Phase portraits of Abel quadratic differential systems of the second kind

ABSTRACT We provide normal forms and the global phase portraits on the Poincaré disk of some Abel quadratic differential equations of the second kind. Moreover, we also provide the bifurcation diagrams for these global phase portraits.


Introduction and statements of the main results
Neils Henrick Abel, one of the most active mathematicians in his time, devoted himself to several topics characteristic of the mathematics of his time (among others, for example, functional equations, algebraic equations, hyper-elliptic integrals). In particular, he studied the integral equations. He defined the Abel integral and worked on methods to solve special integral equations, which were later known as Abel integral equations [12]. His research on integral equations and their solutions led him work on differential equations, where he verified the importance of the Wronskian determinant for a differential equation of order two [9,12]. His study of the theory of elliptic functions was what got him involved into the analysis of special differential equations, which are a generalized Riccati differential equation. Due to his seminal work on these differential equations, they are named Abel differential equations. The Abel differential equations of the first and second kinds are both non-homogeneous differential equations of first order and are related between them by a substitution. Despite the fact that Abel equations of the first kind are very well studied, this is not the case of Abel differential equations of the second kind. In this paper, we will focus on them.
An Abel differential equation of the second kind has the form with A(x), B(x), C(x) ∈ R(x, y). This differential equation can be written equivalently as the polynomial differential systeṁ , with polynomials a(x), b(x), c(x) and d (x). In this paper, we are interested in studying the Abel quadratic polynomial differential systems, i.e. the differential systems of the forṁ coming from the Abel differential equation of the second kind (1). Here, all the parameters are real. We assume thatẋ andẏ do not have a common factor, that is c 2 0 + c 2 1 + c 2 2 = 0. Moreover, a 0 ࣔ 0, otherwise this is not the Abel equation of the second kind, and b 2 1 + c 2 1 + c 2 2 + d 2 1 = 0 to have a quadratic system, otherwise the system would not depend on x and hence would not be of our interest.
In this work, we provide the global phase portraits of all the Abel differential systems of the second kind of degree two given by the family (2) with d 1 = 0. We assume then that d 0 ࣔ 0.
We shall use the well-known Poincaré compactification of polynomial vector fields. For more details, if needed, see chapter V in [6]. We recall that two polynomial vector fields X and Y on R 2 are topologically equivalent if there exists a homeomorphism on S 2 preserving the infinity S 1 and carrying orbits of the flow induced by p(X) into orbits of the flow induced by p(Y). Before stating our theorem, we provide normal forms for the family (2). where S 2 = 1, B 0 , B 1 , r, a ∈ R and b > 0. Additionally, B 1 ࣙ 0 in all cases; rS > 0 in case (K1); and B 1 ࣔ 0 in case (K5).
Observe that once in normal form, (2) is a Kukles quadratic differential system. We note that there are no intersections among the normal forms because the number and the multiplicity of the finite singular points is different from one normal form to the other. For (K3) and (K5), some phase portrait may coincide because the number of real finite singular points is the same.
The main result in this paper is the following one. The global phase portraits on the Poincaré disk of all the families of Proposition 1.1 are provided.
We also remark that the curve C 1 = 0 controlling the bifurcation of the global phase portraits described in Theorem 1.3 for family (K4) is not algebraic. The rest of the curves controlling the bifurcation of the global phase portraits in Theorem 1.3 are algebraic.
In Section 2, we state some basic results that we shall need later on. Proposition 1.1 is proved in section 3. The proof of Theorem 1.2 follows directly from the proof of Theorem 1.3, which is given in Section 4.

Preliminary results
First, we introduce the basic definitions, notations and results that we need for the analysis of the local phase portraits of the finite and infinite singular points of the quadratic differential systems and afterwards we define the Poincaré compactification. The results of Sections 2.1 and 2.3 can be found in [4]. The results of Section 2.2 can be found in [5]. The results of Section 2.4 can be found in [7].

