This is a preprint of: “Phase portraits of Bernoulli quadratic polynomial differential systems”, Jaume Llibre, Weber F. Pereira, Claudio Pessoa, Electron. J. Differential Equations, vol. 2020(48), 1–19, 2020. PHASE PORTRAITS OF BERNOULLI QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEMS JAUME LLIBRE, WEBER F. PEREIRA, CLAUDIO PESSOA Abstract. In this paper we study a new class of quadratic polynomial dif- ferential systems. We classify all global phase portraits in the Poincaré disk of Bernoulli quadratic polynomial dierential systems in R2 . 1. Introduction Quadratic polynomial dierential systems appear frequently in many areas of ap- plied mathematics, electrical circuits, astrophysics, in population dynamics, chem- istry, neural networks, laser physics, hydrodynamics, etc. Although these dieren- tial systems are the simplest nonlinear polynomial systems, they are also important as a basic testing ground for the general theory of the nonlinear dierential systems. There are more than one thousand papers written on the quadratic polynomial dierential systems. For example there is a bibliography of some of these compiled by Reyn which has 426 items plus 55 preprints and 10 Reports published in TUDelft series of reports in 1989. See the books of Ye Yanqian et al. [24], Reyn [20], and Artés, Llibre, Schlomiuk and Vulpe [2] dedicated to the quadratic polynomial dierential systems. See also the classical surveys on these systems by Coppel [6], and Chicone and Jinghuang [5]. Consider the dierential equation dy = A(x)y k + B(x)y, (1.1) dx with k ∈ R \ {0, 1} and A, B non zero real functions. This dierential equation is called Bernoulli dierential equation. Associated to the Bernoulli dierential equation we can dene the Bernoulli dierential system given by ẋ = p(x), (1.2) ẏ = a(x)y k + b(x)y. Note that system (1.2) is equivalently equation (1.1). In this paper we consider Bernoulli polynomial dierential system of degree 2 in R2 , i.e. p(x) is a polynomial with degree at most 2, k = 2, a(x) is a constant non zero, and b(x) is a non zero polynomial of degree at most 1 (otherwise the 2010 Mathematics Subject Classication. Primary 34C35, 58F09; Secondary 34D30. Key words and phrases. Bernoulli equation, Poincaré disk, phase portrait. 1 system (1.2) will be of separable variables). Thus our objective is to classify all phase portraits of the system ẋ = ax2 + bx + c, (1.3) ẏ = dy 2 + (ex + f )y, with d(e2 + f 2 ) 6= 0. The topological phase portraits in the Poincaré of many classes of quadratic polynomial dierential systems have classied. One of the rst classes analyzed was the classication of the quadratic centers which started with the works of Dulac [8], Kapteyn [11, 12], Bautin [4], Schlomiuk [21], ›oªadek [26], Ye and Ye [25], Artés, Llibre and Vulpe [3], ... The class of the homogeneous quadratic systems by Lyagina [14], Markus [15], Korol [13], Sibirskii and Vulpe [22], Newton [17], Date [7] and Vdovina [23],... The class of Hamiltonian quadratic systems, see Artés and Llibre [1], Kalin and Vulpe [10] and Artés, Llibre and Vulpe [3], etc. Our main result is the following one. Theorem 1.1. The phase portraits in the Poincaré disk of system (1.3) are topo- logically equivalent to one of the 22 phase portraits presented in Figures 1 (except Figure 1 (d)), 2 (except Figure 2 (b)), 3 and 4. The proof of above theorem is given in the end of Section 6. 2. Definitions and useful results Let U an open subset of R2 and X : U → R2 a vector eld. If (x0 , y0 ) ∈ U is a singular point of X , we say that (x0 , y0 ) is a hyperbolic singular point when the real part of both eigenvalues of DX(x0 , y0 ) are dierent of zero. If DX(x0 , y0 ) has exactly one of the eigenvalues dierent of zero, we say that (x0 , y0 ) is semi-hyperbolic singular point of X . The point (x0 , y0 ) is called a elementary singular point of X if (x0 , y0 ) is a hyperbolic or a semi-hyperbolic singular point of X , otherwise (x0 , y0 ) is called a non-elementary singular point of X. In this work to classify topologically the singular points of X , we use the deni- tions of node and saddle points (with their stability), also elliptic, hyperbolic and parabolic sectors (attracting or repelling) as in [19]. For analyzing the topological behavior of the ow near a hyperbolic singular point of X , we use the classical theory of dynamical systems and if we want to analyze the behavior of the ow near a semi-hyperbolic singular point we use Theorem 1 of page 151 from [19]. Now we say that a non-elementary singular point (x0 , y0 ) is a nilpotent singularity of X if DX(x0 , y0 ) has both the eigenvalues equals to zero, but DX(x0 , y0 ) is not zero. Information on this nilpotent singular points can be nd in Theorem 3.5 of [9]. Now, if DX(x0 , y0 ) is the null matrix then (x0 , y0 ) is a linearly zero singularity. To study the local phase portraits of the linearly zero singular points, we do blow-ups consisting of a change of coordinates of the form x 7→ x, y 7→ xy, and x 7→ xy, y 7→ y ( for more details, see page 91 of [9]). 