This is a preprint of: “Phase portraits of Abel quadratic differential systems of second kind with symmetries”, Antoni Ferragut, Johanna Denise Garcı́a Saldaña, Clàudia Valls, Dyn. Syst., vol. 34(2), 301–333, 2019. DOI: [10.1080/14689367.2018.1530732] PHASE PORTRAITS OF ABEL QUADRATIC DIFFERENTIAL SYSTEMS OF SECOND KIND WITH SYMMETRIES ANTONI FERRAGUT, JOHANNA D. GARCÍA-SALDAÑA, CLAUDIA VALLS 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. 1. Introduction and statements of the main results Abel differential equations of the second kind are named in honor of Niels Henrik Abel because by a direct substitution they are related with 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 modeling of real problems such as big picture modeling in oceanic circulation (see [1] and the references therein). An Abel differential equation of second kind has the form dy y = A(x)y 2 + B(x)y + C(x), (1) dx with A(x), B(x), C(x) ∈ R(x, y). This differential equation can be written equivalently as the polynomial differential system ẋ = d(x)y, ẏ = a(x)y 2 + b(x)y + c(x), (2) 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 quadratic polynomial differential systems, i.e. the differential systems (2) of degree two: ẋ = (d0 + d1 x)y, ẏ = a0 y 2 + (b0 + b1 x)y + c0 + c1 x + c2 x2 . (3) All the parameters in (3) are real. We assume that ẋ and ẏ do not have a common factor; in particular, we assume that c20 + c21 + c22 6= 0. Moreover, we take a0 6= 0, otherwise this is not the Abel equation of second kind. We also assume that b21 + c21 + c22 + d21 6= 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 second kind of degree two given by system (3) with d1 6= 0. The case d1 = 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. Proposition 1. System (3) with d1 6= 0 leads, after an affine change of the variables and a scaling of the time, to the differential system ẋ = xy, ẏ = Q(x, y), where Q(x, y) is one of the following five quadratic polynomials: (K1) Q(x, y) = S(x − r1 )(x − r2 ) + (B0 + B1 x)y + A0 y 2 ; 2010 Mathematics Subject Classification. 34A34, 34C20, 34C23, 34C05. Key words and phrases. Abel differential equation of second kind, quadratic differential system, phase http://www.gsd.uab.cat portrait, Poincaré disk. 1 2 A. FERRAGUT, J.D. GARCÍA-SALDAÑA, C. VALLS (K2) Q(x, y) = S(x − r)2 + (B0 + B1 x)y + A0 y 2 ; (K3) Q(x, y) = S((x − a)2 + b2 ) + (B0 + B1 x)y + A0 y 2 ; (K4) Q(x, y) = x − T + (B0 + B1 x)y + A0 y 2 ; (K5) Q(x, y) = S + (B0 + B1 x)y + A0 y 2 ; where S ∈ {−1, 1}, T ∈ {−1, 0, 1}, a, B0 , B1 , r, r1 , r2 ∈ R, A0 ∈ R \ {0}, r1 6= r2 and b > 0. Next proposition splits the families in Proposition 1 into several subfamilies, considering the following three symmetries, mentioned above: (Sa ) x 7→ −x; (Sb ) y 7→ −y; (Sc ) (x, y) 7→ −(x, y); (4) and the possible change of sign of the independent variable, t 7→ −t. Proposition 2. After applying the three symmetries (Sa ), (Sb ) and (Sc ), the five families in Proposition 1 can be written, after an affine change of the variables and a scaling of the time, as ẋ = xy, ẏ = Q(x, y), where Q(x, y) is one of the following thirteen quadratic polynomials: (K1a) Q(x, y) = S(x2 − 1) + B0 y + A0 y 2 , B0 > 0; (K1b) Q(x, y) = S(x − r)(x − 1) + A0 y 2 , r 6= 1; (K1c) Q(x, y) = S(x2 − 1) + B1 xy + A0 y 2 , B1 ≥ 0; (K2a) Q(x, y) = Sx2 + y + A0 y 2 ; (K2b) Q(x, y) = S(x − r)2 + A0 y 2 , r ∈ {0, 1}; (K2c) Q(x, y) = Sx2 + B1 xy + A0 y 2 , B1 > 0; (K3a) Q(x, y) = S(x2 + b2 ) + B0 y + A0 y 2 , B0 > 0; (K3b) Q(x, y) = S((x − a)2 + 1) + A0 y 2 , a ∈ R; (K3c) Q(x, y) = S(x2 + 1) + B1 xy + A0 y 2 , B1 > 0; (K4a) Q(x, y) = x − T + A0 y 2 , T 2 ∈ {0, 1}; (K5a) Q(x, y) = S + B0 y + A0 y 2 , B0 > 0; (K5b) Q(x, y) = S + A0 y 2 ; (K5c) Q(x, y) = S + xy + A0 y 2 . In all cases, we have S ∈ {−1, 1} and A0 ∈ R \ {0}. Propositions 1 and 2 are proved in section 3. The main result in this paper is the following theorem, where the global phase portraits on the Poincaré disk of all the families of Proposition 2 are provided. Theorem 3 follows after Theorem 4, in which we relate the global phase portraits of Figure 1 and the families in Proposition 2 and the conditions on the parameters of Q for each case. We say that two polynomial vector fields on the Poincaré disk are topologically equivalent if there exists a homeomorphism from one onto the other which sends orbits to orbits preserving or reversing the direction of the flow. Theorem 3. The families of system (3) with d1 6= 0 appearing in Proposition 2 provide 84 non-topologically equivalent phase portraits. All these phase portraits are shown in Figure 1. The phase portraits provided by each family are: (K1a) (1)–(12). (K1b) (10)–(24). (K1c) (10)–(13), (18), (20), (25)–(32). (K2a) (33)–(37). (K2b) (38)–(46). (K2c) (38)–(40), (43), (44), (47), (48). (K3a) (33)–(37), (49)–(52). (K3b) (34), (35), (37), (50), (51). (K3c) (34), (35), (37), (50), (51), (53)–(58). (K4a) (59)–(70). (K5a) (71)–(83). (K5b) (71), (74)–(76), (82)–(84). (K5c) (35), (37), (51), (53)–(55), (57), (58). PHASE PORTRAITS OF ABEL SYSTEMS OF SECOND KIND 3 Let R+0 = {x ∈ R : x ≥ 0} and R = {x ∈ R : x > 0}. Theorem 3 follows directly after + the next theorem. Theorem 4. Regarding Theorem 3, the parameter spaces modulo the group action and the corresponding bifurcation diagrams are as follows: (K1a) For the family (K1a) the parameter space is (S, A0 , B0 ) ∈ {−1, 1} × (R \ {0}) × R+ . Let C1 := B02 + 4A0 S. When S = −1 the global phase portrait is: (1) for A0 < 0; (2) for 0 < A0 ≤ 1 and C1 > 0; (3) for 0 < A0 ≤ 1 and C1 = 0; (4) for 0 < A0 ≤ 1 and C1 < 0; (5) for A0 > 1 and C1 > 0; (6) for A0 > 1 and C1 = 0; (7) for A0 > 1 and C1 < 0. The bifurcation diagram is shown in Figure 2(a). When S = 1 the global phase portrait is: (8) for A0 < 0 and C1 > 0; (9) for A0 < 0 and C1 = 0; (10) for A0 < 0 and C1 < 0; (11) for 0 < A0 < 1; (12) for A0 ≥ 1. The bifurcation diagram is shown in Figure 2(b). (K1b) For the family (K1b) the parameter space is (S, A0 , r) ∈ {−1, 1}×(R\{0})×(R\{1}). When S = −1 the global phase portrait is: (13) for either A0 = 1 and r < 0, or 0 < A0 < 1 and r < 0; (14) for either A0 < 0 and 0 ≤ r < 1, or A0 < 0 and r > 1; (15) for either 0 < A0 ≤ 1 and 0 < r < 1, or 0 < A0 ≤ 1 and r > 1; (16) for either A0 > 1 and 0 < r < 1, or A0 > 1 and r > 1; (17) for A0 > 1 and r = 0; (18) for A0 > 1 and r < 0; (19) for 0 < A0 ≤ 1 and r = 0; (20) for A0 < 0 and r < 0. The bifurcation diagram is shown in Figure 2(c). When S = 1 the global phase portrait is: (10) for A0 < 0 and r ≤ 0; (11) for 0 < A0 < 1 and r < 0; (12) for A0 ≥ 1 and r < 0; (14) for either A0 ≥ 1 and 0 < r < 1 or A0 ≥ 1 and r > 1; (21) for either A0 < 0 and 0 < r < 1 or A0 < 0 and r > 