Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 57, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LINEAR TYPE GLOBAL CENTERS OF CUBIC HAMILTONIAN SYSTEMS SYMMETRIC WITH RESPECT TO THE x-AXIS LUIS BARREIRA, JAUME LLIBRE, CLAUDIA VALLS 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 [4, 10]. 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 ; and ẋ = P (x, y), ẏ = Q(x, y) called degenerated center, where P (x, y) and Q(x, y) are real analytic functions 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 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 [?] 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 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. c©2020 Texas State University. Submitted January 10, 2019. Published June 8, 2020. 1 2 L. BARREIRA, J. LLIBRE, C. VALLS EJDE-2020/57 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 1.1. Theorem 1.1. Any Hamiltonian vector field having at the origin of coordinates 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: (i) ẋ = −dx− d2+ω2 c y − 3αµx2y − αy3, ẏ = cx+ dy + αx3 + 3αµxy2; (ii) ẋ = −dx− d2+ω2 c y, ẏ = cx+ dy + αx3; (iii) ẋ = −dx− d2+ω2 c y − 3αx2y, ẏ = cx+ dy + 3αxy2; (iv) ẋ = −dx− d2+ω2 c y − 3αx2y − αy3, ẏ = cx+ dy + 3αx2, 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 in the next theorem. Theorem 1.2. 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) x′ = y, y′ = −x3, (ii) x′ = y + y3, y′ = −x3, (iii) x′ = y + x2y + ay3, y′ = −x3 − xy2 with a ≥ 0, (iv) x′ = y − x2y + ay3, y′ = −x3 + xy2 with a ≥ 1, (v) x′ = y + 2xy + ax2y + by3, y′ = −x3 − y2 − axy2 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 [2, Chapter 5]. We also EJDE-2020/57 LINEAR TYPE GLOBAL CENTERS 3 introduce some notation. a1 = −ã1 A0 , a0 = ã0 A0 , b2 = − b̃2 B0 , b1 = − b̃1 B0 , b0 = b̃0 B0 , A0 = 9c2(a12 − a30)2, B0 = c3(3µ− 1)2(3µ+ 1)2α4, ã1 = 8a412c 2 − 12a312a30c 2 − 2a212c 3α− 10a212cαω 2 + 6a12a30c 3α + 24a12a30cαω 2 − 18a230cαω 2 − (c2 − ω2)2, a0 = ω2 ( 4a212c 3 − 6a12a30c 3 − 6a12a30cω 2 + 9a230cω 2 + α(c2 − ω2)2 ) , b̃2 = 3 ( 9a212cµ 3 − 10a212cµ+ 36a12a30cµ 2 + 2a12a30c− 9a230cµ − 18c2µ3α+ 2c2µα− 27µ4αω2 + 12µ2αω2 − αω2 ) , b1 = 8a412c 2 − 36a312a30c 2µ+ 18a212c 3µ2α− 4a212c 3α− 30a212cµαω 2 + 18a12a30c 3µα+ 108a12a30cµ 2αω2 + 12a12a30cαω 2 − 54a230cµαω 2 − 9c4µ2 − 54c2µ3ω2 + 12c2µω2 + 18µ2α2ω4 − 3α2ω4, b0 = ω2 ( 4a212c 3 − 18a12a30c 3µ− 6a12a30cω 2 + 27a230cµω 2 + 9c4µ2α − 6c2µαω2 + αω4 ) , (1.1) and ∆ = c7∆2 1∆2, where ∆1 = 36a312cµ 2 + 8a312c− 162a212a30cµ 3 − 72a212a30cµ+ 162a12a 2 30cµ 2 + 81a12c 2µ4α+ 18a12c 2µ2α− 81a12µ 3αω2 − 81a30c 2µ3α + 243a30µ 4αω2, (1.2) and ∆2 = 36a612c 3µ2 + 32a612c 3 − 216a512a30c 3µ+ 36a412a 2 30c 3 + 108a412c 4µ2α − 48a412c 4α− 324a412c 2µ3αω2 − 180a412c 2µαω2 + 216a312a30c 4µα + 1296a312a30c 2µ2αω2 + 108a312a30c 2αω2 − 36a212a 2 30c 4α+ 24a212c 5 − 864a212a 2 30c 2µαω2 − 135a212c 5µ2 − 648a212c 3µ3ω2 + 18a212c 3µω2 + 972a212cµ 4ω4 + 216a212cµ 2ω4 − 27a212cω 4 + 108a12a 3 30c 2αω2 − 54a12a30c 5µ+ 810a12a30c 3µ2ω2 − 54a12a30c 3ω2 − 1944a12a30cµ 3ω4 + 162a12a30cµω 4 + 9a230c 5 − 54a230c 3µω2 + 81a230cµ 2ω4 + 36c6µ2α− 4c6α− 324c4µ3αω2 + 36c4µαω2 + 972c2µ4αω4 − 108c2µ2αω4 − 972µ5αω6 + 108µ3αω6. (1.3) Theorem 1.3. 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: ẋ = −ω 2 c y − 2a12xy − 3µαx2y − αy3 ẏ = cx+ 3a30x 2 + a12y 2 + αx3 + 3µαxy2, (1.4) where α = ±1 and c, ω, µ ∈ R with c 6= 0, ω > 0, µ > −1/3, a212 + a230 6= 0, and: 4 L. BARREIRA, J. LLIBRE, C. VALLS EJDE-2020/57 (a) either µ = 1/3, a12 = a30, a230(3cω2 − 2c3) + α(c2 − ω2)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, b2b1 > b0 and 9a230 − 4cα < 0. The proof of the above theorem 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 1.1). Note that there values satisfying the conditions in Theorem 1.3. 