This is a preprint of: “Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line”, Jaume Llibre, Mauricio Fronza da Silva, C. R. Acad. Sci. Paris Ser. I Math., 2018. DOI: [10.1016/j.crma.2018.12.008] PHASE PORTRAITS OF INTEGRABLE QUADRATIC SYSTEMS WITH AN INVARIANT PARABOLA AND AN INVARIANT STRAIGHT LINE JAUME LLIBRE1 AND MAURÍCIO FRONZA DA SILVA2 Abstract. We classify the phase portraits of the quadratic polynomial differential systems having an invariant parabola, an invariant straight line, and a Darboux first integral produced by these two invariant curves. 1. Introduction and statement of main results Consider planar polynomial differential systems of the form (1) ẋ = P (x, y), ẏ = Q(x, y), where P and Q are real polynomials defined in R2 . The dot denotes deriv- ative with respect to the independent variable t. The degree of system (1) is the maximum of the degrees of the polynomials P and Q. When a poly- nomial differential system (1) has degree two we call it simply a quadratic system. Let U be a dense and open subset of R2 . A first integral of a differential system (1) is a non–locally constant C 1 function H : U → R which is constant on the orbits of the system (1) contained in U , i.e. H satisfies ∂H ∂H (2) (x, y)P (x, y) + (x, y)Q(x, y) = 0 ∂x ∂y in the points (x, y) ∈ U . We say that a quadratic system is integrable if it has a first integral H : U → R. After the linear differential systems in the plane, i.e. the polynomial differential systems of degree one, the quadratic systems are the easier ones and they have been studied intensively, and more than one thousand papers have been published on those systems, see the references quoted in the books of Ye [24, 25] and Reyn [20]. But the classification of all the integrable quadratic system is an open problem. The phase portrait of a differential system is the decomposition of its domain of definition as union of all its oriented orbits. The phase portraits 2010 Mathematics Subject Classification. Primary 34C05. Key words and phrases. quadratic system, phase portraits, invariant parabola, invari- ant straight line. 1 2 J. LLIBRE AND M.F. DA SILVA of a polynomial differential system is drawn in the Poincaré disc which, roughly speaking, is the closed disc centered at the origin of coordinates with radius one, the interior of this disc is diffeomorphic to R2 and its boundary S1 corresponds to the infinity of R2 , each point of S1 provides a direction for going or coming from infinity. For more details see section 3 or Chapter 5 of [7]. Many classes of integrable quadratic systems have been studied, and for these classes the topological phase portraits of their quadratic systems have been classified in the Poincaré disc. A relatively easy class of integrable quadratic systems is formed by the homogeneous quadratic systems studied by Lyagina [16], Markus [17], Korol [11], Sibirskii and Vulpe [22], Newton [18], Date [5] and Vdovina [23],... An important class of integrable qua- dratic systems are the ones with centers studied by many authors see for instance Dulac [6], Kapteyn [9, 10], Bautin [3], Lunkevich and Sibirskii [15], Schlomiuk [21], Vulpe [27], Żola̧dek [28], Ye and Ye [26], Artés, Llibre and Vulpe [2], ... In general all polynomial differential systems having a center with purely imaginary eigenvalues are integrable, see for instance Poincaré [19] and Liapunov [12], and one example in [13]. Another interesting class is the one formed by the Hamiltonian quadratic systems, see Artés and Llibre [1], Kalin and Vulpe [8] and Artés, Llibre and Vulpe [2]. We say that the algebraic curve f (x, y) = 0 is invariant by the polynomial system (1) if ∂f ∂f (3) P (x, y) + Q(x, y) = K(x, y)f (x, y), ∂x ∂y for some polynomial K(x, y), this polynomial is called the cofactor of the invariant algebraic curve f (x, y) = 0. Note that an invariant algebraic curve of system (1) is formed by orbits of that system. Recently the topological phase portraits in the Poincaré disc of a new class of integrable quadratic systems having an invariant ellipse and an invariant straight line have been classified in [16]. The objective of this paper is to classify the topological phase portraits in the Poincaré disc of a class of integrable quadratic systems having an invariant parabola and an invariant straight line. After an affine change of variables we may suppose without loss of generality that the equation of the invariant parabola and the one of the invariant straight line are y = x2 and ax + by + c = 0 respectively, where a, b and c are constants such that a2 + b2 6= 0. More precisely we shall classify the topological phase portraits in the Poincaré disc of the class of quadratic systems ẋ = bλ1 (y − x2 ) − λ2 (ax + by + c), (4) ẏ = −aλ1 (y − x2 ) − 2λ2 x(ax + by + c), PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 3 a b c d e f g h i j k l m n o p q r s t u v x y z w Figure 1. Phase portraits of systems (4) in the Poicaré disc. where a, b, c, λ1 and λ2 are real parameters such that a2 + b2 6= 0, and λ21 + λ22 6= 0 otherwise we will have the null differential system. Note that system (4) depends on five parameters. 4 J. LLIBRE AND M.F. DA SILVA It is easy to check using the definition of invariant algebraic curve (3) that the quadratic systems (4) have the invariant parabola y − x2 = 0 and the invariant straight line ax + by + c = 0. At the end of section 2 we will show that the quadratic systems (4) is a subclass of all quadratic systems having an invariant parabola and an invariant straight line. Using the Darboux theory (see Theorem 8.7 of [7]) it is easy to obtain that (5) H(x, y) = (y − x2 )−λ2 (ax + by + c)λ1 . is a first integral of the system (4) in the open and dense subset of R2 where it is defined. In fact it is immediate to check that it is a first integral using the definition (2). The existence of this first integral shows that the quadratic systems have no limit cycles. Theorem 1. The phase portraits in the Poincaré disc of systems (4) with (λ21 + λ22 )(a2 + b2 ) 6= 0 are topologically equivalent to one of the 26 phase portraits shown in Figure 1. The correspondence between systems (4) and the phase portraits of Figure 1 are provided in Table 1. This work is organized as follows. In section 2 mainly we present the def- initions concerning the classification of singularities of planar vector fields. In section 3 we introduce the Poincaré compactification. We perform the study of the singularities of system (4) in sections 4 and 5. Finally in section 6 we prove Theorem 1. 2. Preliminary definitions and results Consider a vector field X : U → R2 defined on the open subset U of R2 and denote the Jacobian matrix of X at (x, y) ∈ U by DX(x, y). Let (x0 , y0 ) ∈ U be a singularity of X. We say that (x0 , y0 ) is elementary if one of the following three conditions holds: either the real part of both eigenvalues of DX(x0 , y0 ) are different of zero, in this case (x0 , y0 ) is a hyperbolic singular point; or the eigenvalues of DX(x0 , y0 ) are purely imaginary and (x0 , y0 ) is a focus or a center; or only one of the eigenvalues of DX(x0 , y0 ) is different of zero, in this case (x0 , y0 ) is a semi-hyperbolic singular point. If both eigenvalues of DX(x0 , y0 ) are zero then (x0 , y0 ) is a non-elementary singular point of X. The topological classification of the flow near a hyperbolic or semi-hyperbolic singular point of X is given, for instance in Theorems 2.15 and 2.19 of [7], respectively. Suppose that (x0 , y0 ) is a non-elementary singular point of X. If the matrix DX(x0 , y0 ) is not zero, then (x0 , y0 ) is a nilpotent singularity. If DX(x0 , y0 ) is the zero matrix then (x0 , y0 ) is a linearly zero singularity. The topological classification of a nilpotent singular point is given, for instance in Theorem PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 5 Global Phase Portrait Parameters of system (4) of Figure 1 a λ1 = 0 and bλ2 6= 0. b λ1 = 0, b = 0 and aλ2 6= 0. c λ2 = 0 and bλ1 6= 0. d λ2 = 0, b = 0 and aλ1 6= 0. aλ1 > 0, aλ2 > 0, b = 0 and a(λ1 − 2λ2 ) > 0 or e aλ1 < 0, aλ2 < 0, b = 0 and a(λ1 − 2λ2 ) < 0. aλ1 > 0, aλ2 > 0, b = 0 and a(λ1 − 2λ2 ) < 0 or f aλ1 < 0, aλ2 < 0, b = 0 and a(λ1 − 2λ2 ) > 0. aλ1 > 0, aλ2 > 0, b = 0 and λ1 = 2λ2 or g aλ1 < 0, aλ2 < 0, b = 0 and λ1 = 2λ2 . aλ1 > 0, aλ2 < 0, b = 0 and a(λ1 − 2λ2 ) > 0 or h aλ1 < 0, aλ2 > 0, b = 0 and a(λ1 − 2λ2 ) < 0. i λ1 = λ2 6= 0, b 6= 0 and a2 − 4bc > 0. j λ1 = λ2 6= 0, b 6= 0 and a2 − 4bc = 0. k λ1 = λ2 6= 0, b 6= 0 and a2 − 4bc < 0. 