1; (22) for either 0 < A0 < 1 and 0 < r < 1 or 0 < A0 < 1 and r > 1; (23) for 0 < A0 < 1 and r = 0; (24) for A0 ≥ 1 and r = 0. The bifurcation diagram is shown in Figure 2(d). (K1c) For the family (K1c) the parameter space is (S, A0 , B1 ) ∈ {−1, 1} × (R \ {0}) × R+ 0. Let C2 := B1 − 4A0 S + 4S. 2 When S = −1 the global phase portrait is: (13) for either 0 < A0 < 1 and C2 < 0 or 0 < A0 ≤ 1 and B1 = 0; (18) for A0 > 1 and B1 = 0; (20) for A0 < 0 and B1 = 0; (25) for A0 < 0 and C2 < 0; (26) for A0 < 0 and C2 = 0; (27) for A0 < 0 and C2 > 0; 4 A. FERRAGUT, J.D. GARCÍA-SALDAÑA, C. VALLS (28) for 0 < A0 < 1 and C2 > 0; (29) for 0 < A0 < 1 and C2 = 0; (30) for A0 > 1 and B1 > 0; (31) for A0 = 1 and B1 > 0. The bifurcation diagram is shown in Figure 2(e). When S = 1 the global phase portrait is: (10) for A0 < 0 and B1 ≥ 0; (11) for either 0 < A0 < 1 and B1 ≥ 0 or A0 > 1 and C2 > 0; (12) for either A0 ≥ 1 and B1 = 0 or A0 > 1 and C2 < 0; (32) for A0 = 1 and B1 > 0. The bifurcation diagram is shown in Figure 2(f ). (K2a) For the family (K2a) the parameter space is (S, A0 ) ∈ {−1, 1} × (R \ {0}). When S = −1 the global phase portrait is: (33) for A0 < 0; (34) for 0 < A0 ≤ 1; (35) for A0 > 1. When S = 1 the global phase portrait is: (33) for A0 ≥ 1; (36) for 0 < A0 < 1; (37) for A0 < 0. (K2b) For the family (K2b) the parameter space is (S, r, A0 ) ∈ {−1, 1} × {0, 1} × (R \ {0}). When S = −1 and r = 0 the global phase portrait is: (38) for A0 < 0; (39) for 0 < A0 ≤ 1; (40) for A0 > 1; When S = −1 and r = 1 the global phase portrait is: (38) for A0 < 0; (41) for 0 < A0 ≤ 1; (42) for A0 > 1. When S = 1 and r = 0 the global phase portrait is: (38) for A0 ≥ 1; (43) for 0 < A0 < 1; (44) for A0 < 0. When S = 1 and r = 1 the global phase portrait is: (38) for A0 ≥ 1; (45) for 0 < A0 < 1; (46) for A0 < 0. (K2c) For the family (K2c) the parameter space is (S, A0 , B1 ) ∈ {−1, 1} × (R \ {0}) × R+ . Let C2 := B12 − 4A0 S + 4S When S = −1 the global phase portrait is: (38) for A0 < 0 and C2 < 0; (39) for 0 < A0 < 1 and C2 < 0; (40) for either 0 < A0 < 1 and C2 > 0 or A0 > 1; (43) for A0 < 0 and C2 > 0; (47) for A0 < 0 and C2 = 0; (48) for either 0 < A0 < 1 and C2 = 0 or A0 = 1. The bifurcation diagram is shown in Figure 2(g). When S = 1 the global global phase portrait is: (38) for A0 > 1 and C2 < 0; (43) for either 0 < A0 < 1 or A0 > 1 and C2 > 0; (44) for A0 < 0; (47) for either A0 = 1 or A0 > 1 and C2 = 0; The bifurcation diagram is shown in Figure 2(h). PHASE PORTRAITS OF ABEL SYSTEMS OF SECOND KIND 5 (K3a) For the family (K3a) the parameter space is (S, A0 , B0 ) ∈ {−1, 1} × (R \ {0}) × R+ . Let C3 := B02 − 4A0 S When S = −1 the global phase portrait is: (33) for A0 < 0 and C3 > 0; (34) for 0 < A0 ≤ 1; (35) for A0 > 1; (49) for A0 < 0 and C3 = 0; (50) for A0 < 0 and C3 < 0. The bifurcation diagram is shown in Figure 2(i). When S = 1 the global global phase portrait is: (33) for A0 ≥ 1 and C3 > 0; (36) for 0 < A0 < 1 and C3 > 0; (37) for A0 < 0; (49) for A0 ≥ 1 and C3 = 0; (50) for A0 ≥ 1 and C3 < 0; (51) for 0 < A0 < 1 and C3 < 0; (52) for 0 < A0 < 1 and C3 = 0. The bifurcation diagram is shown in Figure 2(j). (K3b) For the family (K3b) the parameter space is (S, A0 ) ∈ {−1, 1} × (R \ {0}). When S = −1 the global phase portrait is: (34) for 0 < A0 ≤ 1; (35) for A0 > 1; (50) for A0 < 0. When S = 1 the global phase portrait is: (37) for A0 < 0; (50) for A0 ≥ 1; (51) for 0 < A0 < 1. (K3c) For the family (K3c) the parameter space is (S, A0 , B1 ) ∈ {−1, 1} × (R \ {0}) × R+ . Let C2 := B12 − 4A0 S + 4S When S = −1 the global phase portrait is: (34) for 0 < A0 < 1 and C2 < 0; (35) for A0 > 1; (50) for A0 < 0 and C2 < 0; (53) for either 0 < A0 < 1 and C2 = 0 or A0 = 1; (54) for 0 < A0 < 1 and C2 > 0; (55) for A0 < 0 and C2 > 0; (56) for A0 < 0 and C2 = 0. The bifurcation diagram is shown in Figure 2(k). When S = 1 the global global phase portrait is: (37) for A0 < 0; (50) for A0 > 1 and C2 < 0; (51) for 0 < A0 < 1; (56) for A0 > 1 and C2 = 0; (57) for A0 = 1; (58) for A0 > 1 and C2 > 0. The bifurcation diagram is shown in Figure 2(l). (K4a) For the family (K4a) the parameter space is (T, A0 ) ∈ {−1, 0, 1} × (R \ {0}). When T = −1 the global phase portrait is: (59) for A0 < 0; (60) for 0 < A0 < 1; (61) for A0 = 1; (62) for A0 > 1. When T = 0 the global phase portrait is: 6 A. FERRAGUT, J.D. GARCÍA-SALDAÑA, C. VALLS (63) for A0 < 0; (64) for 0 < A0 < 1; (65) for A0 = 1; (66) for A0 > 1. When T = 1 the global phase portrait is: (67) for A0 < 0; (68) for 0 < A0 < 1; (69) for A0 = 1; (70) for A0 > 1. (K5a) For the family (K5a) the parameter space is (S, A0 , B0 ) ∈ {−1, 1} × (R \ {0}) × R+ . Let C3 := B02 − 4A0 S When S = −1 the global phase portrait is: (71) for A0 < 0 and C3 < 0; (72) for A0 < 0 and C3 = 0; (73) for A0 < 0 and C3 > 0; (74) 0 < A0 < 1; (75) for A0 = 1; (76) for A0 > 1; The bifurcation diagram is shown in Figure 2(m). When S = 1 the global phase portrait is: (71) for A0 > 1 and C3 < 0; (72) for A0 > 1 and C3 = 0; (73) for A0 > 1 and C3 > 0; (77) for A0 < 0; (78) for 0 < A0 < 1 and C3 > 0; (79) for A0 = 1 and C3 > 0; (80) for A0 = 1 and C3 = 0; (81) for 0 < A0 < 1 and C3 = 0; (82) for A0 = 1 and C3 < 0; (83) for 0 < A0 < 1 and C3 < 0; The bifurcation diagram is shown in Figure 2(n). (K5b) For the family (K5b) the parameter space is (S, A0 ) ∈ {−1, 1} × (R \ {0}). When S = −1 the global phase portrait is: (71) for A0 < 0; (74) for 0 < A0 < 1; (75) for A0 = 1; (76) for A0 > 1. When S = 1 the global phase portrait is: (71) for A0 > 1; (82) for A0 = 1; (83) for 0 < A0 < 1; (84) for A0 < 0. (K5c) For the family (K5c) the parameter space is (S, A0 ) ∈ {−1, 1} × (R \ {0}). When S = −1 the global phase portrait is: (35) for A0 > 1; (53) for A0 = 1; (54) for 0 < A0 < 1; (55) for A0 < 0. When S = 1 the global phase portrait is: (37) for A0 < 0; (51) for 0 < A0 < 1; (57) for A0 = 1; (58) for A0 > 1. PHASE PORTRAITS OF ABEL SYSTEMS OF SECOND KIND 7 Remark 1. The families appearing in Proposition 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. 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 (because ẋ = xy). Cases with the symmetries Sa or Sb 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 Sc . 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 α- or 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 and 2. Finally we prove Theorem 4 in section 4. 2. 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]. 