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 eigenvalues 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. 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, semihyperbolic 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. Let ∆3 = 9a212c 3(4a612c 3 − 15a212c 5 + 12a412c 4α+ 4c6α− 72a212c 3ω2 − 36a412c 2αω2 − 36c4αω2 + 108a212cω 4 + 108c2αω4 − 108αω6), b̃2 = −a 2 12c− 2c2α− 3αω2 3c , b̃1 = −2a212c− c2α− 6αω2 9α , b̃0 = cω2 9α . (1.5) Theorem 1.4. Any Hamiltonian planar polynomial vector field of degree 3 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 (i) x′ = −ω2y/c, y′ = cx + 3a30x 2 + αx3 with ω > 0, cα > 0, α = ±1 and 9a230 − 4cα < 0; (ii) x′ = −ω2y/c−2a12xy−3αx2y, y′ = cx+a12y 2+3αxy2 with ω > 0, cα > 0, α = ±1 and c(ca12 − 3αω2) < 0; EJDE-2020/57 LINEAR TYPE GLOBAL CENTERS 5 (iii) ẋ = −ω 2 c y − 2a12xy − 3αx2y − αy3, ẏ = cx + a12y 2 + 3αxy2, with ω > 0, α = ±1, cα > 0 and either: (iii.1) ∆3 < 0 and b̃0 > 0 (see (1.5)); (iii.2) ∆3 ≥ 0, b̃2 > 0, b̃0 > 0 and b̃2b̃1 > b̃0 (see (1.5)). The above theorem is proved in Section 3. Note that the conditions in statement (iii.1) are fulfilled for example when α = 1, c = 1/17, ω = 1/8 and a12 = −1, and the conditions in statement (iii.2) are fulfilled for example when α = 1, c = 2, ω = 1 and a12 = 1/16. 2. Proof of Theorem 1.3 First we will state and prove several propositions needed later. Proposition 2.1. Any Hamiltonian planar polynomial vector field of degree 3 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 system ẋ = −ω 2 c y − 2a12xy − 3αµx2y − αy3, ẏ = cx+ 3a30x 2 + a12y 2 + 3αµxy2, (2.1) 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 vari- able, planar cubic homogeneous differential systems which has no infinite singular points can be classified in the following class, see [?, Theorem 3.2]: ẋ = p1x 3 + (p2 − 3αµ)x2y + p3xy 2 − αy3, ẏ = αx3 + p1x 2y + (p2 + 3αµ)xy2 + p3y 3, (2.2) 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 obtain the system (2.2) 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 so because the linear changes of variables that are done to obtain the class (2.2) 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 + ω2)/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 = a30x 3 + a21x 2y + a12xy 2 + a03y 3. Doing so, we obtain the system ẋ = −dx− d2 + ω2 c y − a21x2 − 2a12xy − 3a03y 3 − 3αµx2y − αy3, ẏ = cx+ dy + 3a30x 2 + 2a21xy + a12y 2 + 3αµxy2. 6 L. BARREIRA, J. LLIBRE, C. VALLS EJDE-2020/57 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 obtain the system (2.1) 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 (2.1). Note that on y = 0 we have the solution x = −3a30 ± √ 9a230 − 4cα 2α . 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 polynomials x′ and y′. 