This is a preprint of: “Linear type global centers of cubic Hamiltonian systems symmetric with respect to the x-axis”, Luis Barreira, Jaume Llibre, Clàudia Valls, Electron. J. Differential Equa- tions, vol. 2020(57), 1–14, 2020. LINEAR TYPE GLOBAL CENTERS OF CUBIC HAMILTONIAN SYSTEMS SYMMETRIC WITH RESPECT TO THE x-AXIS LUIS BARREIRA1 , JAUME LLIBRE2 AND CLÀUDIA VALLS1 Abstract. A polynomial differential system of degree 2 has no global centers (that is, centers defined in all the plane except the fixed point). In this paper we characterize the global centers of cubic Hamiltonian systems symmetric with respect to the x-axis, and such that the center has purely imaginary eigenvalues. 1. Introduction and statement of the results The notion of center goes back to Poincaré and Dulac, see [10, 4]. They defined a center for a vector field on the real plane as a singular point having a neighborhood filled of periodic orbits with the exception of the singular point. The problem of distinguishing when a monodromic singular point is a focus or a center, known as the focus-center problem started precisely with Poincaré and Dulac and is still active nowadays with many questions still unsolved. If an analytic system has a center, then it is known that after an affine change of variables and a rescaling of the time variable, it can be written in one of the following three forms: ẋ = −y + P (x, y), ẏ = x + Q(x, y), called linear type center, which has a pair of purely imaginary eigenvalues, ẋ = y + P (x, y), ẏ = Q(x, y) called nilpotent center ẋ = P (x, y), ẏ = Q(x, y) called degenerated center, where P (x, y) and Q(x, y) are real analytic func- tions without constant and linear terms defined in a neighborhood of the origin. We recall that a global center for a vector field on the plane is a singular point p having R2 filled of periodic orbits with the exception of the singular 2010 Mathematics Subject Classification. 34C05, 34C07, 34C08. Key words and phrases. center, global center, Hamiltonian system, symmetry with respect to the x-axis, cubic polynomial differential system. 1 2 L. BARREIRA, J. LLIBRE AND C. VALLS point. The easiest global center is the linear center ẋ = −y, ẏ = x. It is known (see [11, 1]) that quadratic polynomial differential systems have no global centers. The global degenerated (or homogeneous) centers were characterized in [3] while the global quasihomogeneous centers were studied in [5]. However the characterization of the global centers in the cases that the center is either nilpotent or a linear-type center has been done for very particular cases. In the case in which the system is of the form linear plus cubic homogeneous terms with a linear-type center at the origin was done in [8] and with a nilpotent center at the origin was done in [9]. The case in which the system is a real cubic one (in the sense that it contains also quadratic terms) has never been done because the difficulties grow brutally with the appearance of new coefficients. That is the reason for which in this paper we will focus in the case in which the system is Hamiltonian and it is symmetric with respect to the x-axis. This is the first paper in which such classification is done for these systems. We will focus in the case in which there are quadratic terms, since the case of linear+cubic was done before in [6] and we state it here as Theorem 1. Theorem 1. Any Hamiltonian vector field having at the origin of coordi- nates a singular point with purely imaginary eigenvalues of the form linear plus cubic homogeneous terms has a global center at the origin if and only if, after a linear change of variables and a rescaling of its independent variables, can be written as one of the following systems: d2 +ω 2 2 3 3 (i) ẋ = −dx − c y − 3αµx y − αy , ẏ = cx + dy + αx + 3αµxy 2 ; 2 d +ω 2 (ii) ẋ = −dx − 3 c y, ẏ = cx + dy + αx ; d2 +ω 2 2 2 (iii) ẋ = −dx − c y − 3αx y, ẏ = cx + dy + 3αxy ; d2 +ω 2 2 3 2 (iv) ẋ = −dx − c y − 3αx y − αy , ẏ = cx + dy + 3αx , with α = ±1, c, d ∈ R, c 6= 0, ω > 0, µ > −1/3 and cα > 0. We also focus in the case in which there is a linear-type center at the origin since the case in which there is a nilpotent center at the origin was done in [7] (see Theorems 1, 2 and the global phase portraits in Figure 1 of that paper) and we state it here as Theorem 2. Theorem 2. Any Hamiltonian planar polynomial vector