This is a preprint of: “Global phase portraits of the quadratic systems having a singular and irreducible invariant curve of degree 3”, Jaume Llibre, Chara Pantazi, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 33(1), 2350003, 2023. DOI: [10.1142/S0218127423500037] GLOBAL PHASE PORTRAITS OF THE QUADRATIC SYSTEMS HAVING A SINGULAR AND IRREDUCIBLE INVARIANT CURVE OF DEGREE 3 JAUME LLIBRE1 AND CHARA PANTAZI2 Abstract. Any singular irreducible cubic after an affine transformation can be written as either y 2 = x3 , or y 2 = x2 (x + 1), or y 2 = x2 (x − 1). We classify the phase portraits of all quadratic polynomial differential systems having the invariant cubic y 2 = x2 (x + 1). We prove that there are 65 different topological phase portraits for such quadratic polynomial differential systems. 1. Introduction and statement of the main result Quadratic polynomial differential systems (or simply quadratic systems) are systems that can be written into the form ẋ = P (x, y) = P0 + P1 + P2 , ẏ = Q(x, y) = Q0 + Q1 + Q2 , (1) where Pi and Qi are real polynomials of degree i in the variables (x, y) and P22 + Q22 6= 0. An extensive literature is dedicated to the study of the quadratic systems these last years. For a good survey see the book of Reyn [37] or the book of Artes et.al [4], and references therein. For example, the following families of quadratic systems have been studied: homogeneous [16], semi-homogeneous [12], bounded [18], reversible [24, 17], Hamiltonian [2, 15], Lienard [19], integrable using Carleman and Painlevé tools [25], rational integrable [5, 6, 7], the ones having a star nodal point [10], a center [42, 31, 17, 42], one focus and one antisaddle [3], with a semi– elementary triple node [8], chordal [21, 22], with four infinite singular points and one invariant straight line [38], with invariant lines [41], and so on. There is also an extensively literature about Hilbert’s sixteen problem and quadratic systems, see for example [13, 23, 29, 30, 31, 44], and the notion of ciclicity [45, 26, 15], and so on. For the study of some geometric properties of quadratic systems see [39, 40], and others. In particular we pay attention on reference [28] where the authors present a classification of all quadratic systems having one real reducible invariant algebraic curve of degree 3. In [11] it is proved that a cubic algebraic curve (or simply a cubic) is singular and irreducible if and only if it can be written after affine transformations into one of the forms y 2 = x3 , y 2 = x2 (x + 1), y 2 = x2 (x − 1). See Figure 1. -1 1 f1 = y 2 − x3 f2 = y 2 − x2 (x + 1) f3 = y 2 − x2 (x − 1) Figure 1. Singular and irreducible algebraic curves of degree 3. 1 The goal of this paper is to continue the classification of the phase portraits in the Poincaré disc of the quadratic systems having some invariant cubic. Thus our main result is to provide all distinct topological phase portraits of the quadratic systems having the invariant cubic y 2 = x2 (x + 1). Let f = f (x, y) = 0 be a real polynomial. We say that f = 0 is an invariant algebraic curve of system (1) if it satisfies ∂f ∂f P +Q = Kf, ∂x ∂y for some polynomial K called the cofactor of the curve f = 0. Note that an invariant algebraic curve is formed by orbits of system (1). It is easy to check that the quadratic systems (1) having f = y 2 − x2 (x + 1) = 0 as invariant algebraic curve are ẋ = ax + by + ax2 + (3b − 2c)xy,   2 3 9 (2) ẏ = bx + ay + cx + axy + b − 3c y 2 , 2 2 See the appendix for a summary of about the Poincaré compactification of a polynomial differential system and the definition of the Poincaré disc D, which roughly speaking is to identify the interior of the unit closed disc D centered at the origin with the plane R2 and its boundary S1 with the infinity of R2 , in the plane we can go to infinity in as many as directions as points has the circle S1 . Then the Poincaré compactification consists in extend the quadratic differential system from the interior of D to its boundary S1 , i.e. to the infinity of R2 . In this way we can control the orbits of a polynomial differential in a neighborhood of the infinity, and in particular of a quadratic system. Our main result is the following. Theorem 1. For the quadratic systems (2) there are 65 non topological equivalent phase por- traits in the Poincaré disc. The rest of the paper is dedicated to proof Theorem 1. 2. Quadratic systems with the invariant algebraic curve y 2 − x2 (x + 1) = 0 All quadratic systems admitting y 2 − x2 (x + 1) = 0 as the invariant cubic y 2 − x2 (x + 1) = 0 can be written as systems (2). We distinguish the following cases. 2.1. Case a = 0 and c = 0. Then b 6= 0 otherwise the system is not quadratic, and without loss of generality we can consider b = 1. Then system (2) becomes 1  ẋ = y(1 + 3x), ẏ = 2x + 9y 2 , (3) 2 and has the rational first integral 27 y 2 + 9 x + 1 H(x, y) = . (3 x + 1)3 System (3) has the three invariant algebraic curves: f2 = 0, g1 = 3x + 1 = 0 and√g3 = 27 y 2 + 9 x + 1. Additionally, has the three finite singular points P0 (0, 0), P− −1/3, − 6/9 √  and P+ −1/3, 6/9 . P0 is a saddle, P− is an unstable node and P+ is a stable node. In what follows we use the notation introduced in the Appendix for studying the infinite sin- gular points using the Poincaré compactification. The origin of the chart (U1 , F1 ) is a nilpotent singular point and using Theorem 3.5 of [20] is a saddle, and doing blows ups we get its local phase portrait as it shows in Figure 2. Moreover, the origin of the local chart U2 is a hyperbolic stable node. The local and the global phase portraits of system (3) are given in Figure 2. 2 .. O2 .. c0 . . .. P . . . .. . O´1 . + P- P0 O1 . ..´ O2 .. S=17, R=4 Figure 2. The local and the global phase portraits of system (3). Here a = c = 0, b = 1 in system (2). 2.2. Case a = 0 and c 6= 0. Without loss of generality we can consider c = 1, and system (2) can be written as ẋ = by + (3b −2)xy,  9 (4) ẏ = bx + x2 + b − 3 y2, 2 where this b is the old b/c. For b = 2/3 system (4) is a Hamiltonian system with the first integral H = x3 + x2 − y 2 . System (4) for b 6= 2/3 has the rational first integral   b3 + 3 b2 (3 b − 2) x + 2 9 b2 − 12 b + 4 x2 + 27 b3 − 54 b2 + 36 b − 8 y 2 H(x, y) = . (3 bx + b − 2 x)3   The curve g2 = b3 + 3 b2 (3 b − 2) x + 2 9 b2 − 12 b + 4 x2 + 27 b3 − 54 b2 + 36 b − 8 y 2 = 0 is a conic for b 6= 2/3 and it is classified as follows: For b ∈ (−∞, 0) ∪ (0, 2/3) is a hyperbola, for b ∈ (2/3, 8/9) an imaginary ellipse, and for b > 8/9 is a real ellipse. For b = 0 we obtain two real invariant straight lines that intersect into a point. For b = 8/9 the conic is formed by two parallel imaginary straight lines. System (4) has the finite singular points (whenever they exist) √ ! √ ! b b 2b − 2 b b 2b − 2 P0 (0, 0), P1 (−b, 0), P− ,− , P+ , . 2 − 3 b (3 b − 2)3/2 2 − 3 b (3 b − 2)3/2 Note that the points P− and P+ are the intersection points of the three curves f2 = 0, g2 = 0 and g1 = (3 b − 2)x + b = 0. For b ∈ (0, 2/3) ∪ (1, ∞) the point P1 is on the left hand side of the straight line passing through the points P− and P+ . For b ∈ (−∞, 0) ∪ (0, 2/3) ∪ (1, +∞) the points P± exist. For b = 0 the four points collide into P0 . For b = 1 the three points P+ , P− and P1 collide between them. For 2/3 ≤ b < 1 only exist the finite singular points P0 and P1 . The point P0 has the Jacobian matrix ! 0 b , b 0 and its eigenvalues are ±b. For b = 0 the point P0 is linearly zero and doing blow–ups we obtain that its local phase portrait is the union of two elliptic and two parabolic sectors, see Figure 4. The point P1 has the Jacobian matrix ! 0 −3 b2 + 3 b , −b 0 3 b −∞ 0 2/3 8/9 1 ∞ P0 S pepep S S S S S S S P1 C − C C C C C peph S P+ Ns − Ns − − − − − Nu P− Nu − Nu − − − − − Ns Q− S S S − − − − − − Q+ S S S − − − − − − O2 Nu N u Nu Ns Ns N s Ns Ns Ns Table 1. The finite singular points of system (4) √ and its eigenvalues are ±b 3b − 3. For b = 0 the point P1 coincides with P0 . For b = 1 the point P1 is a nilpotent point and by Theorem 3.5 of [20] we have that P1 is the union of one elliptic and one hyperbolic sectors separated by two parabolic sectors, see Figure 9. The eigenvalues of the point P+ are   √ √ q 2 √ 6 2 b − 2b + 2 (b − 1) (3 b − 2) − 4 2 b − 2 b λ1 = ,  (3 b − 2)3/2q  √ √ b (−6 b + 4) 2 b − 2 + 2 (b − 1) (3 b − 2)2 λ2 = − . (3 b − 2)3/2 The eigenvalues of the point P− are   √ √ q 2 √ −6 2 b − 2b + 2 (b − 1) (3 b − 2) + 4 2 b − 2 b λ1 = 3/2 ,  (3 b − 2)  √ √ q √ 6 2 b − 2b + 2 (b − 1) (3 b − 2)2 − 4 2 b − 2 b λ2 = − . (3 b − 2)3/2 For both points P± the product of its eigenvalues is 6 b2 (b − 1)/(3 b − 2). In the chart (U1 , F1 ) for b < 2/3 we obtain the infinite singular points r ! r ! 2 2 Q+ , 0 , Q− − , 0 . 2 − 3b 2 − 3b √ The eigenvalues of the points Q± are ± 4 − 6b. The singular point at the origin of the chart (U2 , F2 ) has the Jacobian matrix   1 − 3b/2 b , 0 3 − 9b/2 and its eigenvalues are λ1 = 1 − 3/2 b, λ2 = 3 − 9/2 b. Thus, for b 6= 2/3 it is a node, whereas for b = 2/3 it is a nilpotent singular point, and by Theorem 3.5 of [20] it is a stable node. We note that the family (2) is invariant under the symmetry (x, y, a, t) → (x, −y, −a, −t). (5) 4 .. O2 .. c1 . . .. Q´- . Q+ .. . .. . . .. P + . P0 P 1 . P- . . Q´+ . Q- . .. O´2 .. S=23, R=5 Figure 3. The local and the global phase portraits of system (4). Here a = 0, b < 0 and c = 1. The curve g2 = 0 is a hyperbola. .. O2 .. c2 Q´- . . Q+ . . .. P0 .. Q´+ . . Q- . . .. O´2 .. S=19, R=6 Figure 4. The local and the global phase portraits of system (4). Here a = b = 0 and c = 1. The curve g2 = 0 are two intersecting real lines .. .. . . . .c 3 . . . .. P P0 P+ . .. 1 . P - . . . . . .. .. S=23, R=5 Figure 5. The local and the global phase portraits of system (4). Here a = 0, b ∈ (0, 2/3) and c = 1. The curve g2 = 0 is a hyperbola. 2.3. Case a 6= 0. Due to the symmetry (5) without loss of generality we can restrict our study to a > 0. We distinguish the following cases. Case 1: c = 0. Then system (2) becomes 3 9 ẋ = ax + by + ax2 + 3 bxy, axy + by 2 .