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)2y6 + 3c2(2a12a30c− 10cµa212 − 9cµa230 + 2c2αµ + 36a12a30cµ 2 + 9a212cµ 3 − 18c2αµ3 − αω2 + 12αµ2ω2 − 27αµ4ω2)y4 + c(8a412c 2 − 4a212c 3α− 36a312a30c 2µ+ 18a12a30c 3αµ+ 18a212c 3αµ2 − 9c4µ2 + 12a12a30cαω 2 − 30a212cαµω 2 − 54a230cαµω 2 + 12c2µω2 + 108a12a30cαµ 2ω2 − 54c2µ3ω2 − 3ω4 + 18µ2ω4)y2 + ω2(4a212c 3 − 18a12a30c 3µ+ 9αc4µ2 − 6a12a30cω 2 + 27a230cµω 2 − 6c2αµω2 + αω4), p3 = cαy2(2a12 − 9a30µ+ 9a12µ 2) + (2a12 − 9a30µ)ω2 + x ( 4a212c− 18a12a30cµ+ 9c2αµ2 − 3αµω2 + 3cµ(3µ− 1)(3µ+ 1)y2 ) , and p4 = −c2y4(3µ− 1)(3µ+ 1)− ω2(3c2µ− ω2) + cy2(6a212cµ− 3c2αµ+ 2αω2 − 9αµ2ω2) + x ( c2α(2a12 − 9a30µ+ 9a12µ 2)y2 + c(2a12 − 9a30µ)ω2 ) . Note that for any value of y which is a solution of p1 = 0 we obtain a unique value of x which is a solution of p3 = 0 or p4 = 0, namely x = (9a30cαµ− 9a12cαµ 2 − 2a12cα)y2 − 2a12ω 2 + 9a20µω 2 4a212c− 18a12a30cµ− 3αµω2 + 9c2αµ2 + (3cµ(3µ+ 1)(3µ− 1))y2 , or x̄ = c2(9µ2 − 1)y4 − c(6a212cµ− 3αµc2 + 2αω2 − 9αµ2ω2)y2 + ω2(3c2µ− ω2) c(cα(2a12 − 9a30µ+ 9a12µ2)y2 + 2a12ω2 − 9a30µω2) . 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 obtain that all the polynomials have p1 as a factor in them. In short, in order to show that there are no solutions of system (2.1) with x 6= 0 we must guarantee that there are no real solutions of p1 = 0. Since p1 depends in the variable y2 we introduce the new variable z = √ y and we must have that z > 0 (note that z = 0 yields y = 0). We consider several cases. EJDE-2020/57 LINEAR TYPE GLOBAL CENTERS 7 Case 1: µ = 1/3 and a12 = a30. In this case p1 = cz(−4a430c 2 + 4a230c 3α− 4a230cαω 2 − c4 + 2c2ω2 − ω4) − ω2(−2a230c 3 + 3a230cω 2 + c4α− 2c2αω2 + αω4). Solving for the variable z we obtain z = −ω 2(a230(3cω2 − 2c3) + α(c2 − ω2)2) c(2a230c+ α(ω2 − c2))2 (2.3) whenever the denominator is not zero. Note that the denominator is equal to zero (solving in ω which is positive) whenever ω = √ c2α− 2a230c α . But then introducing ω into p1 we obtain p1 = −a230c3(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 (2.3) is well-defined. Taking into account that it must be negative we must have a230(3cω2 − 2c3) + α(c2 − ω2)2 < 0 In short, the condition so that there are no finite singular points besides the origin in this case is a230(3cω2 − 2c3) + α(c2 − ω2)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 = −ω2(4a212c 3 − 6a12a30c 3 − 6a12a30cω 2 + 9a230cω 2 + c4α− 2c2αω2 + αω4) + cz(8a412c 2 − 12a312a30c 2 − 2a212c 3α− 10a212cαω 2 + 6a12a30c 3α+ 24a12a30cαω 2 − 18a230cαω 2 − c4 + 2c2ω2 − α2ω4) − 9c3z2(a12 − a30)2. (2.4) Note that now p1 is quadratic in z. We need to investigate when this quadratic equation has either no real roots or both real roots are negative. We state and prove an auxiliary result. Proposition 2.2. Any quadratic polynomial of the form P (z) = z2 + a1z + a0 (2.5) 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. 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 > 0 and a0 > 0. This concludes the proof of the proposition. � We can write p1 in (2.4) as in (2.5) with (note that a12 6= a30) a1 and a0 as in (1.1). 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). 8 L. BARREIRA, J. LLIBRE, C. VALLS EJDE-2020/57 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)2z3 + 3c2(2a12a30c− 10cµa212 − 9cµa230 + 2c2αµ+ 36a12a30cµ 2 + 9a212cµ 3 − 18c2αµ3 − αω2 + 12αµ2ω2 − 27αµ4ω2)z2 + c(8a412c 2 − 4a212c 3α− 36a312a30c 2µ+ 18a212c 3αµ2 + 18a12a30c 3αµ− 9c4µ2 + 12a12a30cαω 2 − 30a212cαµω 2 + 12c2µω2 − 54a230cαµω 2 + 108a12a30cαµ 2ω2 − 54c2µ3ω2 − 3ω4 + 18µ2ω4)z + ω2(4a212c 3 − 18a12a30c 3µ+ 9αc4µ2 − 6a12a30cω 2 + 27a230cµω 2 − 6c2αµω2 + αω4) (2.6) are negative. For that we state and prove an auxiliary result that characterize when this happens for a general cubic equation. Proposition 2.3. Any cubic polynomial of the form P (z) = z3 + b2z 2 + b1z + b0 (2.7) has only negative real roots if and only if: either • ∆ := b21b 2 2 − 4b32 − 4b31b0 + 18b1b2b0 − 27b20 < 0 and b0 > 0, or • ∆ ≥ 0, b2 > 0, b0 > 0 and b2b1 > 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 b2b1 > b0. This concludes the proof of the proposition. � We can write p1 in (2.6) as in (2.7) with b2, b1 and b0 as in (1.1). So, the conditions so that there are no finite singular points are either ∆ < 0 (see (1.2)– (1.3)), b0 > 0 and 9a230 − 4cα < 0, or ∆ ≥ 0, b2 > 0, b0 > 0, b2b1 > 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 1.4 First we state and prove several propositions. Proposition 3.1. 