0 < λ2 < λ1 < 2λ2 , b 6= 0 and a2 − 4bc > 0 or l 2λ2 < λ1 < λ2 < 0, b 6= 0 and a2 − 4bc > 0. 0 < 2λ2 = λ1 , b 6= 0 and a2 − 4bc > 0 or m λ1 = 2λ2 < 0, b 6= 0 and a2 − 4bc > 0. 0 < 2λ2 < λ1 , b 6= 0 and a2 − 4bc > 0 or n λ1 < 2λ2 < 0, b 6= 0 and a2 − 4bc > 0. 0 < λ1 < λ2 , b 6= 0 and a2 − 4bc > 0 or o λ2 < λ1 < 0, b 6= 0 and a2 − 4bc > 0. p λ1 λ2 < 0, b 6= 0 and a2 − 4bc > 0. 0 < λ2 < λ1 < 2λ2 , b 6= 0 and a2 − 4bc = 0 or q 2λ2 < λ1 < λ2 < 0, b 6= 0 and a2 − 4bc = 0. 0 < 2λ2 = λ1 , b 6= 0 and a2 − 4bc = 0 or r λ1 = 2λ2 < 0, b 6= 0 and a2 − 4bc = 0. 0 < 2λ2 < λ1 , b 6= 0 and a2 − 4bc = 0 or s λ1 < 2λ2 < 0, b 6= 0 and a2 − 4bc = 0. 0 < λ1 < λ2 , b 6= 0 and a2 − 4bc = 0 or t λ2 < λ1 < 0, b 6= 0 and a2 − 4bc = 0. u λ1 λ2 < 0, b 6= 0 and a2 − 4bc = 0. v λ1 λ2 < 0, b 6= 0 and a2 − 4bc < 0. 0 < λ1 < λ2 , b 6= 0 and a2 − 4bc < 0 or x λ2 < λ1 < 0, b 6= 0 and a2 − 4bc < 0. 0 < λ2 < λ1 < 2λ2 , b 6= 0 and a2 − 4bc < 0 or y 2λ2 < λ1 < λ2 < 0, b 6= 0 and a2 − 4bc < 0. 0 < 2λ2 = λ1 , b 6= 0 and a2 − 4bc < 0 or z λ1 = 2λ2 < 0, b 6= 0 and a2 − 4bc < 0. 0 < 2λ2 < λ1 , b 6= 0 and a2 − 4bc < 0 or w λ1 < 2λ2 < 0, b 6= 0 and a2 − 4bc < 0. Table 1. Classification of the phase portraits in the Poincaré disc of the quadratic systems (4). 6 J. LLIBRE AND M.F. DA SILVA 3.5 of [7]. The main tool used for studying the local phase portraits of the the linearly zero singularities are the changes of variables called blow-ups. In general a quasi-homogeneous blow-up is a change of variables of the form (x, y) 7→ (xα , xβ y), positive x-direction, (x, y) 7→ (−xα , xβ y), negative x-direction, (x, y) 7→ (xy α , y β ), positive y-direction, (x, y) 7→ (xy α , −y β ), negative y-direction, where α, β are positive integers. For quasi-homogeneous blow-ups in the x-direction (respectively y-direction), when α (respectively β) is odd, the information obtained in the positive x-direction (respectively y-direction) is useful to the negative x-direction (respectively y-direction). A suitable choice of α and β avoids successive homogeneous blow-up’s (see p. 104 of [7]). In this work, all isolated singularities are hyperbolic, nilpotent, or linearly zero. The classification of nilpotent singularities could be perform using Theorem 3.5 of [7]. But in order to obtain information on the localization of the separatrices of the nilpotent singularities we will use quasi-homogeneous blow-ups (see Lemmas 10 and 15). For showing that the class of quadratic systems (4) having the invariant parabola y = x2 , the invariant straight line ax + by + c = 0, and the first integral (5) are not all the quadratic systems having the invariant parabola y = x2 and the invariant straight line ax + by + c = 0, it is sufficient to provide an example. Consider the invariant straight line L = ax + by + c = 8x + 16y + 1 = 0, and the subclass of quadratic systems (4) having this invariant straight line are ẋ = 16λ1 (y − x2 ) − λ2 (8x + 16y + 1), (6) ẏ = −8λ1 (y − x2 ) − 2λ2 x(8x + 16y + 1). But the quadratic system ẋ = 1 + 3x + 33y − 36x2 + 4xy, ẏ = 2(x − 4y + 7x2 − 3xy + 4y 2 ). also has the invariant parabola y = x2 with cofactor kP = −8(1 + 9x − y) and the invariant straight line L with cofactor kL = 8(1 − x + y), it is not contained in the subclass and it has no a first integral of the form (5), because according statement (i) of Theorem 8.7 of [7] such a first integral only exists if there are real numbers µ1 and µ2 not all zero such that µ1 kp + µ2 kL = −8µ1 (1 + 9x − y) + 8µ2 (1 − x + y) = 0, but these µ1 and µ2 do not exist. PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 7 3. The Poincaré compactification The Poincaré compactification is a helpful technique to study the behavior near the infinity of a polynomial vector field defined in R2 . In this section X is the planar polynomial vector field defined by the polynomial differential system (1) of degree d. We shall use the notation S2 = {(z1 , z2 , z3 ) ∈ R3 ; z12 + z22 + z32 = 1} and S1 = {(z1 , z2 , z3 ) ∈ S2 ; z3 = 0} for the two-dimensional sphere and its equator, respectively. We identify R2 with the tangent plane π of S2 at the point (0, 0, 1). Considering the central projection from π into S2 , we obtain a vector field X ′ on S2 \ S1 such that the infinity points of π are projected in S1 . Observe that the vector field X ′ is symmetric with respect to the center of S2 . X ′ is an unbounded vector field near S1 , but after a multiplication by an appropriate factor, the resultant vector field p(X), called the Poincaré compactification of X, is analytic and defined in the whole sphere S2 . The symmetry means that it is sufficient to consider p(X) defined only in the closed northern hemisphere H of S2 . We project the vector field p(X) on H using the ortogonal projection into a vector field on the disc {(z1 , z2 , z3 ) ∈ R3 ; z12 + z22 ≤ 1, z3 = 0}, called the Poincaré disc. It is on this disc where we draw the phase portraits of the polynomial differential systems, its interior is diffeomorphic to R2 and its boundary S1 corresponds to the infinity of R2 . To obtain an explicit expression of the Poincaré compatification p(X), the northern and the southern hemisphere of S2 are denoted by H + = {(z1 , z2 , z3 ) ∈ S2 ; z3 > 0} and H − = {(z1 , z2 , z3 ) ∈ S2 ; z3 < 0}, respectively. We consider the central projections of π := {(z1 , z2 , z3 ) ∈ R3 ; z3 = 1} in H + and H − , which are defined by f+ : π → H+ f − : π → H− 1 and 1 , z 7→ (z1 , z2 , 1) z 7→ (−z1 , −z2 , −1) ∆(z) ∆(z) p respectively. Here ∆(z) = z12 + z22 + 1, z ∈ π. Define the vector field X ′ on S2 \ S1 by  ′ Df + (z)X(z) if w = f + (z), X (w) = Df − (z)X(z) if w = f − (z). The vector field X ′ is unbounded in a neighborhood of S1 , but the vector field w3d X ′ (w) is an analytical extension of X ′ to the whole S2 . This extension is the Poincaré compactification of X that we have denoted by p(X). In general, the vector field p(X) is C ω -equivalent, but not C ω -conjugated, to X in each hemisphere of S2 . Then we study the trajectories of X near a singularity using the correspondent singularities of p(X). Since p(X) may 8 J. LLIBRE AND M.F. DA SILVA has singularities in S1 , a singular point of p(X) which belongs to S2 \ S1 or to S1 is called finite or infinite singular point of X, respectively. We can prove that S1 is invariant under the flow of p(X). Using the expressions of f + and f − we obtain that p(X) is symmetric with respect to the origin, then it is sufficient to study the trajectories of p(X) in H + ∪ S1 . We will obtain the expressions of p(X) in the local charts of S2 . For j = 1, 2, 3 consider Uj = {(z1 , z2 , z3 ) ∈ S2 ; zj > 0}, Vj = {(z1 , z2 , z3 ) ∈ S2 ; zj < 0} and ϕj : Uj → R2 , ψj : Vj → R2 defined by (z2 , z3 ) (z1 , z3 ) (z1 , z2 ) ϕ1 (z) = −ψ1 (z) = , ϕ2 (z) = −ψ2 (z) = , ϕ3 (z) = . z1 z2 z3 We denote by (u, v) the value of ϕj or ψj at the point z. The expression of p(X) in the chart (U1 , ϕ1 ) is        d 1 u 1 u d+1 1 u u̇ = v −uP , +Q , , v̇ = −v P , . v v v v v v The expression of p(X) in the chart (U2 , ϕ2 ) is        d u 1 u 1 d+1 u 1 u̇ = v P , − uQ , , v̇ = −v Q , , v v v v v v and the expression of p(X) in the chart (U3 , ϕ3 ) is u̇ = P (u, v), v̇ = Q(u, v). The expression of p(X) in the chart (Vj , ψj ) is the expression of p(X) in the chart (Uj , ϕj ) multiplied by (−1)d−1 , for j = 1, 2 and 3. Notice that (u, v) ∈ Uj is an infinite singular point of X if and only if the expression of p(X) in the chart (Uj , ϕj ) vanishes at (u, v) and v = 0. Due to the symmetry of p(X) with respect to the origin of coordinates, if z is an infinite singular point of X then −z is also an infinite singular point of X. Then we only need to study p(X) in the charts (Uj , ϕj ), j = 1, 2, 3. If z is an infinite singular point of X with z ∈ U2 and z 6= (0, 0, 0) then z ∈ U1 ∪ V1 . Then, to study all the infinite singular points of p(X), it is sufficient to study the singularities of p(X) in U1 and the origin of U2 . From the definition of the Poincaré compactification follows Remarks 2-4. Remark 2. Suppose that k : R2 → R is a polynomial function of degree d1 and X e is a polynomial planar vector field with degree d2 . Define X = e Let pj (X) and pj (X) kX. e the expression of X and X e in the chart (Uj , ϕj ), respectively, for j = 1 and 2. Then     1 u e and p2 (X) = v d1 k u , 1 p2 (X).e p1 (X) = v d1 k , p1 (X) v v v v PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 9 Remark 3. If b 6= 0 then the points of R2 of the straight line ax + by + c = 0 of R2 are mapped into the points of {a+bu+cv = 0}∩U1 with v 6= 0. Observe that the straight line a + bu + cv = 0 cuts the line of the infinity v = 0 at the point (−a/b, 0). Remark 4. If b = 0 then the points of R2 of the straight line ax + by + c = 0 of R2 are mapped into the points {au + cv = 0} ∩ U2 with v 6= 0. Observe that the straight line au + cv = 0 cuts the line of the infinity v = 0 at the point (0, 0). Remark 5. For each k ∈ R the points of R2 of the parabola y = x2 + k contained in the half-planes y > 0 or y < 0 are mapped into the points of the form v = u2 + kv 2 of U2 , or into the points of the form −v = u2 + kv 2 of V2 , respectively. 4. Phase portraits of system (4) for the case λ1 λ2 = 0 and λ21 + λ22 6= 0. In this section we present the phase portrait in the Poincaré disc of system (4) for the case λ1 λ2 = 0. If λ1 = λ2 = 0 and λ21 + λ22 6= 0. Lemma 6. Suppose that λ1 = 0 and λ2 6= 0. If b 6= 0 or b = 0 then the phase portrait in the Poincaré disc of system (4) is shown in Figures 1-a or 1-b, respectively. Proof. After the rescaling given by ds = λ2 (ax+by+c) dt system (4) becomes (7) x′ = −1, y ′ = −2x, where the prime denotes the derivative with respect to the variable s. The phase portrait in R2 of system (7) is given in Figure 2-a. The expression of the Poincaré compactification of system (7) in the charts (U1 , ϕ1 ) and (U2 , ϕ2 ) is (8) u̇ = uv − 2, v̇ = v 2 , and (9) u̇ = 2u2 − v, v̇ = 2uv, respectively. We present the phase portrait of system (8) in a sufficiently small neighborhood of the axis v = 0 in Figure 2-b. Each orbit of system (7) is contained in a parabola given by y = x2 + k, for an appropriated choice of k ∈ R. From Remark 5 it follows that the phase portrait of system (9) is the one of Figure 2-c. Combining the informations of Figure 2-a-c we obtain the phase portrait in the Poincaré disc of system (7) in Figure 2-d. Using Remark 2 with e = (−1, −2x) we obtain the conclusions of k(x, y) = λ2 (ax + by + c) and X Lemma 6.  10 J. LLIBRE AND M.F. DA SILVA a b c d Figure 2. (a) Phase portrait in R2 of system (7). (b) Phase portrait of system (8) in a neighborhood of the axis v = 0. (c) Phase portrait of system (9). (d) Phase portrait in the Poincaré disc of system (7). Lemma 7. Suppose that λ1 6= 0 and λ2 = 0. If b 6= 0 or b = 0 then the phase portrait in the Poincaré disc of system (4) is given in Figures 1-c or 1-d, respectively. Proof. After the rescaling given by ds = λ1 (y − x2 ) dt system (4) becomes (10) x′ = b, y ′ = −a. The phase portrait in R2 of system (10) is shown in Figure 3. a b Figure 3. Phase portrait of system (10). (a) b > 0; if b < 0 then the orientation of the orbits is reversed. (b) b = 0 and a > 0; if b = 0 and a < 0 then the orientation of the orbits is reversed. The expression of the Poincaré compactification of system (10) in the charts (U1 , ϕ1 ) and (U2 , ϕ2 ) is (11) u̇ = −(a + bu), v̇ = −bv, and (12) u̇ = au + b, v̇ = av, respectively. We present the phase portrait of systems (11) and (12) in Figures 4-a-d. Combining the informations of Figures 3 and 4 we obtain the local and the phase portrait in the Poincaré disc of system (10) in Figure 5. Using Remark 2 with k(x, y) = λ1 (y − x2 ) and X e = (b, −a) we conclude the proof.  PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 11 a b c d Figure 4. (a) Phase portrait of system (11) with b > 0; if b < 0 then the orientation of the orbits is reversed. (b) Phase portrait of system (11) with a > 0 and b = 0; if a < 0 and b = 0 then the orientation of the orbits is reversed. (c) Phase portrait of system (12) with b > 0; if b < 0 then the orientation of the orbits is reversed. (d) Phase portrait of system (12) with a > 0 and b = 0; if a < 0 and b = 0 then the orientation of the orbits is reversed. a b c d Figure 5. (a) Local phase portrait in the Poincaré disc of system (10) with b > 0; if b < 0 then the orientation of the orbits is reversed. (b) Local phase portrait in the Poincaré disc of system (10) with b = 0 and a > 0; if b = 0 and a < 0 then the orientation of the orbits is reversed. (c) Phase portrait in the Poincaré disc of system (10) with b > 0; if b < 0 then the orientation of the orbits is reversed. (d) Phase portrait in the Poincaré disc of system (10) with b = 0 and a > 0; if b = 0 and a < 0 then the orientation of the orbits is reversed. 5. Local phase portraits of system (4) for the case λ1 λ2 6= 0 In this section we will study the local phase portraits in the Poincaré disc of system (4) when λ1 λ2 6= 0. 5.1. Finite singularities of system (4). We denote the parabola y = x2 and the straight line ax + by + c = 0 by p and l, respectively. In this subsection we consider the cases b = 0; b 6= 0 and λ1 = λ2 ; b 6= 0 and λ1 6= λ2 in Lemmas 8-10, respectively. Lemma 8. Suppose that λ1 λ2 6= 0 and b = 0. Then system (4) has a unique singularity (x0 , y0 ) and the local phase portrait in R2 of system (4) at (x0 , y0 ) is given in Figure 6. Proof. From the hypothesis a2 + b2 6= 0 it follows that a = 6 0. The unique singularity of system (4) is (x0 , y0 ) where x0 = −c/a and y0 = c2 /a2 . Observe 12 J. LLIBRE AND M.F. DA SILVA l p l p a b Figure 6. Local phase portrait of system (4). (a) b = 0, aλ1 > 0 and aλ2 > 0. If b = 0, aλ1 < 0 and aλ2 < 0 then the orientation of the orbits is reversed. (b) b = 0, aλ1 > 0 and aλ2 < 0. If b = 0, aλ1 < 0 and aλ2 > 0 then the orientation of the orbits is reversed. that (x0 , y0 ) ∈ p∩l and l is given by the equation x = x0 . Moreover ẏ|x=x0 = −aλ1 (y − y0 ). The eigenvalues of the Jacobian matrix of the vector field de- fined by system (4) evaluated at (x0 , y0 ) are −aλ2 and −aλ1 . Then (x0 , y0 ) is a hyperbolic singular point. Since (x0 , y0 ) ∈ p ∩ l, for the saddle point case it follows the separatrices are contained in p ∪ l. We presented the local phase portrait of system (4) in Figure 6.  