be T -periodic functions. Denote Z T Z T Z T b a= a(t)dt, bb = b(t)dt, bc= c(t)dt 0 0 0 and assume that a 6= 0, b bb2 − 4 b abc > 0. Then there exists ε > 0 such that, for each ε ∈ (−ε, ε), the equation x00 = εa(t)x + εb(t)x2 + εc(t)x3 has at least two isolated non-null T -periodic solutions. MANY PERIODIC SOLUTIONS 5 Proof. We write the given second order equation as the first order system, x0 = y, y 0 = ε[a(t)x + b(t)x2 + c(t)x3 ], which has the form (2) with       x y 0 n = 2, u = , f (t, x, y) = , g(t, x, y) = . y 0 a(t)x + b(t)x2 + c(t)x3 The unperturbed system is x0 = y, y 0 = 0, which is linear, and whose adjoint system is x0 = 0, y 0 = −x. Let   θ k = 1, ξ : R → R , ξ(θ) = 2 , for any θ ∈ R. 0 It is easy to see that, for all t, θ ∈ R,     θ 0 u(t, θ) = , w1 (t) = , 0 1 which are constant functions, thus, in particular, T -periodic. Hence Z = ξ(R) is an 1-dimensional T -period manifold, which, by Proposition 2, is normally non-degenerate. In this case, the Malkin bifurcation function is Z T M : R → R, M (θ) = [a(t)θ + b(t)θ2 + c(t)θ3 ]dt. 0 Then a θ + bb θ2 + b M (θ) = b c θ3 . The conditions ba 6= 0, bb2 − 4 b abc > 0 from the hypothesis, assure that M has two non-null simple zeros. From here, by Theorem 1, the conclusion follows.  Remark 5. Let m ≥ 1 be an integer. One can check that Theorem 4 remains valid for the differential equation of arbitrary order x(m) = εa(t)x + εb(t)x2 + εc(t)x3 , since the Malkin bifurcation function is the same. 6 A. BUICĂ AND A. GASULL 3.2. Other examples with two non-null periodic solutions. For complete- ness and because it looks similar to our previous result, we state without proof the main result of [1], that also concludes the existence of at least two non-null periodic solutions for some equations of the form (1). Theorem 6. [1] Assume that there exist four positive constants a, a, b, b, c, c > 0 such that a ≤ a(t) ≤ a, b ≤ −b(t) ≤ b, c ≤ −c(t) ≤ c, and 2 2 b − b2 < 4ac, b + 4ac > (b − c)2 . Then there exists T > 0, depending on a, a, b, b, c, c, such that if T < T, the equation (1) has at least two non-null T- periodic solutions. 3.3. Examples with three non-null periodic solutions. Theorem 7. Let T > 0 and ϕi ∈ C 2 (R), i ∈ 1, 2, 3 be three T -periodic functions. Assume that 0 < ϕ1 (t) < ϕ2 (t) < ϕ3 (t) for all t ∈ R and consider ∆1 (t) ∆2 (t) ∆3 (t) a(t) = , b(t) = , c(t) = , ∆(t) ∆(t) ∆(t) where ∆ = ϕ1 ϕ2 ϕ3 (ϕ3 − ϕ1 )(ϕ2 − ϕ1 )(ϕ3 − ϕ2 ), ϕ001 ϕ21 ϕ31 ϕ1 ϕ001 ϕ31 ϕ1 ϕ21 ϕ001 ∆1 = ϕ002 ϕ22 ϕ32 , ∆2 = ϕ2 ϕ002 ϕ32 , ∆3 = ϕ2 ϕ22 ϕ002 . ϕ003 ϕ23 ϕ33 ϕ3 ϕ003 ϕ33 ϕ3 ϕ23 ϕ003 Then a, b, c ∈ C(R) are T -periodic and the differential equation x00 = a(t)x + b(t)x2 + c(t)x3 has at least three non-null T -periodic solutions, namely ϕ1 , ϕ2 , ϕ3 . Proof. Given ϕ1 , ϕ2 , ϕ3 as in the hypothesis, it is easy to see that a, b, c are chosen such that a(t)ϕ1 (t) + b(t)ϕ21 (t) + c(t)ϕ31 (t) = ϕ001 (t) a(t)ϕ2 (t) + b(t)ϕ22 (t) + c(t)ϕ32 (t) = ϕ002 (t) a(t)ϕ3 (t) + b(t)ϕ23 (t) + c(t)ϕ33 (t) = ϕ003 (t). Thus ϕ1 , ϕ2 , ϕ3 are the three solutions of the given equation.  Notice that although in most cases the above construction gives a differential equation for which the three functions ϕj (t), j = 1, 2, 3 are isolated T -periodic solutions, in some cases they can belong to a continuum of periodic solutions. MANY PERIODIC SOLUTIONS 7 Remark 8. Let m ≥ 1 be an integer. One can check that Theorem 7 remains valid for the differential equation of arbitrary order x(m) = a(t)x + b(t)x2 + c(t)x3 , (m) by changing each ϕ00j by ϕj . 