ẏ = bx + ay +(6) 2 2 In order to study this family we distinguish the following two subcases according to the param- eter b. Subcase c = b = 0. Then system (2) becomes 1 ẋ = ax(x + 1), ẏ = ay(2 + 3x), (7) 2 5 .. .. c4 . .. P 1 P0 . .. .. .. S=9, R=3 Figure 6. The local and the global phase portraits of system (4). Here a = 0, b = 2/3 and c = 1. The system is Hamiltonian with H(x, y) = −x3 − x2 + y 2 . F = ex is an exponential factor. .. .. . .. P 1 P0 . .. .. .. S=9, R=3 Figure 7. The local and the global phase portraits of system (4). Here a = 0, b ∈ (2/3, 8/9] and c = 1. For b ∈ (2/3, 8/9) the curve g2 = 0 is an imaginary ellipse whereas for b = 8/9 are two parallel imaginary straight lines. .. .. . .. P 1 P0 . .. .. .. S=9, R=3 Figure 8. The local and the global phase portraits of system (4). Here a = 0, b ∈ (8/9, 1) and c = 1. The curve g2 = 0 is a real ellipse. and it has the rational first integral H = x2 (x+1)/y 2 . Without loss of generality we can consider that a = 1. The unique finite singular points are P0 = (0, 0) and P = (−1, 0). The point P0 is an unstable node. The point P ∈ {f2 = 0} is a stable node. The origin O2 of the chart (U2 , F2 ) has a Jacobian matrix identically zero. Doing blow ups O2 is the union of one parabolic and one hyperbolic sector. The local and the global phase portrait of system (7) is given in Figure 11. 6 .. .. c5 . P1 .. P0 . .. .. .. S=12, R=5 Figure 9. The local and the global phase portraits of system (4). Here a = 0, b = c = 1. The curve g2 = 0 is a real ellipse. O2 .. .. c6 . .. P+ . . .. . P1 . P- P0 . .. .. S=16, R=5 Figure 10. The local and the global phase portraits of system (4). Here a = 0, b > 1 and c = 1. The curve g2 = 0 is a real ellipse. .o2 . A1 . . . P. P 0 . . o1 . . . . . . . . S=16, R=5 Figure 11. The local and the global phase portraits of system (7). Here a = 1 and b = c = 0. Subcase c = 0, b 6= 0. Since b 6= 0 without loss of generality we can assume that b = 1. System (6) can be written as 3 9 ẋ = ax + y + ax2 + 3 xy, ẏ = x + ay + axy + y 2 , (8) 2 2 where now a is the previous a/b. System (8) has three finite singular points:   1 1 2 1 1 −9 a2 ± 3 A − a4 ∓ a2 A P0 = (0, 0), P± = − + a ± A, , 3 18 18 54 a √ with A = a 24 + a2 . Note that P0 , P± are points of the curve f2 = 0. For a = 1 the point √ P+ collides with P0 . The point P− is always on the left hand side of the point P+ . For a > 3 the 7 point P− is upper the point P+ , here left hand side or upper are with respect to the x and y axes. The linear part at the origin P0 has eigenvalues a + 1 and a − 1. So, for a ∈ (0, 1) the origin P0 is a saddle. For a > 1 it is an unstable node. For a = 1 we have that P0 is semi–hyperbolic and using Theorem 2.19 of [20] we obtain that is a saddle–node. Now we set  B− = a2 a6 − a4 A + 30 a4 − 18 a2 A + 378 a2 + 1728 − 72 A , and note that for a > 0 we have that B− > 0. At the point P− the eigenvalues are √ 1 −24 A − 42 a2 − a4 + a2 A ± 2B− λ1,2 = < 0, 72 a and consequently that P− is a stable node. We additionally set  B+ = a2 a6 + 30 a4 + a4 A + 378 a2 + 18 a2 A + 1728 + 72 A , and for a > 0 we have that B+ > 0. The eigenvalues associated to the point P+ are √ 1 42 a2 + a4 + a2 A − 24 A ± 2B+ λ1,2 = − . 72 a We have that λ1 > 0 and λ2 > 0 if a ∈ (0, 1), and λ1 λ2 < 0 if a > 1. So for a ∈ (0, 1) the point P+ is an unstable node and for a > 1 is a saddle. In the chart (U1 , F1 ) we obtain the two infinite singular points O1 (0, 0) and Q(−1/3, 0). The origin O1 is a saddle, and the point Q+ has eigenvalues 0 and −a/2, so it is semi-hyperbolic. By Theorem 2.19 of [20] we obtain that the point Q is a saddle–node. The origin of the chart (U2 , F2 ) is a stable node. The local and the global phase portraits of system (7) are given in Figures 12, 13, 14, 15 and 16. . O2 . A2 . .P+ . . . . P0 . O1 . . . . . . P- . Q . . S=22, R=5 Figure 12. The local and the global phase portraits of system (7). Here c = 0, b = 1 and a ∈ (0, 1). For a = 1 system (8) has only two singular points. The origin has eigenvalues 0, 2 and using Theorem 2.19 of [20] it is a saddle–node. The singular point (−5/9, −10/27) is a stable node and is on the curve f2 = 0, see Figure 13. Case 2: c 6= 0. Without loss of generality we can consider c = 1. System (2) becomes ẋ = ax + by + ax2 + (3b − 2)xy,   3 9 (9) ẏ = bx + ay + x2 + axy + b − 3 y2. 2 2 8 . O2 . A3 . . . . P0 . O1 . . . . . . P- . Q . . S=20, R=5 Figure 13. The local and the global phase portraits of system (7). Here c = 0, a = b = 1. . O2 . . . . .. P0 . O1 . .. . . . . P- P+ . Q . . S=22, R=5 Figure 14. The √ local and the global phase portraits of system (7). Here c = 0, b = 1 and a ∈ (1, 3). . . A4 . . . . P0 . . . . . . . . . P- P+ . . . S=21, R=4 Figure 15. The √ local and the global phase portraits of system (7). Here c = 0, b = 1 and a = 3. System (9) has the following finite singular points (whenever are defined):  a P0 = (0, 0), P1 = −b, , P± = (x0 , y0 ) , 3 with √ a2 − 6 b2 + 4 b ± a 24 b2 − 40 b + a2 + 16 x0 = , 2 (3 b − 2)2 √ −9 ab2 + 18 ab − a3 − 8 a ± (3 b2 − a2 − 2 b) 24 b2 − 40 b + a2 + 16 y0 = . 2 (3 b − 2)3 9 . . . . A5 . . P0 . . . . . . . P- . . . P+ . . S=22,R=5 Figure 16. The √ local and the global phase portraits of system (7). Here c = 0, b = 1 and a > 3. In the chart (U1 , F1 ) system (9) has the infinite singular points (whenever they exist) √ ! −a ± a2 − 24 b + 16 Q∓ = , 0 . 2(3 b − 2) The origin of the chart (U2 , F2 ) is an infinite singular point of system (9). We define the following bifurcations curves g1 = a2 + 16 − 24b, g5 = a + b = 0, g2 = 3b − 2 = 0, g6 = 9a2 b2 − 24a2 b + 64a2 − 432b2 + 432b3 = 0, g3 = 24 b2 − 40 b + a2 + 16 = 0, g7 = a2 + 9b3 − 9b2 . g4 = a − b = 0, We also consider the curves √ 1 h = a2 + g1 a − 6 b2 + 4 b, j1 = b − a. (10) 3 2.4. Finite Singular Points. Lemma 2. The number of finite singular points of system (9) is given in Figure 18. Proof. Note that the points P± are not defined whenever g2 = 0 and also when g3 < 0. Addi- cionally, on g3 = 0 the two points P± collide between them. Moreover, on the curve g7 = 0 for b ∈ (−∞, 0) the point P1 collide with the point P+ and for b ∈ (0, 1) the point P1 collide with P− . For a = b = 8/9 the point P+ collide with P0 whereas the point P− collide with P1 . On the straight line g5 = 0 the point P+ colapse with P0 . On g4 = 0 for a = b > 4/5 the point P+ colapse with P0 , whereas for 0 < a = b < 4/5 the point P− colapse with P0 . Finally for a = b = 4/5 the points P± collide with the point P0 .  Lemma 3. The local phase portrait at the point P0 is given in Figure 19. Proof. The point P0 has eigenvalues a ± b. Hence for b > a or b < −a we have that P0 is a hyperbolic saddle. For b > a or b > −a the point P0 is a hyperbolic unstable node. Over the straight lines g4 = 0 and g5 = 0 the point P0 is a semi–hyperbolic singular point, and from Theorem 2.19 of [20] we have that for a 6= 4/5 it is a saddle–node whereas for a = 4/5 it is a saddle.  Remark 4. The finite singular points P± are always points of the invariant curve f2 = 0. So cannot be foci or centers. Lemma 5. The local phase portrait at the point P1 is given in Figure 20. 10 g1=0 2 h=0 3( i ) g4=0 i) (i j1 =0 3 ) v 3 (iii) (i 1 3 3( v ) 4 g3=0 6 5 7 10 3( v i ) 9 3( v i i ) 8 11 13 12 g2=0 17( i ) 16 15 14 17( i i ) 17( i i i ) g7=0 h=0 g6=0 19( i ) 20 19 ( 18( i ) ii) g5=0 19( i i i ) 18( i i ) 18( i i i ) 21 Figure 17. c = 1, a > 0. The bifurcation curves define 21 regions. It is a qualitative picture. √ Proof. The point P1 has eigenvalues e± = ab/4 − a/3 ± g6 /12. For g6 < 0 the eigenvalues become comblex with non–zero real part. On g6 = 0 we have that e+ = e− and P1 is a node and for g6 > 0 we have two different real eigenvalues. Note that e− e+ = −g7 /3. Hence, for g7 > 0 we have that P1 is a hyperbolic saddle, whereas for g7 < 0 and g6 > 0 the point P1 is a hyperbolic stable node. On g7 = 0one of the eigenvalues of P1 becomes zero and P1 is semi–hyperbolic, and from Theorem 2.19 of [20] we have that P1 is a saddle–node.  Lemma 6. The local phase portrait at the singular point P+ is given in Figure 21. Proof. We recall that the point P+ is defined for g2 6= 0 and g3 ≥ 0. On the curve g3 = 0 the point P+ coincides with P− . For a = b = 4/5 ∈ {g3 = 0} ∩ {g4 = 0} the points P+ and P− collide with P0 . Additionally, the point P+ has the eigenvalues √ √ a(−a2 − 42 b2 + 76 b − 32 ) + −a2 + 24 b2 − 16 b g3 ± 2A+ , 8 (3 b − 2)2 11 g=0 4 4 b=1 3 g =0 g=0 3 7 4 4 2 3 2 4 3 3 2 4 2 2 2 2 4/5 8/9 g2=0 4 4 4 3 3 3 g =0 3 7 4 4 g =0 5 Figure 18. The number of finite singular points of system (9) in the different regions, lines and points. where √ A± = ±a5 ± 18a3 b2 ± 72ab4 ∓ 12a3 b ∓ 96ab3 ± 32a b2 g3 +a6 + 30 a4 b2 + 378 a2 b4 + 1728 b6 − 32 a4 b − 792 a2 b3 − 5184 b5 + 8 a4 +552 a2 b2 + 5760 b4 − 128 a2 b − 2816 b3 + 512 b2 . We observe that both eigenvalues cannot be zero simultaneously. Note that A+ ≥ 0 due to Remark 4. The product of the eigenvalues det+ is √  a 45 b3 + 2 a2 − 78 b2 + 32 b g3 g3 −9 b3 + 2 a2 + 6 b2 det+ = − − . 4 (3 b − 2)3 4 (3 b − 2)3 If det+ < 0 then P+ is a saddle, see Figure 21. Note that det+ = 0 on g3 = 0, g5 = 0 and for a = b > 4/5 on g4 = 0. For b < 0 on the points of the curve g7 = 0 we have that det+ = 0. In these cases we have that P+ is a semi–hyperbolic singular point and so we apply Theorem 2.19 of [20], see also Figure 21. If det+ > 0 the point P+ is a node, see Figure 21.  Lemma 7. The local phase portrait at the point P− is given in Figure 22. 12 >0 a +b , g4 =0 =0 a -b de no d dle sa rb olic pe a-b<0 -hy emi a+b>0 s a-b>0 P0 =P+ =P- + a+b>0 =P 4/5 . .. P0 saddle P0 hyperbolic saddle e d no dl e sad li c e rbo P0 hyperbolic unstable node yp i- h -s em =P P0 . 4/5 P =P 0 + sem i -hyp erbo lic s add le n ode a+b a-b>0 =0, a-b> a+b<0 0 P0 hyperbolic saddle g5 = 0 Figure 19. The local phase portrait at the origin P0 = (0, 0). Proof. The point P− is defined for g3 ≥ 0 and g2 6= 0, and has eigenvalues √ √ a(−a2 − 42 b2 + 76 b − 32) + a2 − 24 b2 + 16 b g3 ± 2A− , 8 (3 b − 2)2 where A− is defined in Lemma 6. Note that for all the values of the parameters we have that A− ≥ 0, see also Remark 4. The product of the eigenvalues det− is √  a 45 b3 + 2 a2 − 78 b2 + 32 b g3 g3 −9 b3 + 2 a2 + 6 b2 det− = − . 4 (3 b − 2)3 4 (3 b − 2)3 If det− < 0 the point P− is a saddle, see Figure 22. For g3 = 0 we have that det− = 0. Additionally, for b > 0 on the points of g7 = 0 we have that det− = 0. For b > 4/5 and additionally g4 = 0 we also have det− = 0. In these cases the point P− is semi–hyperbolic and so we apply Theorem 2.19 of [20], see Figure 22. If det− > 0 the point P− is a node, see Figure 22.  2.5. Infinite singular points. Lemma 8. The number of infinite singular points of system (9) is given in Figure 23. 13 hyperbolic stable node with e+ e- P1 =P- g =0 g7=0 6 hyperbolic saddle strong stable focus e+,e- complex de le no sa d d - P1 =P g7=0 strong stable hyperbolic focus h g =0 yperbo hyperbolic stable 6 lic stab e+, e- complex le n node with e+ e- od e w it he += e- Figure 20. The local phase portrait at the singular point P1 = (−b, a/3). Proof. In the local chart (U1 , F1 ) system (9) becomes   1 3 ż1 = 1 + bz2 + az1 + b − 1 z1 2 − bz1 2 z2 , 2 2 ż2 = −z2 (az2 + bz1 z2 + a + 3 bz1 − 2 z1 c) , and for g1 ≥ 0 and g2 6= 0 it has the infinite singular points  √  −a ± g1 Q± = , 0 . 