Lemma 9. Suppose that λ1 λ2 6= 0, b 6= 0 and λ1 = λ2 . The local phase portrait of system (4) is shown in Figure 7. p p p l l l a b c Figure 7. Local phase portrait of system (4). (a) λ1 = λ2 , bλ1 > 0 and a2 − bc > 0. If λ1 = λ2 , bλ1 < 0 and a2 − 4bc > 0 then the orientation of the orbits is reversed. (b) λ1 = λ2 , bλ1 > 0 and a2 − 4bc = 0. If λ1 = λ2 , bλ1 < 0 and a2 − 4bc = 0 then the orientation of the orbits is reversed. (c) λ1 = λ2 and bλ1 > 0. If λ1 = λ2 and bλ1 < 0 then the orientation of the orbits is reversed. Proof. Taking the rescaling given by ds = λ1 dt system (4) becomes (13) x′ = −bx2 − ax − c, y ′ = −(2bx + a)y − ax2 − 2cx. √ In system (13) we have x′ = 0 if and only if x = (a ± a2 − 4bc/(2b)). Then we consider three cases. PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 13 Case 1: a2 − 4bc > 0. In this√case system (13) has two singularities, √ (x1 , y1 ) and (x2 , y2 ), where x1 = −( a2 − 4bc + a)/(2b), x2 = ( a2 − 4bc − a)/(2b) and yj = x2j for j = 1, 2. Observe that for j = 1, 2 we have (xj , yj ) ∈ p ∩ l. The eigenvalues of the Jacobian matrix of the vector field √ defined by system (13) evaluated at (xj , yj ) are equal to (−1)j+1 a2 − 4bc. Then (xj , yj ) is a hyperbolic singular point, j = 1, 2. We show the local phase portrait of system (4) at (x1 , y1 ) and (x2 , y2 ) in Figure 7-a. Case 2: a2 − 4bc = 0. In this case we can write system (13) into the form (14) x′ = −b(x − x0 )2 , y ′ = −(2by + ax)(x − x0 ), with x0 = −a/(2b). After the rescaling given by dτ = (x − x0 ) dt system (14) becomes dx dy (15) = −b(x − x0 ), = −(2by + ax). dτ dτ The singularity of system (15) is (x0 , y0 ), where y0 = x20 . Observe that (x0 , y0 ) ∈ p ∩ l and the tangent line to p at (x0 , y0 ) is l. The eigenvalues of the Jacobian matrix of the vector field de- fined by system (15) evaluated at (x0 , y0 ) are −b and −2b. Then (x0 , y0 ) is a hyperbolic singular point of system (15). We give the local phase portrait of system (4) at (x0 , y0 ) in Figure 7-b. Case 3: a2 − 4bc < 0. In this case system (4) has no singularities. We present the phase portrait of system (4) in Figure 7-c.  The α−limit and the ω−limit set of the orbit Γ are denoted by α(Γ) and ω(Γ), respectively. We say that the orbit Γ connects the singularities (x1 , y1 ) and (x2 , y2 ) when α(Γ) = {(x1 , y1 )} and ω(Γ) = {(x2 , y2 )}; or α(Γ) = {(x2 , y2 )} and ω(Γ) = {(x1 , y1 )}. From now on for system (4) we denote by P , N and Σ the subsets of R2 such that ẋ > 0, ẋ < 0 and ẋ = 0 respectively. Lemma 10. Suppose that λ1 λ2 6= 0, b 6= 0 and λ1 6= λ2 . (a) The local phase portrait in R2 of system (4) is given in Figures 8-11. (b) For the case of Figure 11-a, if Γ is a separatrix of the saddle point and Γ ∩ P 6= ∅ (respect. Γ ∩ N 6= ∅) then Γ ⊂ P (respect. Γ ⊂ N ). Proof. (a) Solving the equation ẋ = 0 for y and replacing the result in the equation ẏ = 0, we obtain that a point (x, y) is a singularity of system (4) when x satisfies (16) 2b2 λ1 λ2 x3 + 3abλ1 λ2 x2 + (2bc + a2 )λ1 λ2 x + acλ1 λ2 = 0 and y is given by bλ1 x2 + aλ2 x + cλ2 y=− . b(λ2 − λ1 ) 14 J. LLIBRE AND M.F. DA SILVA p l p l a b Figure 8. Local phase portrait of system (4) with λ1 λ2 6= 0, λ1 6= λ2 , b 6= 0 and a2 − 4bc > 0. (a) λ1 > λ2 > 0 and b > 0 or λ1 < λ2 < 0 and b < 0. If λ1 < λ2 < 0 and b > 0 or 0 < λ2 < λ1 and b < 0 then the orientation of the orbits is reversed. (b) 0 < λ1 < λ2 and b > 0 or λ2 < λ1 < 0 and b < 0. If λ2 < λ1 < 0 and b > 0 or 0 < λ1 < λ2 and b < 0 then the orientation of the orbits is reversed. p l Figure 9. Local phase portrait of system (4) with λ1 λ2 6= 0, λ1 6= λ2 , b 6= 0 and a2 − 4bc > 0. The figure corresponds to cases λ2 < 0 < λ1 and b > 0 or λ1 < 0 < λ2 and b < 0. If λ1 < 0 < λ2 and b > 0 or λ2 < 0 < λ1 and b < 0 then the orientation of the orbits is reversed. Dividing by 2b2 λ1 λ2 and performing the change of variables given by x 7→ x + a/(2b), equation (16) becomes 4bc − a2 (17) x3 + x = 0. 4b2 Then we consider three cases. Case 1: a2 > 4bc. There exist three singularities p (xj , yj ), j = 0, 1, 2, of the system (4), where x0 = −a/(2b), x1 = x0 − a2 /b2 − 4c/b, x2 = x0 + p a2 /b2 − 4c/b, y0 = −(bλ1 x20 + aλ2 x0 + cλ2 )/(bλ2 − bλ1 ) and yj = x2j for j = 1, 2. Notice that (xj , yj ) ∈ p ∩ l for j = 1, 2 and (x0 , y0 ) ∈ / p ∪ l. PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 15 p p l l a b c Figure 10. Local phase portrait of system (4) at (x0 , y0 ) for the case bλ1 λ2 (λ1 − λ2 ) 6= 0 and 4bc = a2 . (a) 0 < λ2 < λ1 and b > 0 or λ1 < λ2 < 0 and b < 0. If λ1 < λ2 < 0 and b > 0 or 0 < λ2 < λ1 and b < 0 then the orientation of the orbits is reversed. The separatrices are contained in the straight line ax + by + c = 0 and in the region y ≥ x2 . (b) 0 < λ1 < λ2 and b > 0 or λ2 < λ1 < 0 and b < 0. If λ2 < λ1 < 0 and b > 0 or 0 < λ1 < λ2 and b < 0 then the orientation of the orbits is reversed. The separatrices are contained in region ax + by + c ≤ 0 and parabola y = x2 . (c) λ1 < 0 < λ2 and b > 0 or λ2 < 0 < λ1 and b < 0. If λ2 < 0 < λ1 and b > 0 or λ1 < 0 < λ2 and b < 0 then the orientation of the orbits is reversed. The separatrices are contained in the straight line ax + by + c = 0 and in the parabola y = x2 . p p p l l l a b c Figure 11. Local phase portrait of system (4) at (x0 , y0 ). (a) λ1 λ2 < 0, b(λ1 − λ2 ) < 0 and a2 − 4bc < 0. If λ1 λ2 < 0, b(λ1 − λ2 ) > 0 and a2 − 4bc < 0 then the orientation of the orbits is reversed. (b) λ1 λ2 > 0, b(λ1 − λ2 ) < 0, λ2 (λ1 − λ2 ) < 0 and a2 − 4bc < 0. If λ1 λ2 > 0, b(λ1 − λ2 ) > 0, λ2 (λ1 − λ2 ) < 0 and a2 − 4bc < 0 then the orientation of the orbits is reversed. (c) λ1 λ2 > 0, b(λ1 − λ2 ) < 0, λ2 (λ1 − λ2 ) > 0 and a2 − 4bc < 0. If λ1 λ2 > 0, b(λ1 − λ2 ) > 0, λ2 (λ1 − λ2 ) > 0 and a2 − 4bc < 0 then the orientation of the orbits is reversed. The eigenvalues of the Jacobian matrix of the vector field de- fined by system (4) evaluated at (x0 , y0 ) are p (a2 − 4bc)λ1 λ2 (18) ± √ . 2 16 J. LLIBRE AND M.F. DA SILVA The eigenvalues of the Jacobian matrix of the vector field defined by system (4) evaluated at (xj , yj ) are √ √ 2 2 j+1 |b|λ1 a − 4bc j+1 |b|λ2 a − 4bc (−1) and (−1) , j = 1, 2. b b It follows that (xj , yj ) is a hyperbolic singular point, for j = 1, 2. Moreover, if λ1 λ2 > 0 then (x0 , y0 ) is a hyperbolic singular point and, if λ1 λ2 < 0 then (x0 , y0 ) is a center. Since (xj , yj ) ∈ p ∩ l for j = 1, 2 we obtain that p and l are separatrices of system (4). We present the local phase portrait of system (4) at the singularities (x0 , y0 ), (x1 , y1 ) and (x2 , y2 ) in Figures 9 and 12. We will prove that the saddle point is connected with the attracting node and with the repelling node in the cases shown in Figure 12. From this we obtain Figure 8. p l p l a b Figure 12. Local phase portrait of system (4) with λ1 λ2 6= 0, λ1 6= λ2 , b 6= 0 and a2 − 4bc > 0. (a) λ1 > λ2 > 0 and b > 0 or λ1 < λ2 < 0 and b < 0. If λ1 < λ2 < 0 and b > 0 or 0 < λ2 < λ1 and b < 0 then the orientation of the orbits is reversed. (b) 0 < λ1 < λ2 and b > 0 or λ2 < λ1 < 0 and b < 0. If λ2 < λ1 < 0 and b > 0 or 0 < λ1 < λ2 and b < 0 then the orientation of the orbits is reversed. We consider only the case given by Figure 12-a, because the case given by Figure 12-b follows in a similar way. The portion of Figure 12-a which corresponds to the region below to l is shown in Figure 13. We will prove that α(Γ1 ) = {(x1 , y1 )}. It is sufficient to prove that Γ1 ⊂ P. The proof of α(Γ3 ) = {(x2 , y2 )} is analogous. In Figure 13 we suppose that a/b ≤ 0. The case a/b > 0 is treated in a similar way. Under the hypotheses of Lemma 16 we obtain that Σ is given by the points of the parabola b(λ1 − λ2 )y = bλ1 x2 + aλ2 x + cλ2 and it is represented as the dashed line in Figure 13. Observe