field of degree three with a global nilpotent center at the origin, symmetric with respect to the x- axis and with all infinite singular points being non-degenerated hyperbolic sectors, after a linear change of variables and a rescaling of its independent variables, can be written as one of the following systems (i) x0 = y, y 0 = −x3 , (ii) x0 = y + y 3 , y 0 = −x3 , (iii) x0 = y + x2 y + ay 3 , y 0 = −x3 − xy 2 with a ≥ 0, (iv) x0 = y − x2 y + ay 3 , y 0 = −x3 + xy 2 with a ≥ 1, LINEAR TYPE GLOBAL CENTERS 3 (v) x0 = y + 2xy + ax2 y + by 3 , y 0 = −x3 − y 2 − axy 2 with b > 0 and either a ≥ 1, or a < 1 with 4(a − 1)2 (a3 − a2 − ab − 8b) − 27b2 > 0. We will now introduce the main results of the paper. For this, we recall that a polynomial differential system can be extended in a unique analytic way to infinity using the Poincaré compactification, for more details see Chapter 5 of [2]. We also introduce some notation. −ã1 ã0 b̃2 b̃1 b̃0 a1 = , a0 = , b2 = − , b1 = − , b0 = , A0 A0 B0 B0 B0 A0 = 9c2 (a12 − a30 )2 , B0 = c3 (3µ − 1)2 (3µ + 1)2 α4 , ã1 = 8a412 c2 − 12a312 a30 c2 − 2a212 c3 α − 10a212 cαω 2 + 6a12 a30 c3 α + 24a12 a30 cαω 2 − 18a230 cαω 2 − (c2 − ω 2 )2 ,  a0 = ω 2 4a212 c3 − 6a12 a30 c3 − 6a12 a30 cω 2 + 9a230 cω 2 + α(c2 − ω 2 )2 , b̃2 = 3 9a212 cµ3 − 10a212 cµ + 36a12 a30 cµ2 + 2a12 a30 c − 9a230 cµ (1)  − 18c2 µ3 α + 2c2 µα − 27µ4 αω 2 + 12µ2 αω 2 − αω 2 , b1 = 8a412 c2 − 36a312 a30 c2 µ + 18a212 c3 µ2 α − 4a212 c3 α − 30a212 cµαω 2 + 18a12 a30 c3 µα + 108a12 a30 cµ2 αω 2 + 12a12 a30 cαω 2 − 54a230 cµαω 2 − 9c4 µ2 − 54c2 µ3 ω 2 + 12c2 µω 2 + 18µ2 α2 ω 4 − 3α2 ω 4 , b0 = ω 2 4a212 c3 − 18a12 a30 c3 µ − 6a12 a30 cω 2 + 27a230 cµω 2 + 9c4 µ2 α  − 6c2 µαω 2 + αω 4 and ∆ = c7 ∆21 ∆2 , where ∆1 = 36a312 cµ2 + 8a312 c − 162a212 a30 cµ3 − 72a212 a30 cµ + 162a12 a230 cµ2 (2) + 81a12 c2 µ4 α + 18a12 c2 µ2 α − 81a12 µ3 αω 2 − 81a30 c2 µ3 α + 243a30 µ4 αω 2 , and ∆2 = 36a612 c3 µ2 + 32a612 c3 − 216a512 a30 c3 µ + 36a412 a230 c3 + 108a412 c4 µ2 α − 48a412 c4 α − 324a412 c2 µ3 αω 2 − 180a412 c2 µαω 2 + 216a312 a30 c4 µα + 1296a312 a30 c2 µ2 αω 2 + 108a312 a30 c2 αω 2 − 36a212 a230 c4 α + 24a212 c5 − 864a212 a230 c2 µαω 2 − 135a212 c5 µ2 − 648a212 c3 µ3 ω 2 + 18a212 c3 µω 2 (3) + 972a212 cµ4 ω 4 + 216a212 cµ2 ω 4 − 27a212 cω 4 + 108a12 a330 c2 αω 2 − 54a12 a30 c5 µ + 810a12 a30 c3 µ2 ω 2 − 54a12 a30 c3 ω 2 − 1944a12 a30 cµ3 ω 4 + 162a12 a30 cµω 4 + 9a230 c5 − 54a230 c3 µω 2 + 81a230 cµ2 ω 4 + 36c6 µ2 α − 4c6 α − 324c4 µ3 αω 2 + 36c4 µαω 2 4 L. BARREIRA, J. LLIBRE AND C. VALLS + 972c2 µ4 αω 4 − 108c2 µ2 αω 4 − 972µ5 αω 6 + 108µ3 αω 6 . Theorem 3. Any Hamiltonian planar polynomial vector field of degree three with a global linear type center at the origin, symmetric with respect to the x-axis and with no infinite singular points in the Poincaré disc, after a linear change of variables and a rescaling of its independent variables, can be written as: ω2 ẋ = − y − 2a12 xy − 3µαx2 y − αy 3 (I) c ẏ = cx + 3a30 x2 + a12 y 2 + αx3 + 3µαxy 2 , where α = ±1 and c, ω, µ ∈ R with c 6= 0, ω > 0, µ > −1/3, a212 + a230 6= 0, and: (a) either µ = 1/3, a12 = a30 , a230 (3cω 2 − 2c3 ) + α(c2 − ω 2 )2 < 0 and 0 < 9a230 < 4cα; or (b) µ = 1/3, a12 6= a30 , a21 − 4a0 < 0 and 9a230 − 4cα < 0, or (c) µ = 1/3, a12 6= a30 , a21 −4a0 ≥ 0 with a1 > 0, a0 > 0 and 9a230 −4cα < 0, or (d) µ 6= 1/3, ∆ < 0, b0 > 0, and 9a230 − 4cα < 0, or (e) µ 6= 1/3, ∆ ≥ 0, b2 > 0, b0 > 0, b2 b1 > b0 and 9a230 − 4cα < 0. The proof of Theorem 3 is given in section 2. We note that if a12 = a30 = 0 then our system becomes of the form linear plus cubic homogeneous terms and these systems are a particular case of the ones studied in [6] (see Theorem 1 above). We recall that the conditions provided in Theorem 3 are not empty. For example, for condition (a) it is sufficient to take µ = 1/3, α = c = 1, ω = 2 and a12 = a30 < 2/3. For condition (b) it is sufficient to take µ = 1/3, α = 1, c = 7/13120, ω = 1/16, a12 = 1, a30 = 0. For condition (c) it is sufficient to take µ = 1/3, α = c = 1, ω = 1/2, a12 = 5/32 and a30 = 0. For condition (d) it is sufficient to take α = c = µ = 1, ω = 1/16, a12 = −13/16 and a30 = 5/8. Finally, for condition (e) it suffices to take α = c = µ = 1, ω = 7/184, a12 = −21/32 and a30 = −1/2. A singular point p of a planar system is called hyperbolic if both eigen- values of the Jacobian matrix at p have real part different from zero. It