3.4. Examples with eight non-null periodic solutions. Theorem 9. There exist a0 , a3 , a4 , b1 , b2 , b5 , b6 , c0 , c3 , c4 ∈ R and ε > 0 such that, for each ε ∈ (−ε, ε), the differential equation x00 = −x + εa(t)x + εb(t)x2 + εc(t)x3 , where a(t) = a0 + a3 cos(2t) + a4 sin(2t) b(t) = b1 cos(t) + b2 sin(t) + b5 cos(3t) + b6 sin(3t) c(t) = c0 + c3 cos(2t) + c4 sin(2t) + cos(4t), has eight isolated non-null 2π-periodic solutions. Proof. We write the given second order equation as the first order system, x0 = y, y 0 = −x + ε[a(t)x + b(t)x2 + c(t)x3 ], which has the form (2) with n = 2, T = 2π,       x y 0 u= , f (x, y) = , g(t, x, y) = . y −x a(t)x + b(t)x2 + c(t)x3 The unperturbed system is x0 = y, y 0 = −x, which is linear, and whose adjoint is again x0 = y, y 0 = −x. Let k = 2, ξ : R2 → R2 , ξ(θ) = θ, for any θ ∈ R2 . It is easy to see that,       θ1 cos t + θ2 sin t cos t sin t u(t, θ) = , w1 (t) = , w2 (t) = . −θ1 sin t + θ2 cos t − sin t cos t Hence Z = R2 is a 2-dimensional T -period manifold which, by Proposition 2, is normally non-degenerate. In this case, the Malkin bifurcation function is Z 2π Z 2π M : R → R , M1 (θ) = − 2 2 m(t, θ) sin tdt, M2 (θ) = m(t, θ) cos tdt. 0 0 where θ = (θ1 , θ2 ), M = (M1 , M2 ) and m(t, θ) = a(t)(θ1 cos t + θ2 sin t) + b(t)(θ1 cos t + θ2 sin t)2 + c(t)(θ1 cos t + θ2 sin t)3 . 8 A. BUICĂ AND A. GASULL We remark that the above bifurcation function can also be obtained as a classical averaged function. Indeed, through the linear change of variables (x, y) 7→ (ϕ, ψ), given by x = ϕ cos t + ψ sin t, y = −ϕ sin t + ψ cos t, our system is transformed into ϕ0 = −ε m(t, ϕ, ψ) sin t, ψ 0 = ε m(t, ϕ, ψ) cos t. After some tedious but straightforward computations we obtain: 8 − M1 (θ) = 4θ1 a4 + (8a0 − 4a3 )θ2 π +(2b2 + 2b6 )θ12 + (4b1 − 4b5 )θ1 θ2 + (6b2 − 2b6 )θ22 +2c4 θ13 + (6c0 − 3)θ12 θ2 + 6c4 θ1 θ22 + (6c0 − 4c3 + 1)θ23 , 8 M2 (θ) = (8a0 + 4a3 )θ1 + 4a4 θ2 π +(6b1 + 2b5 )θ12 + (4b2 + 4b6 )θ1 θ2 + (2b1 − 2b5 )θ22 +(6c0 + 4c3 + 1)θ13 + 6c4 θ12 θ2 + (6c0 − 3)θ1 θ22 + 2c4 θ23 . Since both components of M are polynomials of degree 3, we deduce, by Bézout Theorem, that M can have either a continuum of roots, or at most 3×3 = 9 simple roots. In the sequel our work is focused on finding the coefficients such that M has exactly 9 roots: the (0, 0) and 8 more simple zeros. The 10 parameters appearing in the expression of M are uniquely defined after imposing that M has, for instance, the following 5 zeros: (1, 1), (2, 7), (3, 5), (−2, 6), (−3, 3). Now the expression of M is kM1 (θ) = 2441173θ13 + 16014171θ12 θ2 + 7323519θ1 θ22 + 718869θ23 −65085731θ12 − 54282494θ1 θ2 − 20379971θ22 +41207986θ1 + 72042478θ2 , −kM2 (θ) = 17330805θ13 + 7323519θ12 θ2 + 16014171θ1 θ22 + 2441173θ23 −16270255θ12 − 130171462θ1 θ2 − 27141247θ22 +89265310θ1 + 41207986θ2 , where k = 16/(3494667π). So, M (θ) = 0 is a system of two polynomial equations with two unknowns. Moreover, this system has integer coefficients. For solving such a system we used an algorithm based on resultants, see for instance [18, 11]. We will prove that, beside the 5 solutions that we initially imposed and the (0, 0) solution, the system has other three real solutions. Finally, due to Bézout Theorem we deduce that all these 9 roots are simple. It holds