This is a preprint of: “Linear type global centers of linear systems with cubic homogeneous nonlin- earities”, Johanna Denise Garcı́a Saldaña, Jaume Llibre, Clàudia Valls, Rend. Circ. Mat. Palermo (2), 15 pp., 2019. DOI: [10.1007/s12215-019-00433-0] LINEAR TYPE GLOBAL CENTERS OF LINEAR SYSTEMS WITH CUBIC HOMOGENEOUS NONLINEARITIES JOHANNA D. GARCÍA-SALDAÑA, JAUME LLIBRE AND CLAUDIA VALLS Abstract. A center p of a differential system in R2 is global if R2 \ {p} is filled of periodic orbits. It is known that a polynomial differential system of degree 2 has no global centers. Here we characterize the global centers of the differential systems ẋ = ax + by + P3 (x, y), ẏ = cx + dy + Q3 (x, y), with P3 and Q3 homogeneous polynomials of degree 3, and such that the center has purely imaginary eigenvalues, i.e. a linear type center. 1. Introduction and statement of the main results The notion of center goes back to Poincaré and Dulac, see [10, 6]. 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. These last years also the centers are perturbed for studying the limit cycles bifurcating from their periodic solutions, see for instance [1, 3]. 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. 2010 Mathematics Subject Classification. 34C05, 34C07, 34C08. Key words and phrases. center, global center, cubic polynomial differential system. 1 2 J.D. GARCÍA-SALDAÑA, J. LLIBRE, C. VALLS 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, 2]) that quadratic polynomial differential systems have no global centers. The global degenerated centers of homogeneous or quasi- homogeneous polynomial differential systems were characterized in [4] and [8], respectively. However the characterization of the global centers in the cases that the center is nilpotent or of linear-type has not been done. This is the first paper in which such classification is done for the linear-type centers for the systems having a linear part at the origin with purely imaginary eigenvalues and cubic homogeneous nonlinearities. 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 [7]. We now state our main results. We first provide normal forms for the differential systems to be studied. Theorem 1. Any vector field having at the origin of coordinates a singular point with purely imaginary eigenvalues of the form linear plus cubic homo- geneous terms and no infinite singular points in the Poincaré disc after a linear change of variables and a rescaling of the independent variable can be written as d2 + ω 2 ẋ = −dx − y + p1 x3 + (p2 − 3αµ)x2 y + p3 xy 2 − αy 3 , (1) c ẏ = cx + dy + αx3 + p1 x2 y + (p2 + 3αµ)xy 2 + p3 y 3 , where α = ±1 and c, d, ω, µ, p1 , p2 , p3 ∈ R with c ̸= 0, ω > 0 and µ > −1/3. The proof of Theorem 1 is given in section 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 boundary of the Poincaré disc. 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 [7] 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. LINEAR TYPE GLOBAL CENTERS 3 Theorem 2. Any vector field having at the origin of coordinates a singular point with purely imaginary eigenvalues of the form linear plus cubic ho- mogeneous terms such that all infinite singular points in the Poincaré disc are linearly zero after a linear change of variables and a rescaling of the independent variable can be written as one of the following three systems: d2 + ω 2 ẋ = −dx − y + p1 x3 + p2 x2 y, (2) c ẏ = cx + dy + αx3 + p1 x2 y + p2 xy 2 ; d2 + ω 2 ẋ = −dx − y + (p2 − 3α)x2 y, (3) c ẏ = cx + dy + (p2 + 3α)xy 2 ; d2 + ω 2 ẋ = −dx − y + (p2 − 3α)x2 y + p3 xy 2 − αy 3 , (4) c ẏ = cx + dy + (p2 + 3α)xy 2 + p3 y 3 ; where α = ±1 and c, d, ω, p1 , p2 , p3 ∈ R with c ̸= 0, ω > 0. The proof of Theorem 2 is given in section 3. Theorem 3. Systems (1) have a linear type center at the origin if and only if they can be written as one of the following five systems: d2 + ω 2 ẋ = −dx − y − 3αµx2 y − αy 3 , (I) c ẏ = cx + dy + αx3 + 3αµxy 2 , where α = ±1 and d, ω, µ ∈ R with c ̸= 0, ω > 0 and µ > −1/3. These systems are Hamiltonian with c d2 + ω 2 2 α 4 3 α H(x, y) = x2 + dxy + y + x + αµx2 y 2 + y 4 . 