is called semi-hyperbolic if only one of the eigenvalues of the Jacobian matrix at p is zero, and if both eigenvalues of the Jacobian matrix at p are zero but this matrix is not identically zero it is called nilpotent. Finally, if the Jacobian matrix at p is identically zero then p is said to be linearly zero. Let q be an infinite singular point and let h be a hyperbolic sector of q. We say that h is degenerated if its two separatrices are contained at infinity, that is, are contained in the equator of the Poincaré sphere. LINEAR TYPE GLOBAL CENTERS 5 It follows from Theorem 2.15 (for hyperbolic singular points), Theorem 2.19 (for semi-hyperbolic singular points) and Theorem 3.5 (for nilpotent singular points) in [2] that a singular point which is either hyperbolic, semi- hyperbolic or nilpotent cannot be formed by two degenerated hyperbolic sectors. So, in order that an infinite singular point q can be formed by two degenerated hyperbolic sectors it must be linearly zero. To state the last main theorem in the paper we introduce some notation. ∆3 = 9a212 c3 (4a612 c3 − 15a212 c5 + 12a412 c4 α + 4c6 α − 72a212 c3 ω 2 − 36a412 c2 αω 2 − 36c4 αω 2 + 108a212 cω 4 + 108c2 αω 4 − 108αω 6 ), (4) a212 c − 2c2 α − 3αω 2 2a212 c − c2 α − 6αω 2 b̃2 = − , b̃1 = − , 3c 9α cω 2 b̃0 = . 9α Theorem 4. Any Hamiltonian planar polynomial vector field of degree three with a global linear type center at the origin, symmetric with respect to the x- axis and with all infinite singular points formed by two degenerated hyperbolic sectors, after a linear change of variables and a rescaling of its independent variables, can be written as one of the following systems (ii) x0 = −ω 2 y/c, y 0 = cx + 3a30 x2 + αx3 with ω > 0, cα > 0, α = ±1 and 9a230 − 4cα < 0; (iii) x0 = −ω 2 y/c − 2a12 xy − 3αx2 y, y 0 = cx + a12 y 2 + 3αxy 2 with ω > 0, cα > 0, α = ±1 and c(ca12 − 3αω 2 ) < 0; 2 (iv) ẋ = − ωc y − 2a12 xy − 3αx2 y − αy 3 , ẏ = cx + a12 y 2 + 3αxy 2 , with ω > 0, α = ±1, cα > 0 and either: (iv.1) ∆3 < 0 and b̃0 > 0 (see (4)); (iv.2) ∆3 ≥ 0, b̃2 > 0, b̃0 > 0 and b̃2 b̃1 > b̃0 (see (4)). Theorem 4 is proved in Section 3. Note that the conditions in statement (iv.1) are fulfilled for example when α = 1, c = 1/17, ω = 1/8 and a12 = −1, and the conditions in statement (iv.2) are fulfilled for example when α = 1, c = 2, ω = 1 and a12 = 1/16. 2. Proof of Theorem 3 In order to prove Theorem 3 we will state and prove several propositions. Proposition 5. Any Hamiltonian planar polynomial vector field of degree three with a linear type center at the origin, symmetric with respect to the x-axis and with no infinite singular points, after a linear change of variables and a rescaling of its independent variables it can be written as the following 6 L. BARREIRA, J. LLIBRE AND C. VALLS system ω2 ẋ = − y − 2a12 xy − 3αµx2 y − αy 3 , (i) c ẏ = cx + 3a30 x2 + a12 y 2 + 3αµxy 2 , where α = ±1 and c, ω, a12 , a30 , µ ∈ R with c 6= 0, µ > −1/3 and ω > 0. Proof. Doing a linear change of variables and a rescaling of the independent variable, planar cubic homogeneous differential systems which has no infinite singular points can be classified in the following class, see [3, Theorem 3.2]: ẋ = p1 x3 + (p2 − 3αµ)x2 y + p3 xy 2 − αy 3 , (i’) ẏ = αx3 + p1 x2 y + (p2 + 3αµ)xy 2 + p3 y 3 , with µ > −1/3 and α = ±1. It was proved in [6] that in the case in which the vector field is Hamiltonian then pi = 0 for i = 1, 2, 3, that is we get the system (i’) with pi = 0 for i = 1, 2, 3. For studying the Hamiltonian cubic planar polynomial vector fields having linear, quadratic and cubic terms, it is sufficient to add to the above family (with pi = 0 for i = 1, 2, 3) a linear and quadratic parts being Hamiltonian. This is due to the fact that the linear changes of variables that are done to obtain the class (i’) are not affine, they are strictly linear. For the linear part, we add the linear terms ax + by in ẋ and the linear terms cx + dy in ẏ. Doing so, taking into account that we must have a linear type center at the origin, it is not easy to see that we can add −dx − (d2 + ω 2 )/cy in ẋ and cx + dy in ẏ with c 6= 0 and ω > 0. For the quadratic part, we add the linear terms −∂H3 /∂y in ẋ and ∂H3 /∂x in ẏ with H3 = a30 x3 + a21 x2 y + a12 xy 2 + a03 y 3 . Doing so, we get the systems d2 + ω 2 ẋ = −dx − y − a21 x2 − 2a12 xy − 3a03 y 3 − 3αµx2 y − αy 3 , c ẏ = cx + dy + 3a30 x2 + 2a21 xy + a12 y 2 + 3αµxy 2 . Since this system must be invariant under the transformation (x, y, t) → 7 (x, −y, −t) we must have d = a21 = a03 = 0 and then we get the system (i) in the statement of the proposition. This concludes the proof.  