2g2 Note that neither Q− nor Q+ coincide with the origin of the chart (U1 , F1 ). The two points Q± collided between them over the curve g1 = 0. In the local chart (U2 , F2 ) system (9) becomes   3 1 ż1 = bz2 + − b + 1 z1 − az1 2 − bz1 2 z2 − z1 3 , 2 2 1  ż2 = − z2 2 bz1 z2 + 2 az2 + 2 z1 2 + 3 az1 + 9 b − 6 , 2 and the origin of the local chart (U2 , F2 ) is an infinite singular point.  Lemma 9. The stability of the infinite singular points of system (9) in the local chart (U1 , F1 ) is given in Figure 24. 14 =0 g4 det+ =0 t+ =0 de unstable node det+>0 unstable node P+ =P0 de saddle node no le dd E P+ 0 sa =P P+ det+<0 8/9 saddle node P+ =P- =P0 saddle hyperbolic saddle 4/5 ode det+=0 d dle saddle saddle n saddle sa det+= EP+ g 2=0 8 2/3 stable node 0 t+> de de no stable node ble sta node det+>0 stable node stable det+>0 4/5 8/9 P+ = P0 se m stable node i h yp erbo lic s add le n g7 =0 ode det+<0 hyperbolic saddle det+>0 unstable node det+ =0 g5 = 0 Figure 21. The local phase portrait at the singular point P+ . Proof. We recall that the points Q± are not defined on g2 = 0. For g1 < 0 are not real points, so we will consider them only when g1 ≥ 0. The point Q+ has eigenvalues p √ − a ± 5 a2 + 4 a g1 − 96 b + 64 . 4 Note that on g1 = 0 the point Q+ collide with Q− and is a semi–hyperbolic saddle–node. Also note that both eigenvalues at the point Q+ cannot be zero. The product of the eigenvalues is √ Det+ = −1/4 a2 − 1/4 a g1 + 6 b − 4, and for g1 > 0 we have that Det+ < 0, and so the point Q+ is a saddle. The point Q− has eigenvalues p √ −a ± 5 a2 − 4 a g1 − 96 b + 64 , 4 and on g1 = 0 is a semi–hyperbolic saddle–node. The product of the eigenvalues at the point Q− is √ Det− = −1/4 a2 + 1/4 a g1 + 6 b − 4. The point Q− changes from a saddle to a node when the values of the parameters of the system cross the line g2 = 0, see Figure 24.  Lemma 10. For b < 2/3 the origin of the local chart (U2 , F2 ) is an unstable node. If b > 2/3 is a stable node. If b = 2/3 the origin is the union of an elliptic and a hyperbolic sector separated by two parabolic sectors. 15 =0 g4 0 t- > det- >0 stable node de saddle det-=0 saddle node de no b le E P- st a 8/9 det -<0 stable node P- = P+ =P0 saddle saddle 4/5 no de saddle saddle de unstable node le no det-= EP- g 2 =0 8 0 dd t-= sa o l ic de unstable node perb saddle det- <0 i hy det-=0 m se P0 P1 det->0 stable node -= P- = P sadd le node 4/5 8/9 stable node stab le nod e g7 =0 stable node stable node det- >0 g5 = 0 Figure 22. The local phase portrait at the singular point P− . Proof. The origin O2 = (0, 0) of the chart (U2 , F2 ) has the Jacobian matrix   −3/2 b + 1 b , 0 3 − 9/2 b with eigenvalues −3b/2 + 1 and 3 − 9b/2. For b > 2/3 the point O2 is a stable hyperbolic node whereas for b < 2/3 becomes an unstable hyperbolic node. For b = 2/3 both eigenvalues become zero, (the Jacobian is not identically zero) so O2 is a nilpotent singular point. Using Theorem 3.5 of [20] and the blow up technique we obtain that the point O2 is the union of an elliptic and a hyperbolic sector separated by two parabolic sectors. Note that the straight line of the infinity locally is contained in the two parabolic sectors.  Proposition 11. For g1 > 0 we obtain two distinct infinite singular points Q− and Q+ . On the curve g1 = 0 the two points collided: Q− = Q+ and one eigenvalue of them becomes zero. For g1 < 0 the two infinite singular points Q− and Q+ do not exist. On the curve g2 = 0 the singular points P± and Q± do not exist. The point P+ changes from a saddle to a node when the parameters of the system cross the line g2 = 0. Q− changes from a node to a saddle when the parameters of the system cross this line and at this line the origin of (U2 , F2 ) is a nilpotent singular point, see Figure 24. On the bifurcation curve g3 = 0 the two finite singular points P± collided between them. On g3 = 0 for a = b = 4/5 the points P± collide to P0 . For g3 < 0 the points P± do not exist. For g3 > 0 see Figures 21 and 22. 16 1+0 0 g= 1 1+2 1+1 1+0 g2 =0 1+2 Figure 23. The number of the infinite singular points of system (9). On the line g4 = 0 the point P0 for a 6= 4/5 is a semi–hyperbolic saddle–node, whereas for a = 4/5 is a saddle. For a = b > 4/5 the point P+ colapse with P0 whereas for 0 < a = b < 4/5 the point P− colapse with P0 . For a = b = 4/5 both points P± collide to P0 and is a saddle. For a = b ∈ (4/5, 8/9) the point P− is a saddle whereas for a = b > 8/9 is a stable node. For a = b > 4/5 the point P+ changes from a saddle to a node. For b < 4/5 the point P− changes from a saddle to a node when the parameters of the system cross the line g4 = 0, see Figures 18, 19, 21 and 22. On the line g5 = 0 the P0 is a semi–hyperbolic saddle–node. The point P+ colapse to P0 . The point P+ changes from a saddle to a node when the parameters of the system cross this line, see Figures 19 and 21. On the curve g6 = 0 the point P1 is a hyperbolic stable node. For g6 < 0 we have that P1 has complex eigenvalues and consequently is a strong stable focus. For g6 > 0 we have that P1 has real eigenvalues, see Figure 20. On the curve g7 = 0 the point P− collided with P1 when b > 0 whereas for b < 0 the point P+ collide with P1 . Note that the product of the eigenvalues of P1 is g7 /(−3). On g7 = 0 the point P1 is a semi–hyperbolic saddle–node. P1 changes from a saddle to a node when the parameters of the system cross this line. The point P1 changes from a saddle to a node when the parameters of the system cross the line g7 = 0, see Figure 20. Proof. The proof of Proposition 11 follows directly from Lemmas 2, 3, 5, 6, 7, 8, 9 and 10.  17 et -=0 0, D E Q+, Q- t+ = U1 De Det+<0, Q+ saddle Det- >0 1 Q- stable hyperbolic node - = Q+ semi g=0 2 E Q+, Q- Det+<0, Q+saddle Det- <0, Q- saddle Q+ saddle Q- saddle Figure 24. The infinite singular points of the local chart (U1 , F1 ). 3. Phase portraits In this section we study the global phase portraits of system (9): We draw the local phase portrait of the finite and infinite singular points in the Poincaré disc, see for details on the Poincaré compactification the Appendix 5. Additionally we plot in the Poincaré disc the invari- ant algebraic curve f2 = 0. Finally, we should present all the global phase portraits. Let L be a straight line and let q be a point of L. We say that q is a contact point of the straight line L with a vector field X, if the vector X(q) is parallel to L. For quadratic systems the following two results are well known. Lemma 12. On any straight line which is not invariant the total number of singular points and contact points is two. If there are two such points, P1 and P2 , then the orbits intersecting the line ∞P1 cross in the same sense as the orbits intersecting the line P2 ∞, and in the opposite sense the orbits P1 P2 . For a proof of Lemma 12 see the lemma in page 296 of Coppel [14]. Lemma 13. On any non invariant straight line through a finite singular point P reaching the infinity in a pair of infinite singular points the orbits crossing the segment ∞P have opposite sense to the orbits crossing the segment P ∞. 18 Lemma 13 is equivalent to Lemma 12 when one of the contacts points mentioned in Lemma 12 goes to infinity. For a proof of Lemma 13 see [1]. Remark 14. In what follows when we apply Lemmas 12 or 13 we must check that the straight lines mentioned in these lemmas are not invariant. In case that there are invariant we shall state this fact explicitly. The following theorem also appears in Coppel’s paper [14]. Theorem 15. A singular point in the interior of a closed path of a quadratic system must be either a focus or a center. Here a closed path is an invariant curve of the quadratic system contained in R2 homeomorphic to a circle such that in its neighborhood contained in the bounded region limited by it the Poincaré return map is defined. Remark 16. In fact the proof which appears in Coppel’s paper [14] also works when the closed path of the quadratic system has some piece at infinity. So Theorem 15 also holds for closed paths having some orbit at infinity. Theorem 17. Let X be a vector field of class C 1 on an open set ∆ ⊆ R2 . Consider γ a closed path of X such that the bounded region R limited by γ is contained in ∆. Then there exist a singular point of X in R, inside the region limited by γ. The proof of Theorem 17 is the same as the proof of Theorem 1.31 of [20]. The next result is due to Berlinskiĭ [9]. Theorem 18. Suppose that a quadratic system has four singular points. If the quadrilateral with vertices at these points is convex then two opposite singular points are saddles and the other two are antisaddles (nodes, foci, or centers). But if the quadrilateral is not convex then, either the three exterior vertices are saddles and the interior vertex is an antisaddle, or the exterior vertices are antissaddles and the interior vertex is a saddle. Next we prove that system (9) has no limit cycles. Lemma 19. System (9) has no limit cycles. Proof. Consider system (9) and X = (P, Q) the corresponding vector field. According to Theo- rem 15 a possible limit cycle can appears only surrounding a focus. Note that for system (9) we only have a focus in the interior of the loop of the curve f2 = 0 for the values of the parameters in the regions r6 , r7 , r8 , r9 , r12 , r13 , r14 , r15 , and on the lines L4 , L5 , L6 , L7 , L13 , L14 , L26 , L27 , L28 , L29 . Also we have a focus on the right side of this curve for the values of the parameters in the regions −4/3 −4/3 r19 and r20 and on the line L8 . The divergence of the system (P f2 , Qf2 ) is −4/3 −4/3 ∂(P f2 ) ∂(Q f2 ) a (3 x + 4) a (3 x + 4) D= + =− . =− ∂x ∂y 6 (−x3 − x24/3 6f2 + y 2 )4/3 Note that the vertical straight line 3x + 4 = 0 does not intersect the invariant curve f2 = 0. Hence D does not change sign in the regions containing the focus. Hence, by the Bendixon– Dulac criterium (see Theorem 7.12 of [20]) there are no periodic orbits in the mentioned regions and lines in the (a, b) parameter plane, and so there are no limit cycles.  