2.1. Singular points. Consider an analytic planar differential system ẋ = P (x, y), ẏ = Q(x, y) and its associated vector field X = (P, Q). A point p ∈ R2 is a singular point of X if P (p) = Q(p) = 0. We define, for a singular point p ∈ R2 , ∆ = Px (p)Qy (p)−Py (p)Qx (p) ∈ R and T = Px (p) + Qy (p) ∈ R. They correspond, respectively, to the determinant and the trace of the Jacobian matrix DX(p). The singular point p is non-degenerated if ∆ 6= 0. It is degenerated otherwise. Then, p is an isolated singular point. Moreover p is a saddle if ∆ < 0, a node if T 2 ≥ 4∆ > 0 (stable if T < 0, unstable if T > 0), a focus if 4∆ > T 2 > 0 (stable if T < 0, unstable if T > 0), and either a weak focus or a center 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 centers. A degenerate singular point p such that T 6= 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 5. 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 neighborhood 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 neighborhood of (0, 0). Assume that f (x) = F (x, g(x)) = µxm + · · · , where m ≥ 2 and µ 6= 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]). 8 A. FERRAGUT, J.D. GARCÍA-SALDAÑA, C. VALLS (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) Figure 1. Phase portraits of the differential systems appearing in Theorems 3 and 4 (I). Proposition 6. 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 neighborhood 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) = 0 in a neighborhood of (0, 0). Assume that G(x, f (x)) = Kxκ + · · · and Φ(x) ≡ (∂F /∂x + ∂G/∂y)(x, f (x)) = Lxλ + · · · , with K 6= 0, κ ≥ 2 and λ ≥ 1. The following statements hold. (i) If κ is even and PHASE PORTRAITS OF ABEL SYSTEMS OF SECOND KIND 9 (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73) (74) (75) (76) (77) (78) (79) (80) (81) (82) (83) (84) Figure 1. Phase portraits of the differential systems appearing in Theorems 3 and 4 (and II). (a) κ > 2λ + 1 then the origin is a saddle-node. (b) κ < 2λ + 1 or Φ ≡ 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 L2 + 4K(λ + 1) ≥ 0, or λ even and κ > 2λ + 1 then the origin is a stable (unstable) node if L < 0 (L > 0). 10 A. FERRAGUT, J.D. GARCÍA-SALDAÑA, C. VALLS B0 B0 2 5 6 8 12 2 9 3 3 7 10 11 12 1 4 4 A0 A0 (a) (b) r r 14 15 15 16 21 22 14 14 14 15 15 16 21 22 14 14 A0 0 14 19 19 17 10 23 24 24 20 13 13 18 10 11 12 12 (c) (d) B1 B1 27 31 26 32 11 28 32 29 25 10 11 12 30 13 A0 A0 20 13 13 18 10 11 12 12 (e) (f) Figure 2. Bifurcation diagrams of the families appearing in Theorems 3 and 4 (I): (a) (K1a) in the case S = −1; (b) (K1a) in the case S = 1; (c) (K1b) in the case S = −1; (d) (K1b) in the case S = 1; (e) (K1c) in the case S = −1; (f) (K1c) in the case S = 1. (b) λ odd, κ = 2λ + 1 and L2 + 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 L2 + 4K(λ + 1) < 0, or κ < 2λ + 1 then the origin is a focus or a center, and if Φ(x) ≡ 0 then the origin is a center. 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 PHASE PORTRAITS OF ABEL SYSTEMS OF SECOND KIND 11 B1 B1 43 48 47 47 43 40 47 48 38 44 43 38 40 39 A0 A0 (g) (h) B0 B0 33 33 33 34 49 49 36 49 52 50 34 35 50 37 50 51 A0 A0 (i) (j) B1 B1 55 53 56 57 58 54 56 53 50 37 51 50 35 34 A0 A0 (k) (l) B0 B0 79 73 73 75 72 72 78 80 81 71 74 76 71 77 82 83 A0 A0 (m) (n) Figure 2. Bifurcation diagrams of the families appearing in Theorems 3 and 4 (and II): (g) (K2c) in the case S = −1; (h) (K2c) in the case S = 1; (i) (K3a) in the case S = −1; (j) (K3a) in the case S = 1; (k) (K3c) in the case S = −1; (l) (K3c) in the case S = 1; (m) (K5a) in the case S = −1; (n) (K5a) in the case S = 1. sector which leave the sector with both increasing and decreasing time. A sector such that all curves in a sufficiently small neighborhood 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. 12 A. FERRAGUT, J.D. GARCÍA-SALDAÑA, C. VALLS 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 center and a focus for a singular point of a quadratic differential system having pure imaginary eigenvalues. Theorem 7. Consider the quadratic differential system ẋ = −y + a20 x2 + a11 xy + a02 y 2 , ẏ = x + b20 x2 + b11 xy + b02 y 2 . (5) Let  ω1 = Aα − Bβ, ω2 = γ 5(Aβ + Bα) − α2 − β 2 , ω3 = γδ(Aβ + Bα), where A = a20 + a02 , B = b20 + b02 , α = a11 + 2b02 , β = b11 + 2a20 , γ = b20 A − (a20 − b11 )A B + (b02 − a11 )AB − a02 B 3 , 3 2 2 δ = a202 + b220 + a02 A + b20 B. Then the following statements hold: (1) If ω1 = ω2 = ω3 = 0 then the origin is a center. (2) If there exists k ∈ {1, 2, 3} such that ωj = 0 for all 1 ≤ j < k and ωk 6= 0, then the origin is a focus of order k. The stability of this focus is given by the sign of ωk 6= 0: when ωk < 0 the focus is stable and when ωk > 0 the focus is unstable. 2.2. Separatrices and canonical regions. Consider the planar differential system ẋ = 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 Ip ⊆ R where the unique maximal solution ϕp : Ip → U of the system such that ϕp (0) = p is defined. (ii) If q = ϕp (t) and t ∈ Ip then Iq = Ip − t = {r − t : r ∈ Ip } and ϕq (s) = ϕp (t + s) for all s ∈ Iq . (iii) The set D = {(t, p) : p ∈ U, t ∈ Ip } is open in R3 and the map ϕ : D → U defined by ϕ(t, p) = ϕp (t) is C r . The map ϕ : D → U is a local flow of class C r on U associated to the system. It verifies: (1) ϕ(0, p) = p for all p ∈ U ; (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 ∈ Ip ; (3) ϕp (−t) = ϕ−1 p (t) for all p ∈ U such that t, −t ∈ Ip . We consider C r -local flows on R2 , with r ≥ 0. Of course when r = 0 the flow is only continuous. Two such flows ϕ and ϕ0 are C k -equivalent, with k ≥ 0, if there exists a C k - diffeomorphism that brings orbits of ϕ onto orbits of ϕ0 preserving sense (but not necessarily the parametrization). Let ϕ be a C r -local flow with r ≥ 0 on R2 . We say that ϕ is C k -parallel if it is C k -equivalent to one of the following flows: (i) R2 with the flow defined by x0 = 1, y 0 = 0. (ii) R2 \ {0} with the flow defined (in polar coordinates) by r0 = 0, θ0 = 1. (iii) R2 \ {0} with the flow defined by r0 = r, θ = 0. We call these flows strip, annular and spiral, respectively. Let p ∈ R2 . We denote by γ(p) the orbit of the flow ϕ through p, i.e. γ(p) = {ϕp (t) : t ∈ Ip }. The positive semiorbit of p is γ + (p) = {ϕp (t) : t ∈ Ip , 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 α(p) = cl (γ − (p)) − γ − (p), ω(p) = cl (γ + (p)) − γ + (p), respectively. Here, as usual, cl denotes the closure. PHASE PORTRAITS OF ABEL SYSTEMS OF SECOND KIND 13 Let γ(p) be an orbit of the flow ϕ. A parallel neighborhood of the orbit γ(p) is an open neighborhood 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 neighborhood N satisfying the following two assumptions: (i) for any q ∈ N , α(q) = α(p) and ω(q) = ω(p); (ii) cl (N ) \ N consists of α(p), ω(p) and exactly two orbits γ(a), γ(b) of ϕ, with α(a) = α(p) = α(b) and ω(a) = ω(p) = ω(b). We denote by Σ the union of all separatrices of ϕ. Σ is a closed invariant subset of R2 . A component of the complement of Σ in R2 , with the restricted flow, is a canonical region of ϕ. The following lemma can be found in [10]. Lemma 8. Every canonical region of a local flow ϕ on R2 is C 0 -parallel. 