3. Poincaré Compactification In the study of trajectories of polynomial vector elds, is essential to understand the behavior of solutions escaping to innity and a important tool for this is the compactication technique. In short, this method consists of extend analytically the vector eld to a compact manifold, in fact to a sphere. We identify Rn with northern and southern hemispheres through simple projections, then the vector eld X in Rn can be extended to a vector eld X in Sn . This method is called the Poincaré compactication. We describe below this method when n = 2, more details, see [9]. Let X be the polynomial vector eld dened on R2 by system ẋ = P (x, y), ẏ = Q(x, y), where P and Q are polynomials in the variables x and y with real coecients. The the degree of the polynomial vector eld X is dened by d = max{deg P, deg Q}. We denote by S2 = {(z1 , z2 , z3 ) ∈ R3 ; z12 + z22 + z32 = 1} and S1 = {(z1 , z2 , z3 ) ∈ S ; z3 = 0}. We identify R2 as the plane z3 = 1, i.e., the tangent plane π of S2 2 at the north pole (0, 0, 1), and using the central projection of π in S2 , we obtain a tangent vector eld dened on S2 \S1 such that the innity points of π are projected in S1 . In general, this vector eld is unbounded near S1 and symmetric about the center of S2 . But this vector eld admits an unique analytical extension to S2 , after of a multiplication by an appropriate factor. This analytical extension is called the Poincaré compactication of X and denoted by p(X). For study p(X), due the symmetry, is sucient to consider its restriction to the closed northern hemisphere H of S2 . We call the Poincaré disk the orthogonal projection of H into the disk {(z1 , z2 , z3 ) ∈ R3 ; z12 + z22 ≤ 1, z3 = 0}. In each hemisphere we have that p(X) is C ω -equivalent, but not C ω -conjugated, to X. Then the singular points of X correspondent singularities of p(X), but may be that p(X) has singularities in S1 . A singular point of p(X) which belongs to S2 \ S1 (respectively S1 ) is called nite (respectively innite) singular point of X. Moreover, we have that S1 is invariant under the ow of p(X). To obtain expressions of p(X) in local coordinates, we consider the charts of the sphere S2 . For j = 1, 2, 3 dene Uj = {(z1 , z2 , z3 ) ∈ S2 ; zj > 0}, Vj = {(z1 , z2 , z3 ) ∈ S2 ; zj < 0} and ϕj : Uj → R2 , ψj : Vj → R2 given by (z2 , z3 ) (z1 , z3 ) (z1 , z2 ) ϕ1 (z) = −ψ1 (z) = , ϕ2 (z) = −ψ2 (z) = , ϕ3 (z) = . z1 z2 z3 If we denote by (u, v) the value of ϕj or ψj at the point z we can prove that the expression of p(X) in the chart (U1 , ϕ1 ) is given by        1 u 1 u 1 u u̇ = v d −uP , +Q , , v̇ = −v d+1 P , . v v v v v v The expression of p(X) in the chart (U2 , ϕ2 ) is        u 1 u 1 u 1 u̇ = v d P , − uQ , , v̇ = −v d+1 Q , , v v v v v v and the expression of p(X) in the chart (U3 , ϕ2 ) is u̇ = P (u, v), v̇ = Q(u, v). Finally, for each j = 1, 2, 3, the expression of p(X) in the chart (Vj , ψj ) is the expression of p(X) in the chart (Uj , ϕj ) multiplied by the factor (−1)d−1 . Using this notation we observe that if (u, v) ∈ Uj is an innite singular point of X if, and only if, the expression of p(X) in the chart (Uj , ϕj ) vanishes in (u, v) and v = 0. Observe that if z is an innite singular point of X then −z is also an innite singular point of X. In this case, from the expressions of p(X) in local coordinates it follows that the behavior of the ow near −z can be determined by the behavior of the ow near z, because the ow near −z diers by the ow near z by the factor (−1)d−1 . Then the study of p(X) in the charts (Vj , ψj ), j = 1, 2, 3, is superuous. Moreover, notice that if z is an innite singular point of X with z ∈ U2 , z 6= (0, 1, 0) then z ∈ U1 ∪ V1 . It follows that to study all the innite singular points of X, it is sucient to study the singularities of p(X) in U1 and the origin of U2 . 4. Markus-Neumann-Peixoto Theorem The study of the phase portrait of a given planar vector elds can be reduced to the determination of the separatrices (see denition below) and a nite number of special orbits. This result is known as Markus-Neumann-Peixoto Theorem, for more details see [15], [16], [19] or p. 33 of [9]. Let X an Y be C 1 -vector elds dened on the open sets U and V of R2 , respec- tively. Denote by (U, Φ) and (V, Ψ) the ow of X and Y, respectively. We say that (U, Φ) and (V, Ψ) are topologically equivalent if there exists a homeomorphism of U in V which carries the orbits of X in orbits of Y, preserving the orientation of the all orbits, and in this case we also say that their phase portraits are topologically equivalent. Consider the following vector elds • V = R2 and Y (x, y) = (1, 0), ∀(x, y) ∈ R2 , • V = R2 \ {(0, 0)} and Y such that, in polar coordinates, is given by ṙ = 0, θ̇ = 1, • V = R2 \ {(0, 0)} and Y such that, in polar coordinates, Y is given by ṙ = r, θ̇ = 0. We call the ow of the three vector elds above of strip ow, annulus ow and nodal ow, respectively. Now, suppose that U = R2 , if the ow (R2 , Φ) is topologically equivalent either to a strip ow or annulus ow or a nodal ow it is called parallel. Denote by γ(p) the orbit of p ∈ U , and by α(p) and ω(p), the respective α-limit and the ω -limit of p. The orbit γ(p) is a separatrix if • γ(p) is a singular point, or • γ(p) is a periodic orbit and there is no neighborhood of γ(p) consisting of periodic orbits, or • γ(p) is homeomorphic to R and there is no neighborhood W of γ(p) with the following two properties:  q ∈ W ⇒ α(q) = α(p) and ω(q) = ω(p),  the boundary of W is composed by α(p), ω(p) and by two another or- bits γ(p1 ), γ(p2 ) such that α(p1 ) = α(p2 ) = α(p) and ω(p1 ) = ω(p2 ) = ω(p). We denote by Σ the union of all separatrices of a given ow (U, Φ) , Σ. is called extended separatrix skeleton,. Note that it is a closed invariant subset of U and each connected component of U \ Σ is an open invariant set, called a canonical region. There exist only three possibilities for the ow in each canonical region, more precisely we have the following