Now we continue with the proof of the theorem. To do so, we compute the finite singular points of system (i). Note that on y = 0 we have the solution p −3a30 ± 9a230 − 4cα x= . 2α LINEAR TYPE GLOBAL CENTERS 7 Since this solution cannot exist we must have 9a230 − 4cα < 0. Now we consider y 6= 0 and we compute the Groëbner basis for the poly- nomials x0 and y 0 . We get a set of six polynomials. The first polynomial is a polynomial of degree six in the variable y while the polynomials p3 and p4 are linear in the variable x. More precisely, p1 = −c3 (3µ − 1)2 (3µ + 1)2 y 6 + 3c2 (2a12 a30 c − 10cµa212 − 9cµa230 + 2c2 αµ + 36a12 a30 cµ2 + 9a212 cµ3 − 18c2 αµ3 − αω 2 + 12αµ2 ω 2 − 27αµ4 ω 2 )y 4 + c(8a412 c2 − 4a212 c3 α − 36a312 a30 c2 µ + 18a12 a30 c3 αµ + 18a212 c3 αµ2 − 9c4 µ2 + 12a12 a30 cαω 2 − 30a212 cαµω 2 − 54a230 cαµω 2 + 12c2 µω 2 + 108a12 a30 cαµ2 ω 2 − 54c2 µ3 ω 2 − 3ω 4 + 18µ2 ω 4 )y 2 + ω 2 (4a212 c3 − 18a12 a30 c3 µ + 9αc4 µ2 − 6a12 a30 cω 2 + 27a230 cµω 2 − 6c2 αµω 2 + αω 4 ) and p3 = cαy 2 (2a12 − 9a30 µ + 9a12 µ2 ) + (2a12 − 9a30 µ)ω 2  + x 4a212 c − 18a12 a30 cµ + 9c2 αµ2 − 3αµω 2 + 3cµ(3µ − 1)(3µ + 1)y 2 , p4 = −c2 y 4 (3µ − 1)(3µ + 1) − ω 2 (3c2 µ − ω 2 ) + cy 2 (6a212 cµ − 3c2 αµ + 2αω 2  − 9αµ2 ω 2 ) + x c2 α(2a12 − 9a30 µ + 9a12 µ2 )y 2 + c(2a12 − 9a30 µ)ω 2 . Note that for any value of y which is a solution of p1 = 0 we get a unique value of x which is a solution of p3 = 0 or p4 = 0, namely (9a30 cαµ − 9a12 cαµ2 − 2a12 cα)y 2 − 2a12 ω 2 + 9a20 µω 2 x= , 4a212 c − 18a12 a30 cµ − 3αµω 2 + 9c2 αµ2 + (3cµ(3µ + 1)(3µ − 1))y 2 or c2 (9µ2 − 1)y 4 − c(6a212 cµ − 3αµc2 + 2αω 2 − 9αµ2 ω 2 )y 2 + ω 2 (3c2 µ − ω 2 ) x̄ = . c(cα(2a12 − 9a30 µ + 9a12 µ2 )y 2 + 2a12 ω 2 − 9a30 µω 2 ) Moreover, both denominators of x and x̄ cannot be zero simultaneously and then for each value of y we always have a value of x. Moreover, setting x (or x̄) into the Groëbner basis we get that all the polynomials have p1 as a factor in them. In short, in order to show that there are no solutions of system (i) with x 6= 0 we must guarantee that there are no real solutions of p1 = 0. Since p1 depends in the variable y 2 we introduce the new variable √ z = y and we must have that z > 0 (note that z = 0 yields y = 0). We consider different cases. Case 1: µ = 1/3 and a12 = a30 . In this case p1 = cz(−4a430 c2 + 4a230 c3 α − 4a230 cαω 2 − c4 + 2c2 ω 2 − ω 4 ) − ω 2 (−2a230 c3 + 3a230 cω 2 + c4 α − 2c2 αω 2 + αω 4 ). 8 L. BARREIRA, J. LLIBRE AND C. VALLS Solving in the variable z we get ω 2 (a230 (3cω 2 − 2c3 ) + α(c2 − ω 2 )2 ) (5) z=− c(2a230 c + α(ω 2 − c2 ))2 whenever the denominator is not zero. Note that the denominator is equal to zero (solving in ω which is positive) whenever r c2 α − 2a230 c ω= . α But then introducing ω into p1 we get p1 = −a230 c3 (2a230 − aα)2 . Setting it to zero, taking into account that aα > 0 and 9a230 − 4aα < 0 we conclude that a30 = 0. But then a12 = 0 which is not possible. Hence, z in equation (5) is well-defined. Taking into account that it must be negative we must have a230 (3cω 2 − 2c3 ) + α(c2 − ω 2 )2 < 0 In short, the condition so that there are no finite singular points besides the origin in this case is a230 (3cω 2 − 2c3 ) + α(c2 − ω 2 )2 < 0, 0 < 9a230 < 4cα. This concludes the proof of statement (a). Case 2: µ = 1/3 and a12 6= a30 . In this case p1 = −ω 2 (4a212 c3 − 6a12 a30 c3 − 6a12 a30 cω 2 + 9a230 cω 2 + c4 α − 2c2 αω 2 + αω 4 ) + cz(8a412 c2 − 12a312 a30 c2 − 2a212 c3 α − 10a212 cαω 2 (6) + 6a12 a30 c3 α + 24a12 a30 cαω 2 − 18a230 cαω 2 − c4 + 2c2 ω 2 − α2 ω 4 ) − 9c3 z 2 (a12 − a30 )2 . Note that now p1 is quadratic in z. We need to investigate when this qua- dratic equation has either no real roots or both real roots are negative. We state and prove an auxiliary result. Proposition 6. Any quadratic polynomial of the form (7) P (z) = z 2 + a1 z + a0 has complex roots if and only if a21 − 4a0 < 0. It has all roots being real and negative if and only if a21 − 4a0 ≥ 0, a1 > 0 and a0 > 0. Proof of Proposition 6. The solutions are real if and only if a21 − 4a0 ≥ 0. In this case, it follows directly from the Routh-Hurwitz criterium that these roots are negative if and only if a1 > 0and a0 > 0. This concludes the proof of the proposition.  