that S(θ1 ) = Resθ2 (M1 , M2 ) =θ1 (θ1 + 3)(θ1 + 2)(θ1 − 1)(θ1 − 2)(θ1 − 3)P3 (θ1 ), T (θ2 ) = Resθ1 (M1 , M2 ) =θ2 (θ2 − 1)(θ2 − 3)(θ2 − 5)(θ2 − 6)(θ2 − 7)Q3 (θ2 ), MANY PERIODIC SOLUTIONS 9 where Resy (U (y), V (y)) denotes the resultant of two polynomials U and V respect to y, and P3 and Q3 are two cubic polynomials with rational coefficients. Since the resultants are not identically zero, it holds that the maximum number of solutions of the system M = 0, taking into account their multiplicities, is 9. In particular, if we prove that there are 9 solutions, all of them must be simple. It is easy to prove that P3 has 3 real roots, say u1 < u2 < u3 and Q3 also, say v1 < v2 < v3 . In fact u1 ≈ −1.037593, u2 ≈ −0.874090, u3 ≈ 38.784916, v1 ≈ −240.095677, v2 ≈ 0.786412, v3 ≈ 5.035431. Hence, each solution (θ1 , θ2 ) of M = 0 is contained in the set of 9 × 9 = 81 couples formed by one solution of S and one of T. In this case, because P3 and Q3 are cubic polynomials, it is not difficult to check that the system has exactly 9 solutions: the 6 ones that we already know, together with (u1 , v3 ), (u2 , v2 ), (u3 , v1 ). For higher degree polynomials, the use of Poincaré-Miranda Theorem helps to decide which of the couples, candidate to be a solution of the system, is an actual solution for it, see [10, 11]. Finally, we give now the exact expressions of the coefficients of the equation that has eight non-null 2π-periodic solutions 40326947 130476 20603993 a(t) = − − cos(2t) − sin(2t) 6989334 105899 3494667 21705751 42732851 32576743 87438611 b(t) = cos t + sin t − cos(3t) + sin(3t) 6989334 6989334 6989334 6989334 7181447 125848 2441173 c(t) = − − cos(2t) − sin(2t) + cos(4t). 6989334 105899 3494667  Remark 10. Let m ≥ 1 be an integer. One can check that Theorem 9 remains valid for the differential equation of arbitrary order x(2m) = (−1)m x + εa(t)x + εb(t)x2 + εc(t)x3 , since the Malkin bifurcation function is the same. 3.4.1. The equation x00 = ã(t)x. In this subsection we will consider the linear equation x00 = ã(t)x, assuming that ã ∈ C(R) is T -periodic and such that all the solutions of this equation are T -periodic. Also, consider its perturbation x00 = ã(t)x + εa(t)x + εb(t)x2 + εc(t)x3 , where a, b, c ∈ C(R) are T -periodic. The Malkin bifurcation theorem can be applied to this equation in the same way as it was applied in the proof of Theorem 9. 10 A. BUICĂ AND A. GASULL Just note that, denoting C(t) and S(t) the two linearly independent solutions of x00 = ã(t)x, that satisfy S(0) = C 0 (0) = 0 and C(0) = S 0 (0) = 1, we have       θ1 C(t) + θ2 S(t) −S 0 (t) −C 0 (t) u(t, θ) = , w1 (t) = , w2 (t) = . θ1 C 0 (t) + θ2 S 0 (t) S(t) C(t) Both components of the Malkin bifurcation function will be, like in Theorem 9, polynomials of degree 3. Thus M can have at most 8 non-null simple zeros. Of course, the realization of this number of non-null simple zeros cannot be decided in this general situation. An interesting aspect to discuss is how to find ã(t) such that all the solutions of 00 x = ã(t)x are T -periodic. An answer is given in the next result. Theorem 11. Let C, S ∈ C 2 (R) be two T -periodic functions with the property that, for all t ∈ R, C(t) S(t) W (t) = > 0. C 0 (t) S 0 (t) Define  2 f 3 W0 W 00 W f= C0 S0 ã = − − , where W . 