that (xj , yj ) ∈ Σ, j = 0, 1, 2 and x1 < x0 < x2 . Since P and N are connected subsets of R2 we obtain that the sets Γ ∩ P and Γ ∩ N are also connected, for all orbit Γ of system (4). PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 17 P (x1 , y1 ) (x2 , y2 ) l Σ P Γ1 Γ3 z Γ2 N (x0 , y0 ) Figure 13. Proof of α(Γ1 ) = {(x1 , y1 )}. The slope of the tangent to Σ at (x0 , y0 ) is −a/b. The slope of the stable subspace of the linearized system at (x0 , y0 ) is √ p a 2 (a2 − 4bc)λ1 λ2 − − , b 2b(λ1 − λ2 ) which is smaller than −a/b, since 2b(λ1 − λ2 ) > 0. From the Stable Manifold Theorem it follows that Γ1 ∩ P 6= ∅ and Γ2 ∩ N 6= ∅. Since ẏ|Σ is given by the polynomial function on the left side of equation (16) divided by b(λ2 − λ1 ), we have (19) ẏ|Σ > 0 if x < x1 or x0 < x < x2 and (20) ẏ|Σ < 0 if x1 < x < x0 or x2 < x. Now we will prove that Γ1 ⊂ P. Suppose that this statement is false. Let φ = (φ1 , φ2 ) the flow of the vector field defined by system (4) and take z ∈ Γ1 ∩ P. Using continuity it follows that there exists t0 ∈ R such that φ(t0 , z) ∈ Σ. Observe that φ(t0 , z) 6= (x0 , y0 ) because z ∈ Γ1 . Then there exist two possibilities: φ1 (t0 , z) > x0 or φ1 (t0 , z) < x0 . Suppose that φ1 (t0 , z) > x0 . Since φ̇1 (t0 , z) = 0, from (19) it follows that the solution φ(t0 , z) is transversal to Σ at φ(t0 , z). Using the Flow Box Theorem there exist t1 , t2 ∈ R such that t1 < t0 < t2 and t1 ≤ t < t0 ⇒ φ(t, z) ∈ N and t0 < t ≤ t2 ⇒ φ(t, z) ∈ P. Since t 7→ φ1 (t, z) is an increasing function on the interval (t0 , t2 ] we have that φ1 (t2 , z) > x0 . But limt→+∞ φ1 (t, z) = x0 , then there exists t3 > t2 such that φ̇1 (t3 , z) < 0, that is φ(t3 , z) ∈ N. Hence we have t1 < t2 < t3 with φ(t1 , z), φ(t3 , z) ∈ N and φ(t2 , z) ∈ / N. This is a contradiction with the fact that the set Γ1 ∩ N is connected. If φ1 (t0 , z) < x0 , using (20) instead (19), we obtain in a similar way a contradiction with the fact that the set Γ1 ∩ P is connected. 18 J. LLIBRE AND M.F. DA SILVA Case 2: a2 = 4bc. The unique solution of equation (17) is x = 0, then the unique singularity (x0 , y0 ) of system (4) is given by x0 = −a/(2b) and y0 = c/b. Observe that (x0 , y0 ) ∈ p ∩ l. Moreover the tangent line to p at (x0 , y0 ) is l. The Jacobian matrix of the vector field defined by system (4) evaluated at (x0 , y0 ) is   a b (λ1 − λ2 ) . −a2 /b −a Then (x0 , y0 ) is a nilpotent singularity of system (4). Doing the translation given by x 7→ x− x0 , y 7→ y − y0 system (4) becomes ẋ = a(λ1 − λ2 )x + b(λ1 − λ2 )y − bλ1 x2 , (21) ẏ = −4c(λ1 − λ2 )x − a(λ1 − λ2 )y − a(2λ2 − λ1 )x2 − 2bλ2 xy. After the linear change of coordinates defined by      x 1/b 0 x 7→ , y a/b2 1/b y system (21) becomes (22) ẋ = b(λ1 − λ2 )y − b2 λ1 x2 , y ′ = −2b2 λ2 xy. In order to obtain information on the localization of the separatrices we take the quasi-homogeneous blow-up in the x−direction given by x = u, y = u2 v. Then system (22) becomes (23) u̇ = b(λ1 − λ2 )u2 v − b2 λ1 u2 , v̇ = −2b(λ1 − λ2 )uv 2 + 2b2 (λ1 − λ2 )uv. Taking the rescaling given by ds = u dt system (23) writes (24) u′ = b(λ1 − λ2 )uv − b2 λ1 u, v ′ = −2b(λ1 − λ2 )v 2 + 2b2 (λ1 − λ2 )v. The singularities (u0 , v0 ) of system (24) with u0 = 0 are (0, 0) and (0, b). The eigenvalues of the Jacobian matrix of the vector field defined by system (24) evaluated at (0, 0) and (0, b) are −b2 λ1 , 2b2 (λ1 − λ2 ) and −b2 , −2b2 (λ1 − λ2 ), respectively. Changing y by −y and v by −v, Figure 14 corresponds also to the case b < 0 and λ1 < λ2 < 0. If b < 0 and 0 < λ2 < λ1 then the orientation of the orbits is reversed. Changing y by −y and v by −v, Figure 15 corresponds also to the case b < 0 and λ2 < λ1 < 0. If b < 0 and 0 < λ1 < λ2 then the orientation of the orbits is reversed. Changing y by −y and v by −v, Figure 16 corresponds also to the case b < 0 and λ1 < 0 < λ2 . If b < 0 and λ2 < 0 < λ1 then the orientation of the orbits is reversed. PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 19 Observe that for system (22) we have ẋ|x=0 = b(λ1 − λ2 )y. System (22) is invariant under the change of coordinates (t, x, y) 7→ (−t, −x, y), then system (22) is symmetric with respect to the y−axis. Then we obtain the following conclusions. If λ1 λ2 > 0 then the sectorial decomposition of system (22) at (0, 0) is composed by one elliptic sector, one hyperbolic sector, one attracting sector and one repelling sector. If λ1 λ2 < 0 then (0, 0) is a saddle point of system (22). v v y u u x a b c Figure 14. Local phase portrait of system (22) at (0, 0) for the case b > 0 and λ1 < λ2 < 0. If b > 0 and 0 < λ2 < λ1 then the orientation of the orbits is reversed. (a) System (24). (b) System (23). (c) System (22). v v y x u u a b c Figure 15. Local phase portrait of system (22) at (0, 0) for the case b > 0 and λ2 < λ1 < 0. If b > 0 and 0 < λ1 < λ2 then the orientation of the orbits is reversed. (a) System (24). (b) System (23). (c) System (22). v v y u u x a b c Figure 16. Local phase portrait of system (22) at (0, 0) for the case b > 0 and λ1 < 0 < λ2 . If b > 0 and λ2 < 0 < λ1 then the orientation of the orbits is reversed. (a) System (24). (b) System (23). (c) System (22). Observe that from the blow-up we obtain that the straight line v = b of system (24) corresponds to the parabola y = bx2 of system (22). Then we have the following conclusions on the separatrices of system (22). For the cases of Figure 14 the separatrices are contained in the axis y = 0 and in the region y ≥ bx2 . For the cases of Figure 15 the separatrices are contained in 20 J. LLIBRE AND M.F. DA SILVA y = bx2 and in the region y ≤ 0. For the cases of Figure 16 the separatrices are contained in y = bx2 and in y = 0. We present the sectorial decomposition of system (4) at (x0 , y0 ) in the Figure 10. Case 3: a2 < 4bc. In this case the unique solution of equation (17) is x = 0, then the unique singularity (x0 , y0 ) of system (4) is given by x0 = −a/(2b) and y0 = ((4bc − a2 )λ2 + a2 (λ1 − λ2 ))/(4b2 (λ1 − λ2 )). Observe that (x0 , y0 ) ∈ / p ∪ l. The eigenvalues of the Jacobian matrix of the vector field de- fined by system (4) evaluated at (x0 , y0 ) are given by (18). If λ1 λ2 < 0 then (x0 , y0 ) is a hyperbolic saddle point of system (4). If λ1 λ2 > 0 then the eigenvalues are purely imaginary and (x0 , y0 ) is a center of system (4). We show the sectorial decomposition of system (4) at (x0 , y0 ) in Figure 11. The proof of statement (a) is complete. The proof of statement (b) is analogous to the proof of α(Γ1 ) = {(x1 , y1 )} done in Case 2.  5.2. Infinite singularities of system (4). The expression of the Poincaré compactification of system (4) in the charts (U1 , ϕ1 ) and (U2 , ϕ2 ) is u̇ = λ2 (a + bu + cv)(uv − 2) + λ1 (1 − uv)(a + bu), (25) v̇ = λ2 (a + bu + cv)v 2 + bλ1 (1 − uv)v, and u̇ = λ2 (au + b + cv)(2u2 − v) + λ1 (v − u2 )(au + b), (26) v̇ = 2λ2 (au + b + cv)uv + aλ1 (v − u2 )v, respectively. 5.2.1. Infinite singularities in U1 . In this section we will determine de sec- torial decomposition of the singularities of system (25) which are of the form (u, 0), with u ∈ R. We consider the cases λ1 − 2λ2 6= 0 and λ1 − 2λ2 = 0 separately. The first one, studied in Lemma 11, follows from the Grobman- Hartman Theorem and the proof is omitted. In the second case v = 0 is a line of singularities and we study the phase portrait of the corresponding system in Lemma 12. Lemma 11. Suppose that λ1 λ2 6= 0 and λ1 − 2λ2 6= 0. (a) If b = 0 then system (25) has no singularities in a neighborhood of the axis v = 0. (b) Suppose b 6= 0. Then the unique singularity (u0 , v0 ) of system (25) with v0 = 0 is given by u0 = −a/b. If bλ1 > 0 and b(λ1 − 2λ2 ) > 0 then (u0 , 0) is a hyperbolic repelling node of system (25). If bλ1 < 0 and b(λ1 − 2λ2 ) < 0 then (u0 , 0) is a hyperbolic attracting node of PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 21 system (25). If λ1 (λ1 − 2λ2 ) < 0 then (u0 , 0) is a hyperbolic saddle point of system (25). Lemma 12. Suppose that λ1 λ2 6= 0 and λ1 − 2λ2 = 0. The phase portrait of system (25) in R2 is shown in Figure 17. a b Figure 17. Phase portrait of system (25). (a) λ1 = 2λ2 and bλ2 > 0. If λ1 = 2λ2 and bλ2 < 0 then the orientation of the orbits is reversed. (b) λ1 = 2λ2 , b = 0 and aλ2 > 0. If λ1 = 2λ2 , b = 0 and aλ2 < 0 then the orientation of the orbits is reversed. Proof. After the rescaling given by ds = λ2 v dt system (25) becomes (27) u′ = cuv − bu2 − au − 2c, v ′ = cv 2 + (a − bu)v + 2b. We consider the cases b 6= 0 and b = 0 separately. Case 1: b 6= 0. In this case system (27) has no singularities (u0 , v0 ) with v0 = 0 and we show the phase portrait of system (25) in a sufficiently small neighborhood of the axis v = 0 in Figure 17-a. Case 2: b = 0. From hypothesis a2 +b2 6= 0 it follows that a 6= 0. In this case the unique singularity (u0 , v0 ) of system (27) with v0 = 0 is u0 = −2c/a. The eigenvalues of the Jacobian matrix of the vector field de- fined by system (27) evaluated at (u0 , 0) are −a and −2c2 /a. Then (u0 , 0) is a hyperbolic saddle point of system (27). We present the phase portrait of system (25) in a sufficiently small neighborhood of the axis v = 0 in Figure 17-b.  5.2.2. The origin of U2 . We denote by p′ the parabola given by v = u2 . From Remark 5 it follows that the points of p \ {(0, 0)} are mapped into the points of p′ . The Jacobian matrix of the vector field defined by system (26) evaluated at (0, 0) is   0 b(λ1 − λ2 ) . 0 0 Then we consider the cases b = 0, b 6= 0 and λ1 = λ2 , and b(λ1 − λ2 ) 6= 0 separately. In the last case the origin is a nilpotent singularity of system 22 J. LLIBRE AND M.F. DA SILVA (26). In order to obtain information on the localization of the separatrices we will use blow-ups to study the singularity (0, 0). Lemma 13. Suppose that λ1 λ2 6= 0 and b = 0. (a) If λ1 6= 2λ2 then the sectorial decomposition of system (26) at the origin is shown in Figure 18. (b) If λ1 = 2λ2 then the sectorial decomposition of system (26) at the origin is shown in Figure 19. p′ p′ a b c Figure 18. Sectorial decomposition of system (26) at (0, 0) for the case b = 0. (a) aλ1 < 0, aλ2 > 0 and a(λ1 − 2λ2 ) < 0. If aλ1 > 0, aλ2 < 0 and a(λ1 − 2λ2 ) > 0 then the orientation of the orbits is reversed. (b) aλ1 < 0, aλ2 < 0 and a(λ1 − 2λ2 ) < 0. If aλ1 > 0, aλ2 > 0 and a(λ1 − 2λ2 ) > 0 then the orientation of the orbits is reversed. (c) aλ1 < 0, aλ2 < 0 and a(λ1 − 2λ2 ) > 0. If aλ1 > 0, aλ2 > 0 and a(λ1 − 2λ2 ) < 0 then the orientation of the orbits is reversed. Figure 19. Sectorial decomposition of system (26) at (0, 0) for the case λ1 λ2 6= 0, b = 0, λ1 = 2λ2 and aλ2 > 0. If λ1 λ2 6= 0, b = 0, λ1 = 2λ2 and aλ2 < 0 then the orientation of the orbits is reversed. Proof. We consider the cases λ1 − 2λ2 6= 0 and λ1 − 2λ2 = 0 separately. Case 1: λ1 − 2λ2 6= 0. We take the quasi-homogeneous blow-up in the positive v−direction given by u = u v, v = v 2 . Then system (26) becomes 1  u̇ = 2cλ2 (u2 − 1)v 3 + a(2λ2 − λ1 )(u2 − 1)u v 2 , 2 (28) 1  v̇ = cλ2 u v + ((2λ2 − λ1 )u2 + aλ1 ) v 3 . 2 PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 23 Taking the rescaling given by ds = v 2 /2 dt system (28) becomes u′ = cλ2 (u2 − 1)v + a(2λ2 − λ1 )(u2 − 1)u, (29)   v ′ = cλ2 u v + (a(2λ2 − λ1 )u2 + aλ1 ) v. The singularities (u0 , v 0 ) of system (29) with v 0 = 0 are (0, 0) and (±1, 0). The eigenvalues of the Jacobian matrix of the vector field defined by system (29) evaluated at (0, 0) are −a(2λ2 − λ1 ) and aλ1 . The eigenvalues of the Jacobian matrix of the vector field defined by system (29) evaluated at (1, 0) or (−1, 0) are 2a(2λ2 − λ1 ) and 2aλ2 . Then (0, 0) and (±1, 0) are hyperbolic singularities of system (29). For system (26) we have that (30) u̇|v=0 = a(2λ2 − λ1 )u3 and v̇|u=0 = aλ1 v 2 . Observe that the straight lines u = ±1 are invariant under the flow of the system (29), then the parabola p′ contain separatrices of system (26) for the cases present in Figures 20 and 21. The quasi-homogeneous blow-up in the positive v−direction is summarized in Figures 20-22. v v v p′ u u u a b c Figure 20. Sectorial decomposition of system (26) at (0, 0) with v ≥ 0, for the case b = 0, a(2λ2 − λ1 ) > 0, aλ2 > 0 and aλ1 < 0. If b = 0, a(2λ2 − λ1 ) < 0, aλ2 < 0 and aλ1 > 0 then the orientation of the orbits is reversed. (a) System (29). (b) System (28). (c) Sectorial decomposition of system (26) at (0, 0) with v ≥ 0. v v v p′ u u u a b c Figure 21. Sectorial decomposition of system (26) at (0, 0) with v ≥ 0, for the case b = 0, a(2λ2 − λ1 ) > 0, aλ2 < 0 and aλ1 < 0. If b = 0, a(2λ2 − λ1 ) < 0, aλ2 > 0 and aλ1 > 0 then the orientation of the orbits is reversed. (a) System (29). (b) System (28). (c) Sectorial decomposition of system (26) at (0, 0) with v ≥ 0. 24 J. LLIBRE AND M.F. DA SILVA v v v u u u a b c Figure 22. Sectorial decomposition of system (26) at (0, 0) with v ≥ 0, for the case b = 0, a(2λ2 − λ1 ) < 0, aλ2 < 0 and aλ1 < 0. If b = 0, a(2λ2 − λ1 ) > 0, aλ2 > 0 and aλ1 > 0 then the orientation of the orbits is reversed. (a) System (29). (b) System (28). (c) Sectorial decomposition of system (26) at (0, 0) with v ≥ 0. We take the quasi-homogeneous blow-up in the negative v−direction given by u = u v, v = −v 2 . Then system (26) becomes 1  u̇ = − 2cλ2 (u2 + 1)v + a(λ1 − 2λ2 )(u3 + u) v 2 , 2 (31) 1  v̇ = − 2cλ2 u v + a((λ1 − 2λ2 )u2 + λ1 ) v 3 . 2 Observe that this change of coordinates maps points (u, v) with v > 0 in points (u, v) with v < 0. Taking the rescaling given by ds = v 2 /2 dt system (31) becomes   u′ = − 2cλ2 (u2 + 1)v + a(λ1 − 2λ2 )(u3 + u) , (32)   v ′ = − 2cλ2 u v + a((λ1 − 2λ2 )u2 + λ1 ) v. The singularity (u0 , v 0 ) of system (32) with v 0 = 0 is (0, 0). The eigen- values of the Jacobian matrix of the vector field defined by system (32) evaluated at (0, 0) are a(2λ2 − λ1 ) and −aλ1 . Then (0, 0) is a hy- perbolic singular point of system (32). Using (30) we summarize the quasi-homogeneous blow-up in the negative v−direction in Figures 23 and 24. The sectorial decomposition of system (26) at (0, 0) shown in Figure 18-a follows from Figure 20-c and Figure 24-c (with reversed orientation). The sectorial decomposition of system (26) at (0, 0) given in Figure 18-b follows from Figure 21-c and Figure 24-c (with reversed orientation). The sectorial decomposition of system (26) at (0, 0) shown in Figure 18-c follows from Figure 22-c and Figure 23-c (with reversed orientation). Case 2: λ1 = 2λ2 . After the rescaling given by ds = λ2 v dt system (26) becomes (33) u′ = −cv + 2cu2 + au, v ′ = 2(cu + a)v. PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 25 v v v u u u a b c Figure 23. Sectorial decomposition of system (26) at (0, 0) with v ≤ 0, for the case λ2 6= 0, b = 0, aλ1 > 0 and a(2λ2 − λ1 ) > 0. If λ2 6= 0, b = 0, aλ1 < 0 and a(2λ2 − λ1 ) < 0 then the orientation of the orbits is reversed. (a) System (32). (b) System (31). (c) Sectorial decomposition of system (26) at (0, 0) with v ≤ 0. v v v u u u a b c Figure 24. Sectorial decomposition of system (26) at (0, 0) with v ≤ 0, for the case λ2 6= 0, b = 0, aλ1 > 0 and a(2λ2 − λ1 ) < 0. If λ2 6= 0, b = 0, aλ1 < 0 and a(2λ2 − λ1 ) > 0 then the orientation of the orbits is reversed. (a) System (32). (b) System (31). (c) Sectorial decomposition of system (26) at (0, 0) with v ≤ 0. The eigenvalues of the Jacobian matrix of the vector field de- fined by system (33) evaluated at (0, 0) are a and 2a. Then (0, 0) is a hyperbolic singular point of system (33). We present the local phase portrait of system (26) at (0, 0) in Figure 19.  