In what follows a heteroclinic loop is formed by two saddles P1 and P2 and two different separatrices connecting these saddles and forming a loop in such a way that at least in one of the two sides of the loop a Poincaré return map is defined. Let µi < 0 < λi be the eigenvalues of the saddles Pi for i = 1, 2. Set µ1 µ2 k= . λ1 λ2 19 If k < 1 then the loop in the region limited by it is unstable, and if k > 1 then the loop is stable, see Poincaré [36] (see Theorem XVII). For regions r19 , r20 and the line L8 we have the following result. Lemma 20. For the values of the parameters in the regions r19 , r20 , and on the line L8 system (9) has no connection between the separatrices of the saddles Q± . Proof. We assume that there is a connection between the separatrices of the saddles Q± of system (9) for the values of the parameters in the regions r19 and r20 . Then there is a heteroclinic loop containing a focus. Note that for the points Q± (see also Lemma 9) p √ ! p √ ! − a − 5 a2 + 4 a g1 − 96 b + 64 − a − 5 a2 − 4 a g1 − 96 b + 64 k= p √ p √ > 1. − a + 5 a2 + 4 a g1 − 96 b + 64 − a + 5 a2 − 4 a g1 − 96 b + 64 Thus the heteroclinic loop is stable. Since the focus in the interior of the heteroclinic loop is also stable it must exist a cicle limit by the Poincaré–Bendixson Theorem (see for instance Corollary 1.30 of [20]). But this is in contradiction with Lemma 19. Now we consider the values of the parameters on the line L8 . If there is a connection between the separatrices of the points Q1 and Q2 , then they form a heteroclinic loop that must contain the focus. Working in a similar way as in Lemma 20 for the saddles Q1 and Q2 we obtain  p √  p √  −a + 5 a2 − 4 a d + 96 a + 64 −a + 5 a2 + 4 a d + 96 a + 64 k= p √  p √  > 1, −a − 5 a2 − 4 a d + 96 a + 64 −a − 5 a2 + 4 a d + 96 a + 64 with d = a2 + 24 a + 16. Thus this heteroclinic loop is stable. Since the focus in the interior of the heteroclinic loop is also stable by the Poincaré–Bendixon Theorem it must exist a limit cycle. But this is in contradiction with Lemma 19.  3.1. Phase portraits in the regions. The bifurcation curves define 21 regions, see Figure 17. Here we are going to present all the phase portraits of system (9) in the Poincaré disc for the values of the parameters in each one of the 21 regions. .. O2 .. r1 P 1. . .. . . .. P + . P0 . P - ..´ O2 .. S=16,R=5 Figure 25. The local and the global phase portraits of system (9) corresponding to the region r1 . For the region r1 we realize the following steps, see also Figure 25. (i) According to Theorem 22 of the Appendix first we draw the separatrices in the Poincaré disc and then we should draw an orbit in each canonical region. This determines com- pletely the global phase portraits in the Poincaré disc. (ii) We first draw the local phase portrait of the finite and infinite singular points in the Poincaré disc, see Figure 25. (iii) Next we study the α − − and the ω − −limits of the separatrices. 20 (iv) Since in the region r1 does not exist any focus we have that no limit cycle exist for the quadratic system (9), see Theorem 15. (v) We should only study the separatrices of the point P1 (saddle). The two unstable separatrices can only reach the infinite stable node O2 or the finite stable node P− . Additionally the ω − −limit of thess two unstable separatrices cannot be the same stable node, otherwise they should define a closed region and it should contain a stable separatrix without its α − −limit. (vi) Finally, we obtain the unique global phase portrait in Figure 25. .. O2 .. O2 .. r2 . Q´- . Q´- . .. P1 .. P1 . . Q+́ . .. Q+́ . .. . .. .P+ P0 . P+ P0 . . P- . P- . . . Q+ . Q+ .. Q- .́.. O2 Q- .́.. O2 Q- S=26,R=7 (a) (b) (c) Figure 26. The local and the global phase portraits of systems (9) correspond- ing to the region r2 . Next we describe the phase portrait of system (9) corresponding to the values of the param- eters in the region r2 , see Figure 26. (i) In the finite region the stable separatrix of Q+ can only have as α − −limit the unstable node O20 . (ii) By similar arguments as in the region r1 we have that the two unstable separatrices of the point P1 can only have as ω − −limit the points O2 and P− . (iii) The unstable separatrix γ of the infinite point Q0+ in the finite region could have as ω − −limit the points P− , P1 or O2 . Consider the straight line passing on the points Q0+ , P− and Q+ . In the region r2 the point P1 is always upper this straight line. According to Lemma 13 the vector field have opposite direction in the two half–lines Q0+ P− and P− Q+ . Additionally note that P1 is always at the same side of the straight line in the region r2 . Therefore the ω − −limit of γ must be the point P− , see Figure 26(b). (iv) The two stable separatrices of P1 have the α − −limits at the points P+ and Q0− . (v) Finally we obtain the unique global phase portrait in Figure 26(c). In what follows we describe the phase portrait of system (9) corresponding to the region r3 , see Figure 27. (i) Consider the straight line passing through the points Q+ , P+ and Q0+ . We distinguish the following cases. (i.1) The point P− is below this straight line, see Figure 27(a). Then the direction of the loop on the curve f2 = 0 determines the direction of the vector field on this line, see also Lemma 13. The separatrices of the points P+ , Q+ and Q0+ are as in Figure 27(a). (i.2) For a and b satisfying the equation h = 0 (see (10)) the point P− belong to this straight line and now the line is invariant for the vector field (9), see Figure 27(b). (i.3) The point P− is above this straight line. Then the direction of the vector field in the segment Q0+ P+ is determined by the unstable node O20 see Figures 27(c),(d),(e1 ), 21 .. O2 .. O2 .. O2 . . Q´- P1 . . Q´- P1 . . Q´- P 1 . Q´+ . Q´+ . Q´+ .. .. . .. P0 P0 P0 . . . . .. . . .. Q+ P - P- P- P + . Q- Q+ P + Q- Q+ P + Q- .. O22́ .. O22́ .. O22́ (a) (b) (c) ..O2 ..O2 . Q´- .. O2 .. Q´- . Q´- .. Q´+ P .. 1 . Q´+ P 1 Q´+ P1 . .. . .. . .. P0 P0 P0 P - . .. . .. Q+ Q+ P - P - P + P + . . .. . Q- Q- Q+ .. O22́ .. O22́ O22́ P + Q- (d) (e1 ) (e2 ) .. r 3( i ) r3( i i ) i( i i i ) .. . .. . . .. .. S=26, R=7 (f) (g) (h) Figure 27. The local and the global phase portraits corresponding to the region r3 . (e2 ). Also note that the unstable separatrix of the saddle Q+ must be upper this straight line, see also Lemma 13 (and also check the stable separatrix of the saddle Q0+ ). For the same reason the unstable separatrices of the point P+ must be as they are shown in Figures 27(c),(d),(e1 ),(e2 ). Now consider the straight line L passing through the points Q− , P0 and Q0− . (j1 ) The point P1 can be upper this straight line L. Then, the unstable separatri- ces of P1 can only have ω–limit the points O2 and P− . This determines the direction of the vector field over the straight line L, see Figures 27(a),(b),(c). (j2 ) For the values of the parameters a and b in the line 3(iv) (so a, b satisfying the equation j1 = 0, see (10)) the point P1 belong to the straight line L and so the line becomes invariant for the vector field (9), see Figure 27(d). (j3 ) The point P1 can be below the straight line L. The unstable separatrices determine the direction of the vector field on this line, see Figure 27(e1 ). 22 (ii) In all the cases the unstable separatrices of the point P1 can have as ω − − limit only the points O2 and P− . (iii) In all the cases the stable separatrices of the point P1 can only have as α − − limit the points P0 and Q0− . (iv) Now consider the straight line L0 passing through the points Q+ , P0 and Q0+ . In general, the point P1 is above this straight line L0 . So in general, in the finite region the unstable separatrix of the point Q0+ must be bellow this straight line and therefore its ω-limit must be the point P− . However, for the values of the parameters a and b in the line 3(vii) (so a, b satisfying the equation j1 = 0, see (10)) the point P1 belongs to this straight line L0 and so becomes invariant, see Figure 27(e2 ). (v) The local phase portrait (a) yields to the global phase portrait (f). The local phase portrait (b) yields to the global phase portrait (g). The local phase portraits (c),(d),(e1 ) and (e2 ) yields to the same global phase portrait (h). We summarize: For the values of the parameters in the regions 3(iii), 3(iv), 3(v), 3(vi) and (3vii) we obtain the global phase portrait (h), see Figure 27. .. O2 .. O2 .. r4 . P1 . P1 . . .. . .. P0 . .. P0 P - . P + P- . P + . .. O´2 .. .. S=16,R=5 Figure 28. The local and the global phase portraits corresponding to the region r4 . Next we describe the phase portrait of system (9) corresponding to the the region r4 , see Figure 28. (i) The unstable separatrices of the point P1 can only have as ω − − limit the points O2 and P− . (ii) Consider the straight line passing through the points O2 , P− and O20 . Then the direction of the loop on the curve f2 = 0 determines the direction of the vector field on this line, see also Lemma 13. Then the stable separatrices of P1 can only have as α − − limit the points P0 and O20 . (iii) The unstable separatrices of P+ have as ω − − limit the points O2 and P− . (iv) Finally we obtain the unique global phase portrait in the region r4 , see Figure 28. Since by Lemma 19 system (9) have no limit cycles the phase portraits corresponding to the regions ri for i = 5, 6, 7, 8, 9, 10 follow immediately from their local phase portraits, see Figures 29, 30, 31, 32, 33 and 34. In what follows we study the phase portrait of system (9) corresponding to the region r11 , see Figure 35. (i) We consider the straight line passing through the points P− and P0 . Note that the points Q+ , Q− are under this straight line, and that the points Q0+ , Q0− are upper. (ii) The unstable separatrices of the points Q0+ and P− can only have as ω − − limit the point O2 . (iii) Then the unstable separatrix A of the point P+ can only have as ω − − limit the point O2 , see also Figure 35(b). 23 .. O2 .. r5 . .. .. P- P+ P0 . .. .. P 1 .. .. S=15,R=4 Figure 29. The local and the global phase portraits corresponding to the region r5 . .. O2 .. r6 . .. .. P- P+ P0 . .. .. P 1 .. .. S=15,R=4 Figure 30. The local and the global phase portraits corresponding to the region r6 . .. O2 .. r7 . .. P0 . .. P1 .. .. S=9,R=3 Figure 31. The local and the global phase portraits corresponding to the region r7 . (iv) Consider the straight line passing through the points Q+ and P+ . Then the unstable separatrix B of the point P+ can only have as ω − − limit the stable node Q− , see also Figure 35(c). (v) The stable separatrix of the point Q+ can only have as α − − limit the point P0 . (vi) Finally we obtain the global phase portrait in the region r11 , see Figure 35(d). The phase portraits corresponding to regions r12 and r13 follow using the same arguments as the ones corresponding to the region r11 and are given in Figures 36 and 37 respectively. Next we describe the phase portrait of system (9) corresponding to the region r14 , see Figure 38. (i) Consider the straight line passing through the points P0 and P1 . Note that Q+ and Q− are situated in the opposite sides of this straight line. 24 .. O2 .. r8 . .. P0 . .. P 1 . . .. . P + .. . P- S=15,R=5 Figure 32. The local and the global phase portrait corresponding to the region r8 . .. O2 P - . . .. P0 P 1 . P + .. S=16, R=5 Figure 33. The local and the global phase portraits corresponding to the region r9 . .. O2 .. r10 . . .. P0 . . .. P- P . 