2.3. 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 S2 as follows (see for instance [8]). Let S2 = {y = (y1 , y2 , y3 ) ∈ R3 : y12 + y22 + y32 = 1} (the Poincaré sphere) and Ty S2 be the tangent plane to S2 at a point y. Identify R2 with T(0,0,1) S2 . Consider the central projection f : T(0,0,1) S2 → S2 . This map defines two copies of X on S2 , one in the northern hemisphere and the other in the southern hemisphere. Denote by X̄ the vector field Df ◦ X defined on S2 except on its equator S1 = {y ∈ S2 : y3 = 0}. Clearly S1 is identified to the infinity of R2 . Usually, when we talk about the circle of the infinity of X we simply talk about the infinity. In order to extend X̄ to a vector field on S2 (including S1 ) 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 y3n−1 X̄ to S2 . On S2 \S1 there are two symmetric copies of X, and knowing the behavior of p(X) around S1 we know the behavior of X in a neighborhood of the infinity. The Poincaré compactification has the property that S1 is invariant under the flow of p(X). The projection of the closed northern hemisphere of S2 on y3 = 0 under (y1 , y2 , y3 ) 7→ (y1 , y2 ) is called the Poincaré disc, and it is denoted by D2 . Two polynomial vector fields X and Y on R2 are topologically equivalent if there exists a homeomorphism on S2 preserving the infinity S1 and carrying orbits of the flow induced by p(X) into orbits of the flow induced by p(Y ). As S2 is a differentiable manifold, for computing the expression of p(X) we can consider the six local charts Ui = {y ∈ S2 : yi > 0}, and Vi = {y ∈ S2 : yi < 0}, i = 1, 2, 3. The diffeo- morphisms Fi : Ui → R2 and Gi : Vi → R2 for i = 1, 2, 3 are the inverses of the central pro- jections 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 Fi (y) or Gi (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):        1 u 1 u 1 u v n ∆(z) Q , − uP , , −vP , (6) v v v v v v on U1 ;        n u 1 u 1 u 1 v ∆(z) P , − uQ , , −vQ , (7) v v v v v v on U2 ; and ∆(z) (P (u, v), Q(u, v)) 1 on U3 , where ∆(z) = (u2 + v 2 + 1)− 2 (n−1) . The expression for Vi is the same as that for Ui except for a multiplicative factor (−1)n−1 . In these coordinates, for i = 1, 2, v = 0 always denotes the points of S1 . 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. 14 A. FERRAGUT, J.D. GARCÍA-SALDAÑA, C. VALLS 3. Proofs of Propositions 1 and 2 We prove in this section Propositions 1 and 2. We recall that these results allow us to split the differential system (3) into the thirteen different families that are discussed in Theorems 3 and 4.   Proof of Proposition 1. By applying the change of variables and time (x, y) → x + dd01 , B2 y , dt B2 dτ = d1 , system (3) becomes ẋ =xy, B22 (c0 d21 − c1 d0 d1 + c2 d20 ) B22 (c1 d1 − 2c2 d0 ) B2 (b0 d1 − b1 d0 ) ẏ = + x+ y 3 d1 2 d1 d21 (8) B22 c2 2 B2 b1 a0 + xy + y 2 . x + d1 d1 d1 We consider first the case c2 6= 0, for which system (8) has at most four singular points. Renaming r r r a0 a0 S a0 c2 c2   d1 = , B2 = , b1 = B1 , b0 = A0 B1 d0 + a0 B0 , A0 A0 c2 A0 S a 0 A0 S we get ẋ =xy, S(A20 c2 d20 − A0 a0 c1 d0 + a22 c0 ) S(a0 c1 − 2A0 c2 d0 ) ẏ = + x + B0 y + Sx2 + B1 xy + A0 y 2 . a20 c2 a0 c2 (9) We focus on the singular points on y = 0. Let ∆ := c21 − 4c0 c2 be the discriminant of ẏ|y=0 . If ∆ > 0 then there are two different real singular points on y = 0, say (r1 , 0) and (r2 , 0); after renaming c2 (A0 d0 − a0 r1 )(A0 d0 − a0 r2 ) c2 (2A0 d0 − a0 (r1 + r2 )) c0 = , c1 = , a20 a0 system (9) leads to the family (K1). If ∆ = 0 then there is a double real singular point on y = 0, say (r, 0); after renaming c2 (A0 d0 − a0 r)2 2c2 (A0 d0 − a0 r) c0 = , c1 = , a20 a0 system (9) leads to the family (K2). If ∆ < 0 then there are two complex conjugate singular points on y = 0; after renaming c2 (A0 d0 − aa0 )2 + a20 bc2 2c2 (A0 d0 − a0 a) c0 = , c1 = , a20 a0 where ẏ|y=0 = (x − a)2 + b2 , with b > 0, system (9) leads to the family (K3). Now we consider  2 the case c2 = 0. If c1 6= 0 then applying the change of variables and B2 c1 B22 c1 d0 time (x, y) → d1 x + d2 , B2 y , dτ dt = Bd12 , system (3) becomes 1 B22 (c0 d1 − c1 d0 ) B2 (b0 d1 − b1 d0 ) b1 a0 ẋ = xy, ẏ = +x+ y+ xy + y 2 . d21 2 d1 B2 c1 d1 Renaming a0 A20 B1 B22 c1 d0 + a20 B0 d1 = , b0 = , b1 = B1 B2 c1 , A0 a0 A0 B2 with s T B2 = a0 and T ∈ {−1, 0, 1}, A0 (A0 c1 d0 − a0 c0 ) PHASE PORTRAITS OF ABEL SYSTEMS OF SECOND KIND 15 the above system becomes the family (K4).   A1 d0 If c1 = 0 then, applying the change of variables and time (x, y) → A1 x + d1 , B2 y , dt B2 dτ = d1 , system (3) becomes B22 c0 B2 (b0 d1 − b1 d0 ) b1 B2 a0 ẋ = xy, ẏ = + y+ xy + y 2 . d1 d21 A1 d1 d1 Renaming a0 A0 A1 B1 d0 + a0 B0 a 0 A1 B 1 d1 = , b0 = , b1 = , A0 A0 B 2 A0 B2 with r a0 S B2 = and S ∈ {1, −1}, A0 c0 the above system becomes the family (K5). Then the proposition follows.  Proof of Proposition 2. The proof follows by applying the symmetries (Sa ), (Sb ) and (Sc ) given in (4) and possibly the change of sign of the independent variable t 7→ −t to the families given in Proposition 1. Applying (Sa ) to the family (K1) we obtain −ẋ = −xy and ẏ = S(−x − r1 )(−x − r2 ) + (B0 − B1 x)y + A0 y 2 . So we get again (K1) if and only if r1 + r2 = 0 and B1 = 0. (K1a) is obtained after additional affine changes which allow us to consider B0 > 0 and to take r1 = −r2 = 1. Applying (Sb ) to the family (K1) we obtain ẋ = −xy and −ẏ = S(x − r1 )(x − r2 ) − (B0 + B1 x)y + A0 y 2 . We need a change of time t 7→ −t and B0 = B1 = 0 to obtain again (K1). Moreover, after additional affine changes we know that we can take r1 = 1 and r2 = r 6= 1. We get (K1b). Finally, applying (Sc ) to the family (K1) we obtain ẋ = xy and −ẏ = S(−x − r1 )(−x − r2 ) − (B0 − B1 x)y + A0 y 2 . We need to apply a change of time t 7→ −t and set r1 + r2 = 0 and B0 = 0. Additionally we can take r1 = −r2 = 1 and B1 ≥ 0. We get (K1c). The other families are obtained in an analogous way. The proposition follows.  4. Proof of Theorem 4 We recall that Theorem 3 follows after Theorem 4. Following Proposition 2, we prove Theorem 4 separately for the thirteen different families. Proof of Theorem 4(K1a). We deal with the differential system ẋ = xy, ẏ = S(x2 − 1) + B0 y + A0 y 2 , (10) where S ∈ {−1, 1}, A0 6= 0, and B0 > 0.  