2 2c 4 2 4 ω2 ẋ = − y + (p2 − 3µα)x2 y − αy 3 , (II) c ẏ = cx + αx3 + (p2 + 3µα)xy 2 , where α = ±1 and c, ω, µ ∈ R with c ̸= 0, ω > 0 and µ > −1/3; ẋ = −dx − cy + p1 x3 − 3αµx2 y − p1 xy 2 − αy 3 , (III) ẏ = cx + dy + αx3 + p1 x2 y + 3αµxy 2 − p1 y 3 , where α = ±1 and c, d, µ ∈ R with |c| > |d|, p1 ̸= 0 and µ > −1/3; (IV) d2 + ω 2 αcd + p1 (c2 − d2 − ω 2 ) 2 ẋ = −dx − y − p 1 x3 − x y − p1 xy 2 − αy 3 , c cd αcd + p1 (d2 + ω 2 − c2 ) 2 ẏ = cx + dy + αx3 + p1 x2 y + xy − p1 y 3 , cd 4 J.D. GARCÍA-SALDAÑA, J. LLIBRE, C. VALLS where α = ±1 and c, d, ω ∈ R with cd ̸= 0, p1 ̸= 0 and ω > 0; ẋ = −cy + p1 x3 + (p2 − α)x2 y − p1 xy 2 − αy 3 , (V) ẏ = cx + αx3 + p1 x2 y + (α + p2 )xy 2 − p1 y 3 , where α = ±1, c ∈ R \ {0} and p1 p2 ̸= 0. The proof of Theorem 3 is given in section 4. In order to state the following theorem we introduce the notation (5) R = (2d(p1 + α) + c(2p1 + α(1 − 3µ)))(2d(p1 − α) − c(2p1 − α(1 − 3µ))), S1 = −c3 d4 p41 (dα((c2 + d2 )p1 + cdα) + p1 (cp1 + dα)ω 2 ), S2 = c4 + 2c2 (d2 − ω 2 ) + (d2 + ω 2 )2 . Theorem 4. Under the assumptions of Theorem 3 the following statements hold. (a) A system (I) has a global center at the origin if and only if cα > 0. (b) A system (II) has a global center at the origin if and only if cα > 0 and either ((p2 + 3µα)ω 2 − c2 α)c(p22 + α2 (1 − 9µ2 )) ≤ 0, or (c2 (p2 − 3µα) + αω 2 )c(p22 + α2 (1 − 9µ2 )) ≥ 0. (c) A system (III) has a global center at the origin if and only if (c ± d)α > 0 and if p1 = −dα/c then cα > 0, but if√p1 ̸= −dα/c then either R < 0, or R > 0 and ((2dp1 +cα(1−3µ))± R)(2p21 +α2 (3µ− 1)) ≤ 0. (d) A system √ (IV) has a global center at the origin if and only if cα((d2 + ω ) S2 + (c2 d2 + d4 − c2 ω 2 + 2d2 ω 2 + ω 4 )) ≤ 0 and if p1 = −dα/c 2 then√ cα > 0, but if p1 ̸= −dα/c then either S1 < 0, or S1 > 0 and −2 S1 + cdp21 (c3 p1 + c2 dα + cp1 (d2 − ω 2 ) − dα(d2 + ω 2 )) ≤ 0. (e) √ A system (V) has a global center √ at the origin if and only if cα(p2 + 4p1 + p2 ) ≥ 0 and cα(p2 − 4p21 + p22 ) ≤ 0. 2 2 The proof of Theorem 4 is given in section 6. Theorem 5. Systems (2) have a linear type center at the origin and no more finite singular points with all infinite singular points formed by two degenerated hyperbolic sectors if and only if they can be written as one of the following systems d2 + ω 2 (VI) ẋ = −dx − y, ẏ = cx + dy + αx3 , c with α = ±1, d, c, ω ∈ R, c ̸= 0, ω > 0 and cα > 0. Note that these systems are Hamiltonian with c d2 + ω 2 2 α 4 H(x, y) = x2 + dxy + y + x ; 2 2c 4 LINEAR TYPE GLOBAL CENTERS 5 d2 + ω 2 d2 + ω 2 x′ = −dx − y + p 1 x3 + p1 x2 y, (VII) c cd d2 + ω 2 y ′ = cx + dy + αx3 + p1 x2 y + p1 xy 2 . cd with α = ±1, d, c, ω ∈ R, ω > 0, cα > 0 and dp1 < 0; ω2 (VIII) x′ = − y + p2 x2 y, y ′ = cx + αx3 + p2 xy 2 . c with α = ±1, c, ω ∈ R, ω > 0, cα > 0 and cp2 < 0; d2 + ω 2 αd(d2 + ω 2 ) 3 α(d2 + ω 2 )2 2 x′ = −dx − y+ x + 2 2 x y, c c(d2 − ω 2 ) c (d − ω 2 ) (IX) α(d2 + ω 2 ) 2 α(d2 + ω 2 ) 2 y ′ = cx + dy + αx3 + x y + xy . c(d2 − ω 2 ) c2 (d2 − ω 2 ) with α = ±1, d, c, ω ∈ R, ω > 0, |d| > |w| and cα > 0. The proof of Theorem 5 is given in section 6. Theorem 6. Systems (3) have a linear type center at the origin and no more finite singular points with all infinite singular points formed by two degenerated hyperbolic sectors if and only if they can be written as one of the following systems d2 + ω 2 (X) ẋ = −dx − y − 3αx2 y, ẏ = cx + dy + 3αxy 2 , c with α = ±1, c, d ∈ R, c ̸= 0, ω > 0 and cα > 0. These systems are Hamiltonian with c d2 + ω 2 2 3 2 2 H(x, y) = x2 + dxy + y + αx y ; 2 2c 2 ω2 (XI) ẋ = − y + (p2 − 3α)x2 y, ẏ = cx + (p2 + 3α)xy 2 , c with α = ±1, c ∈ R \ {0}, p2 ̸= 0, ω > 0, cα > 0 and p2 ∈ [−3, 3]. The proof of Theorem 6 is given in section 7. Theorem 7. Systems (4) have a linear type center at the origin and no more finite singular points with all