result. Proposition 4.1. In each canonical region the ow is parallel. The union of the extended separatrix skeleton with one orbit in each canonical region is called completed separatrix skeleton. Consider the extended separatrix skeleton C1 and C2 of the ows (R2 , Φ) and (R2 , Ψ), respectively. Then, if there exist a homeomorphism of R2 in R2 which map orbits of C1 into orbits of C2 preserving the orientation, we say that C1 and C2 are topologically equivalent. Now we can to present the Markus-Newmann-Peixoto theorem which implies that, to draw the phase portrait of a given planar vector eld, it is sucient deter- mine its completed separatrix skeleton. Theorem 4.2 (Markus-Neumann-Peixoto). Consider the continuous ows (R2 , Φ) and (R2 , Ψ) and suppose that they have only isolated singular points. Then (R2 , Φ) and (R2 , Ψ) are topologically equivalent if, and only if, its completed separatrices skeleton are topologically equivalent. 5. Local Phase Portrait of Finite and Infinite Singular Points In this section we determinate the local local phase portrait of the nite and innite singular points of system (1.3). As in section 3 we denote by p(X) the Poincaré compactication of system (1.3). Here the singular points of p(X) in S1 will be denoted by qi . Remember that, if qi is a singular point of p(X), then −qi is also. Moreover, as the degree of system (1.3) is two, the behavior of the ow near −qi is the same of near qi but reversing the sense of the orbits. Thus we will describe the local phase portrait of the innite singular points qi . In terms of the number, multiplicity and type of the roots of the polynomial p(x) = ax2 + bx + c of system (1.3), we distinguish ve cases. 5.1. Case 1: p(x) has two distinct reals roots. In this case, we can write system (1.3) as ẋ = (x − α)(x − β), ẏ = dy 2 + (ex + f )y, (5.1) 2 2 with d(e + f ) 6= 0 and α 6= β . The singular points of system (5.1) are:     eα + f eβ + f p1 = (α, 0); p2 = (β, 0); p3 = α, − and p4 = β, − . d d Denote by λi , µi , i = 1, . . . , 4, the eigenvalues of the linear parts of system (5.1) at the singular point pi . The next three results determine the local phase portrait of the nite singular points. Proposition 5.1. Suppose that system (5.1) has four singular points, i.e., (eα + f )(eβ + f ) 6= 0. (a) If eα + f > 0, α − β > 0 and eβ + f > 0, then p1 is an unstable node, p3 and p2 are saddles, and p4 is a stable node; (b) If eα + f > 0, α − β > 0 and eβ + f < 0, then p1 is an unstable node, p3 and p4 are saddles, and p2 is a stable node; (c) If eα + f < 0, α − β > 0 and eβ + f > 0, then p3 is an unstable node, p1 and p2 are saddles, and p4 is a stable node; (d) If eα + f < 0, α − β > 0 and eβ + f < 0, then p3 is an unstable node, p1 and p4 are saddles, and p2 is a stable node; (e) If eα + f > 0, α − β < 0 and eβ + f > 0, then p3 is a stable node, p1 and p4 are saddles, and p2 is an unstable node; (f) If eα + f > 0, α − β < 0 and eβ + f < 0, then p3 is a stable node, p1 and p2 are saddles, and p4 is an unstable node; (g) If eα + f < 0, α − β < 0 and eβ + f > 0, then p1 is a stable node, p3 and p4 are saddles, and p2 is an unstable node; (h) If eα + f < 0, α − β < 0 and eβ + f < 0, then p1 is a stable node, p3 and p2 are saddles, and p4 is an unstable node. Proof. We have that λ1 = eα+f , µ1 = α−β , λ2 = eβ+f , µ2 = β−α, λ3 = −eα−f , µ3 = α − β , λ4 = −eβ − f and µ4 = β − α. Therefore, the rest of the proof follows of the fact that all the singular points are hyperbolic, and then its local phase portraits are known.  Proposition 5.2. Suppose that system (5.1) has exactly three singular points, i.e., (eα + f )(eβ + f ) = 0 and (eα + f )2 + (eβ + f )2 6= 0. (a) If eα + f = 0, α − β > 0 and eβ + f > 0, then p1 is a saddle-node, p2 is a saddle, and p4 is a stable node; (b) If eα + f = 0, α − β > 0 and eβ + f < 0, then p1 is a saddle-node, p2 is a stable node, and p4 is a saddle; (c) If eα + f = 0, α − β < 0 and eβ + f > 0, then p1 is a saddle-node, p2 is an unstable node, and p4 is a saddle; (d) If eα + f = 0, α − β < 0 and eβ + f < 0, then p1 is a saddle-node, p2 is a saddle, and p4 is an unstable node; (e) If eα + f > 0, α − β > 0 and eβ + f = 0, then p1 is an unstable node, p2 is a saddle-node, and p3 is a saddle; (f) If eα + f < 0, α − β > 0 and eβ + f = 0, then p1 is a saddle, p2 is a saddle-node, and p3 is an unstable node; (g) If eα + f > 0, α − β < 0 and eβ + f = 0, then p1 is a saddle, p2 is a saddle-node, and p3 is a stable node; (h) If eα + f < 0, α − β < 0 and eβ + f = 0, then p1 is a stable node, p2 is a saddle-node, and p3 is a saddle. Proof. First we suppose that eα + f = 0, so the eigenvalues associated to singular points p1 =(α, 0) are λ1 = 0and µ1 = α − β . Now, doing the change of coordinates s (x, y, t) 7→ u + α, v, , system (5.1) becomes α−β 1 u0 = u + u2 = u + P (u, v), α−β e d v0 = uv + v 2 = Q(u, v), α−β α−β and so p1 correspond to origin. Note that u ≡ 0 is the solution of equation u + d P (u, v) = 0 and Q(0, v) = v 2 . Hence, by Theorem 1 from the page 151 of α−β [19], we have that p1 is a saddle-node. For the others two singularities p2 and p4 , it follows that λ2 = eβ + f , µ2 = β − α, λ4 = −eβ − f and µ4 = β − α. Therefore, the rest of the proof of statements (a), (b), (c) and (d) follows taking into account the signs of the eigenvalues because these points are hyperbolic. The proof of case eβ + f = 0, i.e., statements (e), (f), (g) and (h) is analogous to the previous case.  