Observe that in Lemma 14-e, we will present a finite number of possi- bilities for the sectorial decomposition of system (26). But in the proof of Theorem 1 we will prove that the sectorial decomposition of system (26) at the origin, for this case, is composed by two elliptic sectors. Lemma 14. Suppose that λ1 λ2 6= 0, b 6= 0 and λ1 = λ2 . (a) If a = c = 0 then the sectorial decomposition of system (26) at the origin is shown in Figure 30. (b) If c = 0 and a 6= 0 then the sectorial decomposition of system (26) at the origin is given in Figures 25-c and 26-c. 26 J. LLIBRE AND M.F. DA SILVA (c) If c 6= 0 and a2 > 4bc then the sectorial decomposition of system (26) at the origin is shown in Figures 27-c, 28-c and 29-c. (d) If c 6= 0 and a2 = 4bc then the sectorial decomposition of system (26) at the origin is shown in Figures 31-d and 32-d. (e) If c 6= 0 and a2 < 4bc then the sectorial decomposition of system (26) at the origin is given in Figure 33-c. Proof. Taking the rescaling given by ds = λ1 dt system (26) becomes (34) u′ = 2cu2 v + au3 − cv 2 + bu2 , v ′ = 2cuv 2 + au2 v + av 2 + 2buv. We take the blow-up in the u−direction given by u = u, v = u v, which maps points of the form u < 0, v > 0 into points of the form u < 0, v < 0; and points of the form u < 0, v < 0 into points of the form u < 0, v > 0. Then system (34) becomes (35) u̇ = −cu2 v 2 + 2cu3 v + au3 + bu2 , v̇ = c u v 3 + au v 2 + b u v. Taking the rescaling given by ds = u dt system (35) becomes (36) u′ = −c u v 2 + 2cu2 v + au2 + bu, v ′ = cv 3 + av 2 + bv. The singularities (u0 , v 0 ) of system (36) with u0 = 0 are (0, 0) and (0, v 0 ), where v 0 is a solution of the equation cv 2 + av + b = 0. The eigenvalues of the Jacobian matrix of the vector field defined by system (36) evaluated at (0, 0) are equal to b, then (0, 0) is a hyperbolic singular point of system (36). In order to study the other singularities of system (36) we consider three cases. Case 1: a2 − 4bc > 0. Case 1.1: c = 0. Then a 6= 0. The singularities (u0 , v 0 ) of system (36) with u = 0 are (0, 0) and (0, −b/a). The eigenvalues of the Jacobian matrix of the vector field defined by system (36) evaluated at (0, −b/a) are b and −b. Observe that the straight line v = −b/a is invariant for system (36), then the straight line v = −(b/a)u is invariant for system (26). Moreover, since λ1 = λ2 and c = 0, the axis u = 0 is invariant for system (26). We summarize the blow-up in Figures 25 and 26. Case 1.2: c 6= 0. The singularities (u0 , v 0 ) of√system (36) with u = 0 2 √ (0, v 1 ) and (0, v 2 ), where v 1 = (−a + a − 4bc)/(2c) and v 2 = are (0, 0), 2 −(a + a − 4bc)/(2c). The eigenvalues of the Jacobian matrix of the √vector field defined by system (36) evaluated at (0, v j ) are ±v j a2 − 4bc, for j = 1, 2. Observe that the straight line v = v j is in- variant for system (36), then the straight line v = v j u are invariant for system (26), for j = 1, 2. The blow-up is summarized in Figures 27-29. Case 2: a2 = 4bc. PSfrag PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 27 v v p′ v u u u a b c Figure 25. Sectorial decomposition of system (26) at (0, 0) for the case λ1 = λ2 > 0, a > 0, b > 0 and c = 0; or λ1 = λ2 < 0, a < 0, b < 0 and c = 0. If λ1 = λ2 > 0, a < 0, b < 0 and c = 0; or λ1 = λ2 < 0, a > 0, b > 0 and c = 0 then the orientation of the orbits is reversed. (a) System (36). (b) System (35). (c) Sectorial decomposition of system (26) at (0, 0). v v p′ v u u u a b c Figure 26. Sectorial decomposition of system (26) at (0, 0) for the case λ1 = λ2 > 0, a < 0, b > 0 and c = 0; or λ1 = λ2 < 0, a > 0, b < 0 and c = 0. If λ1 = λ2 < 0, a < 0, b > 0 and c = 0; or λ1 = λ2 > 0, a > 0, b < 0 and c = 0 then the orientation of the orbits is reversed. (a) System (36). (b) System (35). (c) Sectorial decomposition of system (26) at (0, 0). v v v p′ u u u a b c Figure 27. Sectorial decomposition of system (26) at (0, 0) for the case λ1 = λ2 > 0, a > 0, b > 0, c > 0 and a2 − 4bc > 0; or λ1 = λ2 < 0, a < 0, b < 0, c < 0 and a2 − 4bc > 0. If λ1 = λ2 < 0, a > 0, b > 0, c > 0 and a2 − 4bc > 0; or λ1 = λ2 > 0, a < 0, b < 0, c < 0 and a2 − 4bc > 0 then the orientation of the orbits is reversed. (a) System (36). (b) System (35). (c) Sectorial decomposition of system (26) at (0, 0). Case 2.1: a = c = 0. In this case the blow-up is superfluous. Taking the rescaling given by ds = u dt in system (34), we obtain the sectorial decomposition of system (26) at (0, 0) shown in Figure 30. 28 J. LLIBRE AND M.F. DA SILVA v v v p′ u u u a c b Figure 28. Sectorial decomposition of system (26) at (0, 0) for the case λ1 = λ2 > 0, a < 0, b > 0, c > 0 and a2 − 4bc > 0; or λ1 = λ2 < 0, a > 0, b < 0, c < 0 and a2 − 4bc > 0. If λ1 = λ2 < 0, a < 0, b > 0, c > 0 and a2 − 4bc > 0; or λ1 = λ2 > 0, a > 0, b < 0, c < 0 and a2 − 4bc > 0 then the orientation of the orbits is reversed. (a) System (36). (b) System (35). (c) Sectorial decomposition of system (26) at (0, 0). v v v p′ u u u a b c Figure 29. Sectorial decomposition of system (26) at (0, 0) for the case λ1 = λ2 > 0, b > 0, c < 0 and a2 − 4bc > 0; or λ1 = λ2 < 0, b < 0, c > 0 and a2 − 4bc > 0. If λ1 = λ2 < 0, b > 0, c < 0 and a2 − 4bc > 0; or λ1 = λ2 > 0, b < 0, c > 0 and a2 − 4bc > 0 then the orientation of the orbits is reversed. (a) System (36). (b) System (35). (c) Sectorial decomposition of system (26) at (0, 0). Figure 30. Sectorial decomposition of system (26) at (0, 0) for the case λ1 = λ2 6= 0, a = c = 0 and bλ1 > 0. If λ1 = λ2 6= 0, a = c = 0 and bλ1 < 0 then the orientation of the orbits is reversed. Case 2.2: ac 6= 0. The singularities (u0 , v 0 ) of system (36) with u0 = 0 are (0, 0) and (0, v 1 ), where v 1 = −a/(2c). Taking the rescaling given by PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 29 dτ = (v − v 1 ) ds system (36) becomes du au dv (37) = −cu v + 2cu2 + , = cv(v − v 1 ). dτ 2 dτ The eigenvalues of the Jacobian matrix of the vector field de- fined by system (37) evaluated at (0, 0) are equal to a/2. The eigen- values of the Jacobian matrix of the vector field defined by system (37) evaluated at (0, v 1 ) are a and −a/2. Since λ1 = λ2 , for system (26) we observe that u̇|u=0 = −λ1 cv 2 , u̇|v=0 = λ1 u2 (au + b) and v̇|u=0 = λ1 av 2 . The blow-up is summarized in Figures 31 and 32. v v u u a b v v u u c d Figure 31. Sectorial decomposition of system (26) at (0, 0) for the case λ1 = λ2 > 0, a > 0, b > 0, c > 0 and a2 − 4bc = 0; or λ1 = λ2 < 0, a < 0, b < 0, c < 0 and a2 − 4bc = 0. If λ1 = λ2 < 0, a > 0, b > 0, c > 0 and a2 − 4bc = 0; or λ1 = λ2 > 0, a < 0, b < 0, c < 0 and a2 − 4bc = 0 then the orientation of the orbits is reversed. (a) System (37). (b) System (36). (c) System (35). (d) Sectorial decomposition of system (26) at (0, 0). Case 3. a2 < 4bc. The unique singularity (u0 , v 0 ) of system (36) with u0 = 0 is (0, 0). The blow-up is summarized in Figure 33.  