infinite singular points formed by two degenerated hyperbolic sectors if and only if they can be written as one of the following systems d2 + ω 2 (XII) ẋ = −dx − y − 3αx2 y − αy 3 , ẏ = cx + dy + 3αx2 , c with α = ±1, c, d ∈ R, c ̸= 0, ω > 0 and cα > 0. These systems are Hamiltonian with c d2 + ω 2 2 3α 2 2 α 4 H(x, y) = x2 + dxy + y + x y + y ; 2 2 2 4 6 J.D. GARCÍA-SALDAÑA, J. LLIBRE, C. VALLS ω2 (XIII) ẋ = −y + (p2 − 3α)x2 y − αy 3 , ẏ = cx + (p2 + 3α)xy 2 , c with α = ±1, c ∈ R \ {0}, p2 ̸= 0, ω > 0, cα > 0 and c(p2 + 3α) ≥ 0. The proof of Theorem 7 is given in section 8. An immediate consequence of Theorems 4, 5, 6 and 7 is the following. Corollary 8. Any polynomial 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 it satisfies the assumptions of Theorems 4, or 5, or 6, or 7. 2. Proof of Theorem 1 Doing a linear change of variables and a rescaling of the independent variable, planar cubic homogeneous differential systems can be classified into the following ten classes, see [4]: ẋ = x(p1 x2 + p2 xy + p3 y 2 ), (i) ẏ = y(p1 x2 + p2 xy + p3 y 2 ), whose infinity in the Poincaré disc is formed by singular points; ẋ = p1 x3 + p2 x2 y + p3 xy 2 , (ii) ẏ = αx3 + p1 x2 y + p2 xy 2 + p3 y 3 , where α = ±1 and whose infinite singular points in the Poincaré disc are the real solutions of αx4 = 0 at infinity. ẋ = (p1 − 1)x3 + p2 x2 y + p3 xy 2 , (iii) ẏ = (p1 + 3)x2 y + p2 xy 2 + p3 y 3 , whose infinite singular points in the Poincaré disc are the real solutions of 4x3 y = 0 at infinity. ẋ = p1 x3 + (p2 − 3α)x2 y + p3 xy 2 , (iv) ẏ = p1 x2 y + (p2 + 3α)xy 2 + p3 y 3 , where α = ±1 and whose infinite singular points in the Poincaré disc are the real solutions of 6αx2 y 2 = 0 at infinity. ẋ = p1 x3 + (p2 − α)x2 y + p3 xy 2 − αy 3 , (v) ẏ = αx3 + p1 x2 y + (p2 + α)xy 2 + p3 y 3 , where α = ±1 and without infinite singular points; ẋ = p1 x3 + (p2 − 3α)x2 y + p3 xy 2 + y 3 , (vi) ẏ = p21 y + (p2 + 3α)xy 2 + p3 y 3 , LINEAR TYPE GLOBAL CENTERS 7 whose infinite singular points in the Poincaré disc are the real solutions of y 2 (6x2 − y 2 )2 = 0 at infinity; ẋ = p1 x3 + (p2 − 3α)x2 y + p3 xy 2 − αy 3 , (vii) ẏ = p1 x2 y + (p2 + 3α)xy 2 + p3 y 3 , where α = ±1 and whose infinite singular points in the Poincaré disc are the real solutions of αy 2 (6x2 + y 2 ) = 0 at infinity; ẋ = p1 x3 + (p2 − 3µ)x2 y + p3 xy 2 + y 3 , (viii) ẏ = x3 + p1 x2 y + (p2 + 3µ)xy 2 + p3 y 3 , with µ ∈ R and whose infinite singular points in the Poincaré disc are the real solutions of x4 + 6µx2 y 2 − y 4 = 0 at infinity; ẋ = p1 x3 + (p2 − 3αµ)x2 y + p3 xy 2 − αy 3 , (ix) ẏ = αx3 + p1 x2 y + (p2 + 3αµ)xy 2 + p3 y 3 , with α = ±1, µ > −1/3, µ ̸= 1/3 and without infinite singular points; ẋ = p1 x3 + (p2 − 3µ)x2 y + p3 xy 2 − y 3 , (x) ẏ = x3 + p1 x2 y + (p2 + 3µ)xy 2 + p3 y 3 , with µ < −1/3 and whose infinite singular points in the Poincaré disc are the solutions of x4 + 6µx2 y 2 + y 4 = 0 at infinity. Note that when µ = 1/3 system (v) becomes system (ix) and from now on we consider system (ix) with µ > −1/3 and forgot system (v). In this theorem we are interested in systems without infinite singular points. The only classes of systems that have no infinite singular points are systems (ix) with µ > −1/3. For studying the cubic planar polynomial vector fields having linear and cubic terms being the origin a singular point, it is sufficient to add to the above family in (ix) a linear part. This is due to the fact that the linear changes of variables that are done to obtain the classes (i)–(x) are not affine, they are strictly linear. So a linear plus a cubic vector field being the origin a singular point with no infinite singular points in the Poincaré disc can be written as ẋ = ax + by + p1 x3 + (p2 − 3αµ)x2 y + p3 xy 2 − αy 3 , (6) ẏ = cx + dy + αx3 + p1 x2 y + (p2 + 3αµ)xy 2 + p3 y 3 , for some real constants a, b, c, d and with α = ±1, µ > −1/3. The eigenval- ues of the linear part of system (6) at the origin are √ a + d ± (a + d)2 − 4(ad − bc) λ1,2 = . 