Proposition 5.3. Suppose that system (5.1) has exactly two singular points, i.e., eα + f = 0 and eβ + f = 0. Then the singular points are saddle-nodes. Proof. The eigenvalues associated to singularities p1 = (α, 0) and p2 = (β, 0) are λ1 = 0, µ1 = α − β , λ2 = 0 and µ2 = β − α, respectively. In thiscase, since α 6=  β, s we have e = 0. Now, doing the change of coordinates (x, y, t) 7→ u + α, v, , α−β system (5.1) if e = 0, becomes 1 d u0 = u + u2 = u + P (u, v), v0 = v 2 = Q(u, v), α−β α−β and so p1 corresponds to the origin. Note that u ≡ 0 is the solution of equation d u + P (u, v) = 0 and Q(0, v) = v 2 . Hence, by Theorem 1 of page 151 of [19], α−β we have that p1 is a saddle-node. Analogously we have p2 is a saddle-node.  The next result determine the local phase portrait of the innite singular points. Proposition 5.4. Let p(X) be the Poincaré compactication of system (5.1). (a) If 1 − e 6= 0, then p(X) has six singularities ±q1 , ±q2 and ±q3 in the equator S1 . Moreover, q1 is a saddle (resp. stable node) and q2 is a stable node (resp. saddle) if 1 − e < 0 (resp. 1 − e > 0), and q3 is either a stable node when d > 0, or an unstable node when d < 0. (b) If 1 − e = 0, then p(X) has four singularities ±q1 and ±q3 in the equator S1 . Moreover q1 is a saddle-node and q3 is either a stable node when d > 0 or an unstable node when d < 0. Proof. The system associated to p(X) in the charts U1 and U2 are u0 = (−1 + e) u + du2 + (β + α + f ) uv − αβuv 2 , (5.2) v0 = −v + (α + β)v 2 − αβv 3 , and u0 = −du + (1 − e) u2 − (β + α + f ) uv + αβv 2 , (5.3) v0 = −dv − euv − f v 3 , respectively.  In the chart U1 for v = 0 we have the singular points q1 = (0, 0) and q2 = 1−e , 0 of system (5.2). The eigenvalues associated to q1 and q2 are λ11 = e − 1, d λ12 = −1 and λ21 = 1−e, λ22 = −1, respectively. Now in the chart U2 , q3 = (0, 0) is a singular points of system (5.3), and its eigenvalues are λ31 = λ32 = −d. Therefore, the proof of the statement (a) follows by studying the signs of the eigenvalues. For case 1 − e = 0, system (5.2) becomes, after a time rescaling, u0 = −du2 − (β + α + f ) uv + αβuv 2 = P (u, v), (5.4) v0 = v − (α + β)v 2 + αβv 3 = v + Q(u, v). Note that in this case q1 = q2 and, in the chart U1 , q1 correspond to the singular point at the origin of system (5.4) with eigenvalues λ11 = 0 and λ12 = 1. As v ≡ 0 is the solution of equation v + Q(u, v) = 0 and P (u, 0) = −du2 . By Theorem 1 of page 151 of [19], we have that q1 is a saddle-node. Hence statement (b) follows.  5.2. Case 2: p(x) has a double real root. In this case we can write system (1.3) as ẋ = (x − α)2 , ẏ = dy 2 + (ex + f )y. (5.5) The singular points of system (5.5) are:   eα + f p1 = (α, 0) and p3 = α, − . (5.6) d Denote by λi , µi , i = 1, 3, the eigenvalues of the linear parts of system (5.5) at the singular point pi . The next result determines the local phase portrait of the nite singular points. Proposition 5.5. Consider system (5.5). (a) If eα + f 6= 0, then the singular points p1 and p3 are distinct and both are saddle-nodes. (b) If eα + f = 0, then p1 = p3 and it is a singular point with two parabolic sectors and two hyperbolic sectors. Proof. First we suppose that eα + f 6= 0, then λ1 = 0, µ1 =  eα + f , λ3 = 0 and  s µ3 = −(eα + f ). Doing the change of variables (x, y, t) 7→ u + α, v, , f + αe system (5.5) becomes u2 u0 = = P (u, v), f + αe e d v0 = v+ uv + v 2 = v + Q(u, v), f + αe f + αe and so p1 corresponds to the origin. Note that v ≡ 0 is the solution of equation 1 v + Q(u, v) = 0 and P (u, 0) = u2 . Hence, by Theorem 1 of page 151 of [19], f + αe we have that p1 is a saddle-node.   f + αe s Now doing the change of variables (x, y, t) 7→ −dv + α, u + ev − , − , d f + αe system (5.5) becomes d ed ed u0 = u− u2 − uv − v 2 = u + P (u, v), f + αe f + αe f + αe d v0 = v 2 = Q(u, v), f + αe and so p3 corresponds to origin. Analogously to the previous case, we have that p3 is a saddle-node. When eα + f = 0, by (5.6) we have p1 = p3 and λ1 = µ1 = 0. Hence, doing the change of variables (x, y) 7→ (u + α, v), system (5.5) with f = −αe becomes u0 = u2 , v 0 = euv + dv 2 . (5.7) As (0, 0) is a linearly zero singular point of system (5.7), we will doing a blow-up in the direction u. More precisely, doing u = x̃ and v = x̃ỹ in system (5.7) and after a time rescaling, we obtain x̃0 = x̃, ỹ 0 = (e − 1)ỹ + dỹ 2 . (5.8)   1−e When e−1 6= 0 system (5.8) has two singularities p̃1 = (0, 0) and p̃3 = 0, d with respective eigenvalues λ̃1 = 1, µ̃1 = e − 1, λ̃3 = 1 and µ̃3 = 1 − e. If e − 1 > 0 (resp. e − 1 < 0), then p̃1 is an unstable node (resp. saddle), and p̃3 is a saddle (resp. unstable node). For e − 1 = 0, p̃1 is the unique singularity of system (5.8), and by Theorem 1 of page 151 of [19], we have that p̃1 is a saddle-node. Now we do a blow-up in the direction v . More precisely, doing u = x̃ỹ and v = ỹ in system (5.7) and after a time rescaling, we obtain x̃0 = −dx̃ + (1 − e)x̃2 , ỹ 0 = dỹ + ex̃ỹ. (5.9) We have to study only the singular point q̃3 = (0, 0) of system (5.9). This singular point has eigenvalues ±d, and so q̃3 is a saddle. In summary, going back through the blow ups, p1 is a singular point with two hyperbolic sectors and two parabolic sectors.  