p   The finite singular points of system (10) are (±1, 0) and 0, −B0 ± B02 + 4A0 S /2A0 . p  The eigenvalues of the singular points (±1, 0) are B0 ± B02 + 8S /2. If S = 1 then both √ singular points are saddles. If S = −1 then we √ have two cases: if B0 ≥ 2 2 then the singular points are unstable nodes whereas if B0 < 2 2 then the singular points are unstable foci. √  Let C1 := B02 +4A0 S. Then the other two singular points are 0, (−B0 ± C1 )/2A0 and they are real√only if C1≥ 0. √Suppose first that√C1 > 0. The eigenvalues of the singular point 0, (−B0 + C1 )/2A0 are C1 and (−B0 + C1 )/2A0 . Direct computations show that if S = 1 then the point is a unstable √ node and if S = −1 then √ it is a saddle point. √ On the other hand, the point 0, (−B0 − C1 )/2A0 has eigenvalues − C1 , and (−B0 − C1 )/2A0 . Then if A0 < 0 it is a saddle whereas if A0 > 0 it is a stable node. If C1 = 0 then the two points become the singular point (0, 2S/B0 ) which is semi- hyperbolic. We move the point to the origin and we apply the change of variables and time (x, y) → (y, x), dτ dt = B2S , to bring the system to the form given in Proposition 5. So, 0 F (x, f (x)) = −(B0 /8S )x + . . . and the singular point is saddle-node. 3 2 2 16 A. FERRAGUT, J.D. GARCÍA-SALDAÑA, C. VALLS According to expression (7), system (10) at infinity can be studied from the system u̇ = −u(Su2 − Sv 2 + B0 v + A0 − 1), v̇ = −v(Su2 − Sv 2 + B0 v + A0 ), on U2 . We do not need to study the expression (6) because all the infinite singular points can be studiedp from the above differential system. We have three infinite singular points: (0, 0) and ± (1 − A0 )/S, 0 . The origin on U2 has eigenvalues −A0 , 1 − A0 , thus it is an unstable node if A0 < 0, a saddle if 0 < A0 < 1 or a stable node if A0 > 1. When A0 = 1 we apply Proposition 5 because the point is a semi-hyperbolic point. So we have F (u, g(u)) = f (u) = Su3 + · · · and therefore if S = −1 the point is a saddle and if S = 1 the point is a stable node, because ofpa change of time before applying Proposition 5. The other two infinite singular points, (± (1 − A0 )/S, 0), only exist if S(1 − A0 ) > 0. Both of them have eigenvalues −1, −2(1 − A0 ). Thus, for S = 1 and A0 < 1 the points are stable nodes and for S = −1 and A0 > 1 the points are saddles. Statement (K1a) of Theorem 4 follows. Table 1 summarizes the different behaviors of the singular points depending on the values of the parameters. Singular point Conditions Type of point S=1 Saddles (±1, 0) √ S = −1, B0 < 2 2 Unstable Foci √ S = −1, B0 ≥ 2 2 Unstable Nodes  √  S = 1, C1 > 0, Unstable Node 0, −B02A + 0 C1 S = −1, C1 > 0 Saddle C1 = 0 Saddle–Node  √  A0 < 0, C1 > 0 Saddle 0, −B02A − 0 C1 A0 > 0, C1 > 0 Stable Node A0 < 0 Unstable Node 0 < A0 < 1 Saddle (0, 0) ∈ U2 A0 > 1 Stable Node A0 = 1, S = 1 Stable Nodes A0 = 1, S = −1 Saddle  q  1−A0 S = 1, A0 < 1 Stable Nodes ± S , 0 ∈ U2 S = −1, A0 > 1 Saddles Table 1. Behavior of the singular points for family (K1a). Here C1 = B02 + 4A0 S. According to the above mentioned when S = 1 we have the following behaviours: there are three cases for A0 < 1; if C1 < 0 then we obtain the phase portrait (10), if C1 = 0 we have the phase portrait (9) and if C1 > 0 the phase portrait is (8). For 0 < A0 < 1 the phase portrait is (11) and for A0 ≥ 1 we have (12). When S = −1 the behaviours are as follow: there are three cases for A0 > 1; if C1 < 0 then the phase portrait is (7), if C1 = 0 we have the phase portrait (6) and if C1 > 0 the phase portrait is (5). For 0 < A0 ≤ 1 there are also three cases: if C1 < 0 the phase portrait is (4), if C1 = 0 then we have the phase portrait (3) and if C1 > 0 the phase portrait is (2). If A0 < 1 we have the phase portrait (1). Finally, if A0 = 1 and B0 ≤ 2 we have again the phase portrait (3). The phase portraits are given in Figure 1.  PHASE PORTRAITS OF ABEL SYSTEMS OF SECOND KIND 17 Proof of Theorem 4(K1b). We deal here with the differential system ẋ = xy, ẏ = S(x − r)(x − 1) + A0 y 2 , (11) where S ∈ {−1, 1}, A0 6= 0 and r 6= 1.  p  System (11) has the following finite singular points: (1, 0), (r, 0), and 0, ± −S r/A0 . p 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 7. We first write system (11) in the form given by the Theorem, for that we move the point p to the origin, p 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) becomes x2 ẋ = −y − xy, ẏ = x − − A0 y 2 . (r − 1) Then we have ω1 = ω2 = ω3 = 0, so we have a center. p For the point (r, 0) the eigenvalues are ± −S r(1 − r). So, when either S = 1 and r 6∈ (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 6∈ (0, 1) we apply again Theorem 7. By moving the point to the origin and  p p applying the change of variables and time (x, y, t) 7→ x, y r/S(1 − r), −t/ Sr(1 − r) system (11) becomes 1 x2 A0 2 ẋ = −y − xy, ẏ = x + − y . r r−1 r It is easy to see that ω1 = ω2 = ω3 = 0, so the point is a center. 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 5, so y = g(x) = (A0 /S)x2 + . . . , thus we obtain f (x) = (−A0 /S 2 )x3 + . . . , so if A0 > 0 the point is a saddle and if A0 < 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. p  The last two points exist only if A0 r S < 0. The point 0, −S r/A0 has eigenvalues p p −S r/A0 and 2A0 −S r/A0 . Thus, when A0 0 we  have an unstable node. Similarly, the point 0, − −S r/A0 has eigenvalues − −S r/A0 p and −2A0 −S r/A0 . Thus, when A0 < 0 we have a saddle and when A0 > 0 we have a stable node. According to the expression (7), system (11) at infinity can be studied from the system u̇ = −u(Su2 + Srv 2 − S(r + 1)uv + A0 − 1), v̇ = −v(Su2 + Srv 2 − S(r + 1)uv + A0 ). 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) p and two more points at ± (1 − A0 )/S, 0 . The origin on U2 has eigenvalues −A0 , 1 − A0 thus we have an unstable node for A0 < 0; a saddle point for 0 < A0 < 1 and a stable node for A0 > 1. When A0 = 1 the point is semi-hyperbolic. With a change of time, the above system for A0 = 1 is written in the form given in Proposition 5 and we obtain f (u) = Su3 + . . . , 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−A0 ) > 0. Concerning their stability, the eigenvalues are −1, −2(1 − A0 ) so if S = 1 and A0 < 1 the points are stable nodes and if S = −1 and A0 > 1 are saddles. Table 2 summarizes the different possibilities of behavior of the singular points of sys- tem 11 depending on the values of the parameters.  