2 In order to have a linear type center at the origin, these eigenvalues must be ±ωi, for some ω > 0. So a = −d and d2 + bc = −ω 2 . We see that bc ̸= 0, 8 J.D. GARCÍA-SALDAÑA, J. LLIBRE, C. VALLS otherwise the left hand side would be non-negative. Then we can solve for b and we get b = −(d2 + ω 2 )/c. So system (6) becomes d2 + ω 2 ẋ = −dx − y + p1 x3 + (p2 − 3αµ)x2 y + p3 xy 2 − αy 3 , (7) c ẏ = cx + dy + αx3 + p1 x2 y + (p2 + 3αµ)xy 2 + p3 y 3 . This completes the proof of the theorem. 3. Proof of Theorem 2 The proof of Theorem 2 is the same as the proof of Theorem 1 taking into account that system (ii) has only the origin of the local chart U2 as the infinite singular points which is linearly zero if and only if p3 = 0. This gives system (2) in Theorem 2. Moreover, system (iv) has only the origin of the local charts U1 and U2 as infinite singular points and they are both linearly zero if and only if p1 = p3 = 0. This gives system (3) in Theorem 2. System (vii) has an infinite singular point which is the origin of the local chart U1 and is linearly zero if and only if p1 = 0. This gives system (4) in Theorem 2. Any other of the systems (i)–(x) either the infinite is formed by singular points, or they do not provide infinite singular points in the Poincaré disc, or they provide at least one infinite singular point which is not linearly zero. After this, the proof is exactly the same as in the proof of Theorem 1 (that is, adding the linear part). 4. Proof of Theorem 3 We will use the following result, proved in [9]. Theorem 9. Any planar vector field of the form: linear plus cubic ho- mogeneous terms being the origin a singular point with purely imaginary eigenvalues can be written in the form ẋ = y + F x3 + Gx2 y + (H − 3P )xy 2 + Ky 3 , (8) ẏ = −x + Lx3 + (M − H − 3F )x2 y + (N − G)xy 2 + P y 3 . The origin of system (8) is a center if and only if one of the following conditions hold: (a) H = M = N = 0; (b) M = 0, HL = −2F N − HK − HN + 2N P and H 2 F = GHN − HKN + 2N 2 P − H 2 P − HN 2 ; (c) M = 0, H = 5(P − F )/2, L = −G − 4K, N = 5(G + 3K), and 3F 2 + 10F P + 16GK + 64K 2 + 3P 2 = 0. LINEAR TYPE GLOBAL CENTERS 9 First we write the linear part of system (7) into its real Jordan normal form doing the change of variables x = (ωY − dX)/c, y = X and then we do the rescaling τ = ωt. Doing so, the new system can be written as in (8) with c3 p3 − 3αc2 dµ − c2 dp2 + cd2 p1 − αd3 F = , c3 ω 3αc2 µ + c2 p2 − 2cdp1 + 3αd2 4p1 ω ω(cp1 + αd) G= 3 , H= 2 , P = , c ( ) c c3 αω 2 α c4 + 6c2 d2 µ + d4 K= 3 , L=− , c( c3 ω 2 ) 4 c2 p3 − cdp2 + (d2 + ω 2 )p1 2(cp2 − 2dp1 ) M= 2 , N= . c ω c2 Condition (a) in Theorem 9 yields p1 = p2 = p3 = 0 and so we obtain system (I) in the statement of the theorem. Note that in this case system (I) is Hamiltonian with the Hamiltonian stated in the statement of the theorem. Condition (b) in Theorem 9 yields the following real solutions in which cα ̸= 0, ω ̸= 0 and µ > −1/3: • d = p1 = p3 = 0. This condition yields system (II) in the statement of the theorem; √ • p3 = −p1 , p2 = 0, ω = c2 − d2 (and so |c| > |d|). Moreover p1 ̸= 0 because otherwise the system becomes a system (I). This condition yields system (III) in the statement of the theorem; • µ = 1/3, p3 = −p1 , p2 = p1 (d2 + ω 2 − c2 )/(cd) (and so cd ̸= 0). Moreover p1 ̸= 0 because otherwise the system becomes a system (I). This condition yields system (IV) in the statement of the theorem; • µ = 1/3, p3 = −p1 , ω = ±c, d = 0. Note that p1 p2 ̸= 0 because if p2 = 0 then the system becomes a system (III), and if p1 = 0 then the system becomes a system (II). This condition yields system (V) in the statement of the theorem. We will show now that condition (c) is not satisfied. Indeed conditions (c) yield that the unique possible solution with α ̸= 0 is dα 2c3 dp3 + α(d4 − 3c4 ) c4 + d4 p1 = − , p2 = , µ=− . c 2c2 d2 6c2 d2 Note that µ < −1/3 and so this condition is not possible. This completes the proof of the theorem. 