Lemma 15. Suppose that λ1 λ2 6= 0 and b(λ1 − λ2 ) 6= 0. (a) If λ1 6= 2λ2 then the sectorial decomposition of system (26) at the origin is given in Figure 34. (b) If λ1 = 2λ2 then the sectorial decomposition of system (26) at the origin is shown in Figure 35. Proof. We consider the cases λ1 6= 2λ2 and λ1 = 2λ2 separately. Case 1: λ1 6= 2λ2 . In order to obtain information on the localization of the separatrices of the sectorial decomposition of system (26) at (0, 0), we 30 J. LLIBRE AND M.F. DA SILVA v v u u a b v v u u c d Figure 32. Sectorial decomposition of system (26) at (0, 0) for the case λ1 = λ2 > 0, a > 0, b < 0, c < 0 and a2 − 4bc = 0; or λ1 = λ2 < 0, a < 0, b > 0, c > 0 and a2 − 4bc = 0. If λ1 = λ2 > 0, a < 0, b > 0, c > 0 and a2 − 4bc = 0; or λ1 = λ2 < 0, a > 0, b < 0, c < 0 and a2 − 4bc = 0 then the orientation of the orbits is reversed. (a) System (37). (b) System (36). (c) System (35). (d) Sectorial decomposition of system (26) at (0, 0). v v v u u u a b c Figure 33. Sectorial decomposition of system (26) at (0, 0) for the case λ1 = λ2 > 0, b > 0, c > 0 and a2 − 4bc < 0; or λ1 = λ2 < 0, b < 0, c < 0 and a2 − 4bc < 0. If λ1 = λ2 < 0, b > 0, c > 0 and a2 − 4bc < 0; or λ1 = λ2 > 0, b < 0, c < 0 and a2 − 4bc < 0 then the orientation of the orbits is reversed. (a) System (36). (b) System (35). (c) Sectorial decomposition of system (26) at (0, 0). will use blow-up. Morevover, if we use Theorem 3.5 of [7], then the case λ1 = 3λ2 should be treated separately. We take the quasi-homogeneous blow-up in the u−direction given by u = u, v = u2 v. Then system (26) becomes PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 31 p′ p′ a b c Figure 34. Sectorial decomposition of system (26) at the origin. (a) b > 0 and 0 < λ2 < λ1 < 2λ2 ; or b < 0 and 2λ2 < λ1 < λ2 < 0. If b > 0 and 2λ2 < λ1 < λ2 < 0; or b < 0 and 0 < λ2 < λ1 < 2λ2 then the orientation of the orbits is reversed. The separatrices are contained in v = u2 and in the region v ≤ 0. (b) b > 0 and 0 < 2λ2 < λ1 ; or b < 0 and λ1 < 2λ2 < 0. If b > 0 and λ1 < 2λ2 < 0; or b < 0 and 0 < 2λ2 < λ1 then the orientation of the orbits is reversed. The separatrices are contained in v = u2 and in v = 0. (c) b > 0, λ2 > 0 and λ1 < λ2 ; or b < 0, λ2 < 0 and λ1 > λ2 . If b > 0, λ2 < 0 and λ1 > λ2 ; or b < 0, λ2 > 0 and λ1 < λ2 then the orientation of the orbits is reversed. The separatrices are contained in the region v ≥ u2 and in v = 0. Figure 35. Sectorial decomposition of system (26) at the origin. Case λ1 λ2 6= 0, b(λ1 − λ2 ) 6= 0, λ1 = 2λ2 and bλ2 > 0. If λ1 λ2 6= 0, b(λ1 − λ2 ) 6= 0, λ1 = 2λ2 and bλ2 < 0 then the orientation of the orbits is reversed. u̇ = −cλ2 u4 v 2 + (2cλ2 u4 + a(λ1 − λ2 )u3 + b(λ1 − λ2 )u2 )v+ a(2λ2 − λ1 )u3 + b(2λ2 − λ1 )u2 , (38) v̇ = 2cλ2 u3 v 3 + (−2cλ2 u3 + a(2λ2 − λ1 )u2 + 2b(λ2 − λ1 )u)v 2 + (a(λ1 − 2λ2 )u2 + 2b(λ1 − λ2 )u)v. After the rescaling given by ds = u dt system (38) becomes. 32 J. LLIBRE AND M.F. DA SILVA u′ = −cλ2 u3 v 2 + (2cλ2 u3 + a(λ1 − λ2 )u2 + b(λ1 − λ2 )u)v+ a(2λ2 − λ1 )u2 + b(2λ2 − λ1 )u, (39) v ′ = 2cλ2 u2 v 3 + (−2cλ2 u2 + a(2λ2 − λ1 )u + 2b(λ2 − λ1 )u)v 2 + (a(λ1 − 2λ2 )u + 2b(λ1 − λ2 ))v. The singularities (u0 , v 0 ) of system (39) with u0 = 0 are (0, 0) and (0, 1). The eigenvalues of the Jacobian matrix of the vector field defined by system system (39) evaluated at (0, 0) and (0, 1) are b(2λ2 − λ1 ), 2b(λ1 − λ2 ) and bλ2 , −2b(λ1 − λ2 ), respectively. Observe that the straight line v = 1 is an invariant of system (39), then the parabola p′ is an invariant of system (26). If u = 0 and |v| is sufficiently small then the sign of u̇ in system (26) is the same than b(λ1 −λ2 )v. Then we will present only the figures of the blow-up for the case b > 0. We summarize the quasi-homogeneous blow-up in the u−direction in Figures 36-38. v v v p′ u u u a b c Figure 36. Local phase portrait of system (26) at (0, 0) for the case b > 0 and 0 < λ2 < λ1 < 2λ2 ; or b < 0 and 2λ2 < λ1 < λ2 < 0. If b > 0 and 2λ2 < λ1 < λ2 < 0; or b < 0 and 0 < λ2 < λ1 < 2λ2 then the orientation of the orbits is reversed. (a) System (39). (b) System (38). (c) System (26). v v v p′ u u u a b c Figure 37. Local phase portrait of system (26) at (0, 0) for the case b > 0 and 0 < 2λ2 < λ1 ; or b < 0 and λ1 < 2λ2 < 0. If b > 0 and λ1 < 2λ2 < 0; or b < 0 and 0 < 2λ2 < λ1 then the orientation of the orbits is reversed. (a) System (39). (b) System (38). (c) System (26). PHASE PORTRAITS OF A CLASS OF INTEGRABLE QUADRATIC SYSTEMS 33 v v v u u u a b c Figure 38. Local phase portrait of system (26) at (0, 0) for the case b > 0, λ2 > 0 and λ1 < λ2 ; or b < 0, λ2 < 0 and λ2 < λ1 . If b > 0, λ2 < 0 and λ2 < λ1 ; or b < 0, λ2 > 0 and λ1 < λ2 then the orientation of the orbits is reversed. (a) System (39). (b) System (38). (c) System (26). Σ′ Γ′1 Γ′2 N′ P′ B′ A′ Figure 39. Lemma 16. Case 3.2: λ1 = 2λ2 . After the rescaling given by ds = λ2 v dt system (26) becomes u′ = −cv + 2cu2 + au + b, v ′ = 2(cu + a)v + 2bu. Then u′ = b when u = v = 0 and v ′ = 2bu when v = 0. We present the sectorial decomposition of system (26) at the origin in Figure 35.  6. Proof of Theorem 1 Combining the informations of Lemmas 6-15 we obtain the local phase portrait of all the singularities finite and infinite in the Poincaré disc of systems (4) with λ21 + λ22 6= 0, it is shown in Figure 40. The correspondence between the systems (4) and the local phase portrait in the Poincaré disc shown in Figure 40 is given in Table 1. Using the Poincaré-Bendixson Theorem, the Remarks 3-5 and that we have the invariant parabola y = x2 and the invariant straight line ax + by + c = 0, we obtain that each local phase portrait in the Poincaré disc given in Figure 40-a-w, determines a unique global phase portrait in the Poincaré disc of Figure 1-a-w respectively, except for the case given by figures 40-v and 1-v. The proof of the phase portrait in the Poincaré disc of Figure 40-v is given by Figure 1-v follows from the next result. 34 J. LLIBRE AND M.F. DA SILVA Lemma 16. Consider the local phase portrait given in Figure 40-v, which portion corresponding to the region above the line l is shown in Figure 39. Then ω(Γ′1 ) and α(Γ′2 ) are the origin of U2 . Proof. As before for system (4) we denote by P, N and Σ the subsets of R2 such that ẋ > 0, ẋ < 0 and ẋ = 0, respectively. The corresponding subsets in the Poincaré disc are denoted by P ′ , N ′ and Σ′ . In Figure 39 we suppose that a/b ≤ 0. The case a/b > 0 is treated in a similar way. Since Σ is defined by the equation b(λ1 − λ2 )y = bλ1 x2 + aλ2 x + cλ2 , the equation of Σ′ in the charts U1 and U2 is given by b(λ1 − λ2 )uv = bλ1 + aλ2 v + cλ2 v 2 and b(λ1 − λ2 )v = bλ1 u2 + aλ2 uv + cλ2 v 2 , respectively. Then A ∈/ Σ′ and the origin of U2 is contained in Σ′ . If ω(Γ1 ) is different of the origin of U2 then ω(Γ1 ) = {A}. It follows that Γ1 ∩ N ′ 6= ∅. This is a contradiction with Lemma 10-(b).  References [1] J.C. Artés and J. Llibre, Quadratic Hamiltonian vector fields, J. Differential Equations 107 (1994), 80–95. [2] J.C. Artés, J. Llibre and N. Vulpe, Complete geometric invariant study of two classes of quadratic systems, Electronic J. of Differential Equations 2012, No. 09 (2012), 1–35. [3] N.N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sbornik 30 (1952), 181–196, Amer. Math. Soc. Transl. Vol. 100 (1954), 1–19. 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[26] Wei Yin Ye and Ye Yanqian, On the conditions of a center and general integrals of quadratic differential systems, Acta Math. Sin. (Engl. Ser.) 17 (2001), 229–236. [27] N. I. Vulpe, Affine–invariant conditions for the topological discrimination of qua- dratic systems with a center, Differential Equations 19 (1983), 273–280. [28] H. Żola̧dek, Quadratic systems with center and their perturbations, J. Differential Equations 109 (1994), 223–273. 1 Departament de Matemátiques, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat 2 Departamento de Matemática, Universidade Federal de Santa Maria, 97110-820, Santa Maria, RS, Brazil E-mail address: mauriciofronzadasilva@hotmail.com 36 J. LLIBRE AND M.F. DA SILVA a b c d e f g h i j k l m n o p q r s t u v x y z w Figure 40. Local phase portraits in the Poincaré disc of systems (4).