18 A. FERRAGUT, J.D. GARCÍA-SALDAÑA, C. VALLS Singular point Conditions Type of point S(1 − r) > 0 Saddle (1, 0) S(1 − r) < 0 Center Sr(r − 1) > 0 Saddle Sr(r − 1) < 0 Center (r, 0) S = 1, r = 0, A0 < 0 Stable Node S = −1, r = 0, A0 < 0 Unstable Node r = 0, A0 > 0 Saddle  q  A0 < 0, Sr > 0 Saddles 0, ± −Sr A0 A0 > 0, Sr < 0 Nodes A0 < 0 Unstable Node 0 < A0 < 1 Saddle (0, 0) ∈ U2 A0 > 1 Stable Node S = 1, A0 = 1 Stable Node S = −1, A0 = 1 Saddle  q  1−A0 S = 1, A0 < 1 Stable Nodes ± S ,0 ∈ U2 S = −1, A0 > 1 Saddles Table 2. Behavior of the singular points for the family (K1b). The rest of the cases follow in a similar way as the previous ones. We shall provide the table with the behavior of the singular points and additional important information if needed. Proof of Theorem 4(K1c). We deal now with the differential system ẋ = xy, ẏ = S(x2 − 1) + B1 xy + A0 y 2 , (12) where S2 = 1, A0 6= 0 and B1 ≥ 0. Table 3 shows the behavior of all singular points depending on the parameters. In the case S = −1, B1 = 0, A0 > 0 we have only two finite singular points and they are centres, this can be proved by applying Theorem 7. The phase portrait is determined by the value of A0 . If 0 < A0 ≤ 1 the phase portrait is (13) and if A0 > 1 the phase portrait is (18) of Figure 1. The behavior of the all critical points at infinity of system (12) can be studied on U2 through the system u̇ = −u(Su2 − Sv 2 + B1 u + A0 − 1), v̇ = −v(Su2 − Sv 2 + B1 u + A0 ).  Proof of Theorem 4(K2a). We deal now with the differential system ẋ = xy, ẏ = Sx2 + y + A0 y 2 , (13) where S2 = 1, A0 6= 0. Table 4 shows the behavior of all singular points depending on the parameters. According to the expression (7), system (13) at infinity can be studied from the system u̇ = −u(Su2 + v + A0 − 1), v̇ = −v(Su2 + v + A0 ). We do not need to study the expression (6) because all the infinite singular points can be studied from the above differential system.  PHASE PORTRAITS OF ABEL SYSTEMS OF SECOND KIND 19 Singular point Conditions Type of point S=1 Saddle √ (1, 0) S = −1, B1 ≥ 2 2 Unstable Node √ S = −1, 0 < B1 < 2 2 Unstable Focus S = −1, B1 = 0 Center S=1 Saddle √ (−1, 0) S = −1, B1 ≥ 2 2 Stable Node √ S = −1, 0 < B1 < 2 2 Stable Focus S = −1, B1 = 0 Center  q  S = 1, A0 > 0 Nodes 0, ± AS0 S = −1, A0 < 0 Saddles A0 < 0 Unstable Node (0, 0) ∈ U2 0 < A0 < 1 Saddle A0 > 1 Stable Node A0 = 1 Saddle-Node C2 > 0, A0 < 1 Stable Node  √  −B1 + C2 , 0 ∈ U2 C2 > 0, A0 > 1 Saddle 2S A0 = 1 Saddle-Node C2 = 0 Saddle-Node C2 > 0, S = 1 Stable Node  √  C2 > 0, S = −1 Saddle −B1 − C2 2S , 0 ∈ U2 A0 = 1, S = −1 Saddle A0 = 1, S = 1 Stable Node C2 = 0 Saddle-Node Table 3. Behavior of the singular points for the family (K1c). Here C2 = B12 − 4A0 S + 4S. Proof of Theorem 4(K2b). We deal now with the differential system ẋ = xy, ẏ = S(x − r)2 + y + A0 y 2 , (14) where S 2 = 1, A0 6= 0, r ∈ {0, 1}. In Table 5 we summarize the behavior of all singular points depending on the parameters. We consider two cases: r =  1 and r = 0. For the case r = 1 system (14) has three p finite singular point: (1, 0) and 0, ± −S/A0 ) . The point (1, 0) is semi-hyperbolic. By applying Proposition 6 we have y = f (x) = 0 and G(x, f (x)) = Sx2 + . . . , moreover Φ = 0 then the point is a cusp. The phase portraits corresponding to the cusp are (41), (42), (45) and (46), depending on the value of A0 and S. The infinite singular points of system (14) with r = 1 can be studied from the system u̇ = −u(Su2 − 2Suv + Sv 2 + A0 − 1), v̇ = −v(Su2 − 2Suv + Sv 2 + A0 ), on U2 , and we do not need to study the expression (6) because all the infinite singular points can be studied from the above differential system. 20 A. FERRAGUT, J.D. GARCÍA-SALDAÑA, C. VALLS Singular point Conditions Type of point S=1 Saddle (0, 0) S = −1 Unstable Node   A0 > 0 Stable Node 0, − A10 A0 < 0 Saddle A0 < 0 Unstable Node 0 < A0 < 1 Saddle (0, 0) ∈ U2 A0 > 1 Stable Node A0 = 1, S = 1 Stable Node A0 = 1, S = −1 Saddle  q  1−A0 S = 1, A0 < 1 Stable Nodes ± S , 0 ∈ U2 S = −1, A0 > 1 Saddles Table 4. Behavior of the singular points for the family (K2a). Singular point Conditions Type of point r=0 Degenerate (r, 0) r=1 Cusp  q  r = 1, A0 > 0, S = 1 Saddles 0, ± − AS0 r = 1, A0 < 0, S = −1 Nodes A0 < 0 Unstable Node 0 < A0 < 1 Saddle (0, 0) ∈ U2 A0 > 1 Stable Node A0 = 1, S = 1 Stable Node A0 = 1, S = −1 Saddle  q  1−A0 S = 1, A0 < 1 Stable Nodes ± S , 0 ∈ U2 S = −1, A0 > 1 Saddles Table 5. Behavior of the singular points for the family (K2b). For the case r = 0 system (14) has only one finite singular point: (0, 0). This point is degenerate so we determined its behavior 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) ṗ = −p(Sp2 + A0 − 1), ẏ = y(Sp2 + A0 ).  p  This new system has three singular points on y = 0: (0, 0), ± (1 − A0 )/S, 0 . The eigenvalues of (0, 0) are A0 and 1 − A0 so either if A0 < 0 or A0 > 1 then the point is a saddle, if 0 < A0 < 1 then the point is an unstable node and if A0 = 1 the point is semi- hyperbolic. Applying proposition (5) we prove that the point is unstable node if S = −1 and a saddle if S = 1. PHASE PORTRAITS OF ABEL SYSTEMS OF SECOND KIND 21  p  On the other hand, the points ± (1 − A0 )/S, 0 have eigenvalues 1, −2(1 − A0 ) so they are saddles if A0 < 1 and unstable nodes if A0 > 1. Finally, undoing the blow-up we obtain the local behaviour of the point (0, 0) of system (14) with r = 0. According to the expression (7) the infinite singular points of system (14) for the case r = 0 can be studied from the system u̇ = −u(Su2 + A0 − 1), v̇ = −v(Su2 + A0 ), on U2 . 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 4(K2c). We deal now with the differential system ẋ = xy, ẏ = Sx2 + B1 xy + A0 y 2 , (15) where S 2 = 1, A0 6= 0 and B1 > 0. Table 6 shows the behavior of all singular points depending on the parameters. Singular point Conditions Type of point (0, 0) Degenerate A0 < 0 Unstable Node (0, 0) ∈ U2 0 < A0 < 1 Saddle A0 > 1 Stable Node A0 = 1 Saddle-Node C2 > 0, A0 < 1 Stable Node  √  −B1 + C2 , 0 ∈ U2 C2 > 0, A0 > 1 Saddle 2S A0 = 1 Saddle-Node C2 = 0 Saddle-Node C2 > 0, S = 1 Stable Node  √  C2 > 0, S = −1 Saddle −B1 − C2 2S , 0 ∈ U2 A0 = 1, S = −1 Saddle A0 = 1, S = 1 Stable Node C2 = 0 Saddle-Node Table 6. Behavior of the singular points for the family (K2c). Here again C2 = B12 − 4A0 S + 4S. System (15) has only one finite singular point: (0, 0). This point is degenerate so, again as in case (K2b), we use the blow-up technique to known its behavior. System (15) at infinity can be studied from the system u̇ = −u(Su2 + B1 u + A0 − 1), v̇ = −v(Su2 + B1 u + A0 ). √  This system has three singular points: (0, 0) and two more points at (−B1 ± C2 )/2S, 0 where C2 := B12 − 4A0 S + 4S. These points only exist if C2 ≥ 0. When C2 = 0 the point (−B1 /2S, 0) is a semi-hyperbolic singular point. Applying Proposition 5 we have v = f (u) = 0 so G(u, f (u)) = −(B1 /2)u2 + . . . thus the point is a saddle-node. When A0 = 1 the point (0, 0) on U2 is a semi-hyperbolic singular point. By using Proposition 5 we prove that the point is a saddle-node.  Proof of Theorem 4(K3a). We deal now with the differential system ẋ = xy, ẏ = S(x2 + 1) + B0 y + A0 y 2 , (16) 22 A. FERRAGUT, J.D. GARCÍA-SALDAÑA, C. VALLS where S 2 = 1, A0 6= 0 and B0 > 0. Table 7 shows the behavior of all singular points depending on the parameters. Singular point Conditions Type of point  √  C3 > 0, S = 1 Saddle 0, − B02A + C3 0 C3 > 0, S = −1 Unstable Node  √  A0 < 0, C3 > 0 Saddle − C3 0, − B02A 0 A0 > 0, C3 > 0 Stable Node C3 = 0 Saddle-Node A0 < 0 Unstable Node 0 < A0 < 1 Saddle (0, 0) ∈ U2 A0 > 1 Stable Node A0 = 1, S = 1 Stable Node A0 = 1, S = −1 Saddle  q  1−A0 A0 < 1 Stable Nodes ± S , 0 ∈ U2 A0 > 1 Saddles Table 7. Behavior of the singular points for the family (K3a). Here C3 := B02 − 4A0 S. p  System (16) has two singular points 0, (−B0 ± B02 − 4A0 S)/2A0 . Let C3 := B02 −4A0 S be so the points only exist if C3 ≥ 0. For the case C3 = 0 the critical point becomes (0, −2S/B0 ) with eigenvalues 0 and −2S/B0 so is a semi-hyperbolic singular point. Applying Proposition 5 we have y = f (x) = 0 so G(x, f (x)) = −(B03 /8S 2 )x2 + . . . , thus the point is a saddle-node. The phase portrait in this case corresponds to (49) and (52) of Figure 1 depending of the value of A0 . The behavior of system (16) at infinity can be studied from the system u̇ = −u(Sv 2 + Su2 + B0 v + A0 − 1), v̇ = −v(Sv 2 + Su2 + B0 v + A0 ).  Proof of Theorem 4(K3b). We deal now with the differential system ẋ = xy, ẏ = S((x − a)2 + 1) + A0 y 2 , (17) where S 2 = 1, A0 6= 0 and a ∈ R. Table 8 shows the behavior of all singular points depending on the parameters. According to the expression (7), system (16) at infinity can be studied from the system u̇ = −u(S(av − u)2 + Sv 2 + A0 − 1), v̇ = −v(S(av − u)2 + Sv 2 + A0 ). The behavior of the infinite singular points is the same as in the case (K3a).  Proof of Theorem 4(K3c). We deal now with the differential system ẋ = xy, ẏ = S(x2 + 1) + B1 xy + A0 y 2 , (18) where S2= 1, A0 6= 0 and B1 > 0. Table 9 shows the behavior of all singular points depending on the parameters. The behavior of system (18) at infinity can be studied from the system u̇ = −u(Su2 + Sv 2 + B1 u + A0 − 1), v̇ = −v(Su2 + Sv 2 + B1 u + A0 ). √  The above system has at most three singular points: (0, 0) and (−B1 ± C2 )/2S, 0 where C2 := B12 − 4A0 S + 4S. If A0 = 1 the point (0, 0) is semi-hyperbolic and applying Propo- sition 5 it can be proved that it is a saddle-node. For the case C2 = 0, again by using PHASE PORTRAITS OF ABEL SYSTEMS OF SECOND KIND 23 Singular point Conditions Type of point  q  2 +1) A0 < 0, S = 1 Saddle 0, −S(a A0 A0 > 0, S = −1 Unstable Node  q  −S(a2 +1) A0 < 0, S = 1 Saddle 0, − A0 A0 > 0, S = −1 Stable Node A0 < 0 Unstable Node 0 < A0 < 1 Saddle (0, 0) ∈ U2 A0 > 1 Stable Node A0 = 1, S = 1 Stable Node A0 = 1, S = −1 Saddle  q  1−A0 A0 < 1 Stable Nodes ± S , 0 ∈ U2 A0 > 1 Saddles Table 8. Behavior of the singular points for the family (K3b). Proposition 5, we prove that the point (−B1 /2S, 0) is a saddle-node. The phase portrait in this case corresponds to (53) or (56) of Figure 1 depending of the value of A0 .  Singular point Conditions Type of point  q  S = 1, A0 < 0 Saddle 0, − AS0 S = −1, A0 > 0 Unstable Node  q  S = 1, A0 < 0 Saddle 0, − − AS0 S = −1, A0 > 0 Stable Node A0 < 0 Unstable Node (0, 0) ∈ U2 0 < A0 < 1 Saddle A0 > 1 Stable Node A0 = 1 Saddle-Node C2 > 0, A0 < 1 Stable Node  √  −B1 + C2 , 0 ∈ U2 C2 > 0, A0 > 1 Saddle 2S A0 = 1 Saddle-Node C2 = 0 Saddle-Node C2 > 0, S = 1 Stable Node  √  C2 > 0, S = −1 Saddle −B1 − C2 2S , 0 ∈ U2 A0 = 1, S = −1 Saddle A0 = 1, S = 1 Stable Node C2 = 0 Saddle-Node Table 9. Behavior of the singular points for the family (K3c). Here C2 = B12 − 4A0 S + 4S. 24 A. FERRAGUT, J.D. GARCÍA-SALDAÑA, C. VALLS Proof of Theorem 4(K4a). We deal now with the differential system ẋ = xy, ẏ = −T + x + A0 y 2 , (19) where A0 6= 0 and T ∈ {−1, 0, 1}. Table 10 shows the behavior of all singular points depending on the parameters.   p System (19) has at most three finite singular points: (T, 0) and 0, ± T /A0 . For the √ √ case of the point (T, 0) the eigenvalues are T and − T , so the point is a saddle if T = 1 and it is a center if T = −1. It can be proved by moving the singular point to the origin and applying Theorem 7 to the system obtained: ẋ = −y + xy, ẏ = x + A0 y 2 , the phase portraits in this case are (59)-(62) of Figure 1 depending of the value of A0 . If T = 0 then according to Proposition 6 the point is a saddle if A0 > 0 and is an unstable node if A0 < 0. Following the expression (6) and (7), the study of the behavior of the singular points at infinity can be done through the systems u̇ = (A0 − 1)u2 + v − T v 2 , v̇ = −uv, (20) u̇ = −u(−T v 2 + uv + A0 − 1), v̇ = −v(−T v 2 + uv + A0 ), (21) on U1 and U2 respectively. Note that when A0 = 1 the infinity is degenerate. System (20) has a singular point: (0, 0), which is semi-hyperbolic. Applying Proposition 5 we have v = g(u) = (1 − A0 ) + . . . and then F (u, g(u)) = (A0 − 1)u3 + . . . . So if A0 > 1 the point is an unstable node and if A0 < 1 is a saddle.  Singular point Conditions Type of point T = −1 Center (T, 0) T =1 Saddle T = 0, A0 > 0 Saddle T = 0, A0 < 0 Unstable Node  q  T = 1, A0 > 0 Nodes 0, ± AT0 T = −1, A0 < 0 Saddles A0 > 1 Unstable Node (0, 0) ∈ U1 A0 < 1 Saddle A0 < 0 Unstable Node (0, 0) ∈ U2 0 < A0 < 1 Saddle A0 > 1 Stable Node Table 10. Behavior of the singular points for the family (K4a). If A0 = 1 then the infinity is degenerate. Proof of Theorem 4(K5a). We deal now with the differential system ẋ = xy, ẏ = S + B0 y + A0 y 2 , (22) where S 2 = 1, A0 6= 0 and B0 > 0. The behavior of all singular points depending on the parameters is shown in Table 11.  p  System (22) has at most two finite singular point: 0, (−B0 ± B02 − 4A0 S)/2A0 . The existence of the points depend on the value of the discriminant C3 := B02 − 4A0 S. When C3 = 0 by applying Proposition 5 we prove that the point (0, −2S/B0 ) is a saddle-node and depending of the value of A0 we have the phase portrait (72), (80) or (81), given in Figure 1. PHASE PORTRAITS OF ABEL SYSTEMS OF SECOND KIND 25 Following (6) and (7), system (22) at infinity can be studied from the systems u̇ = Sv 2 + B0 uv + (A0 − 1)u2 , v̇ = −uv, (23) u̇ = −u(Sv 2 + B0 v + A0 − 1), v̇ = −v(Sv 2 + B0 v + A0 ), (24) on U1 and U2 respectively. Note that if A = 1 the infinity is degenerate. On the other hand, for A0 6= 1 system (23) has a singular point: (0, 0), such a point is degenerate. To obtain its local behavior we apply the blow-up technique. Singular point Conditions Type of point  √  C3 > 0, S = 1 Saddle + C3 0, − B02A 0 C3 > 0, S = −1 Unstable Node A0 < 0, C3 > 0 Saddle  √  − C3 0, − B02A A0 > 0, C3 > 0 Stable Node 0 C3 = 0 Saddle-Node (0, 0) ∈ U1 A0 6= 1 Degenerate A0 < 0 Unstable Node (0, 0) ∈ U2 0 < A0 < 1 Saddle A0 > 1 Stable Node Table 11. Behavior of the singular points for the family (K5a). Here again C3 := B02 − 4A0 S. If A0 = 1 then the infinity is degenerate.  