ON THE ABEL DIFFERENTIAL EQUATIONS OF THIRD KIND

. Abel equations of the ﬁrst and second kind have been widely studied, but one question that never has been addressed for the Abel polynomial diﬀerential systems is to understand the behavior of its solutions (without knowing explicitly them), or in other words, to obtain its qualitative behavior. This is a very hard task that grows exponentially as the number of parameters in the equation increases. In this paper, using Poincar´e compactiﬁcation we classify the topological phase portraits of a special kind of quadratic diﬀerential system, the Abel quadratic equations of third kind. We also describe the maximal number of polynomial solutions that Abel polynomial diﬀerential equations can have.

1. Introduction and statement of the main results. Generalized polynomial Abel differential equations of the form C(x)y mẏ = A(x) + B(x)y (1) (here the dot denotes derivative with respect to the independent variable x and m ≥ 0), appear in all textbooks of ordinary differential equations as examples of nonlinear equations and in many mathematical and applied problems, see [10,13,14] and the references therein. If m = 0 or m = 1 equations (1) become the well-known Abel equations of the first and second kind that have been widely studied. Here we will focus in the case in which m = 2, known as Abel equations of the third kind, and when the functions A, B, C are polynomials in x. More precisely we will work with the polynomial differential equations or equivalently to the polynomial differential systeṁ where A(x) = a(x)/c(x), B(x) = b(x)/c(x), C(x) = c(x), and the dot denotes derivative with respect to the independent variable t.
The first motivation of this paper comes from works about the existence of polynomial solutions for another kind of polynomial differential systems that appears in applied problems known as Riccati differential systems. Such systems can be written as a(x)ẏ = b 0 (x) + b 1 (x)y + b 2 (x)y 2 , where a, b 0 , b 1 and b 2 are polynomial in the variable x and many papers about them can be found. For instance, Rainville [14] proved the existence of one or two polynomial solutions for a subclass of such systems. Bhargava and Kaufman [2,3] obtained some sufficient conditions for such equations to have polynomial solutions. Campbell and Golomb [4,5] provided some criteria determining the degree of polynomial solutions of such equations. Bhargava and Kaufman [1] also considered a more general form of such equations and got some criteria on the degree of polynomial solutions of the equations. Giné, Grau and Llibre [9] proved that polynomial differential equations of the form a(x)ẏ = b 0 (x) + b 1 (x)y + .. + b n (x)y n with n ≥ 1, b i (x) ∈ R[x], i = 0, 1, ..., n and b n (x) = 0 and a(x) = 1 have at most n polynomial solutions and they also prove that this bound is sharp. More recently, Gasull, Torregrosa and Zhang in [7] proved that the maximum number of polynomial solutions of the equations studied in [9] when a(x) is nonconstant is n + 1 when n ≥ 1 and these bounds are sharp. In short, to be best of our knowledge, the question of knowing the maximum number of polynomial solutions of some polynomial differential equations like (1) when C(x) is nonconstant is open. This is also interesting because it is similar to a question of Poincaré about the degree and number of algebraic solutions of autonomous planar polynomial differential equations in terms of their degrees, when these systems have finitely many algebraic solutions.
The first theorem of the paper is the following. The proof of Theorem 1.1 is given in Section 2. Another question that never has been addressed for the Abel polynomial differential systems is to understand the behavior of its solutions (without knowing explicitly them), or in other words, to obtain its qualitative behavior. This is a very hard task that grows exponentially as the number of parameters in the equation increases. This is the reason why in our second main result among the Abel polynomial differential systems of third kind, we will focus on the quadratic ones. The Abel quadratic differential polynomial systems of third kind are of the forṁ where c = 0 and a 0 , a 1 , a 2 , b 0 , b 1 , b 2 are real parameters with a 2 0 + a 2 1 + a 2 2 = 0. We will obtain the phase portraits of the polynomial vector fields X here studied and will be given in the Poincaré disk D, see Chapter 5 of [6] for the definition of the Poincaré disk and the expressions of the compactified polynomial vector fields p(X ) in the local charts U 1 and U 2 of D that we shall use in the computations.