5. Proof of Theorem 4 We will study each of the systems (I)–(V) separately. 10 J.D. GARCÍA-SALDAÑA, J. LLIBRE, C. VALLS First note that in case of system (I) we get that the system is Hamiltonian. It was proved in [5] that in this case there unique condition so that the unique finite singular point is the origin is cα > 0. Hence statement (a) is proved. The singular points of system (II) are ( √ ω ) ( c ) (0, 0), 0, ±i √ , ±i , 0 , (±x, ±y) cα α with √ √ (p2 + 3µα)ω 2 − c2 α c2 (p2 − 3µα) + αω 2 x= 2 , y=i 2 c(p2 + α (1 − 9µ )) 2 c(p22 + α2 (1 − 9µ2 )) (note that if c(p22 + α2 (1 − 9µ2 )) = 0 the points (±x, ±y) do not exist). So in order that the candidates to be singular points different from the origin do not exist we must have cα > 0 and either ((p2 + 3µα)ω 2 − c2 α)c(p22 + α2 (1 − 9µ2 )) ≤ 0, or (c2 (p2 − 3µα) + αω 2 )c(p22 + α2 (1 − 9µ2 )) ≥ 0. We recall that if (p2 +√ 3µα)ω 2 − c2 α = 0, i.e., p2 = α(c2 − 3µω 2 )/ω 2 , then x = 0 and y = iω/ αc which is non real because cα > 0. On the other hand, if c2 (p√ 2 2 2 2 2 − 3µα) + αω = 0, that is, p2 = α(3c µ − ω )/c then y = 0 and x = i c/α which is also non real. So statement (b) is proved. The singular points of system (III) (which has the condition |c| > |d| and ̸ 0) are, p1 = √ √ c+d c−d (0, 0), ±i (1, 1), ±i (1, 1), ±(x± , −y ± ) α(1 + 3µ) α(1 + 3µ) where √ √ √ 2dp1 + cα(1 − 3µ) ± R 2dp1 + cα(1 − 3µ) ∓ R x± = , y± = − x± , 2(2p21 + α2 (3µ − 1)) 2(cp1 + dα) with R introduced in (5). Note that if 2p21 + α2 (3µ − 1) = 0 then the points ±(x± , −y ± ) do not exist. On the other hand, if cp1 + dα = 0, i.e., p1 = −dα/c then the points ±(x± , −y ± ) become ( √ √ ) c ) ( c ±i ,0 0, ±i . α α So in order that all the candidates to be singular points different from the origin do not exist, we must have (besides the condition |c| > |d|, p1 = ̸ 0), (c ± d)α > 0 and if p1 = −dα/c then cα > 0, but√ if p1 ̸ = −dα/c then either R < 0, or R > 0 and ((2dp1 + cα(1 − 3µ)) ± R)(2p21 + α2 (3µ − 1)) ≤ 0. Hence statement (c) is proved. The singular points of system (IV) (which has the condition cd ̸= 0 and ̸ 0) are, besides the origin, p1 = ( √ ) ( d2 + ω 2 − c2 ± √S ) c2 d3 p31 ± S1 2 ± − 2 2 2 y, y , ± − ŷ, ŷ c d p1 (cp1 + dα) 2cd LINEAR TYPE GLOBAL CENTERS 11 where √ √ −2 S1 + cdp21 (c3 p1 + c2 dα + cp1 (d2 − ω 2 ) − dα(d2 + ω 2 )) y=− , p41 S2 √ √ (d2 + ω 2 ) S2 + (c2 d2 + d4 − c2 ω 2 + 2d2 ω 2 + ω 4 ) ŷ = − √ , 2cα S2 and S1 , S2 were introduced in (5). Note that if cp1 + dα = 0 that is p1 = −dα/c then the singular points ( √ ) c2 d3 p31 ± S1 ± − 2 2 2 y, y c d p1 (cp1 + dα) become √ √ ( c ) c 1 ( 2 ) ±i ,0 , ±i √ d + ω 2 − c2 , −2dc . α α S2 So in order that all the candidates to be singular points different from the origin do not√exist, we must have (besides the condition cd ̸= 0 and p1 = ̸ 0), cα((d2 + ω 2 ) S2 + (c2 d2 + d4 − c2 ω 2 + 2d2 ω 2 + ω 4 )) ≤ 0 and if p1 = −dα/c √ then cα > 0, but if p1 ̸= −dα/c then either S1 < 0, or S1 > 0 and −2 S1 + cdp21 (c3 p1 + c2 dα + cp1 (d2 − ω 2 ) − dα(d2 + ω 2 )) ≤ 0. This completes the proof of statement (d). The singular points of system (V) are, besides the origin, √ (1 − i 1 + i) √ (1 + i 1 − i) c c ±√ √ ,− √ , ∓√ √ ,− √ , 2p1 − ip2 2 2 2p1 + ip2 2 2 (T √ ) √ ( T (p + 4p2 + p2 )T ) 1 2p T 2 2 2 ± √ ,− √ 1 1 , ± √2 , √1 2 p2 + 4p1 + p22 2 2 2 2p1 where √ √ √ √ c(p2 + 4p21 + p22 ) c(p2 − 4p21 + p22 ) T1 = − √ and T2 = √ . α 4p21 + p22 α 4p21 + p22 So taking into account that p1 p2 ̸= 0, in order that all the candidates to be singular √ points different from √ the origin do not exist, we must have cα(p2 + 4p21 + p22 ) ≥ 0 and cα(p2 − 4p21 + p22 ) ≤ 0. This concludes the proof of the theorem. 