Proof of Theorem 4(K5b). We deal now with the differential system ẋ = xy, ẏ = S + A0 y 2 , (25) where S2 = 1, A0 6= 0. In Table 12 we give the behavior of the singular points depending on the parameters. According to (6) and (7), system (25) at infinity can be studied from the systems u̇ = (A0 − 1)u2 + Sv 2 , v̇ = −uv, (26) u̇ = −u(Sv 2 + A0 − 1), v̇ = −v(Sv 2 + A0 ). (27) The behavior at infinity is the same as in case (K5a). Singular point Conditions Type of point  q  S = 1, A0 < 0 Saddle 0, − AS0 S = −1, A0 > 0 Unstable Node  q  S = 1, A0 < 0 Saddle 0, − − AS0 S = −1, A0 > 0 Stable Node (0, 0) ∈ U1 A0 6= 1 Degenerate A0 < 0 Unstable Node (0, 0) ∈ U2 0 < A0 < 1 Saddle A0 > 1 Stable Node Table 12. Behavior of the singular points for the family (K5b). 26 A. FERRAGUT, J.D. GARCÍA-SALDAÑA, C. VALLS  Proof of Theorem 4(K5c). We deal now with the differential system ẋ = xy, ẏ = S + xy + A0 y 2 , (28) where S 2 = 1, A0 6= 0. The behavior of all singular points depending on the parameters are shown in Table 13. As stated by expressions (6) and (7), system (28) at infinity can be studied from systems u̇ = (A0 − 1)u2 + Sv 2 + u, v̇ = −uv, (29) u̇ = −u(Sv 2 + u + A0 − 1), v̇ = −v(Sv 2 + u + A0 ), (30) on U1 and U2 respectively. System (30) has two singular points: (0, 0) and (1 − A0 , 0). When A0 = 1 the point is semi-hyperbolic. Doing a change of time to system (30) and applying Proposition 5 we prove that the point is a saddle-node. So the phase portraits given in Figure 1 that correspond to this case are (53) if S = −1 or (57) if S = 1. With respect to system (29), the origin is the only singular point and it is semi-hyperbolic. Applying Proposition 5 we have v = g(u) = −Su2 + . . . , and hence F (u, g(u)) = f (u) = Su3 + . . . , so if S = 1 then the point is an unstable node and if S = −1 then it is a saddle. Singular point Conditions Type of point  q  S = 1, A0 < 0 Saddle 0, − AS0 S = −1, A0 > 0 Unstable Node  q  S = 1, A0 < 0 Saddle 0, − − AS0 S = −1, A0 > 0 Stable Node S=1 Unstable Node (0, 0) ∈ U1 S = −1 Saddle A0 < 0 Unstable Node (0, 0) ∈ U2 0 < A0 < 1 Saddle A0 > 1 Stable Node A0 = 1 Saddle-Node Table 13. Behavior of the singular points for the family (K5c).  5. Invariant straight lines We compute in this last section the invariant straight lines of the differential systems appearing in Proposition 2. When real, they appear in the phase portraits of Figure 1. We recall that an algebraic curve f = 0, f ∈ C[x, y], is invariant under the flow of a differential system ẋ = P , ẏ = Q of degree m ∈ N if there exists k ∈ C[x, y], deg k < m, such that ∂f ∂f P +Q = kf. (31) ∂x ∂y 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 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. PHASE PORTRAITS OF ABEL SYSTEMS OF SECOND KIND 27 Family Conditions Straight lines p √ √ B0 = (1 − 2A0 ) S/(1 − A0 ) S(1 ± x) − 1 − A0 y = 0 (K1a) p √ √ B0 = −(1 − 2A0 ) S/(1 − A0 ) S(1 ± x) + 1 − A0 y = 0 √ √ A0 = 1/(1 − r) r − 1(x − r) ± rSy = 0 (K1b) p A0 = r/(r − 1) (1 − r)S(x − 1) ± y = 0 p √ √ B1 = (1 − 2A0 ) S/A0 S(x ± 1) − A0 y = 0 (K1c) p √ √ B1 = −(1 − 2A0 ) S/A0 S(x ± 1) + A0 y = 0 √ √ (K2b) r=0 Sx ± 1 − A0 y = 0 p (K2c) (B1 ± B12 + 4S(1 − A0 ))x + 2(A0 − 1)y = 0 p √ √ B0 = (1 − 2A0 )b S/(A0 − 1) S(bi ± x) − 1 − A0 y = 0 (K3a) p √ √ B0 = (1 − 2A0 )b S/(A0 − 1) S(bi ± x) + 1 − A0 y = 0 √ (K3b) A0 = 1/2, a = 0 √2S(x ± i) + y = 0 2S(x ± i) − y = 0 p √ √ B1 = (1 − 2A0 ) S/A0 S(x ± i) − A0 y = 0 (K3c) p √ √ B1 = −(1 − 2A0 ) S/A0 S(x ± i) + A0 y = 0 √ (K4a) A0 = 1, T 2 = 1 x − T ± Ty = 0 p (K5a) (B0 ± B02 − 4SA0 ) + 2A0 y = 0 √ √ (K5b) S ± i A0 y = 0 Table 14. Complex invariant straight lines of each family of differential systems appearing in Proposition 2 besides x = 0. A family not appearing in this table is a family with only one invariant straight line, x = 0. Acknowledgements A. Ferragut is supported by the Ministerio de Economía, Industria y Competitividad - Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER) and by the Universitat Jaume I grant P1-1B2015-16. J. D. García-Saldaña is supported by FONDECyT grant number 11171115/2017. C. Valls is supported by Portuguese National Funds through FCT - Fundação para a Ciência e a Tecnologia within the project PTDC/MAT/117106/2010 and by CAMGSD. References [1] G. Alobaidi amd R. Mailler, On the Abel equation of the second kind with sinusoidal forcing, Nonlinear Analysis: Modeling and Control, 12 (2007), 33–44. [2] M.J. Álvarez, A. Ferragut and X. Jarque A survey on the blow up technique, Int. J. Bifurcation and Chaos 21 (2011), 3103–3118. [3] A.A. Andronov, E.A. Leontovich, I.I. Gordon and A.G. Maier, Qualitative theory of second- order dynamic systems, John Wiley and Sons, 1973. [4] V. I. Arnol’d and Y. S. Ilyashenko, Dynamical Systems I, Ordinary Differential Equations. Ency- clopaedia of Mathematical Sciences, Vol. 1–2, Springer-Verlag, Heidelberg, 1988. [5] L. Cairó and J. Llibre, Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 2, Nonlinear Anal., Ser. A: Theory Methods 67 (2007), 327–348. [6] J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, On the integrability of two-dimensional flows, J. Differential Equations 157 (1999), 163–182. [7] A. Ferragut and C. Valls, Phase portraits of Abel quadratic differential systems of second kind, Dyn. Syst. (2017), DOI 10.1080/14689367.2017.1402296. [8] E.A.V. Gonzales, Generic properties of polynomial vector fields at infinity, Trans. Amer. Math. Soc. 143 (1969), 201–222. [9] C. Li, Two problems of planar quadratic systems, Scientia Sinica (Series A) 26 (1983), 471–481. [10] D.A. Neumann, Classification of continuous flows on 2-manifolds, Proc. Amer. Math. Soc. 48 (1975), 73–81. 28 A. FERRAGUT, J.D. GARCÍA-SALDAÑA, C. VALLS [11] N. Rozov Encyclopeida of Mathematics. Available online at http:///www.encyclopediaofmath.org/index.php?title=Abel_differential_equation(and)oldid=18925. [12] J. Sotomayor, Liçóes de Equaçóes Diferenciais Ordinárias, IMPA, Rio de Janerio, 1979. [13] Ye Yanqian et al., Theory of Limit Cycles, Transl. Math. Monographs 66, Amer. Math. Soc., Providence, 1984. A. Ferragut: Departament de Matemàtiques and IMAC; Universitat Jaume I, Av. de Vicent Sos Baynat, s/n, Campus del Riu Sec, 12071 Castelló de la Plana, Spain E-mail address: ferragut@uji.es J.D. García-Saldaña: Departamento de Matemática y Física Aplicadas, Universidad Cató- lica de la Santísima Concepción, Alonso de Ribera 2850, Concepción, Chile E-mail address: jgarcias@ucsc.cl C. Valls: Departamento de Matemática, Instituto Superior Técnico, Universidade de Lis- boa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal E-mail address: cvalls@math.ist.utl.pt