We say that two polynomial vector fields p(X ) and p(Y) in the Poincaré disk are topologically equivalent if there exists a homeomorphism from one onto the other sending orbits to orbits preserving or reversing the direction of the flow. The separatrix configuration of p(X ) is formed by all the separatrices of p(X ) plus an orbit in each one of its canonical regions. Recall that a canonical region is a connected component of the Poincaré disk after removing from it all the separtrices of the vector field. Moreover, if p(X ) and p(Y) are two compactified Poincaré polynomial vector fields with finitely many separatrices, then they are topologically equivalent if and only if their separatrix configurations are homeomorphic (see [12] for a proof). This last result will be used for obtaining the phase portraits in the Poincaré disk of our polynomial differential system (3).
Our second main result is the following one.
3. Normal forms given in Theorem 1.2. In this section we describe the normal forms given in Theorem 1.2, that is, systems (i)-(iv).
Proposition 1. An Abel quadratic polynomial differential system (3) after a linear change of variables and a rescaling of its independent variable, can be written as Proof. Doing a linear change of variables and a rescaling of the independent variable (the time) of the form where the dot denotes derivative with respect to the new time T . Since c = 0 we take α = cβ 2 γ. Now we consider different cases.
If a 2 = 0 we take σ = −a 1 /(2a 2 ) and γ = 1/(a If a 2 = 0 and a 1 = 0 we take σ = −a 0 /a 1 , β = 1/(a 1 cγ 2 ) and we get system (ii) with On the other hand, if b 1 = 0 then taking γ = β/a 0 we get system (iv) with In short, we obtain system (iv) with k 1 ∈ {0, 1}. This concludes the proof of the proposition. 4. Global behavior of system (i) of Theorem 1.2. In this section we describe the global phase portraits of system (i). We will separate the proof in different subsections. First we study the finite and infinite singular points and then we joint the obtained information to describe the distinct global phase portraits.
If k 0 > 0 there are no finite singular points. If k 0 = 0 the unique finite singular point is the origin and if k 0 < 0 there are two finite singular points which are = 0 they are semi-hyperbolic and using Theorem 2.19 in [6] both of them are saddle-nodes. If k 0 < 0, is semi-hyperbolic. Using Theorems 3.5 and 2.19 in [6] we conclude that ( is semi-hyperbolic. Using Theorems 3.5 and 2.19 in [6] we conclude that (− √ −k 0 , 0) is a cusp and ( √ −k 0 , 0) is a saddlenode. If k 0 < 0 and k 1 = k 2 = 0 then both (± √ −k 0 , 0) are nilpotent and by Theorem 3.5 in [6] we get that they are both cusps.
If k 0 = 0 and k 1 = 1 the origin is the unique finite singular point. It is semihyperbolic. Applying Theorem 2.19 in [6] we conclude that it is a saddle-node. If k 0 = k 1 = 0 the origin is again the unique finite singular point which in this case is linearly zero. Applying a blow up in the direction (x, y) → (x, w) where w = y/x and a rescaling ds = xdt, we get the following systeṁ This system either has one or three singular points (counted with multiplicities). These singular points are of the form (0, w) where w is a real root of 1+k 2 w−w 3 = 0. The discriminant of the cubic equation is 4k 3 2 − 27. If k 2 < 3/4 1/3 there is a unique real solution (positive) which is hyperbolic. If k 2 = 3/4 1/3 there are two real solutions, w = −1/2 1/3 (a saddle-node) and w = 2 2/3 (a saddle). If k 2 > 3/4 1/3 there are three real different solutions that are hyperbolic. To study the topological type of these solutions we will study the polynomial f (w) = 1 + k 2 w − w 3 . If k 2 < 0 then f is strictly decreasing and intersects the w-axes in a unique point (the real root of 1 + k 2 w − w 3 ). The eigenvalues of the Jacobian matrix at the point (0, w) are w 2 and k 2 − 3w 2 . Note that at this root w = 0 and k 2 − 3w 2 < 0, so this point is a saddle. If k 2 > 0 then f has a maximum at w = k 2 /3 and a minimum at w = − k 2 /3. If k 2 ∈ (0, 3/4 1/3 ) then f (− k 2 /3) and f ( k 2 /3) are both positive. Since lim w→∞ f (w) = −∞ and lim w→−∞ f (w) = ∞, we get that f intersects the w-axes in a unique point (the real root). Again by the eigenvalues of the Jacobian matrix at this point we get that it is a saddle. If k 2 > 3/4 1/3 then f (− k 2 /3) is negative and f ( k 2 /3) is positive. Since lim w→∞ f (w) = −∞ and lim w→−∞ f (w) = ∞ we get that f intersects the w-axes in three points w 0 , w 1 and w 2 (the real roots), where w 0 < − k 2 /3, w 1 ∈ − k 2 /3, k 2 /3) and w 2 > k 2 /3. Again by the eigenvalues of the Jacobian matrix at these points we get that w 0 and w 2 are saddles and w 1 is an unstable node.