Singular points
Consider an analytic planar differential systemẋ = P(x, y),ẏ = Q(x, y) and its associated vector field X = (P, Q).
They correspond, respectively, to the determinant and the trace of the Jacobian matrix DX(p).
The singular point , and either a weak focus or a centre if T = 0 < ; for more details see [2].
The singular point p is called hyperbolic if the two eigenvalues of the Jacobian matrix DX(p) have nonzero real part. So the hyperbolic singular points are the non-degenerate ones except the weak foci and the centres.
A degenerate singular point p such that T ࣔ 0 is called semi-hyperbolic, and p is isolated in the set of all singular points. Next we summarize the results on semi-hyperbolic singular points; see Theorem 65 of [2]. Proposition 2.1: Let (0, 0) be an isolated singular point of the vector field (F(x, y), y + G(x, y)), where F and G are analytic functions in a neighbourhood of the origin starting at least with quadratic terms in x and y. Let y = g(x) be the solution of the equation y + G(x, y) = 0 in a neighbourhood of (0, 0). Assume that f(x) = F(x, g(x)) = μx m + , where m ࣙ 2 and μ ࣔ 0. If m is odd, then the origin is either an unstable node or a saddle depending on whether μ > 0 or μ < 0, respectively. If m is even, then (0, 0) is a saddle-node, i.e. the singular point is formed by the union of two hyperbolic sectors with one parabolic sector.
The singular points which are non-degenerate or semi-hyperbolic are called elementary. When = T = 0 but the Jacobian matrix at p is not the zero matrix and p is isolated in the set of all singular points, we say that p is nilpotent. Next, we summarize some results on nilpotent singular points (see Theorems 66 and 67 and the simplified scheme of Section 22.3 of [2]).

Proposition 2.2: Let (0, 0) be an isolated singular point of the vector field (y + F(x, y), G(x, y)), where F and G are analytic functions in a neighbourhood of the origin starting at least with quadratic terms in x and y. Let y = f(x) be the solution of the equation y + F(x, y)
The following statements hold: (i) If κ is even and (a) κ > 2λ + 1 , then the origin is a saddle-node. (b) κ < 2λ + 1 or ࣕ 0, then the origin is a cusp, i.e. a singular point formed by the union of two hyperbolic sectors. (ii) If κ is odd and K > 0, then the origin is a saddle. (iii) If κ is odd, K < 0 and (a) λ even, κ = 2λ + 1 and L 2 + 4K(λ + 1) ࣙ 0, or λ even and κ > 2λ + 1, then the origin is a stable (unstable) node if L < 0 (L > 0). (b) λ odd, κ = 2λ + 1 and L 2 + 4K(λ + 1) ࣙ 0, or λ odd and κ > 2λ + 1, then the origin is an elliptic saddle, i.e. a singular point formed by the union of one hyperbolic sector and one elliptic sector. (c) κ = 2λ + 1 and L 2 + 4K(λ + 1) < 0, or κ < 2λ + 1, then the origin is a focus or a centre, and if (x) ࣕ 0, then the origin is a centre.
Finally, if the Jacobian matrix at the singular point p is identically zero and p is isolated inside the set of all singular points, then we say that p is linearly zero. The study of its local phase portrait needs a special treatment: the directional blow-ups (see for more details [1,3]). But if a quadratic vector field has a finite linearly zero singular point, then it is equivalent to a homogeneous quadratic differential system doing if necessary a translation of the linearly zero singular point to the origin, and the global phase portraits of the quadratic homogeneous vector fields are well known (see [14]).
The definitions of hyperbolic, parabolic and elliptic sectors near a singular point can be found in [2]. Roughly speaking, in a hyperbolic sector, there are curves through points of the sector which leave the sector with both increasing and decreasing time. A sector such that all curves in a sufficiently small neighbourhood of the singular point tend to it as t → +Ý (t → −Ý) and leave the sector as t → −Ý (t → +Ý) is known as a parabolic sector. Finally, a sector containing loops, and moreover only nested loops, is known as an elliptic sector.
The number of elliptic sectors and the number of hyperbolic sectors in a neighbourhood of a singular point are denoted by e and h, respectively. The rest of the sectors are parabolic. The index of a singular point p is defined as i(p) = (e − h)/2 + 1 ∈ Z. For a proof of this formula, see [2].
Next, we state a result of [10] that allows to distinguish between a centre and a focus for a singular point of a quadratic differential system having pure imaginary eigenvalues. Theorem 2.3: Consider the quadratic differential system: Let Then the following statements hold: (1) If ω 1 = ω 2 = ω 3 = 0, then the origin is a centre.
(2) If there exists k ࢠ {1, 2, 3} such that ω j = 0 for all 1 ࣘ j < k and ω k ࣔ 0, then the origin is a focus of order k. The stability of this focus is given by the sign of ω k ࣔ 0: when ω k < 0, the focus is stable, and when ω k > 0, the focus is unstable.