LINEAR TYPE GLOBAL CENTERS 9 We can write p1 in (6) as in (7) with (note that a12 6= a30 ) a1 and a0 as in (1). So, the conditions so that there are no finite singular points are either a21 − 4a0 < 0 and 9a230 − 4cα < 0, or a21 − 4a0 ≥ 0 with a1 > 0, a0 > 0 and 9a230 − 4cα < 0. This concludes the proof of statements (b) and (c). Case 3: µ 6= 1/3. or in other words, that all the solutions of the cubic equation p1 = −c3 (3µ − 1)2 (3µ + 1)2 z 3 + 3c2 (2a12 a30 c − 10cµa212 − 9cµa230 + 2c2 αµ + 36a12 a30 cµ2 + 9a212 cµ3 − 18c2 αµ3 − αω 2 + 12αµ2 ω 2 − 27αµ4 ω 2 )z 2 + c(8a412 c2 − 4a212 c3 α − 36a312 a30 c2 µ + 18a212 c3 αµ2 (8) + 18a12 a30 c3 αµ − 9c4 µ2 + 12a12 a30 cαω 2 − 30a212 cαµω 2 + 12c2 µω 2 − 54a230 cαµω 2 + 108a12 a30 cαµ2 ω 2 − 54c2 µ3 ω 2 − 3ω 4 + 18µ2 ω 4 )z + ω 2 (4a212 c3 − 18a12 a30 c3 µ + 9αc4 µ2 − 6a12 a30 cω 2 + 27a230 cµω 2 − 6c2 αµω 2 + αω 4 ) are negative. For that we state and prove an auxiliary result that charac- terize when this happens for a general cubic equation. Proposition 7. Any cubic polynomial of the form (9) P (z) = z 3 + b2 z 2 + b1 z + b0 has only negative real roots if and only if: (1) either ∆ := b21 b22 − 4b32 − 4b31 b0 + 18b1 b2 b0 − 27b20 < 0 and b0 > 0, or (2) ∆ ≥ 0, b2 > 0, b0 > 0 and b2 b1 > a0 . Proof. Note that ∆ < 0 if and only if the cubic equation P (z) has one real root and two complex conjugate roots. Since P (z) → −∞ as z → −∞, the unique real root is negative if and only if P (0) = b0 > 0. On the other hand, ∆ ≥ 0 if and only if the cubic equation P (z) has three (counted with multiplicity) real roots. Using the Routh-Hurwitz criterium these roots are negative if and only if b2 > 0, b0 > 0 and b2 b1 > b0 . This concludes the proof of the proposition.  We can write p1 in (8) as in (9) with b2 , b1 and b0 as in (1). So, the conditions so that there are no finite singular points are either ∆ < 0 (see (2)–(3)), b0 > 0 and 9a230 − 4cα < 0, or ∆ ≥ 0, b2 > 0, b0 > 0, b2 b1 > b0 and 9a230 − 4cα < 0. This concludes the proof of statements (d) and (e) and finishes the proof of the theorem. 3. Proof of Theorem 4 In order to prove Theorem 4 we will state and prove several propositions. 10 L. BARREIRA, J. LLIBRE AND C. VALLS Proposition 8. Any Hamiltonian planar polynomial vector field of degree three with a linear type center at the origin, symmetric with respect to the x-axis and with all infinite singular points being linearly zero, after a linear change of variables and a rescaling of its independent variables it can be written as one of the following systems 2 (II) ẋ = − ωc y − 2a12 xy, ẏ = cx + 3a30 x2 + a12 y 2 + αx3 ; 2 (III) ẋ = − ωc y − 2a12 xy − 3αx2 y, ẏ = cx + 3a30 x2 + a12 y 2 + 3αxy 2 ; 2 (IV) ẋ = − ωc y − 2a12 xy − 3αx2 y − αy 3 , ẏ = cx + 3a30 x2 + a12 y 2 + 3αxy 2 where α = ±1 and c, ω, a12 , a30 ∈ R with c 6= 0, a212 + a230 6= 0 and ω > 0. Proof. The proof is very similar to the one of Proposition 5. Doing a linear change of variables and a rescaling of the independent variable, planar cubic homogeneous differential systems which have infinite singular points being linearly zero can be classified in the following three classes, see [3]: (ii’) ẋ = p1 x3 + p2 x2 y, ẏ = αx3 + p1 x2 y + p2 xy 2 ; (iii’) ẋ = (p2 − 3α)x2 y, ẏ = (p2 + 3α)xy 2 ; (iv’) ẋ = (p2 − 3α)x2 y + p3 xy 2 − αy 3 , ẏ = (p2 + 3α)xy 2 + p3 y 3 , where α = ±1. System (ii’) comes from systems (IX’) of Theorem 3.2 of [?] of taking into account that the unique pair of infinite singular points at the origins of the local chart U2 and V2 (see the Poincaré compactification in [2]) is linearly zero if and only if p3 = 0). System (iii’) comes from system (VII’) of [?] taking into account that the unique pair of infinite singular points are the origins of Ui and Vi for i = 1, 2, and all of them are linearly zero if and only if p1 = p3 = 0. System (iv’) comes from system (IV’) of [?]. Here the unique pair of infinite singular points is the origins of U1 and V1 , which are linearly zero if and only if p1 = 0. It was proved in [6] that if systems (ii’), (iii’) and (iv’) are Hamiltonian then pi = 0 for i = 1, 2, 3. So, we get systems (ii’)–(iv’) with pi = 0 for i = 1, 2, 3. For studying the Hamiltonian cubic planar polynomial vector fields having linear, quadratic and cubic terms, it is sufficient to add to the above families (with pi = 0 for i = 1, 2, 3) linear and quadratic parts being Hamiltonian. This is due to the fact that the linear changes of variables that are done to obtain the classes (ii’), (iii’) and (iv’) are strictly linear, see [?]