6. Proof of Theorem 5 First we write the linear part of system (2) into its real Jordan normal form doing the change of variables x = (ωY − dX)/c, y = X and then we do the rescaling τ = ωt. Doing so the new system can be written as in (8) 12 J.D. GARCÍA-SALDAÑA, J. LLIBRE, C. VALLS with d 2 2 c2 p2 + 3αd2 − 2cdp1 F = (cdp1 − c p 2 − αd ), G = , c3 ω c3 αω 2 d4 α (cp1 + dα)ω K = 3 , L = − 3 2, P = , c c ω c3 4p1 ω 2(cp2 − 2dp1 ) 4 H= 2 , N= , M = 2 (p1 (d2 + ω 2 ) − cdp2 ). c c2 c ω Condition (a) in Theorem 9 yields p1 = p2 = 0 and so we obtain system (VI). Note that system (VI) is Hamiltonian with Hamiltonian c d2 + ω 2 2 α 4 H = x2 + dxy + y + x . 2 2c 4 It was proved in [5] that in this case the unique infinite singular point which is the origin of the local chart U2 is formed by two degenerated hyperbolic sectors if and only if cα > 0 and that in this case there are no more finite singular points besides the origin. Condition (b) in Theorem 9 with p21 + p22 + p23 ̸= 0 yields the conditions (b.1) p2 = p1 (d2 + ω 2 )/(cd) with cd ̸= 0 and p1 ̸= 0 because otherwise this system becomes a system (VI); (b.2) p1 = 0, d = 0 and p2 ̸= 0 because otherwise this system becomes a system (VI); dα(d2 + ω 2 ) α(d2 + ω 2 )2 (b.3) p1 = 2 2 , p2 = 2 2 with |d| ̸= |ω|. c(d − ω ) c (d − ω 2 ) System (2) with the conditions (b.1) becomes d2 + ω 2 d2 + ω 2 x′ = −dx − y + p 1 x3 + p1 x2 y, (9) c cd d2 + ω 2 y ′ = cx + dy + αx3 + p1 x2 y + p1 xy 2 . cd The singular points of system (9) are (√ 1/2 √ ) d cd3/2 p1 ± i cd(d3 α + (cp1 + dα)ω 2 ) (0, 0), ± ,− , p1 p1 (d2 + ω 2 ) √ ( c dωc ) ±i ω, . α(d2 + ω 2 ) d2 + ω 2 In order that all the candidates to be singular points different from the origin do not exist, and taking into account that condition cα > 0, dp1 > 0 and cd(d3 α + (cp1 + dα)ω 2 ) < 0 never hold, we must have cα > 0 and dp1 < 0. Now we study the infinite singular points. We already know that the unique infinite singular point is the origin of the local chart U2 that is linearly LINEAR TYPE GLOBAL CENTERS 13 zero. On the local chart U2 we have d2 + ω 2 2 u̇ = − v − 2duv 2 − αu4 − cu2 v 2 , c ( d2 + ω 2 ) v̇ = −v p1 u + p1 u2 + dv 2 + αu3 + cuv 2 . cd Doing two horizontal blow ups and one vertical blow up and taking into account that cα > 0 and dp1 < 0 we conclude that the origin of the local chart U2 is formed by two degenerated hyperbolic sectors. So we obtain system (VII). System (2) with the conditions (b.2) becomes ω2 x′ = − y + p2 x2 y, (10) c y ′ = cx + αx3 + p2 xy 2 . The singular points of system (10) are ( √ ) (√ c ) ω c2 p2 + αω 2 (0, 0), ±i ,0 , ± √ , ±i . α cp2 cp22 In order that all the candidates to be singular points different from the origin do not exist, and taking into account that condition cα > 0, cp2 > 0 and c(c2 p2 + αω 2 ) < 0 is null, we must have cα > 0 and cp2 < 0. Now we study the infinite singular points. We already know that the unique infinite singular point is the origin of the local chart U2 that is linearly zero. On the local chart U2 we have ω2 2 u̇ = − v − αu4 − cu2 v 2 , c v̇ = uv(p2 + αu2 + cv 2 ). Doing one horizontal and one vertical blow up, taking into account that cα > 0 and cp2 < 0 we conclude that the origin of the local chart U2 is formed by two degenerated hyperbolic sectors. So we obtain system (VIII). System (2) with the conditions (b.3) becomes d2 + ω 2 αd(d2 + ω 2 ) 3 α(d2 + ω 2 )2 2 x′ = −dx − y+ x + 2 2 x y, c c(d2 − ω 2 ) c (d − ω 2 ) (11) αd(d2 + ω 2 ) 2 α(d2 + ω 2 ) 2 y ′ = cx + dy + αx3 + x y + xy . c(d2 − ω 2 ) c2 (d2 − ω 2 ) The singular points of system (11) are √ √ cω 2 ( dc ) c(d2 − ω 2 ) ( cd(1 ± i) ) (0, 0), ±i 1, − , ± 1, − . α(d2 + ω 2 ) d2 + ω 2 α(d2 + ω 2 ) d2 + ω 2 14 J.D. GARCÍA-SALDAÑA, J. LLIBRE, C. VALLS In order that all the candidates to be singular points different from the origin do not exist we must have cα > 0. Now we study the infinite singular points. We already know that the unique infinite singular point is the origin of the local chart U2 that is linearly zero. On the local chart U2 we have ω 2 + d2 2 u̇ = − v − 2duv 2 − αu4 − cu2 v 2 , c ( ( )( ) ( )( )) v c2 d2 − ω 2 v 2 (cu + d) + αu3 + αu d2 + ω 2 cdu + d2 + ω 2 v̇ = − . c2 (d2 − ω 2 ) Doing one horizontal and one vertical blow up, taking into account that cα > 0 we see that if |d| > |ω| the origin of the local chart U2 is formed by two degenerated hyperbolic sectors while if |d| < |ω| there are parabolic sectors arriving to the origin of the local chart U2 and so it is not formed by two degenerated hyperbolic sectors. In short, for having the origin as the unique finite singular point and that the origin of the local chart U2 is formed by two degenerated hyperbolic sectors we must have cα > 0 and |d| > |ω|. So we obtain system (IX). For condition (c) of Theorem 9 with p21 + p22 + p23 ̸= 0 systems (2) have no linear type center at the origin. This completes the proof of Theorem 5. 7. Proof of Theorem 6 First we write the linear part of system (3) into its real Jordan normal form doing the change of variables x = (ωY − dX)/c, y = X and then we do the rescaling τ = ωt. Doing so, the new system can be written as in (8) with d(p2 + 3α) p2 + 3α 6d2 α F =− , G= , K = 0, L=− , cω c cω 2 2p2 4dp2 P = 0, H = 0, N= , M =− . c cω Condition (a) in Theorem 9 yields p2 = 0 and so we obtain system (X). Note that system (X) is Hamiltonian with Hamiltonian c d2 + ω 2 2 3 2 2 H(x, y) = x2 + dxy + y + αx y . 2 2c 2 It was proved in [5] that in this case the unique infinite singular points are the origin of the local charts U1 and U2 and they are formed by two degenerated hyperbolic sectors if and only if cα > 0 and that in this case there are no more finite singular points besides the origin. LINEAR TYPE GLOBAL CENTERS 15 Condition (b) in Theorem 9 with p2 ̸= 0 yields the condition d = 0. So system (3) becomes ω2 (12) ẋ = − y + (p2 − 3α)x2 y, ẏ = cx + (p2 + 3α)xy 2 . c First we study the finite singular points. They are ( √ ) ω c (0, 0), ±√ , ±i , c(p2 − 3α) p2 + 3α where they do not exist if and only if either c(p2 − 3α) ≤ 0, or c(p2 − 3α) > 0 and c(p2 + 3α) ≥ 0. Now we study the infinite singular points of this system. We already know that they are the origins of the local charts U1 and U2 . On the local chart U1 system (12) becomes ω2 2 2 ( ω2 2) u̇ = 6αu2 + cv 2 + u v , v̇ = uv 3α − p2 + v . c c The only infinite singular point of this system is the origin which is linearly zero. Doing one horizontal and one vertical blow up, we get that if c(p2 + 3α) > 0 and α(p2 +3α) > 0 or p2 +3α = 0 and cα > 0, then the origin of the local chart U1 is formed by two degenerated hyperbolic sectors. Otherwise there are parabolic sectors arriving at the origin of the local chart U1 and so it is not formed by two degenerated hyperbolic sectors. These conditions are equivalent to: either p2 + 3α > 0, c > 0 and α = 1; or p2 + 3α < 0, c < 0 and α = −1; or p2 + 3α = 0 and cα > 0. On the local chart U2 system becomes ω2 2 u̇ = −6αu2 − v − cu2 v 2 , v̇ = −uv(p2 + 3α + cv 2 ). c Doing one horizontal and one vertical blow up, we get that if c(p2 − 3α) < 0 and α(p2 − 3α) < 0, or p2 − 3α = 0 and cα > 0, then the origin of the local chart U2 is formed by two degenerated hyperbolic sectors. Otherwise, there are parabolic sectors arriving to the origin of the local chart U2 and so it is not formed by two degenerated hyperbolic sectors. These conditions are equivalent to: either p2 − 3α > 0, c < 0 and α = −1; or p2 − 3α < 0, c > 0 and α = 1; or p2 − 3α = 0 and cα > 0. In short, for having the origin as the unique finite singular point and that both the origins of the local charts U1 and U2 are formed by two degenerated hyperbolic sectors we must have cα > 0 and p2 ∈ [−3, 3]. Hence we obtain system (XI). For condition (c) of Theorem 9 with p2 ̸= 0 systems (3) have no linear type center at the origin. This completes the proof of Theorem 6. 16 J.D. GARCÍA-SALDAÑA, J. LLIBRE, C. VALLS 8. Proof of Theorem 7 First we write the linear part of system (4) into its real Jordan normal form doing the change of variables x = (ωY − dX)/c, y = X and then we do the rescaling τ = ωt. Doing so the new system can be written as in (8) with cp3 − dp2 − 3dα p2 + 3α F = , G= , K = 0, cω c α(c2 + 6d2 ) 2p2 4(cp3 − dp2 ) L=− 2 , P = 0, H = 0, N = , M= . cω c cω Condition (a) in Theorem 9 yields p1 = p2 = 0 and so we obtain sys- tem (XII). Note that system (XII) is Hamiltonian with Hamiltonian c d2 + ω 2 2 3α 2 2 α 4 H(x, y) = x2 + dxy + y + x y + y . 