This is a preprint of: “Quadratic systems with an invariant algebraic curve of degree 3 and a Darboux invariant”, Jaume Llibre, Regilene D. S. Oliveira, Camila A. B. Rodrigues, Electron. J. Differential Equations, vol. 2021(69), 1–52, 2021. QUADRATIC SYSTEMS WITH AN INVARIANT ALGEBRAIC CURVE OF DEGREE 3 AND A DARBOUX INVARIANT JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES Abstract. The planar quadratic systems having a Darboux invariant defined by invari- ant straight lines of total multiplicity two or by an invariant conic have been studied in [13] and [14], respectively. Here we shall present the normal forms of the planar qua- dratic systems having an invariant cubic. Moreover we classify the phase portraits in the Poincare disc of all planar quadratic polynomial differential systems with invariant cubic curve and having a Darboux invariant defined by it. 1. Introduction and statements of the results Even after hundreds of studies on the topology of real planar quadratic vector fields the complete characterization of their phase portraits is a quite complex task. This family of systems depends on twelve parameters but, after affine transformations and time rescaling, we arrive at families with five parameters, which is still a big number of parameters. Many subclasses have been considered. Denote by R[x, y] the ring of the real polynomials in the variables x and y. Consider the differential system in R2 given by (1) ẋ = P (x, y), ẏ = Q(x, y), where P, Q ∈ R[x, y]. Here the dot denotes derivative with respect to the time t and the degree of system (1) is m = max{deg P, deg Q}. When m = 2 we say that system (1) is a quadratic polynomial differential system or simply a quadratic system. More than one thousand papers have been published about quadratic systems, see for instance [15] for a bibliographical survey. The quadratic sys- tems appear in the modeling of many natural phenomena described in different branches of science, in biological and physical applications. Besides the applications the quadratic systems became a matter of interest for the mathematicians. Considering algebraic in- variant curves, some authors have published on the subject, for example, [3] and [12]. In the first one the authors studied cubic systems with invariant straight lines of total multi- plicity eight that have three distinct infinite singularities. The second paper is dedicated to study the normal forms and global phase portraits of quadratic and cubic integrable systems when they have two nonconcentric circles as invariant algebraic curves. In this paper we assume that the polynomials P and Q are coprime, otherwise system (1) can be reduced to a linear or constant system doing a rescaling of the time variable. The first objective of this paper is to characterize all quadratic systems having invariant cubics. Then using the normal forms obtained, we investigate which systems have a Darboux invariant of the form est f1λ1 f2λ2 f3λ3 if the cubic is the product of three straight Key words and phrases. quadratic vector fields, algebraic invariant curve, Darboux invariant, global phase portrait. 2010 Mathematics Subject Classification: Primary 34C05, 34A34. 1 2 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES lines fi = 0 for i = 1, 2, 3, of the form est f1λ1 f2λ2 if the cubic is the product of one straight line f1 = 0 and an irreducible conic f2 = 0, and of the form est f1λ1 if f1 = 0 is an irreducible cubic. The paper is organized as follows. In Section 2 we present our main results. They are divided in two subsections. In section 3 we present definitions and results that will be used for proving our main results. Finally in Sections 4, 5 and 6 we prove the main results. 