The local phase portraits of the innite singular points in this case are the same obtained in Case 1. In fact, the Poincaré compactication of system (5.5) in the charts U1 and U2 are given by systems (5.2) and (5.3) doing α = β , respectively. Therefore, we have the same result as Proposition 5.4 whose the proof is analogous. The result is the following. Proposition 5.6. Let p(X) be the Poincaré compactication of system (5.5). (a) If 1 − e 6= 0, then p(X) has six singularities ±q1 , ±q2 and ±q3 in the equator S1 . Moreover, q1 is a saddle (resp. stable node) and q2 is a stable node (resp. saddle) if 1 − e < 0 (resp. 1 − e > 0), and q3 is either a stable node when d > 0, or an unstable node when d < 0. (b) If 1 − e = 0, then p(X) has four singularities ±q1 and ±q3 in the equator S1 . Moreover q1 is a saddle-node and q3 is either a stable node when d > 0 or an unstable node when d < 0. 5.3. Case 3: p(x) has only one real root. In this case, we can write system (1.3) as ẋ = x − α, ẏ = dy 2 + (ex + f )y. (5.10) The singular points of system (5.10) are:   eα + f p1 = (α, 0) and p2 = α, − . (5.11) d Denote by λi , µi , i = 1, 2, the eigenvalues of the linear parts of system (5.10) at the singular point pi . The next result determines the local phase portrait of the nite singular points. Proposition 5.7. Consider system (5.10). (a) If eα + f > 0 (resp. eα + f < 0), then the singular point p1 is an unstable node (resp. saddle) and p2 is saddle (resp. unstable node). (b) If eα + f = 0, then p1 = p2 and it is a saddle-node. Proof. When eα + f 6= 0, we have λ1 = 1, µ1 = eα + f , λ2 = 1, µ2 = −(eα + f ). Therefore the proof of statement (a) follows from the signs of the eigenvalues. Now if eα + f = 0, by (5.11) we have p1 = p2 and λ1 = 1 and µ1 = 0. Hence doing the change of variables (x, y) 7→ (u + α, v), system (5.10) with f = −αe becomes u0 = u, v 0 = euv + dv 2 . Hence by Theorem 1 of page 151 of [19], we have that p̃1 is a saddle-node. Statement (b) is proved.  The next result determines the local phase portrait of the innite singular points. Proposition 5.8. Let p(X) be the Poincaré compactication of system (5.10). (a) If e 6= 0, then p(X) has six singularities ±q1 , ±q2 and ±q3 in the equator S1 . Moreover, ±q1 , ±q2 are saddle-nodes and q3 is stable (resp. unstable) node when d > 0 (resp. d < 0). (b) If e = 0, then p(X) has four singularities ±q1 and ±q3 in the equator S1 . Moreover q1 is a singular point with two hyperbolic sectors and two parabolic sectors, and q3 is either a stable node when d > 0, or an unstable node when d < 0. Proof. The system associated to p(X) in the charts U1 and U2 are u0 = eu + du2 + (f − 1)uv + αuv 2 , (5.12) v0 = −v 2 + αv 3 , and u0 = −du − eu2 + (1 − f )uv − αv 2 , (5.13) v0 = −dv − euv − f v 2 , respectively. In the chart U2 , q3 = (0, 0) is a singular point of system (5.13), and its eigenvalues are λ31 = λ32 = −d. Therefore q3 is stable (resp. unstable) node when d > 0 (resp. d < 0). We suppose e 6= 0. In the  chart U1 for v = 0 we have the singular points e q1 = (0, 0) and q2 = − , 0 of system (5.12). The eigenvalues associated to q1 d and q2 are λ11 = e, λ12 = 0 and λ21 = −e, λ22 = 0, respectively. By Theorem 1 of page 151 of [19], we have  that q1 is a saddle-node.  Analogously, after the change of 1−f e s variables (u, v, t) 7→ x + y − , y, − applied to system (5.12), we obtain d d e that q2 is a saddle-node. This proves statement (a). For the case e = 0 in the chart U1 we have that q1 = q2 = (0, 0) is a linearly zero singular point. We do a blow-up in the direction u. More precisely, doing u = x̃ and v = x̃ỹ in system (5.12) and after a time rescaling, we obtain x̃0 = dx̃ + (f − 1)x̃ỹ + αx̃2 ỹ 2 , ỹ 0 = −dỹ − f ỹ 2 . (5.14)   d When f 6= 0, system (5.14) has two singularities q̃1 = (0, 0) and q̃2 = 0, − f d with respective eigenvalues λ̃1 = d, µ̃1 = −d, λ̃2 = and µ̃2 = d. Note that q̃1 is f always a saddle. Now q̃2 is either a saddle if f < 0, or an unstable (resp. stable) node if f > 0 and d > 0 (resp. f > 0 and d < 0). Now when f = 0, q̃1 is a unique singularity of system (5.14) ,and as in the previous case it is a saddle. We do a blow-up in the direction v . More precisely, doing u = x̃ỹ and v = ỹ in system (5.12) and after a time rescaling, we obtain x̃0 = f x̃ + dx̃2 , ỹ 0 = −ỹ + αỹ 2 . (5.15) We study only the singular point q̃3 = (0, 0) of system (5.15). This singular point has eigenvalues λ̃3 = f and µ̃3 = −1, and so q̃3 is either a saddle if f > 0, or a stable node if f < 0, or (by Theorem 1 of page 151 of [19]) a saddle-node if f = 0. In short, going back through the blow-ups we get that q1 is a singular point with two hyperbolic sectors and two parabolic sectors. So statement (b) is proved.  5.4. Case 4: p(x) is constant. In this case, we can write system (1.3) as ẋ = 1, ẏ = dy 2 + (ex + f )y. (5.16) Note that system (5.16) does not have nite singular points. The next result determines the local phase portrait of the innite singular points. Proposition 5.9. Let be p(X) be in the equatorthe Poincaré compactication of system (5.16). (a) If e 6= 0, then p(X) has six singularities ±q1 , ±q2 and ±q3 in the equator S1 . Moreover, q1 is a topological saddle (resp. stable node) if e > 0 (resp. e < 0), q2 is a topological saddle (resp. stable node) if e < 0 (resp. e > 0), and q3 is stable (resp. unstable) node when d > 0 (resp. d < 0). (b) If e = 0 and f 6= 0, then p(X) has four singularities ±q1 and ±q3 in the equator S1 . Moreover q1 is a saddle-node and q3 is either a stable node when d > 0, or an unstable node when d < 0. (c) If e = 0 and f = 0, then p(X) has four singularities ±q1 and ±q3 in the equator S1 . Moreover q1 is a singular point with two hyperbolic sectors and two parabolic sectors and q3 is either a stable node when d > 0, or an unstable node when d < 0. Proof. The system associated to p(X) in the charts U1 and U2 are u0 = eu + du2 + f uv − uv 2 , v 0 = −v 3 , (5.17) and