1 . P + .. .. S=16,R=5 Figure 34. The local and the global phase portraits corresponding to the region r10 . (ii) The unstable separatrices of the points Q+ and Q0− can only have as ω − − limit the point P+ . (iii) The stable separatrices of the points Q− and Q0+ can only have as α − − limit the point P− . (iv) Finally we obtain the global phase portrait in the region r14 , see Figure 38. The global phase portraits of system (9) in the regions r15 and r16 follow by similar arguments as in the region r14 and are given in Figures 39 and 40. Next we describe the phase portrait of system (9) corresponding to the region r17 , see Figure 41. (i) Consider the straight line passing through the points Q+ , P0 and Q0+ . In the finite region the unstable separatrix γ of the point Q+ must be over the straight line, otherwise it must have as ω − −limit one of the points O20 or Q− . If the point Q− is the ω − −limit 25 .. O2 .. O2 . Q´- . . Q´+ . . . .. P - P0 . . .. P - P0 P1 . .. P1 . .. Q+ P + Q+ P + Q- .. O´2 .. A B Q- (a) .. O2 (b) .. . . r11 . . . . .. P - P0 . . .. . .. P1 . .. Q+ P + .. Q- .. S=26,R=7 (c) (d) Figure 35. The local and the global phase portraits corresponding to the region r11 . .. O2 .. . . r12 . . . . P- . .. . .. P0 P1 . ..P + Q+ . .. .. Q- .. S=26,R=7 Figure 36. The local and the global phase portraits in the region r12 . of γ then there is a closed path without a singular point in its interior, a contradiction. If the ω − −limit of γ is the point O20 then the stable separatrix of the point Q− cannot have an α − −limit, a contradiction. So the direction of the vector field on the straight line is determined, see Figure 41(a)). Additionally the ω − −limit of γ is the point P+ . Moreover the stable separatrix of the point Q0+ must be on the upper side of the straight line. (ii) Consider the straight line passing through the points Q0− , P0 and Q− . We distinguish the following cases for the position of the point P1 with respect to this line: (ii.1) The point P1 is below this straight line. The direction of the vector field on this straight line is as in Figure 41(b), otherwise one of the unstable separatrices of the point P1 should have as ω − − limit the point O20 . This is a contradiction because then the stable separatrix of the point Q0+ should not have an α limit. Hence in this case the global phase portrait is given in Figure 41(c). 26 .. O2 .. O2 .. . . . r13 . . . . .. . .. . .. P0 P0 . P1 .P- . .. P + Q+ P1 . P- . .. P + Q+ . .. .. Q- .. Q- .. S=25,R=7 Figure 37. The local and the global phase portraits corresponding to the region r13 . .. O2 .. r14 Q´- . . . P+ Q+ . . . . .. P1 P0 . .. . . . P- Q- . . . Q´+ .. O2́ .. S=23,R=5 Figure 38. The local and the global phase portraits corresponding to the region r14 . .. .. O2 . . . . r.15 Q´+ . P + Q+ P . . .. P- P0 . . .. . 1 Q´- . . Q-- . .. O´2 .. S=24,R=5 Figure 39. The local and the global phase portraits corresponding to the region r15 . (ii.2) The point P1 belongs to this straight line and so the straight line is invariant by the vector field (9). This happen when the parameters a and b satisfy equation j1 = 0, see relation (10). Then the unstable separatrix of the point Q0− can only have as ω − −limit the point P1 , see Figure 41(d). Then the global phase portrait is given in Figure 41(e). (ii.3) The point P1 is upper this straight line. Then in the finite region the unstable separatrix γ̃ of the point Q0− must be below this straight line (see Figure 41(f)), otherwise γ̃ should have as ω − −limit one of the points P+ or P1 , see Figure 41(g). If P+ is the ω − −limit of γ̃ then the unstable separatrices of P1 must go to P+ but then the stable separatrix of P1 has no α − −limit. So the correct direction of the vector field on the mentioned straight line is the one of Figure 41(f). Now γ̃ cannot have as ω − −limit the point Q0+ . So the only possible ω − −limit of γ̃ is the point P− . Finally we obtain the global phase portrait, see Figure 41(h). 27 .. O2 .. . P + . . Q+ . r . . 16 . . .. . P- P0 . .. P1 . . . . Q-- . .. R=24,S=5 Figure 40. The local and the global phase portraits corresponding to the region r16 . .. .. .. .. O2 O2 . . . . . r17(i) Q´- . . P + Q+ Q´- . P+ Q+ . P 1 .. P1 .. .. P- P0 P- .. P0 .. . . . . . Q´+ .. Q- Q´+ . .. ´ O´2 .. Q- S=24, R=5 (a) .. .. (b) (c) O2 . . . . . r17(ii) Q´- . P+ Q+ .. P1 .. P- .. P0 .. . . . . Q´+ .. Q- .. S=23, R=4 (d) .. (e) .. .. . . O2 . . O2 . . . r17(iii) Q´- . P+ Q+ Q´- . P+ Q+ . . . . . P1 P1 . .. .. P- P0 .. P- P0 . . Q´+ . . Q´+ . . Q- .. .. Q- .. S=24,R=5 (f ) (g) (h) Figure 41. The local and the global phase portraits corresponding to the region r17 . Next we describe the phase portrait of system (9) corresponding to the region r18 , see Figure 42. 28 .. O2 .. O2 .. O2 . Q+ . Q+ . Q+ .. Q´- . P- .. P+ P .. Q´- . P- .. P+ P .. Q´- . P- . . P+ P1 .. .. .. 1 1 P0 P0 P0 . . . ´ Q+ . Q- ´ Q+ . Q- ´ Q+ . Q- .. O´2 .. O´2 .. O´2 (a) (b) (c) .. O2 .. O2 .. O2 . Q+ . Q+ . Q+ . Q´- . . . . Q´- P- . . Q´- . . . ... P+ P1 P1 P- P1 . P- .. P+ .. P+ P0 . P0 . P0 . . Q´+ Q- . Q´+ Q- . Q´+ Q- .. O´2 .. O´2 .. O´2 (d) (e) (f ) .. r18(i) .. r18(ii) .. r18(iii) . . . . . . . . . . . . . . .. ... .. . . . . . . .. S=24,R=5 .. S=23,R=4 .. S=24,R=5 (g) (h) (i) Figure 42. The local and the global phase portraits corresponding to the region r18 . (i) Consider the straight line passing through the points Q− , P0 and Q0− . Note that the points P1 , P− and P+ in the region r18 are always upper this straight line, see Figure 42(a). The direction of the closed loop of the curve f2 = 0 determines the direction of the vector field over this straight line, see Lemma 13. The unstable separatrix of the point Q0− is upper this straight line and only can have as ω − − limit the point P− . Additionally the stable separatrix of the point Q− is upper this straight line. (ii) Now we prove that there is no connection between the separatrices of the saddles Q+ and Q− . If there is a conexion then in its interior should contain a singular point which must be a focus or a center, and this is a contradiction because P1 is a node, see Theorems 15, 17 and Remark 16. (iii) Consider the straight line passing through the points Q+ , P0 and Q0+ . Note that the points P− and P+ are always upper this straight line. We consider the following cases for the point P1 : (iii.1) The point P1 is upper this straight line, see Figure 42(a). Then, the unstable separatrix γ of Q+ must be upper this straight line, otherwise either should 29 have O20 as ω − −limit, but then the stable separatrix of the point Q− could not have an α − −limit, a contradiction; or γ connects with the stable separatrix of Q− , but as in (ii) this is a contradiction. This determines the direction of the vector field over this last straight line. Then γ can only have as ω − −limit the point P1 . (iii.2) The point P1 belongs to this straight line, see Figure 42(b). Since this straight line is now invariant, the unstable separatix of Q+ belongs to this line and can only have as ω−−limit the point P1 . (iii.3) The point P1 is below this straight line, see Figure 42(c). Then the unstable separatrix γ of the point Q+ can only be below this straight line. This determines the direction of the vector field over this straight line. Then again the ω − −limit of γ is the point P1 . (iv) We consider the straight line passing through the points Q− , P+ and Q0− . We note that on the points of the curve h = 0 (see relation (10)) the point P− belongs to the straight line and system (9) has this straight line invariant. Now we distinguish three cases: (iv.1) The point P− is below this straight line. Then the unstable separatrices of the point P+ have as ω − − limit the points P1 and P− , see Figure 42(d). (iv.2) The point P− belongs to this straight line. Then the unstable separatrices of the point P+ have as ω − − limit the points Q− and P− , see Figure 42(e). (iv.3) The point P− is upper this straight line. Then the unstable separatrices of the point P+ have as ω − − limit the points O20 and P− , see Figure 42(f). (v) P0 is the α − − limit of the unstable separatrice of the point Q0+ , see Figure 42(a). (vi) Finally, we obtain the three global phase portraits in the region r18 see Figures 42(g),(h) and (i). Now we study the phase portrait of system (9) corresponding to the region r19 , see Figure 43. (i) Consider the straight line passing through the points Q+ , P0 and Q0+ see Figure 43. Note that in the region r19 the points P− and P+ are always upper this straight line and the point P1 is always below the straight line. In the finite region, the unstable separatrix γ of the point Q+ is below the straight line, otherwise could not have an ω − −limit. This determines the direction of the vector field over the straight line. So the ω − −limit of γ can only be the point P1 . The point O20 cannot be the ω limit of γ because in this case a stable separatrix of Q− would be without an α − −limit. Additionally there is no connection between the separatrices of Q+ and Q− , see Lemma 20. (ii) Consider the straight line passing through the points Q− , P0 and Q0− , see Figure 43(c). Note that the point P− is upper the straight line. Then the direction of the closed loop of the invariant curve f2 = 0 determines the direction of the vector field over this straight line. The unstable separatrix of Q0− must be over the straight line and can only have as ω − −limit the point P− . (iii) Consider the straight line passing through the points Q+ , P+ and Q0+ . For the values of a and b that satisfy equation h = 0 (see relation (10)) system (9) has this straight line invariant. Hence we distinguish three cases: (iii.1) The point P− is below the straight line. Then the unstable separatrices of the point P+ have as ω − − limit the points P1 and P− . (iii.2) The point P− is on the straight line. Then the unstable separatrices of the point P+ have as ω − − limit the points Q− and P− . (iii.3) The point P− is upper the straight line. Then the unstable separatrices of the point P+ can only have as ω − − limit the points O20 and P− . (iv) Finally we obtain the three global phase portraits in the region r19 see Figures 43(a),(b) and (c). Next we describe the phase portrait of system (9) corresponding to the region r20 , see Figure 44. 30 .. O2 .. O2 .. O2 . . Q+ . . Q+ . Q+ Q´- . . . P- P0 P P Q´- P- . . . P0 P P . Q´- . . P- . P1 .. .. .. + 1 + 1 P+ . . P0 . Q´+ . Q- Q´+ . Q- Q´+ . Q- .. O´2 .. O´2 .. O´2 .. O2 .. O2 .. O2 . Q+ . Q+ . Q+ Q´- . . . . . Q´- P- . . Q´- . . . . ... P+ P1 P1 P- P1 P- .. P+ .. P+ P0 . P0 . P0 . Q´+ . Q- . Q´+ Q- Q´+ . Q- .. O´2 .. O´2 .. O´2 .. r19(i) .. r19(ii) .. r19(iii) . . . . .. . . . . . . . . . .. .. .. . . . . . . .. S=24,R=5 .. S=23,R=4 .. S=24,R=5 (a) (b) (c) Figure 43. The local and the global phase portraits corresponding to the region r19 . .. O2 .. O2 .. r20 . . . . Q´- . . P- . Q+ Q´- . .P- . Q+ . . . .. P P . .. . .. P0 1 P0 1 . P + . Q- . P + . Q- . . Q´+ Q´+ .. O´2 .. O´2 .. S=24, R=5 Figure 44. The local and the global phase portraits corresponding to the region r20 . (i) Consider the straight line passing through the points Q− , P0 and Q0− . Note that in the region r20 the point P− is always upper the straight line. The direction of the closed loop of the invariant curve f2 = 0 determines the direction of the vector field on this 31 straight line. The unstable separatrix of Q0− is upper the straight line and can only have as ω − − limit the point P− . The stable separatrix of the point Q− is upper this straight line. (ii) Consider the straight line passing through the points Q+ , P0 and Q0+ . Note that in the region r20 the points P+ and P1 are always below this straight line. The direction of the closed loop of the invariant curve f2 = 0 determines the direction of the vector field on this straight line. The stable separatrix of the point Q0+ is below the straight line and can only have as α − −limit the point P+ . (iii) By Lemma 20 we have that there is no connection between the separatrices of Q+ and Q− . Consider the unstable separatrix γ of Q+ . Then γ cannot have O20 as ω − − limit because then the stable separatrix of Q− will not have an α − −limit. Hence the ω − −limit of γ can only be the point P1 . (iv) Finally we obtain the global phase portrait in the region r20 , see Figure 44. .. O2 .. r21 Q´- . . .. P- P0 .. . . .. Q+ .. . . P P 1 . . . . Q´+ + Q- O´2 .. .. S=24,R=5 Figure 45. The local and the global phase portraits corresponding to the region r21 . In what follows we present the phase portrait of system (9) corresponding to the the region r21 , see Figure 45. (i) Consider the straight line passing through the points Q+ , P0 and Q0+ . Then in the region r21 the points P1 and P+ are always below this straight line. The stable separatrix of Q0+ can only have as α − − limit the point P+ . (ii) Consider also the straight line passing through the points Q− , P0 and Q0− . Then in the region r21 the points P− and P1 are always upper this straight line. The unstable separatrix γ of Q0− can have as ω − − limit the point P− . (iii) The unstable separatrix γ̃ of Q+ cannot have as ω − − limit the point Q− because of Theorem 15. If γ̃ has as ω − − limit the point O20 then in the finite region a stable separatrix of Q− would be without an α − − limit. Therefore the only possibility that remains is that γ̃ has as ω − − limit the point P1 . (iv) In the finite region the stable separatrix of Q− must have as α − − limit the point O2 . (v) Finally we obtain the global phase portrait in the region r21 , see Figure 45. 3.2. Phase portraits on the lines. The bifurcation curves define 31 lines, see Figure 46. In this section we are going to present the phase portraits of system (9) in each line. We should provide the details for obtaining the phase portraits on the lines L8 , L9 , L11 , L12 , L24 and L25 . Since the arguments used in the study of the phase portraits corresponding to these lines are the same for studying the remaining lines we only provide their phase portraits in the corresponding figures. First we explain the phase portrait on the line L8 , see Figure 54. By Lemma 20 there is no connection between the separatrices of the points Q1 and Q2 . Note that the unstable separatrix of the point Q1 cannot have as ω − − limit the point O20 . In the opposite case using the 32 g1=0 g4=0 L10 L1 h=0 L11( i ) i) 3 (i ) j1=0 v (i 3 L11( i i ) L11( i i i ) L2 L11( i v ) L26 L15 L11( v ) L20 g3=0 ) L21 vi 3( L 12 L27 L13 L22 L17 L14 L31( ii ) L28 L29 L30 L31( i ) L31( iii ) g2=0 L23 L18 17( i i ) j1 =0 L19 L24( i ) g7=0 L8 19 ( ii) g6=0 h=0 g5=0 L24(iii) 18( i i ) L9 L25 Figure 46. c = 1, a > 0. The bifurcation curves define 31 lines. It is a qualita- tive picture. .. O2 .. . . P1 . . L1 . . . .. . .. P2 P0 . . Q+ . . Q- .. .. S=24,R=7 Figure 47. The local and the global phase portraits corresponding to the line L1 . continuity we should have a connection between the separatrices of the saddles. Hence, the unstable separatrix of the point Q1 must have as ω − − limit the point P1 . So on the line L8 we obtain a unique global phase portrait, see Figure 54. For the line L9 note that there is no connection between the separatrices of the saddles Q1 and Q2 . If there is a conexion then in its interior should contain a singular point which must be a focus or a center, and this is a contradiction because P1 is a node, see Theorems 15 and 33 .. O2 .. L2 . P1 . . .. . .. P2 P0 .. .. S=14,R=5 Figure 48. The local and the global phase portraits corresponding to the line L2 . .. O2 .. L3 . . .. . . .. P P1 2 P0 .. .. S=13,R=4 Figure 49. The local and the global phase portraits corresponding to the line L3 . .. O2 .. L4 . P . 2 .. ..P1 P0 .. .. .. .. S=13,R=4 Figure 50. The local and the global phase portraits corresponding to the line L4 . 17. Now we apply similar arguments as the ones for studying the phase portrait on the line L8 and we obtain the unique global phase portrait on the line L9 given in Figure 55. Now we will describe the phase portrait on the line L11 , see Figure 57. √ (i) For a > 4 5 the point P− is below the straight line passing through the points Q,P+ and Q0 , see Figure 57, L11 (i). (ii) The 0 √ straight line passing through the points Q, P+ and Q becomes invariant for a = 4 5. In this case √ the point P− belong to this line, see Figure 57, L11 (ii). (iii) For 4 < a < 4 5 the point P− is upper the straight line passing through the points Q,P+ and Q0 , see Figure 57, L11 (iii). (iv) Consider the straight line passing through the points P0 , Q and Q0 . For a > 4 the point P1 is upper this straight line, see Figure 57, L11 (i), L11 (ii), L11 (iii). For a = 4 this straight line becomes invariant. In this case the point P1 belong to this line, see Figure 57, L11 (iv). For a < 4 the point P1 is below this straight line, see Figure 57, L11 (v). 34 .. O2 .. L5 . . .P . P0 1 .P2 . . . S=13,R=5 Figure 51. The local and the global phase portraits corresponding to the line L5 . . .. O2 . . .. O2 L6 . . .. P0 . .. P1 . .. .. . . P 2 Q- Q+ .. . S=23,R=7 Figure 52. The local and the global phase portraits corresponding to the line L6 . . .. O2 . L P2 . . P1 . P 0 . . . S=21, R=5 Figure 53. The local and the global phase portraits corresponding to the line L7 . For the line L12 we consider the straight line passing through the points Q, P+ and Q0 . Note that the unstable separatrix of the point Q0 determines the direction of the vector field on this line. The unique global phase portrait is given in Figure 58. Now we are going to explain the phase portrait on the line L24 see Figure 70. Note that there is no connection between the separatrices of the saddles Q1 and Q2 because the point P1 is a node, see Theorems 15, 17 and Remark 16. Now consider the straight line passing through the points Q2 , P+ and Q02 . There are three possibilities: (i) The point P− is bellow the straight line. Then the unstable separatrix of Q02 must be bellow the straight line, otherwise cannot have an ω–limit. This determines the direction of the vector field on the line and we obtain the global phase portrait L24i . 35 .. O2 .. L8 . . . . . Q1 . . . P 2 P 1 .. .. P0 . . . . Q2 .. .. S=22,R=5 Figure 54. The local and the global phase portraits corresponding to the line L8 . .. O2 .. L9 . . . . . Q1 . . . P2 P 1 .. .. P0 . . . . Q2 .. .. S=22,R=5 Figure 55. The local and the global phase portraits corresponding to the line L9 . .. O2 .. L10 . .. P1 P + . . . .. Q´ .. P0 . . . P- . Q .. .. S=22, R=7 Figure 56. The local and the global phase portraits corresponding to the line L10 . (ii) The point P− belongs to the straight line and the line is invariant. Then we obtain the global phase portrait L24ii . (iii) The point P− is upper the straight line. Then the direction of the loop of the curve f2 = 0 determines the direction of this straight line. Then we obtain the global phase portrait L24iii . Now we are going to explain the phase portrait on the line L25 , see Figure 71. If there is a connection between the separatrices of the points Q1 and Q2 then they form a heteroclinic loop that must contain the node, a contradiction, see Theorems 15, 17 and also Remark 16. Hence the unstable separatrix of the point Q1 must have as ω − − limit the point P1 . Note that the unstable separatrix of the point Q1 cannot have as ω − − limit the point O20 . In the opposite case using the continuity we should have a connection between the separatrices of the saddles. So on the line L25 we obtain a unique global phase portrait, see Figure 71. 36 3.3. Phase portraits on the intersection points. The bifurcation curves intersect into 11 points, see Figure 78. The intersection points are described in what follows, see also Figure 78. √ √ p1 = {g4 = 0, g1 = 0} =  (12 + 8 2, 12 + 8 2), 8 8 p2 = {g4 = 0, g7 = 0} = , , 9 9 √ √ ! −68 + 8 78 −68 + 8 78 p3 = {g4 = 0, g6 = 0} = , , 3 3   4 4 p4 = {g4 = 0, g3 = 0} = , , 5 5 √ √ (12 − 8 2, 12 − 8 2), p5 = {g4 = 0, g1 = 0} =  2 2 p6 = {g4 = 0, g2 = 0} = , , 3  q 3 √ √ √ 4 (37 + 9 318)1/3 (2(37 + 9 318)2/3 − (37 + 9 318)1/3 − 58) p7 = {g1 = 0, g6 = 0} =  √ , 3 (37 + 9 318)( 1/3) √ 2/3 √ 1/3 1/3  2 2 37 + 9 318 − 37 + 9 318 − 58 2 √ + , 27 37 + 9 318 3  rp √ 3 √  √ 2/3 p 3 √   2 13 + 16 2 13 + 16 2 − 13 + 16 2 − 7 p8 = {g1 = 0, g7 = 0} =  2 p 3 √ , 13 + 16 2 √ 2/3 p √  3 13 + 16 2 − 13 + 16 2 − 7 2 1/3 p √ + , 3 13 + 16 2 3   2√ 2 p9 = {g2 , g7 } = 3, , 3 3 ! √ 4 13 2 p10 = {g2 , g6 } = , , 13 3   68 8√ 68 8 √ p11 = {g5 , g6 } = 3 + 78, − − 78 . 3 3 3 We present the phase portraits in each of these intersection points in Figures 79, 80 and 81. 4. Topological classification of the global phase portraits Proof of Theorem 1. In what follows we denote by S the number of separatrices and by R the number of the canonical regions. In order to present the topological classification of all global phase portraits of system (2) we apply Theorem 22 of the Appendix due to Markus, Neumann and Peixoto, see [32, 33, 34] and the notion of separatrix configuration that appears there. We recall that two global phase portrait are not topological equivalent when does not exist a homeomorphism to bring the separatrix configuration of one to the separatrix configuration of the other, see again Theorem 22. For a = c = 0 system (2) becomes system (3) and there is the global phase portrait C0 with S = 17 and R = 4. Now for a = 0 and c 6= 0 system (2) becomes system (4) and we obtain the following phase portraits S R Phase portraits 19 6 C2 23 5 C1 , C3 9 3 C4 12 5 C5 16 5 C6 37 Note that the phase portrait C1 is not topological equivalent to the phase portrait C3 because their separatrix configurations are not homeomorphic. Now we consider the case where a 6= 0. Because of the symmetry (5) of system (2) we can restrict our study to a > 0. First we consider the case where c = 0. So we work with system (6) and we obtain S R Phase portraits 16 5 A1 22 5 A2 20 5 A3 21 4 A4 22 5 A5 Note that the phase portrait A2 is not topological equivalent to the phase portrait A5 because their separatrix configurations are not homeomorphic. For the same reason the phase portrait of A1 is not topologically equivalent to the phase portrait C6 , or simply A1 6= C6 . S R Phase portraits 9 3 p4 = r7 , C4 10 3 p2 12 5 C5 13 4 p3 = L3 = L4 = L26 , L15 13 5 L5 = L27 14 5 L2 , L16 15 4 r5 = r6 = L20 15 5 r8 16 5 r9 = r10 = L21 , r1 = C6 , r4 , A1 16 6 p6 17 4 C0 17 6 p9 18 5 L31i , L31ii 18 6 L28 19 6 L30 = L29 , L31iii , C2 , L7 , p10 19 7 p5 20 5 A3 20 7 p1 , p8 21 4 A4 21 5 L7 21 6 L11iv , L11ii 21 7 L14 22 5 p11 = L8 = L9 , L19 , L18 , A2 , A5 22 7 p7 = L12 = L13 , L11i , L11iii = L11v , L12 , L10 23 4 r18ii = r19ii = L24ii , r17ii 23 5 C1 , C3 , r14 23 7 L6 24 5 L23 = r15 = r16 , r17iii = r18i = r19i = L24i , r18iii = r19iii , r20 = L24iii = L25 = r21 , r17i 24 7 L1 , L17 25 6 r3ii , L22 25 7 r13 26 7 r2 , r3iii , r3i , r11 = r12 Case S = 9 and R = 3. Then p4 = r7 and p4 6= C4 because in C4 in the interior of the loop of the curve f2 = 0 we have a center whereas in p4 we have a focus. Case S = 13 and R = 4. We have p3 = L3 = L4 = L26 , L15 6= L26 . 38 Case S = 13 and R = 5. We have L27 = L5 . Case S = 14 and R = 5. L2 6= L16 . Case S = 16 and R = 5. We obtain r9 = r10 = L21 ,C6 = r1 , r9 6= r1 r9 6= r4 , r9 6= A1 r4 6= r1 , A1 6= r1 , A1 6= r4 . Case S = 18 and R = 5 There are two phase potraits L31i 6= L31ii . Case S = 19 and R = 5. We have p10 6= C1 . Case S = 19 and R = 6 We have L29 = L30 , L29 6= L31iii , L29 6= C2 , L29 6= C2 , L29 6= L7 , L29 6= p10 , L31iii 6= C2 , L31iii 6= L7 , L31iii 6= p10 , C2 6= L7 , C2 6= p10 , L7 6= p10 . Case S = 20 and R = 7 The two phase portraits are p1 6= p8 . Case S = 21 and R = 6. We have two different phase portraits L11iv 6= L11ii . Case S = 22 and R = 5. We have L8 = L9 = p11 , L8 6= L19 , L8 6= L18 , L8 6= A2 , L8 6= A5 , L19 6= L18 , L19 6= A2 , L19 6= A5 , A2 6= L18 , A5 6= L18 , A5 6= A2 . Case R = 22 and S = 7. We have L12 = L13 = p7 , L11iii = L11v , L11i 6= L11iii , L11i 6= L12 , L11i 6= L10 , L11iii 6= L10 , L11iii 6= L12 , L12 6= L10 . Case S = 23 and R = 4. We have r18ii = r19ii = L24ii , r18ii 6= r17ii . Case S = 23 and R = 5. We have C1 6= C3 , C1 6= r14 , C3 6= r14 . Case R = 24 and S = 5. We have L23 = r15 = r16 , r17iii = r18i = r19i = L24i , r18iii = r19iii , r20 = L24iii = L25 = r21 , r15 6= r18i , r15 6= r18iii , r15 6= r20 , r15 6= r17i r18i 6= r18iii , r18i 6= r20 , r18i 6= r17i , r18iii 6= r20 , r18iii 6= r17i , r17i 6= r20 . Case R = 24 and S = 7. We have L1 6= L17 . Case S = 25 and R = 6. We have r3ii 6= L22 . Case S = 26 and R = 7 We have r11 = r12 , r2 6= r3i , r2 6= r3iii , r2 6= r12 , r3i 6= r3iii , r3i 6= r12 , r3iii 6= r12 . In summary, we can compute 65 different topological phase portraits in the Poincaré disc for the quadratic systems (2). This completes the proof of the theorem.  5. Appendix This appendix has two subsections. 5.1. Poincaré compactification. We consider the polynomial differential system (1) of degree m and its corresponding vector field X . In order to plot the global phase portrait of system (1) we need to control the orbits that come or escape at infinity. For doing this control we consider the Poincaré compactification of system (1). For more details on this compactification see Chapter 5 of [20]. Let R2 be the plane in R3 defined by (y1 , y2 , y3 ) = (x1 , x2 , 1). We define the Poincaré sphere S2 = {y = (y1 , y2 , y3 ) ∈ R3 : y12 + y22 + y32 = 1} and we denote by T(0,0,1) S2 the tangent space to S2 at the point (0, 0, 1) (see [35]). We consider the central projection f : T(0,0,1) : R2 → S2 . Note that f defines two copies of X , one in the northern hemisphere {y ∈ S2 : y3 > 0} and the other in the southern hemisphere. Let X̂ = Df ◦ X and note that X̂ is defined on S2 except on its equator S1 . Then the points at infinity of R2 are in bijective correspondence with S1 = {y ∈ S2 : y3 = 0}, ( the equator of S2 ). Hence S1 is identified to be the infinity of R2 . Then the Poincaré compactified vector field p(X ) of X will be analytic vector field induced on S2 as follows. If we multiply X̂ by the factor y3m , the vector field y3m X̂ defined in the whole S2 . 39 Note that on S2 \ S1 there are two symmetric copies of X . Hence the behavior of p(X ) around S1 gives the behavior of X near the infinity. The Poincaré disc D is the projection of the closed northern hemisphere of S2 on y3 = 0 under (y1 , y2 , y3 ) 7−→ (y1 , y2 ). Moreover, S1 is invariant under the flow of p(X ). Two polynomial vector fields X and Y on R2 are topologically equivalent if there exists a homeomorphism on S2 preserving the infinity S1 carrying orbits of the flow induced by p(X ) into orbits of the flow induced by p(Y). Note that the homeomorphism should preserve or reverse simultaneously the sense of all orbits of the two compactified vector fields p(X ) and p(Y). Since S2 is a differentiable manifold we can consider the six local charts Ui = {y ∈ S2 : yi > 0}, and Vi = {y ∈ S2 : yi < 0} for i = 1, 2, 3 with the diffeomorphisms Fi : Vi −→ R2 and Gi : Vi −→ R2 , which are the inverses of the central projections from the planes tangent at the points (1, 0, 0), (−1, 0, 0), (0, 1, 0), (0, −1, 0), (0, 0, 1) and (0, 0, −1), respectively. Let z = (z1 , z2 ) be the value of Fi (y) or Gi (y) for any i = 1, 2, 3. Then the expressions of the compactified vector field ,p(X ) of X are ! 1 z  1 z  1 z  1 1 1 z2m ∆(z) Q , − z1 P , , −z2 P , in U1 , z2 z2 z2 z2 z2 z2 ! z 1  z 1  z 1  m 1 1 1 z2 ∆(z) P , − z1 Q , , −z2 Q , in U2 , z2 z2 z2 z2 z2 z2  ∆(z) P (z1 , z2 ), Q(z1 , z2 ) in U3 , 1 − where ∆(z) = (z12 + z22 + 1) 2(m−1) . The expressions of the vector field p(X ) in the local chart Vi is the same as in the chart Ui multiplying by the factor (−1)m−1 . In these coordinates z2 = 0 denotes the points of S1 . We omit the factor ∆(z) by rescaling the vector field p(X ), and so we obtain a polynomial vector field in each local chart. The infinity S1 is invariant with p(X ). 5.2. Separatrix configuration. Let p(X ) be the Poincaré compactification in S2 of a polyno- mial vector field X in R2 . We consider the definition of parallel flows given by Markus [32] and Neumann in [33]. Let φ be a C ω local flow on the two dimensional manifold R2 or R2 \ {0}. The flow (M, φ) is C k parallel if it is C ω -equivalent to one of the following ones: strip: (R2 , φ) with the flow φ defined by ẋ = 1, ẏ = 0; annular: (R2 \ {0}, φ) with the flow φ defined (in polar coordinates) by ṙ = 0, θ̇ = 1; spiral: (R2 \ {0}, φ) with the flow φ defined by ṙ = r, θ̇ = 1. The separatrices of the vector field p(X ) in the Poincaré disc D are (i) all the orbits of p(X ) which are in the boundary S1 of the Poincaré disc (recall that S1 is the infinity of R2 ); (ii) all the finite singular points of p(X ); (iii) all the limit cycles of p(X ); and (iv) all the separatrices of the hyperbolic sectors of the finite and infinite singular points of p(X ). We denote by Σ the union of all separatrices of the flow (D, φ) defined by the compactified vector field p(X ) in the Poincaré disc D. Then Σ is a closed invariant subset of D. Every connected component of D \ Σ, with the restricted flow, is called a canonical region of φ. For a proof of the following result see [27] and [33]. 40 Theorem 21. Let φ be a C ω flow in the Poincaré disc with finitely many separatrices, and let Σ be the union of all its separatrices. Then the flow restricted to every canonical region is C ω parallel. The separatrix configuration Σc of a flow (D, φ) is the union of all the separatrices Σ of the flow together with an orbit belonging to each canonical region. The separatrix configuration Σc of the flow (D, φ) is said to be topologically equivalent to the separatrix configuration Σ̃c of the flow (D, φ̃) if there exists a homeomorphism from Σc to Σc which transforms orbits of Σc into orbits of Σ̃c , and orbits of Σ into orbits of Σ̃. Theorem 22. Let (D, φ) and (D, φ̃) be two compactified Poincaré flows with finitely many separatrices coming from two polynomial vector fields (1). Then they are topologically equivalent if and only if their separatrix configurations are topologically quivalent. For a proof of Theorem 22 see [32, 33, 34]. In sort, in order to classify the phase portraits in the Poincaré disc of a planar polynomial differential system having finitely many separatrices, it is enough to describe their separatrix configuration. Acknowledgments J. Llibre is partially supported by the Agència de Gestió d’Ajuts Universitaris i de Re- cerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017- 777911. J. Llibre and C. Pantazi are also supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER). C. Pan- tazi is additionally partially supported by the Catalan Grant 2017SGR1049 and the Spanish MINECO-FEDER Grant PGC2018-098676-B-I00/AEI/FEDER/UE. References [1] J.C. Artés, R.E. Kooij and J. Llibre, Structurally stable quadratic vector fields, Mem. Amer. Math. Soc. 134 (1998), 108 pp. [2] J.C. Artés and J. Llibre, Hamiltonian quadratic systems, J. Diff. Eqns. 107 (1994), 80–95. [3] J.C. Artés and J. Llibre, Phase portraits for quadratic systems having a focus and one antisaddle, Rocky Mount.J. Math. 24 (1994), 875–889. [4] J.C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe, Geometric configurations of singularities of planar polynomial differential systems, to appear in Birkhäuser, 2020. [5] J.C. Artés, J. Llibre and N. Vulpe, Quadratic systems with a rational first integral of degree two: a complete classification in the coefficient space R12 , Rend. Circ. Mat. di Palermo 56 (2007), 417–444. [6] J.C. Artés, J. Llibre and N. Vulpe, Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space R12 , Rend. Circ. Mat. di Palermo 59 (2010), 419–449. [7] J.C. Artés, J. Llibre and N. Vulpe, Quadratic systems with a polynomial first integral: A complete classifi- cation in the coefficient space R12 , J. Diff. Eqns. 246 (2009), 3535–3558. [8] J.C. Artés, A.C. Rezende and R.D.S. Oliveira, Global phase portraits of quadratic polynomial differential systems with a semi–elemental triple node, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 23 (2013), no. 8, 1350140, 21 pp. [9] A. N. Berlinskiĭ, On the behavior of the integral curves of a differential equation, (Russian). Izv. Vysš. Učebn. Zaved. Matematika, (1960), 15, 3–18. [10] A.N. Berlinskiĭ, Qualitative study of the differential equation ẋ = x + b0 x2 + b1 xy + b2 y 2 , ẏ = y + a0 x2 + a1 xy + a2 y 2 J. Diff. Eqns, (1966), 2, 174–178. [11] R. Bix, Conics and cubics. A concrete introduction to algebraic curves, Second Edition, Undergraduate Texts in Math., Springer, 2006 [12] L. Cairó and J. Llibre, Phase portraits of planar semi–homogeneous vector fields, Nonlinear Analisis, Th. Meth. & App. (1997) 29, 783–811. [13] F. Chen, C. Li, J. Llibre and Z. Zhang, A uniform proof on the weak Hilbert’s 16th problem for n = 2, J. Diff. Eqns. 221 (2006), 309–342. [14] W.A. Coppel, A survey of quadratic systems, J. Diff. Eqns. 2 (1966), 293–304. 41 [15] S.-N. Chow, C. Li and Y. Yi, The cyclicity of period annulus of degenerate quadratic Hamiltonian system with elliptic segment loop, Erg. Th. & Dyn. Syst. 22 (2002), 1233–1261. [16] B. Coll, A. Gasull and J. Llibre, Some theorems on the existence, uniqueness and nonexistence of limit cycles for quadratic systems, J. Diff. Eqns. 67 (1987), 372–399. [17] B. Coll, C. Li and R. Prohence, Quadratic perturbations of a class of quadratic reversible systems with two centers, Disc. & Contin. Dyn. Sys. 24 (2009), 699–729. [18] R.J. Dickson and L.M. Perko, Bounded quadratic systems in the plane, J. Diff Eqns. 6 (1970), 251–273. [19] F. Dumortier and C. Li, Quadratic Lienard equations with quadratic damping, J. Diff. Eqns. 139 (1997), 41–59. [20] F. Dumortier, J. Llibre and J. C. Artés, Qualitative theory of planar polynomial systems, Springer, 2006. [21] A. Gasull, S. Li-Ren and J. Llibre, Chordal quadratic systems, Rocky Mount. J. Math. 16 (1986), 751–782. [22] A. Gasull and J. Llibre, On the nonsingular quadratic differential equations in the plane, Proc. Amer. Math. Soc. 104 (1988), 793–794. [23] L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case, Invent. Math. 143 (2001), 449–497. [24] L. Gavrilov and I. D. Iliev, Second order analysis in polynomially perturbed reversible quadratic Hamiltonian systems, Erg. Th. & Dyn. Syst. 20 (2000), 1671–1686. [25] D.D. Hua, L. Cairo, M.R. Feix, K.S. Govinder and P.G.L. Leach, Connection between the existence of first integrals and the Painlevé property in two-dimensional Lotka-Volterra and quadratic systems, Proc. Roy. Soc. London 452 (1996), 859–880. [26] M. Han and C. Yang, On the cyclicity of a 2-polycycle for quadratic systems, Chaos, Solitons Fract. 23 (2005), 1787–1794. [27] W. Li, J. Llibre, M. Nicolau and X. Zhang, On the differentiability of first integrals of two dimensional flows, Proc. Amer. Math. Soc. 130 (2002), 2079–2088. [28] J. Llibre, J.S. Pérez del Rı́o and J.A. Rodrı́guez, Phase portraits of a new class of integrable quadratic vector fields, Dyn. Cont. Disc. Impulsive Syst. 7 (2000), 595–616. [29] C. Li and J. Llibre, A unified study on the cyclicity of period annulus of the reversible quadratic Hamiltonian systems, J. Dyn. and Diff. Eqns. 16 (2004), 271–295. [30] C. Li and Z. Zhang, Remarks on 16th weak Hilbert problem for n=2, Nonlinearity 15, (2002), 1975–1992. [31] V.A. Lunkevich and K.S. Sibirskii, Integrals of a general quadratic differential system in cases of a center, Diff. Eqns. 18 (1982), 563–568. [32] L. Markus, Global structure of ordinary differential equations in the plane, Trans. Amer. Math Soc. 76 (1954), 127–148. [33] D.A. Neumann, Classification of continuous flows on 2–manifolds, Proc. Amer. Math. Soc. 48 (1975), 73–81. [34] M.M. Peixoto, Dynamical Systems. Proccedings of a Symposium held at the University of Bahia, 389–420, Acad. Press, New York, 1973. [35] H. Poincaré, Sur l’intégration des équations différentielles du premier ordre et du premier degré I, Rend. Circ. Mat. di Palermo 5 (1891), 161–191. [36] H. Poincaré, Sur les courbes définies par une équation différentielle, Oevres compltes, Vol.1, 1928. [37] J.W. Reyn, A bibliography of the qualitative theory of quadratic systems of differential equations in the plane, Third edition. Delft University of Technology, Report 94–02, 1994. [38] I.G. Roset, Nonlocal bifurcation of limit cycles and quadratic differential equations in the plane (Russian), Dissertation Kand. Phys. Mat. Samarkand University, 1991. [39] R. Roussarie and D. Schlomiuk, On the geometric structure of the class of planar quadratic differential systems, Qual. Th. of Dyn. Syst. 3 (2002), 93–121. [40] D. Schlomiuk and N. Vulpe, Geometry of quadratic differential systems in the neighborhood of infinity, J. Diff. Eqns. 215 (2005), 357–400. [41] A. Schlomiuk and N. Vulpe, Integrals and phase portraits of planar quadratic differential systems with invariant lines of at least five total multiplicity, Rocky Mountain J. Math. 38 (2008), 1887–2104. [42] N.I. Vulpe, Affine-invariant conditions for the topological discrimination of quadratic systems with a center, J. Diff. Eqns. 19 (1983), 273–280. [43] Ye, Yan Qian; Cai, Sui Lin; Chen, Lan Sun; Huang, Ke Cheng; Luo, Ding Jun; Ma, Zhi En; Wang, Er Nian; Wang, Ming Shu and Yang, Xin., A Theory of limit cycles, Translated from the Chinese by Chi Y. Lo. Second edition. Translations of Mathematical Monographs. 66. Amer. Math. Soc., Providence, RI, 1986. [44] P. Zhang, On the distribution and number of limit cycles for quadratic systems with two foci, Qual. Theory Dyn. Syst. 3 (2002), 437–463. [45] H. Żola̧dek, The cyclicity of triangles and lines in quadratic systems, J. Diff. Eqns. 122 (1995), 137–159. 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bel- laterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 42 2 Departament de Matemàtiques, Universitat Politècnica de Catalunya, (EPSEB), Av. Doctor Marañón, 44–50, 08028 Barcelona, Spain Email address: chara.pantazi@upc.edu 43 .. O2 .. L11(i) . . P1 . . Q´ ... P0 .. . . P + . Q . . P- .. .. S=22,R=7 .. O2 .. L11(ii) . . P1 . . Q´ .. P0 .. . . . . . P- . P + Q .. .. S=21,R=6 .. O2 .. L11(iii) . . P1 . . Q´ .. P0 .. . . . . P- Q . . P + .. .. .. S=22,R=7 .. O2 L11(iv) .. Q´ P1 .. .. P0 .. . . P- P + .. .. .. Q .. .. O2 .. S=21,R=6 L11(v) . . Q´ P . . 1 .. P0 .. . . P - P .. .. .. + Q .. S=22,R=7 44 Figure 57. The local and the global phase portraits corresponding to the line L11 . .. O2 .. L12 . . Q’ . . P- . .. . .. P0 . . . . P1 Q P .. + .. S=22, R=7 Figure 58. The local and the global phase portraits corresponding to the line L12 . .. O2 .. L13 . . . . P- .. .. .. .. P0 . . . . P 1 Q P .. + .. S=22, R=7 Figure 59. The local and the global phase portraits corresponding to the line L13 . .. . L14 .. .. . .. .. S=21, R=7 Figure 60. The local and the global phase portraits corresponding to the line L14 . .. O2 .. L15 . . .. P + P0 . . .. P- .. .. S=13, R=4 Figure 61. The local and the global phase portraits corresponding to the line L15 . 45 .. O2 .. L16 . . .. P - .. P0 . P + . .. .. S=14, R=5 Figure 62. The local and the global phase portraits corresponding to the line L16 . .. O2 .. L17 . . . . . . .. P- .. P0 .. . P + Q+ .. . .. .. Q- S=24, R=7 Figure 63. The local and the global phase portraits corresponding to the line L17 . .. O2 .. L18 . . . . P + . . Q+ . . .. P- .. P0 . . . . Q- .. .. S=22, R=5 Figure 64. The local and the global phase portraits corresponding to the line L18 . .. O2 .. L19 . . . . . . P + Q+ . . .. P- .. P0 . . . . Q- .. .. S=22, R=5 Figure 65. The local and the global phase portraits corresponding to the line L19 . 46 .. O2 .. L 20 . .. .. P+ P0 . .. .. P- P 1 .. .. S=15,R=4 Figure 66. The local and the global phase portraits on the line L20 . .. O2 .. L 21 . . .. P0 . . .. P- P 1 . P + . .. .. S=16,R=5 Figure 67. The local and the global phase portraits on the line L21 . .. O2 .. . . L 22 . . . . .. P - P0 . . .. .. . P1 . .. Q1 P + .. Q2 .. S=25,R=6 Figure 68. The local and the global phase portraits on the line L22 . .. O2 .. L 23 . . . P Q1 . . . . + . P- . .. . .. P1 P0 . . . . .. Q2 .. S=24,R=5 Figure 69. The local and the global phase portraits on the line L23 . 47 .. O2 .. L 24(i) .. L 24(ii) . . . . . . . Q1 . . Q2´. . .. . .. P+ . . .. . P1 P- . P0 . . . . . Q2 Q1´ .. .. .. .. S=24,R=5 S=23,R=4 .. O2 L 24(iii) . . . Q1 ´. Q2 . . . . . .. . P- P+ P1 .. . P0 . .´ . Q2 Q1 .. .. S=24,R=5 Figure 70. The local and the global phase portraits on the line L24 . .. O2 .. L25 . . . Q1 . . . . P+ P1 . . .. P0 . . .. . . P- Q2 . .. .. S=24, R=5 Figure 71. The local and the global phase portraits on the line L25 . .. O2 .. L26 P- . . .. P1 P0 . . .. .. .. S=13, R=4 Figure 72. The local and the global phase portraits on the line L26 . 48 .. O2 .. L 27 . .. P 1 P0 . .. P- . . .. .. S=13, R=5 Figure 73. The local and the global phase portraits on the line L27 . .. .. . O2 . L 28 P 1 . ..P0 . .. . . P- . .. . Q .. Q S=18, R=6 Figure 74. The local and the global phase portraits on the line L28 . .. O2 . L 29 .. .. P - P0 P 1 . .. Q Figure 75. The local and the global phase portraits on the line L29 . .. O2 L 30 . .. .. P - P0 P 1 . .. Q Figure 76. The local and the global phase portraits on the line L30 . 49 .. O2 .. . . L31(i) . . P1 . . .. P - .. P0 . . .. Q .. .. O2 .. S=18, R=5 . . L31(ii) . P1 . . . .. P - .. P0 . . .. Q .. .. O2 .. S=18, R=5 . . P1 . . L31(iii) P -. .. P0 . .. . . .. Q .. S=19, R=6 Figure 77. The local and the global phase portraits on the line L31 . 50 g g4=0 1=0 h=0 p1 j1 =0 L11(ii) L11( i v ) g3=0 p2 p3 p4 p8 p7 p5 p9 L31( i i ) g2=0 p6 p10 g7=0 h=0 g6=0 g5=0 L24(ii) p11 Figure 78. c = 1, a > 0. The bifurcation curves intersect into 11+4 points. It is a qualitative picture. .. .. p .. p1 2 p 3 . . .. .. . .. .. . . .. S=20,R=7 S=10,R=3 .. S=13,R=4 .. Figure 79. The phase portraits in the points p1 , p2 and p3 . 51 .. p4 .. p .. . 5 . p6 . .. . .. . .. .. . .. .. .. .. S=9,R=3 p7 .. S=19,R=7 p8 S=16, R=6 .. p9 . . . . . . .. .. . .. . . . . . .. S=22, R=7 .. S=20, R=7 .. S=17, R=6 Figure 80. The phase portraits in the points pk for k = 4, ..., 9. .. p10 .. p11 . . . . . . . .. .. . . . .. S=19, R=6 .. S=22,R=5 Figure 81. The phase portraits in the points p1 0 and p11 . 52