Separatrices and canonical regions
Consider the planar differential systemẋ = P(x, y),ẏ = Q(x, y), where P, Q ∈ C r , r ࣙ 1. For this differential system, the following three properties are well known (see for more details [13]): (i) For all p ࢠ U, there exists an open interval I p ⊆ R where the unique maximal solution ϕ p : I p → U of the system such that ϕ p (0) = p is defined.
The map ϕ: D → U is a local flow of class C r on U associated to the system. It verifies: (2) ϕ(t, ϕ(s, p)) = ϕ(t + s, p) for all p ࢠ U and for all s and t such that s, t + s ࢠ I p ; ( We consider C r -local flows on R 2 , with r ࣙ 0. Of course, when r = 0, the flow is only continuous. Two such flows ϕ and ϕ are C k -equivalent, with k ࣙ 0, if there exists a C kdiffeomorphism that brings orbits of ϕ onto orbits of ϕ preserving sense (but not necessarily the parametrization).
Let ϕ be a C r -local flow with r ࣙ 0 on R 2 . We say that ϕ is C k -parallel if it is C k -equivalent to one of the following flows: We call these flows strip, annular and spiral, respectively. Let p ∈ R 2 . We denote by γ (p) the orbit of the flow ϕ through p, i.e.
In a similar way, we define the negative semiorbit γ − (p) of p.
We define the α-limit and the ω-limit of p as (γ ± (p)) and let respectively. Here, as usual, cl denotes the closure.
Let γ (p) be an orbit of the flow ϕ. A parallel neighbourhood of the orbit γ (p) is an open neighbourhood N of γ (p) such that ϕ is C k -equivalent in N to a parallel flow for some k ࣙ 0.
We say that γ (p) is a separatrix of ϕ if it is not contained in a parallel neighbourhood N satisfying the following two assumptions: We denote by the union of all separatrices of ϕ. is a closed invariant subset of R 2 . A component of the complement of in R 2 , with the restricted flow, is a canonical region of ϕ.
The following lemma can be found in [11].

Lemma 2.4:
Every canonical region of a local flow ϕ on R 2 is C 0 -parallel.

The Poincaré compactification
Let X be a real planar polynomial vector field of degree n. The Poincaré compactified vector field p(X) corresponding to X is an analytic vector field induced on S 2 as follows (see for instance [8]). Let S 2 = {y = (y 1 , y 2 , y 3 ) ∈ R 3 : y 2 1 + y 2 2 + y 2 3 = 1} (the Poincaré sphere) and T y S 2 be the tangent plane to S 2 at a point y. Identify R 2 with T (0,0,1) S 2 . Consider the central projection f : T (0,0,1) S 2 → S 2 . This map defines two copies of X on S 2 , one in the northern hemisphere and the other in the southern hemisphere. Denote byX the vector field Df•X defined on S 2 except on its equator S 1 = {y ∈ S 2 : y 3 = 0}. Clearly, S 1 is identified to the infinity of R 2 . Usually, when we talk about the circle of the infinity of X, we simply talk about the infinity.
In order to extendX to a vector field on S 2 (including S 1 ), it is necessary for X to satisfy suitable conditions. If X is a real polynomial vector field of degree n, then p(X) is the only analytic extension of y n−1 3X to S 2 . On S 2 \S 1 , there are two symmetric copies of X, and knowing the behaviour of p(X) around S 1 , we know the behaviour of X in a neighbourhood of the infinity. The Poincaré compactification has the property that S 1 is invariant under the flow of p(X). The projection of the closed northern hemisphere of S 2 on y 3 = 0 under (y 1 , y 2 , y 3 ) →(y 1 , y 2 ) is called the Poincaré disc, and it is denoted by D 2 .
Two polynomial vector fields X and Y on R 2 are topologically equivalent if there exists a homeomorphism on S 2 preserving the infinity S 1 and carrying orbits of the flow induced by p(X) into orbits of the flow induced by p(Y).
As S 2 is a differentiable manifold, for computing the expression of p(X), we can consider the six local charts U i = {y ∈ S 2 : y i > 0}, and V i = {y ∈ S 2 : y i < 0}, i = 1, 2, 3. The diffeomorphisms F i : U i → R 2 and G i : V i → R 2 for i = 1, 2, 3 are the inverses of the central projections from the planes tangent at the points (1, 0, 0), (−1, 0, 0), (0, 1, 0), (0, −1, 0), (0, 0, 1) and (0, 0, −1), respectively. If z = (u, v) is the value of F i (y) or G i (y) for any i = 1, 2, 3 (so z represents different things according to the local charts under consideration), then we obtain the following expressions for p(X): on U 3 , where (z) = (u 2 + v 2 + 1) − 1 2 (n−1) . The expression for V i is the same as that for U i except for a multiplicative factor (−1) n − 1 . In these coordinates, for i = 1, 2, v = 0 always denotes the points of S 1 . We can omit the factor (z) by scaling the vector field p(X). Thus, the expression of p(X) becomes a polynomial vector field in each local chart.