2 2c 2 4 It was proved in [5] that in this case the unique infinite singular point is the origin of the local chart U1 which is formed by two degenerated hyperbolic sectors if and only if cα > 0 and that in this case there are no more finite singular points besides the origin. Condition (b) in Theorem 9 with p22 + p23 ̸= 0 and α ̸= 0 yields the condition p3 = d = 0. So we get the system ω2 (13) ẋ = − y + (p2 − 3α)x2 y − αy 3 , ẏ = cx + (p2 + 3α)xy 2 , c whose finite singular points are ( √ √ ) ( iω ) (p2 + 3α)ω 2 − c2 α c (0, 0), 0, ± √ , ± , ±i . cα c(p22 − 9α2 ) p2 + 3α Note that if p2 = ±3α the last four points do not exist. So in order that they do not exist we must have cα > 0 and either c(p2 +3α) ≥ 0, or c(p2 +3α) < 0 and (p2 − 3α)((p2 + 3α)ω 2 − c2 α) > 0. Now we study the infinite singular points of system (13). We already know that it is the origin of the local chart U1 . On the local chart U1 system (13) becomes ω2 2 2 ( ω2 2) u̇ = 6αu2 + cv 2 + αu4 + u v , v̇ = uv 3α − p2 + αu2 + v . c c Doing one horizontal and one vertical blow up and taking into account that cα > 0, we get that if c(p2 + 3α) ≥ 0 and α(p2 + 3α) ≥ 0, then after the blow-down the origin of the local chart U1 is formed by two degenerated hyperbolic sectors. On the other hand, if the above conditions do not hold, then the origin of the local chart U1 is not formed by two degenerated hyperbolic sectors. In short, in order that system (13) has no more finite singular points besides the origin and the origin of the local chart U1 is LINEAR TYPE GLOBAL CENTERS 17 formed by two degenerated hyperbolic sectors we must have c(p2 + 3α) ≥ 0 and cα > 0. Hence we get system (XIII). For condition (c) of Theorem 9 with p22 + p23 ̸= 0 systems (4) have no linear type center at the origin. This completes the proof of Theorem 7. Acknowledgements J. D. Garcı́a–Saldaña is supported by FONDECyT grant 11171115/2017. J. Llibre is supported by the Ministerio de Economı́a, Industria y Com- petitividad - 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. C. Valls is partially supported by FCT/Portugal through UID/MAT/04459/2013. References [1] A. Algaba, C. Garcı́a and J. Giné, Nondegenerate centers and limit cycles of cubic Kolmogorov systems, Nonlinear Dynam. 91 (2018), 487–496. [2] 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. [3] C. Christopher and C. Li, Limit cycles of differential equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007. [4] 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. [5] I.E. Colak, J. Llibre and C. Valls, Hamiltonian non-degenerate centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations 257 (2014), 1623–1661. [6] H. Dulac, Détermination et integration d’une certaine classe d’équations différentielle ayant par point singulier un centre, Bull. Sci. Math. 32 (1908), 230– 252. [7] F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, New York, 2006. [8] 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. [9] J.M. Pearson, N.G. Lloyd and C.J. Christopher, Algorithm derivation of centre conditions, SIAM Review archive, 38 (196), 619–636. [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. J.D. Garcı́a-Saldaña: Departamento de Matemática y Fı́sica Aplicadas, Uni- versidad Católica de la Santı́sima Concepción, Alonso de Ribera 2850, Con- cepción, Chile E-mail address: jgarcias@ucsc.cl 18 J.D. GARCÍA-SALDAÑA, J. LLIBRE, C. VALLS J. Llibre: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain E-mail address: jllibre@mat.uab.es C. Valls: Departamento de Matemática, Instituto Superior Técnico, Uni- versidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal E-mail address: cvalls@math.ist.utl.pt