Doing the blowing down we get that when k 2 < 3/4 1/3 the origin is formed by two hyperbolic sectors and when k 2 ≥ 3/4 1/3 it is formed by the union of two hyperbolic sectors separated by parabolic sectors. See Figure 4 for details about the blowing down.
). This will help when studying the global behavior of the solutions. Moreover,ẏ = 0 yields the nullcline curve y = −(x 2 + k 0 )/(k 1 + k 2 x). The nulllcline curve separates the phase portrait in two regions such that above it,ẏ is positive and below it,ẏ is negative.
To study the singular points at infinity of a polynomial vector field via the Poincaré compactification we need to study the singular points in the local chart U 1 and the origin of the local chart U 2 .
System (i) in the local chart U 2 is written aṡ So the origin of the local chart U 2 is not an infinite singular point.

4.1.
Global phase portraits for system (i) with k 0 > 0. If k 0 > 0 there are no finite singular points, that is, it is a chordal quadratic system (see [8]). If k 2 < 3/4 1/3 there is a unique stable node in the local chart U 1 and using Remark  1 we get that the unique possible global phase portrait is (1) of Figure 1. This phase portrait is topologically equivalent to 8 in [8], where it is described all the possible global phase portrait for chordal quadratic polynomial differential systems. As described in [8] when there is a unique stable node in the local chart U 1 there is a unique phase portrait. If k 2 = 3/4 1/3 then we have two singular points in the local chart U 1 , a saddle-node and a stable node. In this case the global phase portrait is (2) of Figure 1. This is due the fact thatẏ| y=0 = x 2 + k 0 > 0. This corresponds to 5 in [8]. Although in [8] there are more possible configurations the fact thatẏ| y=0 = x 2 + k 0 > 0 prevent them here. If k 2 > 3/4 1/3 there are three singular points in the local chart U 1 , a saddle and two stable nodes. Again using thatẏ| y=0 = x 2 + k 0 > 0 we get that the unique global phase portrait is (3) of Figure 1. This corresponds to 1 in [8]. As in the previous case although in [8] there are more possible configurations the fact thatẏ| y=0 = x 2 + k 0 > 0 prevent them here.

4.2.