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 (I) ẋ = −ω 2 c y − 2a12xy, ẏ = cx+ 3a30x 2 + a12y 2 + αx3; (II) ẋ = −ω 2 c y − 2a12xy − 3αx2y, ẏ = cx+ 3a30x 2 + a12y 2 + 3αxy2; (III) ẋ = −ω 2 c y − 2a12xy − 3αx2y − αy3, ẏ = cx+ 3a30x 2 + a12y 2 + 3αxy2 where α = ±1 and c, ω, a12, a30 ∈ R with c 6= 0, a212 + a230 6= 0 and ω > 0. EJDE-2020/57 LINEAR TYPE GLOBAL CENTERS 9 Proof. The proof is very similar to the one of Proposition 2.1. 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 [?]: (i’) ẋ = p1x 3 + p2x 2y, ẏ = αx3 + p1x 2y + p2xy 2; (ii’) ẋ = (p2 − 3α)x2y, ẏ = (p2 + 3α)xy2; (iii’) ẋ = (p2 − 3α)x2y + p3xy 2 − αy3, ẏ = (p2 + 3α)xy2 + p3y 3, where α = ±1. System (i’) comes from systems [3, (IX’) of Theorem 3.2] 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 (ii’) comes from [3, system (VII’)] 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 (iii’) comes from [3, system (IV’)]. 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 (i’)–(iii’) are Hamiltonian then pi = 0 for i = 1, 2, 3. So, we obtain systems (i’)–(iii’) 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 (i’), (ii’) and (iii’) are strictly linear, see [3]. 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 easy to see that we can add −dx− d2+ω2 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 = a30x 3 + a21x 2y + a12xy 2 + a03y 3. Doing so, we obtain the system ẋ = −dx− d2 + ω2 c y − a21x2 − 2a12xy − 3a03y 3 + P3, ẏ = cx+ dy + 3a30x 2 + 2a21xy + a12y 2 +Q3, where P3 and Q3 are the vector fields in each of the classes (i’)–(iii’) 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 obtain the systems (II)–(IV) in the statement of the proposition. This concludes the proof. � Proposition 3.2. System (I) has a global linear-type center at the origin and no more finite singular points with all the infinite singular points formed by two de- generated hyperbolic sectors if and only if they can be written as in (i) of Theorem 1.4. Consequently system (I) has a global center at the origin. 10 L. BARREIRA, J. LLIBRE, C. VALLS EJDE-2020/57 Proof. We already know that the pair of infinite singular points of system (I) are the origins of the local charts U2 and V2. Thus on the local chart U2 we obtain u′ = −3a12uv − 3a30u 3v − cu2v2 − αu4 − ω2 c v2, v′ = −v(a12v + cuv2 + 3a30u 2v + αu3). (3.1) The origin of the local chart U2 is linearly zero. We need to do blow-ups to un- derstand the local behavior at this point. We perform the directional blow-up (u, v) 7→ (u,w) with w = v/u and we obtain u′ = −u2 ( 3a12w + 3a30u 2 + cu2w2 + αu2 + ω2 c w2 ) , v′ = uw2 ( 2a12 + ω2 c w ) . We eliminate the common factor u between u′ and w′ and we obtain the system u′ = −u ( 3a12w + 3a30u 2 + cu2w2 + αu2 + ω2 c w2 ) , v′ = w2 ( 2a12 + ω2 c w ) . When u = 0, this system has the singular points (0, 0) and ( 0,−2ca12 ω2 ) . Computing the eigenvalues of the Jacobian matrix at the second singular point we obtain 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 obtain u′ = −u ( α+ 3a30uz + cu2z2 + ω2 c z2 ) , z′ = z ( α+ 3a30uz + cu2z2 + 2ω2 c z2 ) . When u = 0 the possible singular points are (0, 0) and ( 0,± i ω √ cα/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 (3.1) we obtain 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 system (3.1) 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 degen- erated hyperbolic sectors. we must have a12 = 0 and cα > 0. Then system (I) becomes ẋ = −ω 2 c y, ẏ = cx+ 3a30x 2 + αx3 with cα > 0. The finite singular points are (0, 0), (−3a30 ± √ 9a230 − 4cα 2α , 0 ) . EJDE-2020/57 LINEAR TYPE GLOBAL CENTERS 11 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 3.3. System (II) has 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 1.4. Proof. We already know that the unique pairs of infinite singular points of system (II) are the origins of Ui and Vi for i = 1, 2. System (II) on the local chart U1 becomes u′ = 6αu2 +3a30v+3a12u 2v+cv2 + ω2 c u2v2, v′ = uv ( 3α+2a12v+ ω2 c v2 ) . (3.2) Computing the eigenvalues of the Jacobian matrix at the origin we obtain 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 u′ and w′ and we obtain u′ = −u ( 6α+ 3a12uw + cw2 + ω2 c u2w2 ) , w′ = −w ( 3α+ a12uw + cw2 ) . When u = 0, there are three singular points which are (0, 0) and ( 0,±i √ 3α/c ) , if cα < 0. Computing the eigenvalues of the Jacobian matrix at the second and third singular points we obtain 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 degenerated 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 (3.2) we see that locally the origin of U1 consists of two degenerated hyperbolic sectors. On the local chart U2 system (II) is u′ = −6αu2 − 3a12uv − ω2 c v2 − cu2v2, v′ = −v(3αu+ a12v + cuv2). (3.3) 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 obtain u′ = −u ( 6α+ 3a12w + cu2w2 + ω2 a w2 ) , v′ = w ( 3α+ 2a12w + ω2 c w2 ) . When u = 0 the possible singular points are (0, 0) and ( 0, −ca12 ± √ a2a212 − 3cαw2 ω2 ) . If c2a212 − 3cαw2 > 0 the two last singular points exist. Computing the eigenvalues of the Jacobian matrix at these points we obtain that at least one of them is a node. Hence, going back through the changes of variables until system (3.3) we obtain that the origin of the local chart U2 must have parabolic sectors and so it cannot be formed by two degenerated hyperbolic sectors. 12 L. BARREIRA, J. LLIBRE, C. VALLS EJDE-2020/57 If c2a12− 3cαw2 = 0, that is a12 = √ 3αω/ √ c, we have two finite singular points which are the origin (which is a saddle) and the point (0,− √ 3 √ αc/ω) which is linearly zero. First we translate it to the origin 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 obtain u′ = − 1 cω2 ( 3c3αu− 2 √ 3a3 √ cαu2z + c2u3z2ω2 + √ 3 √ cαzω3 + uω4z2 ) , z′ = z cω2 ( 3αc3 − 2 √ 3c3 √ cαωuz + c2ω2u2z2 + 2ω4z2 ) . The singular points are (0, 0) and ( 0, i √ 3c3/2 √ α/( √ 2ω2) ) 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 obtain that it is a saddle. Going back through the changes of variables up to system (3.3) we obtain 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 c2a12 − 3cαω2 < 0 we obtain that the unique finite singular point is the origin which is a saddle. Hence, going back trough the changes of variables up to system (3.3) we obtain 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 be formed by two degenerated hyperbolic sectors we must have a30 = 0, cα > 0 and c(ca12−3αω2) < 0. System becomes x′ = −ω 2 c y − 2a12xy − 3αx2y, y′ = cx+ a12y 2 + 3αxy2. The singular points are (0, 0) and (±x±, y±) where x± = −ca12 ∓ √ c2a12 − 3cαω2 3cα , y± = ± √ cx±√ −a12 − 3αx± . Taking into account that c2a12−3cαw2 < 0 we obtain that the unique finite singular point is the origin. This completes the proof of the proposition. � Proposition 3.4. System (III) has 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.1) or (iii.2) of Theorem 1.4. 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 (III) becomes u′ = 3a30v + 3a12u 2v + cv2 + 6αu2 + αu4 + ω2 c v2, v′ = uv ( 3α+ 2a12v + ω2 c v2 + αu2 ) . (3.4) Computing the eigenvalues of the Jacobian matrix at the origin we obtain that it is nilpotent if a30 6= 0 and linearly zero if a30 = 0. So, a30 = 0. EJDE-2020/57 LINEAR TYPE GLOBAL CENTERS 13 We perform the directional blow-up (u, v) 7→ (u,w) with w = v/u, and eliminat- ing the common factor u we obtain u′ = u(6α+ 3a12uw + cw2 + αu2 + ω2 c u2w2), w′ = −w(3α+ a12uw + cw2). 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 obtain 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 (3.4) we obtain that the origin of the local chart U1 is formed by two degenerated hyperbolic sectors. System (III) becomes ẋ = −ω 2 c y − 2a12xy − 3αx2y − αy3, ẏ = cx+ a12y 2 + 3αxy2. On y = 0 the unique finite singular point is the origin. With y 6= 0 solving ẏ = 0 we obtain x = − a12 c+ 3αy2 y2. (3.5) Note that if y = ± i √ c√ 3α then ẏ has no solution and so we can assume that c+3αy2 6= 0. Substituting (3.5) in ẋ we obtain y c(c+ 3αy2)2 ( − 2c2a212y 2 + c3y2α− 3ca212y 4α+ 6c2y4α2 + 9cy6α3 + c2ω2 + 6cy2αω2 + 9y4α2ω2 ) . We must see that there is no solution of the sextic equation cαy6 + 3α(−ca212 + 2αc2 + 3αω2)y4 + c(−2ca212 + αc2 + 6αω2)y2 + c2ω2 Or in other words, that there are not positive roots of the cubic equation Q = 9cαz3 + 3αx(−ca212 + 2αc2 + 3αω2)z2 + c(−2ca212 + αc2 + 6αω2)z + c2ω2. We can write Q as in (2.7) with b2 = b̃2, b1 = b̃1 and b0 = b̃0 given in (1.5). So, in view of Proposition 2.3, the conditions so that there are no finite singular points are either ∆3 < 0 (see (1.5)) and b̃0 > 0, or ∆3 ≥ 0, b̃2 > 0, b̃0 > 0 and b̃2b̃1 > b̃0. This concludes the proof. � Then the proof of Theorem 1.4 is an immediate consequence of Propositions 3.1– 3.3. Acknowledgements. L. Barreira and C. Valls were partially supported by FCT, Portugal through UID/MAT/04459/2019. J. Llibre, was supported by the Minis- terio 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. 14 L. BARREIRA, J. LLIBRE, C. VALLS EJDE-2020/57 References [1] J. C. Artés, J. Llibre. N. Vulpe; Complete geometric invariant study of two classes of qua- dratic systems, Electronic J. Differential Equations 2012 (2012), No. 9, pp. 1–35. [2] F. Dumortier, J. Llibre, J.C. Artés; Qualitative Theory of Planar Differential Systems, Springer Verlag, New York, 2006. [3] A. Cima, 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. 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Vulpe; Affine-invariant conditions for the topological discrimination of quadratic systems with a center, Differential Equations 19 (1983), 273–280. Luis Barreira Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049-001, Lisboa, Portugal Email address: barreira@math.tecnico.ulisboa.pt Jaume Llibre Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat Claudia Valls Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049-001, Lisboa, Portugal Email address: cvalls@math.tecnico.ulisboa.pt 1. Introduction and statement of the results 2. Proof of Theorem ?? 3. Proof of Theorem ?? Acknowledgements References