4 W 2W W C 00 S 00 Then the functions C(t) S(t) p and p W (t) W (t) are two linearly independent T -periodic solutions of x00 = ã(t)x. Proof. First we make the following notations. C S ϕ ψ fν = ϕ0 ψ 0 ϕ= √ , ψ=√ , Wν = , W . W W ϕ0 ψ 0 ϕ00 ψ 00 By direct but cumbersome computations we obtain that Wν = 1, fν = −ã. W The second order differential equation whose solutions are ϕ and ψ is ϕ ψ x ϕ0 ψ 0 x0 = 0, ϕ00 ψ 00 x00 fν x = 0 and, further, to x00 = ã(t)x.  which is equivalent to Wν x00 − Wν0 x0 + W MANY PERIODIC SOLUTIONS 11 3.5. Examples with an arbitrary number of non-null periodic solutions. Theorem 12. Let N ≥ 1 be an arbitrary integer. We have that there exists ε > 0 such that for each ε ∈ (−ε, ε) the differential equation x00 = −x3 + ε[a(t)x + b(t)x2 + c(t)x3 ], has 2N non-null T -periodic solutions. Proof. We intend to apply Theorem 3. We write the given second order equation as the first order system, x0 = −y, y 0 = x3 − ε[a(t)x + b(t)x2 + c(t)x3 ], which is T -periodic and has the form (5) with   x u= , H(x, y) = x4 /4 + y 2 /2, H(t, x, y) = −a(t)x2 /2 − b(t)x3 /3 − c(t)x4 /4. y The unperturbed system is (7) x0 = −y, y 0 = x3 . It has a unique singularity (0, 0) and all its nontrivial orbits are the level curves of H defined above, and they are Jordan curves. For every h > 0 denote by τ (h) the main period of the orbit H = h. It is known that the period function τ : (0, ∞) → (0, ∞) is such that τ 0 (h) < 0 for all h > 0 and that the image of τ is the whole interval (0, ∞). For these results see [5]. In fact, when the origin is a degenerated center (the eigenvalues of the linear part at the critical point are not purely imaginary) it is known that the period of the orbits tend to infinity when the orbits approach the critical point, see [7]. Consider A ⊂ R2 the period annulus of (7) whose boundaries are the orbits of periods T2 = T + 1 and, respectively, T1 = T /(N + 1). Note that T /j ∈ (T1 , T2 ) for each j = 1, N . Theorem 3 assures that, for each j = 1, N , system (7) has at least two T -periodic solutions, which make exactly j rotations around the origin in the period time T . Thus, system (7) has at least 2N different T-periodic solutions.  Remark 13. The conclusion of Theorem 12 is also valid when, instead of the unperturbed equation x00 = −x3 we consider, for example, x00 = −x + x2 or x00 = −x + x3 . 4. Consequences of the Malkin theorem in the planar case In this section we use the notations of subsection 2.1, considering the particular case when n = 2. We assume that the planar autonomous system (8) u0 = f (u) has a period annulus P ⊂ R2 , i.e. P is nonempty, open and connected, is invariant under the flow of (8) and any orbit of (8) in P is a nontrivial closed curve. Thus, 12 A. BUICĂ AND A. GASULL in particular, f (u) 6= 0 for all u ∈ P. In these hypotheses it is known that there exists a C 1 first integral H : P → R without critical points. Recall that the C 1 , locally nonconstant function H : P → R is a first integral of (8) if and only if h∇H, f i = 0 in P. S Let I ⊂ R be an open nonempty interval such that P = h∈I Γh , where Γh is the closed orbit of (8) in P contained in H = h. Denote by σ(t, h) the solution of (8) whose orbit is Γh . Define also the period function τ : I → (0, ∞), τ (h) being the minimal period of σ(t, h). Fix T > 0 and assume that there exist h∗ ∈ I and an integer j ≥ 1 such that τ (h∗ ) = T /j. Denote by ξ(t) = σ(t, h∗ ). We have that the cycle Z = ξ(R) is an 1–dimensional T -period