A result on heteroclinic connections in planar systems
The following theorem (see [7]) will be used to obtain a curve in the bifurcation diagram of family (K4) when S = −1.

Proof of Proposition 1.1
We consider first the case c 2 ࣔ 0, for which (2) with d 1 = 0 has two finite singular points, which are (−c 1 ± √ )/(2c 2 ), 0), where = c 2 1 − 4c 0 c 2 . We split this case into three subcases depending on the sign of . If > 0, that is we have two real different singular points, then the change of variables and time where r ࣔ 0 is such that c 2 d 0 r − a 0 = 0 and S ࢠ { − 1, 1} is the sign of a 0 c 2 ࣔ 0, leads to the first normal form (K1) after renaming If = 0, that is we have a double real singular point, then the change of variables and time 1} is again the sign of a 0 c 2 ࣔ 0, leads to the second normal form (K2) after renaming If < 0, that is we have two complex conjugate singular points, then the change of variables and time where a ∈ R and b > 0 is such that a 2 0 (c 2 1 − 4c 0 c 2 ) + 4b 2 c 2 2 d 2 0 = 0 and S ࢠ { − 1, 1} is the sign of a 0 c 2 ࣔ 0, leads to the third normal form (K3) after renaming Now we consider the case c 2 = 0. If c 1 ࣔ 0, then we have one single finite singular point (−c 0 /c 1 , 0). If c 1 = 0, then we have no finite singular points. We distinguish between these two cases. If c 1 ࣔ 0, then the change of variables and time We finally consider the case c 1 = 0. We must have c 0 ࣔ 0, otherwise y|(ẋ,ẏ), and b 1 ࣔ 0, otherwise the whole system does not depend on x and is not of our interest. The change of variables and time where S ࢠ { − 1, 1} is the sign of a 0 c 0 ࣔ 0, leads to the fifth normal form (K5) after renaming Finally, we deal with the different restrictions on the parameters shown in the proposition. If B 1 = 0 in (K5), then x is not appearing in the right-hand side of the system, so we remove this case.
The change of variables (x, y, t) →(x, −y, −t) together with B 0 → − B 0 and B 1 → − B 1 allows to assume B 1 ࣙ 0 in all cases. Additionally, in the normal form (K1) after the change of variables (x, y, t) →(x + r, y, t), B 0 →B 0 − B 1 r and r → − r, we can assume that rS > 0.

Proof of Theorem 1.3
After Proposition 1.1, we prove the theorem separately for the five different normal forms, starting from (K5).

Proof of Theorem 1.3(e):
We deal here with the differential systeṁ where S 2 = 1 and B 1 > 0. Systems in normal form (6) have no finite singular points. According to the expressions (4) and (5), systems in normal form (6) at infinity can be studied from the systemṡ respectively. The characteristic equation at infinity is xy(B 1 x + y) = 0, so three different singular points appear at infinity. The first system has a singular point at (0, 0), with eigenvalues −1, −1, thus it is a node. The other two singular points can be studied from