. For the linear part we add the linear terms ax + by in ẋ and the linear terms cx + dy in ẏ. Doing so, taking into account that we must have a linear LINEAR TYPE GLOBAL CENTERS 11 d2 +ω 2 type center at the origin, it is easy to see that we can add −dx − c y in ẋ and cx + dy in ẏ with c 6= 0 and ω > 0. For the quadratic part we add the quadratic terms −∂H3 /∂y in ẋ and ∂H3 /∂x in ẏ with H3 = a30 x3 + a21 x2 y + a12 xy 2 + a03 y 3 . Doing so, we get the systems d2 + ω 2 ẋ = −dx − y − a21 x2 − 2a12 xy − 3a03 y 3 + P3 , c ẏ = cx + dy + 3a30 x2 + 2a21 xy + a12 y 2 + Q3 , where P3 and Q3 are the vector fields in each of the classes (ii’)–(iv’) with pi = 0 for i = 1, 2, 3. Since the above systems must be invariant under the transformation (x, y, t) 7→ (x, −y, −t) we must have d = a21 = a03 = 0 and then we get the systems (II)–(IV) in the statement of the proposition. This concludes the proof.  Proposition 9. Systems (II) have a global linear-type center at the origin and no more finite singular points with all the infinite singular points formed by two degenerated hyperbolic sectors if and only if they can be written as in (ii) of Theorem 4. Consequently systems (II) have a global center at the origin. Proof. We already know that the pair of infinite singular points of system (II) are the origins of the local charts U2 and V2 . Thus on the local chart U2 we get ω2 2 u0 = −3a12 uv − 3a30 u3 v − cu2 v 2 − αu4 − v , (10) c v 0 = −v(a12 v + cuv 2 + 3a30 u2 v + αu3 ). The origin of the local chart U2 is linearly zero. We need to do blow-ups to understand the local behavior at this point. We perform the directional blow-up (u, v) 7→ (u, w) with w = v/u and we get  ω2   ω2  u0 = −u2 3a12 w +3a30 u2 +cu2 w2 +αu2 + w2 , v 0 = uw2 2a12 + w . c c 0 0 We eliminate the common factor u between u and w and we obtain the system  ω2 2  ω2  u0 = −u 3a12 w + 3a30 u2 + cu2 w2 + αu2 + w , v 0 = w2 2a12 + w . c c When u = 0, this system has the singular points  2ca12  (0, 0) and 0, − 2 . ω 12 L. BARREIRA, J. LLIBRE AND C. VALLS Computing the eigenvalues of the Jacobian matrix at the second singular point we get that it is a node. Going back through the changes of variables we see that in this case the origin of U2 must have parabolic sectors, and so the origin of U2 cannot be the union of two degenerated hyperbolic sectors. Hence, a12 = 0. In this case the unique singular point is the origin which is again linearly zero. Hence we need to do another blow up. We do it in the form (u, w) 7→ (u, z) with z = w/u. Doing so, and eliminating the common factor u2 we get  ω2   2ω 2 2  u0 = −u α+3a30 uz +cu2 z 2 + z 2 , z 0 = z α+3a30 uz +cu2 z 2 + z . c c When u = 0 the possible singular points are  r i cα  (0, 0) and 0, ± . ω 2 The sign of cα determines the existence of the last two singular points, and so we analyze both possibilities. If cα < 0, all three singular  points are real. The origin which is a saddle √ and the points 0, ±icα/( 2ω) which are nodes. Again going back through the changes of variables up to systems (10) we get that the origin of U2 contains parabolic sectors. Hence, we must have cα > 0. If cα > 0 the only singular point is the origin which is a saddle. Going back to the changes of variables until systems (10) we see that locally the origin of U2 consists of two degenerated hyperbolic sectors. In short, in order that the origin of the local chart U2 is formed by two degenerated hyperbolic sectors. we must have a12 = 0 and cα > 0. Then system (II) becomes ω2 ẋ = − y, ẏ = cx + 3a30 x2 + αx3 c with cα > 0. The finite singular points are  p  −3a30 ± 9a230 − 4cα (0, 0), ,0 . 2α Since we want that there are no finite singular points among the origin we must have 9a230 − 4cα < 0. This completes the proof of the proposition.  