Global phase portraits for system (i) with k 0 = 0 and k 1 = 1. In this case there is a unique singular point, the origin which is a saddle-node. Using thatẏ| y=0 = x 2 + k 0 > 0 andẋ| y=0 = 0 we have a unique possible global phase portrait for each of the cases depending on whether in the local chart U 1 we have a unique stable node, a saddle-node and a stable node, or a saddle and two stable nodes. In short we have phase portrait (4)
In Case 3 the two saddle-nodes are as in Figure 5 (c). Again using the first statement in Remark 1 we conclude that the unique global phase portrait is (18) of Figure 1 which is attained for example when k 0 = −1, k 1 = 0 and k 2 = 14/10.  Figure 5. The position of the separatrices of the two saddle-nodes (± √ −k 0 , 0) for system (i) with k 0 < 0, k 2 < 3/4 1/3 and (k 1 + √ −k 0 k 2 )(k 1 − √ −k 0 k 2 ) = 0 4.5. Global phase portraits for system (i) with k 0 < 0, k 2 = 3/4 1/3 and (k 1 + √ −k 0 k 2 )(k 1 − √ −k 0 k 2 ) = 0. We will separate again our study in the three cases above. Note that the statement about the nullcline in Remark 1 forces that the α-limit of the separatrix of the saddle-node at infinity must be the finite point ( √ −k 0 , 0). The argument above together with the first statement in Remark 1 imply that the possibilities of the global behavior of the solutions in Case 1 is limited to phase portraits (19) and (20) of Figure 1 and, (21) of Figure 2. Phase portrait (19) is attained at k 0 = −3/20, k 1 = 1 and phase portrait (21) at k 0 = −2/25, k 1 = 1. As in the previous subsection, phase portrait (20) corresponds to the separatrix connection and it is realized due to the continuity of the parameters. We recall that phase portraits (10)-(12) of Figure 1 correspond to the case in which the two infinity singular points at the local chart U 1 coalesce.
The argument at the beginning of this subsection together with the first statement in Remark 1 forbids the existence of global phase portraits in Case 2.
The same arguments as in the above subsection imply that in Case 3, the possible global phase portraits are (22) (28) is the separatrix connection and it is realized due to the continuity of the parameters. We remark that phase portraits (19) and (20) of Figure 1 and, (21) of Figure 2 correspond to the case in which the saddle and stable node in the local chart U 1 coalesce.
As in the previous subsection for Case 2 there are no possible global phase portraits.
In Case 3 the possible global phase portraits are (30)-(34) of Figure 2. Phase portrait (30) is attained at k 0 = −1/2, k 1 = 1, k 2 = 2, phase portrait (32) at k 0 = −1, k 1 = 1, k 2 = 2 and phase portrait (34) at k 0 = −1, k 1 = 0, k 2 = 2. Phase portraits (31) and (33) correspond to separatrix connections and they are realized due to the continuity of the parameters. We remark that phase portraits (22)-(26) of Figure 2 correspond to the case in which the saddle and stable node in the local chart U 1 coalesce. 4.7. Global phase portraits for system (i) with k 0 < 0 and k 1 = − √ −k 0 k 2 . Since k 1 ∈ {0, 1} we get that k 2 ≤ 0. This implies that in the local chart U 1 we have a unique stable node. Moreover, the finite singular point ( √ −k 0 , 0) is a cusp. If k 2 = 0 the finite singular point (− √ −k 0 , 0) is a saddle-node and if k 2 = 0 it is a cusp. We remark that the phase portraits of quadratic systems with a finite cusp were studied by Jager in [11]. Here, using Remark 1 we get that the unique global phase portraits are (36)-(38) of Figure 2 when k 2 = 0 and phase portrait (35) of Figure 2 when k 2 = 0. Phase portrait (36) is attained when k 0 = −2, k 2 = −1/2 and it is topologically equivalent to phase portrait 4 of Figure 14 in [11]. Phase portrait (38) is attained when k 0 = −1, k 2 = −1 and is topologically equivalent to phase portrait 1 of Figure 14 in [11]. Phase portrait (37) corresponds to the separatrix connection, is realized due to the continuity of the parameters and is topologically equivalent to phase portrait 2 of Figure 14 in [11]. 4.8. Global phase portraits for system (i) with k 0 < 0, k 1 = √ −k 0 k 2 and k 2 = 0. In this case k 2 > 0 (the case k 1 = k 2 = 0 is given in subsection 4.7). The explanation about the nullclines and the first statement in Remark 1 yield that the unique global phase portraits are (36) of Figure 2 when k 2 < 3/4 1/3 (there is a unique stable node in the local chart U 1 ), (39) of Figure 2 when k 2 = 3/4 1/3 (there is a stable node and a saddle-node in the local chart U 1 ) and (40) of Figure 2 when k 2 > 3/4 1/3 (there are two stable nodes and a stable node in the local chart U 1 ). Phase portrait (39) is topologically equivalent to phase portrait 7 of Figure 15 in [11] and phase portrait (40) is topologically equivalent to phase portrait 7 of Figure  16 in [11].