manifold. It is known that its nondegeneracy condition can be given in terms of the period function. More precisely, we have the following result, whose proof will be included for completeness. The proof in the particular case of a Hamiltonian system can be found in [9]. Proposition 14. The cycle Z is an 1–dimensional T -period manifold which is normally nondegenerate if and only if τ 0 (h∗ ) 6= 0. Proof. One can check that u(t, θ) = ξ(t + θ), which, of course, is T -periodic. It is clear that the number of independent T -periodic solutions of the variational system v 0 = Df (ξ(t + θ))v does not depend on θ. Thus, we will study only the system (9) v 0 = Df (ξ(t))v. Let us start proving that f (ξ(t)) and ∂σ ∂h (t, h∗ ) are both solutions of (9). Recall that ξ 0 (t) = f (ξ(t)). Hence, derivating with respect to t this equality we get 0 0 f (ξ(t)) = ξ 0 (t) = Df (ξ(t))ξ 0 (t) = Df (ξ(t))f (ξ(t)), as desired. By using that σ(t, h) = f (σ(t, h)) and derivating with respect to h, after some manipulations we get that  ∂σ 0 ∂σ (t, h) = Df (σ(t, h)) (t, h). ∂h ∂h ∗ Plugging h = h we arrive to  ∂σ 0 ∂σ (t, h ) = Df (ξ(t)) (t, h∗ ), ∗ ∂h ∂h as we wanted to prove. We claim that f (ξ(t)) and ∂σ ∂h (t, h∗ ) are linearly independent. Indeed, taking the derivative with respect to h in the relation H(σ(t, h)) = h, h ∈ I, t ∈ R, one gets h∇H(σ(t, h)), ∂σ∂h (t, h)i = 1 for all h ∈ I, t ∈ R. Assuming, by contradic- tion, that ∂h (t, h) = αf (ξ(t)) (with α ∈ R∗ ), from the previous relation we obtain ∂σ h∇H(ξ(t)), f (ξ(t))i = 1/α. This contradicts the fact that H is a first integral of (8). MANY PERIODIC SOLUTIONS 13 Notice also that f (ξ(t)) is a T -periodic solution. Thus, Z is normally degenerate if and only if ∂σ ∂h (t, h∗ ) is also T -periodic, which is equivalent to the condition ∂σ ∂σ (10) (T, h∗ ) = (0, h∗ ). ∂h ∂h By the definition of τ (h), we have σ(τ (h), h) = σ(0, h), h ∈ I. Since the minimal period of ξ is T /j and we are interested in studying T -periodic solutions, we need to work with the relation σ(jτ (h), h) = σ(0, h), h ∈ I. After taking the derivative with respect to h in the above relation, and using that jτ (h∗ ) = T and ∂σ ∂t (T, h∗ ) = ξ 0 (T ) = ξ 0 (0) = f (ξ(0)), we obtain ∂σ ∂σ jτ 0 (h∗ ) f (ξ(0)) +(T, h∗ ) = (0, h∗ ). ∂h ∂h Since f (ξ(0)) is not the null vector, using the above relation we deduce that the degeneracy condition (10) is equivalent to τ 0 (h∗ ) = 0.  The next result will be useful in the construction of the Malkin bifurcation function. Proposition 15. [∇H(ξ(t))]tr is a solution of the linear system v 0 = −[Df (ξ(t))]tr v. Proof. As usual, ∇H is a line vector. Note that, in fact, we have to prove the validity of the following relation for all t ∈ R, d ∇H(ξ(t)) = −∇H(ξ(t))Df (ξ(t)). dt This will follow, on one hand, from d ∇H(ξ(t)) = D(∇H)(ξ(t))ξ 0 (t) = D(∇H)(ξ(t))f (ξ(t)), dt and on the other hand, from the identity D(∇H)f = −∇HDf, valid in P, which is obtained by differentiating h∇H, f i = 0 in P. The proof is done.  