Proposition 10. Systems (III) have a global linear-type center at the origin and no more finite singular points with all the infinite singular points formed by two degenerated hyperbolic sectors if and only if they can be written as in (iii) of Theorem 4. Proof. We already know that the unique pairs of infinite singular points of system (III) are the origins of Ui and Vi for i = 1, 2. System (III) on the LINEAR TYPE GLOBAL CENTERS 13 local chart U1 becomes (11) ω2 2 2  ω2  u0 = 6αu2 + 3a30 v + 3a12 u2 v + cv 2 + u v , v 0 = uv 3α + 2a12 v + v 2 . c c Computing the eigenvalues of the Jacobian matrix at the origin we get that it is nilpotent if a30 6= 0 and linearly zero if a30 = 0. So, in order that it is formed by the union of two degenerated hyperbolic sectors we must have a30 = 0. We need to do blow-ups to understand the local behavior at this point. We perform the directional blow-up (u, v) 7→ (u, w) with w = v/u and we eliminate the common factor u between u0 and w0 and we get  ω2 2 2   u0 = −u 6α + 3a12 uw + cw2 + u w , w0 = −w 3α + a12 uw + cw2 . c When u = 0, there are three singular points which are  r  3α (0, 0) and 0, ±i , c if cα < 0. Computing the eigenvalues of the Jacobian matrix at the second and third singular points we get that they are nodes. Going back through the changes of variables we see that in this case the origin of U1 must have parabolic sectors and so the origin of U1 cannot be the union of two degen- erated hyperbolic sectors. Hence, cα > 0. In this case the unique singular point is the origin which is a saddle. Going back through the changes of variables until systems (11) we see that locally the origin of U1 consists of two degenerated hyperbolic sectors. On the local chart U2 system (III) is ω2 2 (12) u0 = −6αu2 − 3a12 uv − v − cu2 v 2 , v 0 = −v(3αu + a12 v + cuv 2 ). c The origin of the local chart U2 is linearly zero. We perform the directional blow-up (u, v) 7→ (u, w) with w = v/u, and eliminating the common factor u we get  ω2 2  ω2 2 u0 = −u 6α + 3a12 w + cu2 w2 + w , v 0 = w 3α + 2a12 w + w . a c When u = 0 the possible singular points are  −ca ± pa2 a2 − 3cαw2  12 12 (0, 0) and 0, . ω2 If c2 a212 − 3cαw2 > 0 the two last singular points exist. Computing the eigenvalues of the Jacobian matrix at these points we get that at least one of them is a node. Hence, going back through the changes of variables until system (12) we get that the origin of the local chart U2 must have parabolic sectors and so it cannot be formed by two degenerated hyperbolic sectors. √ √ If c2 a12 − 3cαw2 = 0, that is a12 = 3αω/ c, we have two finite singular points which are the origin (which is a saddle) and the point 14 L. BARREIRA, J. LLIBRE AND C. VALLS √ √ (0, − 3 αc/ω) which √ is linearly zero. First we translate it to the ori- gin setting w = − 3cα/ω + W . We perform the directional blow-up (u, W ) 7→ (u, z) with z = W/u, and eliminating the common factor u2 we get 1  √ √ √ √  u0 = − 2 3c3 αu − 2 3a3 cαu2 z + c2 u3 z 2 ω 2 + 3 cαzω 3 + uω 4 z 2 , cω 0 z  √ √  z = 2 3αc3 − 2 3c3 cαωuz + c2 ω 2 u2 z 2 + 2ω 4 z 2 . cω  √ √ √  The singular points are (0, 0) and 0, i 3c3/2 α/( 2ω 2 ) which due to the fact that cα > 0 the last two do not exist. Computing the eigenvalues of the Jacobian matrix at the origin we get that it is a saddle. Going back through the changes of variables up to system (12) we get that the origin of the local chart U2 is not formed by two degenerated hyperbolic sectors, because there are more hyperbolic sectors not in the equator of the Poincaré sphere coming from these last saddles. Finally, if c2 a12 − 3cαω 2 < 0 we get that the unique finite singular point is the origin which is a saddle. Hence, going back trough the changes of variables up to system (12) we get that the origin of the local chart U2 is formed by two degenerated hyperbolic sectors. In short, in order that the origins of the local charts U1 and U2 are formed by two degenerated hyperbolic sectors we must have a30 = 0, cα > 0 and c(ca12 − 3αω 2 ) < 0. System becomes ω2 x0 = − y − 2a12 xy − 3αx2 y, y 0 = cx + a12 y 2 + 3αxy 2 . c The singular points are (0, 0) and (±x± , y± ) where √ √ −ca12 ∓ c2 a12 − 3cαω 2 cx± x± = , y± = ± √ . 3cα −a12 − 3αx± Taking into account that c2 a12 − 3cαw2 < 0 we get that the unique finite singular point is the origin. This completes the proof of the proposition.  