From the previous subsections we conclude the global study of system (i) of Theorem 1.2.
5. Global behavior of system (ii) of Theorem 1.2. System (ii) has the origin as the unique critical point. The eigenvalues of the Jacobian matrix at the origin are 0 and k 1 . So, it is semi-hyperbolic if k 1 = 0 and nilpotent if k 1 = 0. Applying Theorems 2.19 and 3.5 from [6] we get that the origin is a saddle-node if k 1 = 0 and a cusp if k 1 = 0.
The Poincaré compactification p(X ) of system (ii) in the local chart U 1 is given byu If k 2 ∈ {−1, 0} the unique singular point in the local chart U 1 is the origin. If k 2 = 1 the singular points in the local chart U 1 are p 1 = (− √ k 2 , 0), p 2 = (0, 0) and p 3 = ( √ k 2 , 0). If k 2 = −1 the origin is a semi-hyperbolic stable node, if k 2 = 0 the origin is a nilpotent stable node, and if k 2 = 1 the origin is a semi-hyperbolic saddle and the points p 1 and p 3 are hyperbolic stable nodes.
System (ii) in the local chart U 2 is written aṡ So the origin of the local chart U 2 is not an infinite singular point.

5.1.
Global phase portraits for system (ii) with k 2 ∈ {−1, 0}. Taking into account that the unique infinite singular point is the origin of the local chart U 1 , which is a stable node, and that the origin is the unique finite singular point (either a saddle-node or a cusp), we conclude that the unique possible global phase portraits are (4) of Figure 1 (when k 1 = 0) and (41) Figure 3 (when k 1 = 0).

5.2.
Global phase portraits for system (ii) with k 2 = 1. If k 1 = 0 the unique possible global phase portrait is (42) of Figure 3. If k 1 = 0 there are three possible global phase portraits according to the ω− limit of the separatrix of the finite saddle-node. They are (6) of Figure 1 and, (43) and (44) of Figure 3. Phase portrait (6) is attained, for example when k 1 = −2 and phase portrait (44) when k 1 = −1/2. Phase portrait (43) corresponds to the separatrix connection and it is realized due to the continuity of the parameter k 1 . From the previous subsections we conclude the global study of system (ii) of Theorem 1.2.
6. Global behavior of systems (iii) and (iv) of Theorem 1.2. Systems (iii) and (iv) do not have finite singular points so they are chordal quadratic systems.
The Poincaré compactification p(X ) of system (iii) in the local chart U 1 is given byu = k 2 u − u 3 + v 2 ,v = −u 2 v. If k 2 = −1 the unique singular point in the local chart U 1 is the origin which is a semi-hyperbolic unstable node. If k 2 = 1 there are three infinity singular points which are (0, 0) (a semi-hyperbolic saddle) and (0, ±1) which are stable nodes.
System (iii) in the local chart U 2 is written aṡ So the origin of the local chart U 2 is not an infinite singular point. When k 2 = −1 the unique global phase portrait is (1) of Figure 1 and when k 2 = 1 the unique global phase portrait is (3) of Figure 1.
The Poincaré compactification p(X ) of system (iv) in the local chart U 1 is given byu = −u 3 + k 1 uv + v 2 ,v = −u 2 v, and in the local chart U 2 bẏ The unique infinite singular point is the origin of the local chart U 1 which is a linearly zero stable node, so the unique global phase portrait is (1) of Figure 1.
The proof of Theorem 1.2 comes from Sections 3-6.