u0 = −du − eu2 − f uv + v 2 , v 0 = −dv − euv − f v 2 , (5.18) respectively. In the chart U2 , q3 = (0, 0) is a singular points of system (5.18), and its eigenval- ues are λ31 = λ32 = −d. Therefore q3 is stable (resp. unstable) node when d > 0 (resp. d < 0). We suppose e 6= 0. In the  chart U1 for v = 0 we have the singular points e q1 = (0, 0) and q2 = − , 0 of system (5.17). The eigenvalues associated to q1 d and q2 are λ11 = e, λ12 = 0 and λ21 = −e, λ22 = 0, respectively. By Theorem 1 of page 151 of [19], we have that q1 is a topological saddle if e >0, and stable node  if f e s e < 0. Analogously after the change of variables (u, v, t) 7→ x − y − , y, − d d e in system (5.17), we obtain that q2 is a topological saddle if e < 0, and a stable node if e > 0 . For case e = 0 in the chart U1 we have that q1 = q2 = (0, 0) is a linearly zero singular point. We do a blow-up in the direction u. More precisely, doing u = x̃ and v = x̃ỹ in system (5.17) and after a time rescaling, we obtain x̃0 = dx̃ + f x̃ỹ − x̃2 ỹ 2 , ỹ 0 = −dỹ − f ỹ 2 . (5.19)   d When f 6= 0, system (5.19) has two singularities q̃1 = (0, 0) and q̃2 = 0, − f with respective eigenvalues λ̃1 = d, µ̃1 = −d, λ̃2 = 0 and µ̃2 = d. Note that  q̃1 is d s always a saddle. Now doing the change of variables (x̃, ỹ, t) 7→ ũ, ṽ − , to f d system (5.19), it becomes d 2 f 2 1 f 2 ũ0 = − ũ + ũṽ + ũ2 ṽ − ũ2 ṽ 2 , ṽ 0 = ṽ − ṽ . f2 d f d d Hence by Theorem 1 of page 151 of [19], we have that q̃2 is a saddle-node. Now when f = 0, q̃1 is the unique singularity of system (5.19), and as the previous case it is a saddle. We do a blow-up in the direction v . More precisely, doing u = x̃ỹ and v = ỹ in system (5.17) and after a time rescaling, we obtain x̃0 = f x̃ + dx̃2 , ỹ 0 = −ỹ 2 . (5.20) We have to study only the singular point q̃3 = (0, 0) of system (5.20). This singular point has eigenvalues λ̃3 = f and µ̃3 = 0, and so q̃3 is a saddle-node, by Theorem 1 of page 151 of [19], if f 6= 0. If f = 0 we do a new blow-up to system (5.20) in the direction x̃ (x̃ = ũ and ỹ = ũṽ ) obtaining, after a time rescaling ũ0 = dũ, ṽ 0 = −dṽ − ṽ 2 . (5.21) System (5.21) has two singular points (0, 0) and (0, −d) with respective eigenvalues [d, −d] and [d, d], so (0, 0) is a saddle, and (0, −d) is a node (stable if d < 0 and unstable if d > 0). Now doing a blow-up in direction ỹ (x̃ = ũṽ and ỹ = ṽ ), system (5.20) becomes after time rescaling ũ0 = ũ + dũ2 , ṽ 0 = −ṽ. (5.22) We have that (0, 0) is a saddle of system (5.22). Going back through the blow-ups we conclude that q1 is either a saddle-node if f 6= 0, or a singular point with two hyperbolic sectors and two parabolic sectors if f = 0.  5.5. Case 5: p(x) has two complex conjugated roots. In this case we can write system (1.3) as ẋ = x2 − 2αx + α2 + β 2 , ẏ = dy 2 + (ex + f )y. (5.23) 2 2 2 Note that α ± iβ are the roots of x − 2αx + α + β = 0, and so system (5.23) does not have nite singular points. The next result determine the local phase portrait of the innite singular points. Proposition 5.10. Let p(X) be the Poincaré compactication of system (5.23). (a) If e 6= 1, then p(X) has six singularities ±q1 , ±q2 and ±q3 in the equator S1 . Moreover, q1 (resp. q2 ) is either a saddle (resp. stable node) if e > 1, or a stable node (resp. saddle) if e < 1, and q3 is stable (resp. unstable) node when d > 0 (resp. d < 0). (b) If e = 1, then p(X) has four singularities ±q1 and ±q3 in the equator S1 . Moreover q1 is a saddle-node, and q3 is either a stable node when d > 0, or an unstable node when d < 0. Proof. The system associated to p(X) in the charts U1 and U2 are u0 = (e − 1)u + du2 + (2α + f )uv − (α2 + β 2 )uv 2 , (5.24) v0 = −v + 2αv 2 − (α2 + β 2 )v 3 , and u0 = −du + (1 − e)u2 − (2α + f )uv + (α2 + β 2 )v 2 , (5.25) v0 = −dv − euv − f v 2 , respectively. In the chart U2 , q3 = (0, 0) is a singular points of system (5.25), and its eigenval- ues are λ31 = λ32 = −d. Therefore q3 is stable (resp. unstable) node when d > 0 (resp. d < 0). We suppose e 6= 1 . In thechart U1 for v = 0 we have the singular points 1−e q1 = (0, 0) and q2 = , 0 of system (5.24). The eigenvalues associated to d q1 and q2 are λ11 = e − 1, λ12 = −1 and λ21 = 1 − e, λ22 = −1, respectively. Therefore, the proof of statement (a) follows from the signs of the eigenvalues. For case e = 1 in the chart U1 we have that q1 = q2 = (0, 0), and the eigenvalues are λ11 = 0, λ12 = −1. By Theorem 1 of page 151 of [19], we have that q1 is a saddle-node. Hence statement (b) follows.  6. Main Results In this section we classify all global phase portraits in the Poincaré disk of system (1.3). The rst result is about the existence of limit cycles. Proposition 6.1. Systems (1.3) do not have limit cycle. Proof. Observe that the rst equation of system (1.3) does not depend of the vari- able y . Hence, solving this dierential equation, the solutions are not a periodic functions and so system (1.3) does not have periodic solutions.  