Proposition 11. Systems (IV) have a global linear-type center at the origin and no more finite singular points with all the infinite singular points formed by two degenerated hyperbolic sectors if and only if they can be written as in (iv.1) or (iv.2) of Theorem 4. Proof. We already know that the unique pair of infinite singular points are the origins of U1 and V1 . On the local chart U1 system (IV) becomes ω2 2 u0 = 3a30 v + 3a12 u2 v + cv 2 + 6αu2 + αu4 + v , (13) c  ω2 2  v 0 = uv 3α + 2a12 v + v + αu2 . c LINEAR TYPE GLOBAL CENTERS 15 Computing the eigenvalues of the Jacobian matrix at the origin we get that it is nilpotent if a30 6= 0 and linearly zero if a30 = 0. So, a30 = 0. We perform the directional blow-up (u, v) 7→ (u, w) with w = v/u, and eliminating the common factor u we get ω2 2 2 u0 = u(6α + 3a12 uw + cw2 + αu2 + u w ), c w0 = −w(3α + a12 uw + cw2 ). p The singular points are (0, 0) and (0, ±i (3α)/c). The two last points exist if and only if cα < 0. In this case computing the eigenvalues of the Jacobian matrix at these points we get that they are nodes. So this case is not possible. Hence, we must have cα > 0. Then the unique singular point is the origin which is a saddle. Going back through the changes of variables till system (13) we get that the origin of the local chart U1 is formed by two degenerated hyperbolic sectors. System (IV) becomes ω2 ẋ = − y − 2a12 xy − 3αx2 y − αy 3 , ẏ = cx + a12 y 2 + 3αxy 2 . c On y = 0 the unique finite singular point is the origin. With y 6= 0 solving ẏ = 0 we get a12 (14) x=− y2. c + 3αy 2 √ i c Note that if y = ± √ 3α then ẏ has no solution and so we can assume that 2 c + 3αy 6= 0. Substituting (14) in ẋ we get y (−2c2 a212 y 2 + c3 y 2 α − 3ca212 y 4 α + 6c2 y 4 α2 + 9cy 6 α3 + c2 ω 2 c(c + 3αy 2 )2 + 6cy 2 αω 2 + 9y 4 α2 ω 2 ). We must see that there is no solutions of the sextic equation cαy 6 + 3α(−ca212 + 2αc2 + 3αω 2 )y 4 + c(−2ca212 + αc2 + 6αω 2 )y 2 + c2 ω 2 Or in other words, that there are not positive roots of the cubic equation Q = 9cαz 3 + 3αx(−ca212 + 2αc2 + 3αω 2 )z 2 + c(−2ca212 + αc2 + 6αω 2 )z + c2 ω 2 . We can write Q as in (9) with b2 = b̃2 , b1 = b̃1 and b0 = b̃0 given in (4). So, in view of Proposition 7, the conditions so that there are no finite singular points are either ∆3 < 0 (see (4)) and b̃0 > 0, or ∆3 ≥ 0, b̃2 > 0, b̃0 > 0 and b̃2 b̃1 > b̃0 . This concludes the proof of the proposition.  Proof of Theorem 4. The proof of Theorem 4 is an immediate consequence of Propositions 8–10.  16 L. BARREIRA, J. LLIBRE AND C. VALLS Acknowledgements The first and third authors are partially supported by FCT/Portugal through UID/MAT/04459/2013. The second author is supported by the Ministerio de Economı́a, Industria y Competitividad - Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017 SGR 1617, and the European project Dynamics- H2020-MSCA-RISE-2017-777911. References [1] J.C. Artés, J. Llibre and N. Vulpe, Complete geometric invariant study of two classes of quadratic systems, Electronic J. Differential Equations 2012 (2012), No.9, pp. 1–35. [2] F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, New York, 2006. [3] A. Cima and J. Llibre, Algebraic and topological classification of the homogeneous cubic vector fields in the plane, J. Math. Anal. and Appl. 147 (1990), 420–448. [4] H. Dulac, Détermination et integartion d’une certaine classe d’équations différentielle ayant par point singulier un centre, Bull. Sci. Math. 32 (1908), 230– 252. [5] W.P. Li, J. Llibre, J.Yang and Z. Zhang, Limit cycles bifurcating from the period annulus of quasihomogeneous centers, J. Dynam. Differential Equations 21 (2009), 133–152. [6] I.E. Colak, J. Llibre and C. Valls, Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, Advances in Mathematics 259 (2014), 655–687. [7] F.S. Dias, J. Llibre and C. Valls, Polynomial Hamiltonian systems of degree 3 with symmetric nilpotent centers, Math. Comput. Simul. 144 (2018), 60–77. [8] J.D. Garcı́a-Saldaña, J. Llibre and C. Valls, Linear type global centers of linear systems with cubic homogeneous nonlinearities, preprint. [9] J.D. Garcı́a-Saldaña, J. Llibre and C. Valls, Nilpotent global centers of linear systems with cubic homogeneous nonlinearities, preprint. [10] H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, Oeu- vreus de Henri Poincaré, 1, Gauthier–Villars, Paris, 1951, pp. 95–114. [11] N. Vulpe, Affine-invariant conditions for the topological discrimination of quadratic systems with a center, Differential Equations 19 (1983), 273–280. 1 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal Email address: barreira@math.tecnico.ulisboa.pt Email address: cvalls@math.tecnico.ulisboa.pt 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat