Phase portraits of Abel quadratic differential systems of second kind with symmetries

ABSTRACT We provide normal forms and the global phase portraits on the Poincaré disk of the Abel quadratic differential equations of the second kind having a symmetry with respect to an axis or to the origin. Moreover, we also provide the bifurcation diagrams for these global phase portraits.


Introduction and statements of the main results
Abel differential equations of the second kind are named in honour of Niels Henrik Abel because by a direct substitution they are related to the Abel differential equations of the first kind which were obtained by him in his studies of the theory of elliptic functions (see [11]) and after that, he made a crucial research on them. Abel differential equations of the second kind have various applications as they appear to reduce the order of many higher order nonlinear problems. They are also frequently found in the modelling of real problems such as big picture modelling in oceanic circulation (see [1] and the references therein).
An Abel differential equation of 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 where a(x), b(x), c(x) and d(x) are polynomials such that A(x) = a(x)/d(x), B(x) = b(x)/d(x) and C(x) = c(x)/d (x). In this paper, we are interested in studying the Abel CONTACT Antoni Ferragut ferragut@uji.es quadratic polynomial differential systems, i.e. the differential systems (2) of degree two: All the parameters in (3) are real. We assume thatẋ andẏ do not have a common factor; in particular, we assume that c 2 0 + c 2 1 + c 2 2 = 0. Moreover, we take a 0 = 0, otherwise this is not the Abel equation of second kind. We also assume that b 2 1 + c 2 1 + c 2 2 + d 2 1 = 0, otherwise the system does not depend on x and hence it is not of our interest.
In this work, we study the global phase portraits of the Abel differential systems of the second kind of degree two given by system (3) with d 1 = 0. The case d 1 = 0 is completely studied in [7]. Since this is a huge challenge, we restrict our study to the differential systems (3) having a symmetry with respect to an axis or with respect to the origin, see (4) below. We shall provide all the possible global phase portraits for these families. For that purpose, we shall use the well-known Poincaré compactification of polynomial vector fields, see Section 2.3. Before stating our main theorem, we split the family (3) into five different families. where S ∈ {−1, 1}, T ∈ {−1, 0, 1}, a, B 0 , B 1 , r, r 1 , r 2 ∈ R, A 0 ∈ R \ {0}, r 1 = r 2 and b > 0.

Remark 1.1:
The families appearing in Proposition 1.2 do not have limit cycles. Here, we provide an explanation of this fact. Quadratic systems having a straight line can have at most one limit cycle; moreover, this limit cycle must surround a focus. See [13]. In our families, x = 0 is always invariant, hence the system has at most one limit cycle. Moreover, this limit cycle must surround a focus not on x = 0, and hence on y = 0 (becauseẋ = xy). Cases with the symmetries S a or S b cannot have limit cycles, the former because the symmetry would make appear two limit cycles, the latter because no foci can exist on y = 0. So, it remains to explore the cases with the symmetry S c .
The only situation where a focus (actually two because of the symmetry) appears is in family (K1c), concretely in phase portraits (25)-(31). In all these phase portraits, each focus is the αor ω-limit of a separatrix of a saddle, hence no limit cycles apply here.
The paper is organized as follows. Section 2 provides some classical results about phase portraits. Section 3 is devoted to the proof of Propositions 1.1 and 1.2. Finally, we prove Theorem 1.4 in Section 4.

Preliminary results
In this section, 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 Subsections 2.1 and 2.3 can be found in [5]. The results of Subsection 2.2 can be found in [6].

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 matrixDX(p).
The singular point p is non-degenerated if = 0. It is degenerated otherwise. Then, p is an isolated singular point.
, and either a weak focus or a centre if T = 0 < ; for more details see [3].  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 [3]. 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 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 [3]).
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 [2,4]. 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 [13].
The definitions of hyperbolic, parabolic and elliptic sectors near a singular point can be found in [3]. 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 → +∞ (t → −∞) and leave the sector as t → −∞ (t → +∞) 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 [3].
Next, we state a result of [9] 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 systeṁ Let Then the following statements hold: (2) If there exists k ∈ {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 [12]: (i) For all p ∈ 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 ∈ U and for all s and t such that s, t + s ∈ 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 k -diffeomorphism 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. γ (p) = {ϕ p (t) : t ∈ I p }. The positive semiorbit of p isγ + (p) = {ϕ p (t) : t ∈ I p , t ≥ 0}. 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: (i) for any q ∈ N, α(q) = α(p) and ω(q) = ω(p); 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 [10].

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 byD 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 on U 2 ; and 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.

Proofs of Propositions 1.1 and 1.2
We prove in this section Propositions 1.1 and 1.2. We recall that these results allow us to split the differential system (3) We consider first the case c 2 = 0, for which system (8) has at most four singular points. Renaming we geṫ (9) We focus on the singular points on y = 0. Let := c 2 1 − 4c 0 c 2 be the discriminant ofẏ| y=0 . If > 0 then there are two different real singular points on y = 0, say (r 1 , 0) and (r 2 , 0); after renaming Renaming and T ∈ {−1, 0, 1}, the above system becomes the family (K4). If c 1 = 0 then, applying the change of variables and time (x, y) Renaming We need a change of time t → −t and B 0 = B 1 = 0 to obtain again (K1). Moreover, after additional affine changes we know that we can take r 1 = 1 and r 2 = r = 1. We get (K1b).
The other families are obtained in an analogous way. The proposition follows.

Proof of Theorem 1.4
We recall that Theorem 1.3 follows after Theorem 1.4. Following Proposition 1.2, we prove Theorem 1.4 separately for the thirteen different families.
The finite singular points of system (10) is a stable node. If C 1 = 0 then the two points become the singular point (0, 2S/B 0 ) which is semihyperbolic. We move the point to the origin and we apply the change of variables and time (x, y) → (y, x), dt/dτ = B 0 /2S, to bring the system to the form given in Proposition 2.1. So, F(x, f (x)) = −(B 3 0 /8S 2 )x 2 + · · · and the singular point is saddle node. According to expression (7), system (10) at infinity can be studied from the systeṁ on U 2 . We do not need to study the expression (6) because all the infinite singular points can be studied from the above differential system. We have three infinite singular points:  Table 1 summarizes the different behaviors of the singular points depending on the values of the parameters.
According to the above mentioned when S = 1, we have the following behaviours: there are three cases for A 0 < 1; if C 1 < 0 then we obtain the phase portrait (10), if C 1 = 0 we have the phase portrait (9) and if C 1 > 0 the phase portrait is (8). For 0 < A 0 < 1 the phase portrait is (11) and for A 0 ≥ 1 we have (12).
When S = −1, the behaviours are as follows: there are three cases for A 0 > 1; if C 1 < 0 then the phase portrait is (7), if C 1 = 0 we have the phase portrait (6) and if C 1 > 0 the phase portrait is (5). For 0 < A 0 ≤ 1 there are also three cases: if C 1 < 0 the phase portrait is (4), if C 1 = 0 then we have the phase portrait (3) and if C 1 > 0 the phase portrait is (2). If A 0 < 1 we have the phase portrait (1). Finally, ifA 0 = 1 and B 0 ≤ 2 we have again the phase portrait (3). The phase portraits are given in Figures 1 and 2.

Proof of Theorem 1.4(K1b):
We deal here with the differential systeṁ where S ∈ {−1, 1}, A 0 = 0 and r = 1. System (11) has the following finite singular points: (1, 0), (r, 0), and (0, The eigenvalues of the point (1, 0) are ± √ S(1 − r). Thus, for either S = 1 and r < 1 or S = −1 and r > 1 the point is a saddle. When either S = 1 and r > 1 or S = −1 and r < 1 we apply Theorem 2.3. We first write system (11) in the form given by the Theorem, for that we move the point to the origin, after that we apply the change of variables and time (x, y, t) → (x, −y/ √ S(r − 1), t/ √ S(r − 1)) and system (11) becomeṡ Then we have ω 1 = ω 2 = ω 3 = 0, so we have a centre. For the point (r, 0) the eigenvalues are ± √ −S r(1 − r). So, when either S = 1 and r ∈ (0, 1) or S = −1 and r ∈ (0, 1) the point is a saddle. When either S = 1 and r ∈ (0, 1) or S = −1 and r ∈ (0, 1) we apply again Theorem 2.3. By moving the point to the origin and applying the change of variables and time (x, y, t) → (x, y √ r/S(1 − r), −t/ √ Sr(1 − r)) system (11) becomesẋ It is easy to see that ω 1 = ω 2 = ω 3 = 0, so the point is a centre. Finally, if r = 0 the point becomes (0, 0) and is semi-hyperbolic. Indeed, system (11) for r = 0 is written in the form given in Proposition 2.1, so y = g(x) = (A 0 /S)x 2 + · · · , thus we obtain f (x) = (−A 0 /S 2 )x 3 + · · · , so if A 0 > 0 the point is a saddle and if A 0 < 0 is a node. The stability of the node is given by S: if S = 1 it is a stable node and if S = −1 it is an unstable node.
The According to the expression (7), system (11) at infinity can be studied from the systeṁ We do not need to study the expression (6) because all the infinite singular points can be studied from the above differential system. We have three infinite singular points: (0, 0) and two more points at (± The origin on U 2 has eigenvalues −A 0 , 1 − A 0 thus we have an unstable node for A 0 < 0; a saddle point for 0 < A 0 < 1 and a stable node for A 0 > 1. When A 0 = 1 the point is semi-hyperbolic. With a change of time, the above system for A 0 = 1 is written in the form given in Proposition 2.1 and we obtain f (u) = Su 3 + · · · , so if S = 1 the point is a stable node and if S = −1 a saddle point.
The last two points exist only if S(1 − A 0 ) > 0. Concerning their stability, the eigenvalues are −1, −2(1 − A 0 ) so if S = 1 and A 0 < 1 the points are stable nodes and if S = −1 and A 0 > 1 are saddles. Table 2 summarizes the different possibilities of behaviour of the singular points of system 11 depending on the values of the parameters.
The rest of the cases follow in a similar way as the previous ones. We shall provide the table with the behaviour of the singular points and additional important information if needed.

Proof of Theorem 1.4(K1c):
We deal now with the differential systeṁ where S 2 = 1, A 0 = 0 and B 1 ≥ 0. Table 3 shows the behaviour of all singular points depending on the parameters. In the case S = −1, B 1 = 0, A 0 > 0 we have only two finite singular points and they are centres, this can be proved by applying Theorem 2.3. The phase portrait is determined by the value of A 0 . If 0 < A 0 ≤ 1 the phase portrait is (13) and if A 0 > 1 the phase portrait is (18) of Figures 1 and 2.
The behaviour of the all critical points at infinity of system (12) can be studied on U 2 through the systeṁ

Proof of Theorem 1.4(K2a):
We deal now with the differential systeṁ where S 2 = 1, A 0 = 0. Table 4 shows the behaviour of all singular points depending on the parameters. According to the expression (7), system (13) at infinity can be studied from the systeṁ We do not need to study the expression (6) because all the infinite singular points can be studied from the above differential system.
The infinite singular points of system (14) with r = 1 can be studied from the systeṁ on U 2 , and we do not need to study the expression (6) because all the infinite singular points can be studied from the above differential system. For the case r = 0 system, (14) has only one finite singular point: (0, 0). This point is degenerate so we determined its behaviour by applying the blow-up technique (for more details see [2]). Starting from system (14) with r = 0 we apply the change (x, y) → (x, py), where p is a new variable, to obtain (after cancelling a common factor y) p = −p(Sp 2 + A 0 − 1),ẏ = y(Sp 2 + A 0 ).   According to the expression (7), the infinite singular points of system (14) for the case r = 0 can be studied from the systeṁ on U 2 . Since in the above system we can study all the infinite singular points, we do not need to study the expression (6).

Proof of Theorem 1.4(K2c):
We deal now with the differential systeṁ where S 2 = 1, A 0 = 0 and B 1 > 0. Table 6 shows the behaviour of all singular points depending on the parameters. System (15) has only one finite singular point: (0, 0). This point is degenerate so, again asin case (K2b), we use the blow-up technique to known its behaviour.
System (15) at infinity can be studied from the systeṁ This system has three singular points: (0, 0) and two more points at ((−B 1 ± √ C 2 )/2S, 0) where C 2 := B 2 1 − 4A 0 S + 4S. These points only exist if C 2 ≥ 0. When C 2 = 0 the point thus the point is a saddle node. When A 0 = 1, the point (0, 0) on U 2 is a semi-hyperbolic singular point. By using Proposition 2.1, we prove that the point is a saddle node.

Proof of Theorem 1.4(K3a):
We deal now with the differential systeṁ where S 2 = 1, A 0 = 0 and B 0 > 0. Table 7 shows the behaviour of all singular points depending on the parameters.

Proof of Theorem 1.4(K3b):
We deal now with the differential systeṁ where S 2 = 1, A 0 = 0 and a ∈ R. Table 8 shows the behaviour of all singular points depending on the parameters.
According to the expression (7), system (16) at infinity can be studied from the systeṁ The behaviour of the infinite singular points is the same as in the case (K3a).

Proof of Theorem 1.4(K3c):
We deal now with the differential systeṁ where S 2 = 1, A 0 = 0 and B 1 > 0. Table 9 shows the behaviour of all singular points depending on the parameters. The behaviour of system (18) at infinity can be studied from the systeṁ The above system has at most three singular points: (0, 0) and point (0, 0) is semi-hyperbolic and applying Proposition 2.1 it can be proved that it is a saddle node. For the case C 2 = 0, again by using Proposition 2.1, we prove that the point (−B 1 /2S, 0) is a saddle node. The phase portrait in this case corresponds to (53) or (56) of Figures 1 and 2 depending on the value ofA 0 .

Proof of Theorem 1.4(K4a):
We deal now with the differential systeṁ where A 0 = 0 and T ∈ {−1, 0, 1}. Table 10    Following the expression (6) and (7), the study of the behaviour of the singular points at infinity can be done through the systemṡ on U 1 and U 2 , respectively. Note that when A 0 = 1 the infinity is degenerate. System (20) has a singular point: (0, 0), which is semi-hyperbolic. Applying Proposition 2.1 we have v = g(u) = (1 − A 0 ) + · · · and then F(u, g(u)) = (A 0 − 1)u 3 + · · · . So if A 0 > 1 the point is an unstable node and if A 0 < 1 is a saddle.

Proof of Theorem 1.4(K5a):
We deal now with the differential systeṁ where S 2 = 1, A 0 = 0 and B 0 > 0. The behaviour of all singular points depending on the parameters is shown in Table 11.
on U 1 and U 2 , respectively. Note that if A = 1 the infinity is degenerate.    On the other hand, for A 0 = 1 system (23) has a singular point: (0, 0), such a point is degenerate. To obtain its local behaviour, we apply the blow-up technique.

Proof of Theorem 1.4(K5b):
We deal now with the differential systeṁ where S 2 = 1, A 0 = 0. In Table 12 we give the behaviour of the singular points depending on the parameters. According to (6) and (7), system (25) at infinity can be studied from the systemṡ The behaviour at infinity is the same as in case (K5a).

Invariant straight lines
We compute in this last section the invariant straight lines of the differential systems appearing in Proposition 1.2. When real, they appear in the phase portraits of Figures 1  and 2. We recall that an algebraic curve f = 0, f ∈ C[x, y], is invariant under the flow of a differential systemẋ = P,ẏ = Q of degree m ∈ N if there exists k ∈ C[x, y], deg k < m, such that We call this polynomial k the cofactor of f = 0. Table 14 shows the invariant straight lines, that is the invariant algebraic curves f = 0 of degree 1, of each family of differential systems appearing in Proposition 1.2, besides x = 0.
Of course x = 0 is invariant because x|ẋ, and hence equation (31) is satisfied with k = y. A family not appearing in this table is a family without invariant straight lines besides x = 0.

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