We deduce that for the planar system (11) u0 = f (u) + εg(t, u) the Malkin bifurcation function, M : R → R, has the following expression Z T M (θ) = h∇H(ξ(t + θ)), g(t, ξ(t + θ))idt. 0 14 A. BUICĂ AND A. GASULL Note that M is T /j-periodic. Note also that it can be written as Z T (12) M (θ) = h∇H(ξ(t)), g(t − θ, ξ(t))idt. 0 A consequence of Theorem 1 in this case is the following result. Theorem 16. Assume that τ 0 (h∗ ) 6= 0 and that there exists θ∗ ∈ [0, T /j) such that M (θ∗ ) = 0 and M 0 (θ∗ ) 6= 0, where M is given in (12). Then there exists ε > 0 such that for each ε ∈ (−ε, ε), equation (11) has a T -periodic solution uε with lim uε (0) = ξ(θ∗ ) and with the property that it is isolated in the set of T -periodic ε→0 solutions of (11). Without loosing the generality, suppose that the origin is a singularity of u0 = f (u) and that the closed orbits in P encircle the origin. Note that, since uε (t) and ξ(t + θ∗ ) are ”close enough” on the compact interval [0, T ], we deduce that the number of rotations of uε (t) around the origin in the interval [0, T ] is equal to the number of rotations of ξ(t + θ∗ ) around the origin in the same interval. Thus, uε makes exactly j rotations around the origin in the period time T . We present a consequence of Theorem 7 from [4] for the case considered in this section. Theorem 17. Assume that τ 0 (h∗ ) 6= 0 and M is given in (12). Then (i) For any sequences (um )m≥1 from C(R, R2 ) and (εm )m≥1 from [0, 1] such that um (0) → ξ(θ∗ ) ∈ ξ(R), εm → 0 as m → ∞ and um is a T -periodic solution of (11) with ε = εm , we have that M (θ∗ ) = 0. (ii) Assume that there exist 0 ≤ θ1 < θ2 < T /j such that M (θ1 )M (θ2 ) < 0. Then M has at least two zeros θ1∗ ∈ (θ1 , θ2 ), θ2∗ ∈ (θ2 , θ1 + T /j), while system (11) has at least two T -periodic solutions, denoted u1,ε , u2,ε , which make exactly j rotations around the origin in the period time T , and such that the distance between ϕε (0) and ξ(R) goes to 0 as ε → 0. Remarks. Hypothesis (ii) in the theorem above assures that the Brouwer degree of M on both intervals (θ1 , θ2 ) and (θ2 , θ1 + T /j), respectively, is not null. This remark is to justify that the hypotheses of Theorem 7 from [4] are satisfied. For the Brouwer degree theory one can see the book of Dincă-Mawhin [8]. In general, Theorem 17 does not assure that the periodic solutions obtained are isolated in the set of T -periodic solutions. RT If M is not identically null and 0 M (θ)dθ = 0, then there exist 0 ≤ θ1 < θ2 < T /j such that M (θ1 )M (θ2 ) < 0. In the sequel we provide an example of perturbation g in system (11) such that the Malkin bifurcation function associated satisfies the hypothesis in (ii) of Theorem 17. In addition, we provide an example of such system whose Malkin bifurcation function does not have any zero, thus the system does not have T - periodic solutions like in (i) of Theorem 17. The last result assures the existence of 2N periodic solutions for an arbitrary integer N ≥ 1. We will skip the proofs of the next results, since they are not difficult. MANY PERIODIC SOLUTIONS 15 Theorem 18. Assume that τ 0 (h∗ ) 6= 0. Let A ∈ C(R) be T -periodic and consider g(t, u) = A(t)∇H(u). For the planar system (13) u0 = f (u) + εA(t)∇H(u) the Malkin bifurcation function can be computed as Z T M (θ) = A(t − θ)||∇H(ξ(t))||2 dt. 0 RT (i) If M is not identically null and 0 A(s)ds = 0, then there exist 0 ≤ θ1 < θ2 < T /j such that M (θ1 )M (θ2 ) < 0. (ii) If A(t) 6= 0 for all t ∈ R, then M (θ) 6= 0 for all θ ∈ R. Theorem 19. Let N ≥ 1 be fixed. Assume that τ 0 (h) 6= 0 for all h ∈ I and that τ (I) = (0, ∞). For any j ≥ 1 denote by ξj (t) the solution of u0 = f (u) of minimal period T /j and Z T Mj (θ) = A(t − θ)||∇H(ξj (t))||2 dt. 0 (i) If there exist N functions from {Mj : j ≥ 1} that are not identically null RT and 0 A(s)ds = 0, then system (13) has at least 2N different T -periodic solutions. (ii) If A(t) 6= 0 for all t ∈ R, then Mj (θ) 6= 0 for all θ ∈ R, for all j ≥ 1. Acknowledgements This work was supported by Ministerio de Ciencia, Innovación y Universidades of the Spanish Government by grants MTM2016-77278-P (MINECO/AEI/FEDER, UE) and 2017-SGR-1617 from AGAUR, Generalitat de Catalunya. References [1] A.L.A. de Araujo, K.M. Pedroso, Multiple periodic solutions and positive homoclinic solution for a differential equation, Bull. Belg. Math. Soc. Simon Stevin 20 (2013), 535–546. [2] G. Austin, Biomathematical model of aneurysm of the circle of willis I: The Duffing equation and some approximate solutions, Mathematical Biosciences 11 (1971), 163–172. [3] A. Buică, J.-P. Françoise, J. Llibre, Periodic solutions of nonlinear periodic differential sys- tems with a small parameter, Commun. Pur. Appl. Anal. 6 (2007), 103–111. [4] A. Buică, J. Llibre, O. Makarenkov, Bifurcations from nondegenerate families of periodic solutions in Lipschitz systems, J. Differential Equations 252 (2012), 3899–3919. [5] A. Cima, A. Gasull, F. Mañosas, Cyclicity of a family of vector fields, J. Math. Anal. Appl. 196 (1995), 921–937. [6] J. Cronin, Biomathematical model of aneurysm of the circle of Willis: A qualitative analysis of the differential equation of Austin, Math. Biosci. 16 (1973), 209–225. [7] C.J. Christopher, C.J. Devlin, Isochronous centres in planar polynomial systems, SIAM J. Math. Anal. 28 (1997), 162–177. [8] G. Dincă, J. Mawhin, Brouwer degree and applications, Preprint, 2009. [9] A. Fonda, M. Sabatini, F. Zanolin, Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré–Birkhoff Theorem, Topol. Meth. Nonlin. Anal. 40 (2012), 29–52. [10] A. Gasull, M. Llorens, V. Mañosa. Periodic points of a Landen transformation, Commun. Nonlinear Sci. Numer. Simulat. 64 (2018), 232–245. 16 A. BUICĂ AND A. GASULL [11] A. Gasull, V. Mañosa. Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem. Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 651–670. [12] M.R. Grossinho, L. Sanchez, A note on periodic solutions of some nonautonomous differential equations, Bull. Austral. Math. Soc. 34 (1986), 253–265. [13] H. Guggenheimer, Hill equations with coexisting periodic solutions. II, Comment. Math. Helv. 44 (1969), 381–384. [14] Ph. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London- Sydney, 1964. Pn [15] A. Lins Neto, On the number of solutions of the equation dx/dt = j=0 aj (t)xj , 0 ≤ t ≤ 1, for which x(0) = x(1), Invent. Math. 59 (1980), 67–76. [16] I.G. Malkin, On Poincaré’s theory of periodic solutions, Akad. Nauk SSSR. Prikl. Mat. Meh. 13 (1949), 633–646 (in Russian). [17] F. Neuman, Criterion of periodicity of solutions of a certain differential equation with a periodic coefficient, Ann. Mat. Pura Appl. (4) 75 (1967), 385–396. [18] B. Sturmfels, Solving systems of polynomial equations, volume 97 of CBMS Regional Con- ference Series in Mathematics. Published for the Conference Board of theMathematical Sci- ences,Washington, DC; by the American Mathematical Society, Providence, RI (2002). 1 Departamentul de Matematică, Universitatea Babeş–Bolyai, Str. Kogălniceanu 1, 400084 Cluj–Napoca, Romania E-mail address: abuica@math.ubbcluj.ro 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Barcelona, Spain E-mail address: gasull@mat.uab.cat