Proof of Theorem 1.3(d):
We deal here with the differential systeṁ where S 2 = 1 and B 1 ࣙ 0. Systems in normal form (7) have a finite singular point at the origin, with eigenvalues (B 0 ± B 2 0 + 4S)/2. Observe that the sign of B 0 determines the stability of nodes and foci, when it is not zero. To study the case B 0 = 0, S = −1, the only one for which we have a centre-focus, we use Theorem 2.3: we write (7) into the form (3) to obtaiṅ We have ω 1 = −B 1 . Since γ = 0, we have ω 2 = ω 3 = 0, and thus we have a centre if and only if B 1 = 0. Moreover, according to Theorem 2.3, and taking into account that we changed the sign of the time to obtain this system, (7) has an unstable focus when B 1 > 0. Table 1 summarizes the different possibilities of behaviour of this singular point depending on the values of the parameters.
According to expressions (4) and (5), (7) at infinity can be studied from the systemṡ As in (K5), the characteristic equation at infinity is xy(B 1 x + y) = 0, so at most three different singular points appear. The point in the direction x = 0 has eigenvalues −1, −1, so it is a node. In the case B 1 ࣔ 0, we have two more singular points. The point in the direction y = 0 has eigenvalues B 1 , 0. Applying Proposition 2.1 to (0, 0), we have This means an unstable node if S = 1 and a saddle if S = −1. Note that we must consider apart the case B 0 B 1 = S: in this case y = g(x) = 0 and f (x) = B 2 1 Sx 3 , so the same result is obtained. However, the system has an invariant straight line in this case, so we shall have to take this curve into account in the bifurcation diagram.
Still in the case B 1 ࣔ 0, we apply Proposition 2.1 to the third infinite singular point (−B 1 , 0). We have y = g(x) = −((B 0 + B 1 )B 1 − S)Sx 2 + , and hence F (x, g(x)) = f (x) = −B 2 1 x 2 + · · · . Thus, we have a saddle-node. We must consider apart the case C 0 := (B 0 + B 1 )B 1 − S = 0: in this case, we have y = g(x) = 0 and f (x) = −B 2 1 x 2 , so the same result is obtained. However, also in this case, the system has an invariant straight line, so we shall have to take this curve into account in the bifurcation diagram.
When B 1 = 0, the two last singular points are the same (degenerate) singular point. To obtain its local behaviour, we need to apply the blow-up technique (see [1] for more details). Starting from (8) with B 1 = 0 (which has our point at the origin), we apply the change v = u 2 t, where t is a new variable, to obtaiṅ This system has a saddle at the origin and a semi-hyperbolic singular point at (0, −S). Applying Proposition 2.1 to this point, we have y = g(x) = −(2B 2 0 + S)x 2 + · · · , and hence F(x, g(x)) = f(x) = Sx 3 + , so we have an unstable node if S = 1 and a saddle if S = −1. Note that we must consider apart the case 2B 2 0 = −S = 1: in this case, y = g(x) = ±3/ √ 2x 3 + · · · and f(x) = −x 3 + , so the same result is obtained. Although we do not obtain any invariant straight line in this case, we shall take these two points into account in the bifurcation diagram. Table 1 summarizes the different behaviours of the singular points at infinity. Next we draw the bifurcation diagrams. When S = 1, we have the following behaviours: if C 0 < 0, then we obtain the phase portrait (44). If C 0 = 0, then the system has an invariant algebraic curve, given by B 1 x + y = 0. This straight line forms the stable separatrices of the saddle at the origin (and the unstable separatrix of the saddle-node at infinity); we have the phase portrait (45). When C 0 > 0, we are in (46). We note that the curve B 0 B 1 = 1 does not play any role in this bifurcation diagram.
When S = −1 and B 1 ࣔ 0, we apply Theorem 2.5 with m = −B 0 and n = 1/B 1 . According to [7], we have the phase portrait (52) when B 0 ࣙ 0. Next, we have a phase portrait with a hyperbolic limit cycle that is born from a Hopf bifurcation at the origin when B 0 < 0 and C 1 := B 0 + B 1 /4 − B 3 1 /64 − 5B 5 1 /512 + · · · > 0. This is phase portrait (51). This limit cycle disappears when C 1 = 0 giving rise to a heteroclinic connection (see the phase portrait (50)). This heteroclinic connection is a separatrix of an infinite saddle when C 1 < 0 and C 2 := B 0 B 1 + 1 > 0 (see the phase portrait (49)). The two separatrices not through the origin meet when C 2 = 0 (see the phase portrait (48)), and, finally, they separate again when C 2 < 0 (see the phase portrait (47)). We note that the curve C 0 = 0 does not play any role in this bifurcation diagram.
Finally, we consider the case S = −1 and B 1 = 0. When B 0 = 0, we have a centre at the origin, so we have the phase portrait (54). If B 0 ࣔ 0, then we have either a node or a focus, and so the phase portrait is not changing topologically. We have the phase portrait (53). The case 2B 2 0 = −S = 1 is included here. The bifurcation diagrams in Figure 3 show all the different phase portraits depending on the values of B 0 and B 1 ࣙ 0. For S = −1, the bifurcation curves in the second quadrant, C 1 = 0 and C 2 = 0, were computed in [7], where the Hopf bifurcation appearing in this system is studied. Note that we also found C 2 .