Theorem 6.2. Consider system (1.3). If p(x) = ax2 + bx + c has two distinct reals roots, then the phase portrait is topological equivalent to one of the phase portraits of Figure 1. Proof. In this case system (1.3) can be written in the form (5.1). We have that x = α, x = β and y = 0 are invariant straight lines of system (5.1). These three straight lines intersect in the singular points p1 = (α, 0) and p2 = (β, 0), and determine four innite singular points ±q1 and ±q3 corresponding to the origin of the charts U1 , V1 , U2 and V2 in the Poincaré compactication, respectively. By Theorem 5.4, ±q3 are always a nodes. Moreover we can have additionally two innite singular points ±q2 and either one, or two nite singular points p3 and p4 . First we suppose system (5.1) has four nite singular points. By Theorem 5.1, p1 and p2 are saddles or nodes. When they are saddles, p3 and p4 are nodes and, by statements (3) and (6) of Theorem 5.1, these nodes live in opposite half-planes determined by the invariant straight line y = 0, and we obtain that 1 − e > 0. In fact, consider the statements (3) of Theorem 5.1, we have that eα + f < 0 and −eβ − f < 0 and so e(α − β) < 0. Now, as α − β > 0, it follows that (α − β) − e(α − β) = (α − β)(1 − e) > 0, i.e. 1 − e > 0. Since 1 − e > 0, by Theorem 5.4, we always have six innite singular points, ±q1 are nodes and ±q2 are saddles. Therefore in this case using Theorem 4.2 the phase portrait of system (5.1) is equivalent to Figure 1 (a). If p1 and p2 are nodes, as in the previous case, p3 and p4 are saddles and live in opposites half-planes determined by the invariant straight line y = 0. However in this case we can have 1 − e 6= 0 and 1 − e = 0. Hence, by Theorem 5.4, there are either six innite singular points (i.e., ±q1 and ±q2 are nodes or saddles), or four innite singular points (i.e., ±q1 are saddle-nodes). Thus the phase portrait of system (5.1) is equivalent to one of Figure 1 (b)-(c). If p1 is saddle (resp. node) and p2 is node (resp. saddle), then by statements (1), (4), (5) and (8) of Theorem 5.1, p3 is a node (resp. a saddle) and p4 is a saddle (resp. a node) and they live in the same half-plane determined by the invariant straight line y = 0. Moreover as in the previous case there are either six innite singular points (i.e., ±q1 and ±q2 are nodes or saddles), or four innite singular points (i.e., ±q1 are saddle-nodes). Note that when ±q1 is a saddle, we have a heteroclinic connection between a nite saddle and ±q1 . Otherwise we do not have heteroclinic orbits. Thus in this case the phase portrait of system (5.1) is equivalent to one of Figure 1 (d)-(f). Suppose system (5.1) has three nite singular points. By Theorem 5.2 these singular points are a saddle-node, a saddle and a node. Moreover the saddle- node is p1 or p2 . If we have a saddle in the variant straight line y = 0, then by statements (1), (4), (6) and (7) of Theorem 5.2 and by Theorem 5.4, the innite singular points ±q1 are nodes and ±q2 are saddles. Thus the phase portrait is equivalent to Figure 1 (i). Now if we have a node in the invariant straight line y = 0, then by statements (2), (3), (5) and (8) of Theorem 5.2 and by Theorem 5.4, we can have either four or six innite singular points. When there exist only four innite singular points ±q1 are saddle-nodes and the phase portrait is equivalent to Figure 1 (j). When there exist six innite singular points and ±q1 are nodes (resp. saddles), then ±q2 are saddles (resp. nodes) and phase portrait is equivalent to one of Figure 1 (g)-(h). Finally we consider the case that system (5.1) has two nite singular points. By Theorem 5.3 these singular points are saddle-nodes. Now as eα + f = eβ + f = 0 and α 6= β , we obtain e = 0. Hence by Theorem 5.4 system (5.1) has six innite singular points, ±q1 and ±q3 are nodes and ±q2 are saddles, then the phase portrait is equivalent to Figure 1 (k).  Theorem 6.3. Consider system (5.5). If p(x) have one real double root, then the phase portraits are topological equivalent to one of Figure 2. Proof. In this case system (1.3) can be written in the form (5.5). We have that x = α and y = 0 are invariant invariant straight lines of system (5.5). These straight lines intersect at the singular point p1 = (α, 0) and determine four innite singular points ±q1 and ±q3 corresponding to the origin of the charts U1 , V1 , U2 and V2 in the Poincaré compactication. By Theorem 5.4 ±q3 are always nodes. Moreover we can have additionally two innite singular points ±q2 , and one nite singular point p3 . By Proposition 5.5 if eα + f 6= 0, we have two nite singular points, both are saddle-nodes. If 1 − e 6= 0, by Theorem 5.4, we have six innite singular points. (a) (r, s) = (6, 25) (b) (r, s) = (7, 26) (c) (r, s) = (7, 22) (d) (r, s) = (7, 26) (e) (r, s) = (6, 25) (f) (r, s) = (6, 21) (g) (r, s) = (7, 24) (h) (r, s) = (6, 23) (i) (r, s) = (7, 24) (j) (r, s) = (6, 19) (k) (r, s) = (6, 21) Figure 1. Phase Portraits of Case 1. Here r denotes the number of canonical regions of the phase portrait and s its number of separatrices. When 1 − e < 0, ±q1 are saddles, ±q2 are nodes and we have a connection between the separatrices of a hyperbolic sector from p1 with one of these innite saddles and the phase portrait is topologically equivalent to Figure 2 (a). Now if 1 − e > 0, ±q1 are nodes, ±q2 are saddles, and the phase portrait are topologically equivalent to Figure 2 (b). For 1 − e = 0 by Theorem 5.4 we have four innite singular points and ±q1 are saddle-nodes, so the phase portrait is topologically equivalent to Figure 2 (c). In the case eα + f = 0 by Theorem 5.5, p1 is the only nite singular point and it is a singular point with two parabolic sectors and two hyperbolic sectors. Now by Theorem 5.4, we have six innite singular points when 1 − e 6= 0 and four innite singular points otherwise. Hence the phase portrait is equivalent to one of Figure 2 (d)-(e).  (a) (r, s) = (5, 20) (b) (r, s) = (6, 21) (c) (r, s) = (5, 16) (d) (r, s) = (6, 19) (e) (r, s) = (4, 13) Figure 2. Phase Portraits of Case 2. Theorem 6.4. Consider system (5.10). If p(x) have a unique real root, then the phase portraits are topological equivalent to one of Figure 3. Proof. In this case system (1.3) can be written in to the form (5.10). We have that x = α and y = 0 are invariant straight lines of system (5.10). These straight lines intersect at the singular point p1 = (α, 0) and determine four innite singular points ±q1 and ±q3 corresponding to the origin in the charts U1 , V1 , U2 and V2 in the Poincaré compactication, respectively. By Theorem 5.8 q3 is always a node. Moreover we can have additionally two innite singular points ±q2 and one nite singular point p2 . By Theorems 5.7 and 5.8 if eα + f > 0 and e 6= 0, then p1 is a node, p2 is a saddle, ±q1 and ±q2 are saddle-nodes. Hence the phase portrait is topologically equivalent to Figure 3 (a). Analogously if eα + f < 0, p1 is a saddle, p2 is a node and the phase portrait is topologically equivalent to Figure 3 (b). If eα + f = 0, then there exist a unique nite singular point p1 , and by Theorem 5.7 it is a saddle-node. When e 6= 0 by Theorem 5.8 we have six innite singular points, the saddle-nodes ±q1 and ±q2 and the nodes ±q3 . Then the phase portrait is topologically equivalent to Figure 3 (c). Now when e = 0, by Theorem 5.8, we have four innite singular points, i.e., ±q1 are singular points with two hyperbolic sectors and two parabolic sectors and ±q3 are nodes. Hence the phase portrait is given by Figure 3 (d).  Theorem 6.5. Consider system (5.16). Then the phase portraits are topological equivalent to one of Figure 4. (a) (r, s) = (5, 20) (b) (r, s) = (4, 19) (c) (r, s) = (5, 18) (d) (r, s) = (3, 12) Figure 3. Phase Portraits of Case 3. Proof. In this case system (1.3) can be written in to the form (5.16). We have that y = 0 is an invariant straight line of system (5.16). This straight line determines two innite singular points ±q1 corresponding to the origin of the charts U1 and V1 in the Poincaré compactication, respectively. In this case, we do not have nite singular points and by Theorem 5.9, the singular points ±q3 , corresponding to the origin of the charts U2 and V2 , always are nodes. Moreover, when e 6= 0 we have six innite singular points, i.e., we have additionally two innite singular points ±q2 . If e > 0, then ±q1 are topological saddles and ±q2 are nodes. For e < 0, ±q1 are nodes and ±q2 are topological saddles. Hence the phase portrait is topologically equivalent to one of Figure 4 (a)-(b). Now when e = 0 by Theorem 5.9, we have four innite singular pints. Moreover, ±q1 are saddle-nodes if f 6= 0, or singular points with two hyperbolic sectors and two parabolic sectors if f = 0. Therefore the phase portrait s topologically equivalent to one of Figure 4 (c)-(d).  Theorem 6.6. Consider system (5.23). Then the phase portraits are topological equivalent to one of Figures 4 (a), (b) and (d). Proof. The proof of this theorem is analogous to Theorem 6.5. However in this case we do not have an innite singular point with two parabolic and two hyperbolic sectors, and so we have only three phase portraits given in Figures 4 (a), (b) and (d).  Proof of Theorem 1: The proof of Theorem 1 it follows from Theorems 6.3, 6.4, 6.5 and 6.6. By Theorem 4.2, phase portraits with distinct numbers (r, s) are not topologically equivalent. Note that (r, s) are distinct in all Figures 14, except in Figures 1 (a) and (e); Figures 1 (b) and (d); Figures 1 (f), (k) and Figure 2 (b); Figures 1 (g) and (i); Figure 1 (j) and Figure 2 (d); Figure 2 (a) and Figure 3 (a). The phase portraits of Figures 1 (a) and (e) are topologically distinct, because in (a) we have a saddle connection between the nite saddles and in (e) do not. The (a) (r, s) = (2, 13) (b) (r, s) = (3, 14) (c) (r, s) = (3, 10) (d) (r, s) = (2, 9) Figure 4. Phase Portraits of Case 4. phase portraits in Figures 1 (b) and (d) are topologically equivalent by Theorem 4.2. The phase portrait of Figure 1 (f) is topologically distinct of Figures 1 (k) and Figure 2 (b), because Figure 1 (f) we have only four innite singular points. Now, doing a rotation by a angle of π/2 radians, after a reection through the y -axis and reversing the orientation of the orbits, is easy to see that the phase portraits of Figures 1 (k) and Figure 2 (b) are topologically equivalent. The phase portraits of Figure 1 (g) and (i) are topologically distinct, because in Figure 1 (i) we have a connection between a nite and innite saddle, and in Figure 1 (g) do not. The phase portraits of Figure 1 (j) and Figure 2 (d) are topologically distinct, because in Figure 1 (j) we have three nite singular points and in Figure 2 (d) we have one nite singular point. The phase portraits of Figure 2 (a) and Figure 3 (a) are topologically distinct, because in Figure 2 (a) the nite singular points are two saddle-nodes and in Figure 3 (a) the nite singular points are a node and a saddle.  Acknowledgments The rst author is supported by the Ministerio de Economìa, Industria y com- petitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d'Ajusts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The sec- ond and third autors are supported by FAPESP grant 2013/34541-0. 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Jaume Llibre Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address : jllibre@mat.uab.cat Weber F. Pereira and Claudio Pesso Departamento de Matemática, Universidade Estadual Paulista, Campus São José do Rio Preto, IBILCE, R. Cristóvão Colombo, 2265, 15.054-000, São José do Rio Preto, SP, Brazil Email address : weber.pereira@unesp.br, c.pessoa@unesp.br