Proof of Theorem 1.3(c):
We deal now with the differential systeṁ where S 2 = 1, B 1 ࣙ 0 and b > 0.
This system has no finite singular points. According to expression (5), the family of systems (9) can be studied at infinity from the systeṁ We have three infinite singular points: a node at the origin, with eigenvalues −1, −1, and two more points at u = (−B 1 ± B 2 1 − 4S)/2. The existence of the last two points depends on the value of the discriminant B 2 1 − 4S. Concerning their stability, at least one eigenvalue is always zero. When B 2 1 − 4S > 0, we can apply Proposition 2.1 to obtain two saddle-nodes. When B 1 = 2, S = 1 and B 0 ࣔ −(2a + 1), we apply Proposition 2.2 to get a saddle-node. In the case B 1 = 2, S = 1 and B 0 = −(2a + 1), we need to apply the blow-up technique because both eigenvalues are zero. In this case, we have a singular point at (−1, 0). We move the singular point to the origin and do the change v = ut, where t is a new variable. We obtaiṅ This system has, on u = 0, a unique real singular point, located at the origin. It is a saddle. The other two infinite singular points are complex.
We note that we do not need to study expression (4) because all the infinite singular points can be studied from the above differential system. Table 2 shows the behaviour of all singular points depending on the parameters. Only the phase portrait (55) appears in this case for S = −1. Figure 4 shows all the possible phase portraits when S = 1. We have the phase portraits (43) when B 1 < 2, (41) when B 1 = 2 and B 0 ࣔ −(2a + 1), (42) when B 1 = 2 and B 0 = −(2a + 1), and (55) when B 1 > 2. We note that the bifurcation diagram does not depend on b.

Proof of Theorem 1.3(b):
We deal now with the differential systeṁ where S 2 = 1 and B 1 ࣙ 0. This system has a finite singular point at the origin with eigenvalues 0, B 0 . If B 0 ࣔ 0, then according to Proposition 2.1, it is a saddle-node. If B 0 = 0, then according to Proposition 2.2, it is a cusp. Using expression (5), (10) can be studied at infinity from the The behaviour of the infinite singular points is the same as in the case (K3), so Table  2 with a = 0 shows their behaviours depending on the parameters. As in (K3), we do not need to study expression (4) because all the infinite singular points can be studied from the previous differential system.
When applying Proposition 2.1 to the infinite singular points in the case B 2 1 − 4S > 0, we must take into account the case C 1 := B 2 0 + B 0 B 1 + S = 0. This curve in the bifurcation diagram separates two behaviours (see Figure 5). Indeed, straightforward computations show that, in the case C 1 = 0, the system has an invariant straight line given by B 0 x − y = 0.
According to expression (5), (11) at infinity becomeṡ This system has at most three real singular points on v = 0. We note that we do not need to study expression (4) because all the infinite singular points can be studied from the previous differential system. The origin of this system is a stable node: it has eigenvalues −1, −1. The  others are located at u = (−B 1 ± B 2 1 − 4S)/(2S). We have three cases depending on the sign of the discriminant B 2 1 − 4S. If it is positive, then applying Proposition 2.1, we know that we have two saddle-nodes. As in previous cases, we need to take into account some manifolds in the bifurcation diagram: they are C 1 := (B 0 + B 1 )(B 0 + B 1 r) + (r − 1) 2 S = 0 and C 2 := B 0 (B 0 + B 1 + B 1 r) + (r + 1) 2 S = 0.
If the discriminant B 2 1 − 4S is zero, i.e. B 1 = 2 and S = 1, then we must distinguish two subcases: if B 0 ࣔ −(r + 1), then Proposition 2.2 shows that we have a saddle-node, and, if B 0 = −(r + 1), we need to use the blow-up technique. We obtain three singular points, all of them saddles.
Finally, when the discriminant is negative, the two singular points are complex. Table 3 shows the singular points and their behaviour depending on the parameters. Twentynew phase portraits appear for the family (11). The bifurcation diagrams are shown in Figures 6 and 7. The diagram in Figure 6 corresponds to the case S = 1. It is drawn on the half-plane (B 0 , B 1 ), because the value of r > 0 does not affect the diagram. The diagram in Figure 7(a) corresponds to S = −1 with B 1 > 0 and it is drawn on the halfplane (B 0 , r). The diagram in Figure 7(b) corresponds to S = −1 with B 1 = 0 and it is also drawn on the half-plane (B 0 , r).
We note that C 2 = 0 does not play any role in the bifurcation diagram when S = 1. The point with the phase portrait (8) in Figure 6 corresponds to B 1 = 2, B 0 = −(r + 1).

Disclosure statement
No potential conflict of interest was reported by the authors.