2. Statement of the main results The objective of this section is to present the main results of this investigation. Since the cubic curves can be classified as reducible and irreducible curves (according to the polynomial defining the curve admits fatorization or not), we split the obtained results in two subsections. In the first one we consider planar quadratic systems having irreducible cubics and in the second one, the reducible ones. Theorem A. Each quadratic system admitting an irreducible invariant cubic after an affine change of coordinates and a rescaling of the time variable can be written as one of the following systems. (i) ẋ = 2(ax + by + dxy + cx2 ), ẏ = 3(ay + bx2 + cxy + dy 2 ), (ii) ẋ = 2(ax + by + (3b − 2c)xy + ax2 ), ẏ = 2bx + 2ay + 2cx2 + 3axy + (9b − 6c)y 2 , (iii) ẋ = 2(ax − by + (3b + 2c)xy − ax2 ), ẏ = 2bx + 2ay + 2cx2 − 3axy + (9b + 6c)y 2 , (iv) ẋ = 2y(a + bx), ẏ = ar − 2(ar + a + br)x + (3a + br + b)x2 + 3by 2 , (v) ẋ = 2y(b + cx), ẏ = b + 2(br − c)x + (3b − cr)x2 + 3cy 2 . Theorem B. Each quadratic system admitting an irreducible invariant cubic having a Darboux invariant can be written after an affine change of coordinates and a rescaling of the time variable as 3 (2) ẋ = x + y, ẏ = y + x2 . 2 2 3 In this case y = x is the invariant algebraic curve and the Darboux invariant is given by e−6t (y 2 − x3 ). The global phase portrait of such system is given in Figure 1. Figure 1. Phase portrait of system (2). QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 3 Theorems A and B are proved in section 4. 2.1. Reducible invariant cubics. Each reducible cubic can be written as the product of two polynomials one of degree two and the other of degree one (i.e, a conic and a straight line respectively). The conics can be classified in ellipses (E), complex ellipses (CE), hyperbolas (H), parabolas (P), two real straight lines intersecting in a point , two real parallel straight lines (PL), one double invariant real straight line (DL), two complex straight lines intersecting in a real point (p), and two complex parallel straight lines (CL). So the normal forms of the reducible cubics, except to an affine transformation, are (E) (x2 + y 2 − 1)(ax + by + c) = 0, (CE) (x2 + y 2 + 1)(ax + by + c) = 0, (H) (x2 − y 2 − 1)(ax + by + c) = 0, (P) (y − x2 )(ax + by + c) = 0, (LV) xy(ax + by + c) = 0, (PL) (x2 − 1)(ax + by + c) = 0, (DL) x2 (ax + by + c) = 0, (CL) (x2 + 1)(ax + by + c) = 0, (p) (x2 + y 2 )(ax + by + c) = 0. We shall say that a quadratic system is of type (E) if it has a real ellipse and a straight line as invariant irreducible algebraic curves; of type (CE) if it has a complex ellipse and a straight line as invariant irreducible algebraic curves, and respectively with all the nine types of conics described above. The first result of this paper classifies the quadratic systems having a reducible invariant cubic. Theorem C. If a quadratic system (1) has a reducible invariant cubic then it can be written, after an affine change of coordinates, into one of the following forms (CE) ẋ = −(x2 + y 2 + 1) − 2α1 y(y + ax + c), ẏ = a(x2 + y 2 + 1) + 2α1 x(y + ax + c), (E.1) ẋ = −(x2 + y 2 − 1) − 2α1 y(y + ax + c), ẏ = a(x2 + y 2 − 1) + 2α1 x(y + ax + c), (E.2) ẋ = (β1 /2)(x2 + y 2 − 1) − y(β2 y − α2 x + cβ2 ), ẏ = (y + c)(α2 y + β2 c x + α2 ), with α2 (c + 1) = 0, (H.1) ẋ = (β1 /2)(x2 − y 2 − 1) + β2 y(y + c), ẏ = β2 y(y + c), (H.2) ẋ = (x + c)(α2 x + γ2 y + α2 ), ẏ = −(γ1 /2)(x2 − y 2 − 1) + x(γ2 x + α2 y + cγ2 ), with α2 (c + 1) = 0, (H.3) ẋ = (A/2)(x2 − y 2 − 1) − y(α − cβ + x(β − cα) + y(γ − cα)), ẏ = (A/2)(x2 − y 2 − 1) − x(α − cβ + βx + y(γ − cα)) + cα(y 2 + 1), with c(γ + β) = 0, √ √ (H.4) ẋ = (A/2)(x2 − y 2 − 1) + y(aα − β d√ + x(aβ − α d) + βy), √ ẏ = (−Aa/2)(x2 − y 2 − 1) + x(aα − β d + aβx + βy) − α d(y 2 + 1), with d = a2 − 1, 4 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES (H.5) ẋ = −(x2 − y 2 − 1) + 2α1 y(y + ax + c), ẏ = a(x2 − y 2 − 1) + 2α1 x(y + ax + c), with c2 6= a2 − 1, (P.1) ẋ = x(α2 + β2 x + γ2 y), ẏ = α1 (y − x2 ) + 2α2 x2 + 2y(β2 x + γ2 y), (P.2) ẋ = −β1 (y − x2 ) + y(β2 + γ2 x) + (α2 + γ2 c)x + cβ2 , ẏ = 2(y + c)(α2 + β2 x + γ2 y), with c α2 = 0, (P.3) ẋ = −(y − x2 ) − α(y + ax + c), ẏ = a(y − x2 ) − 2α x(y + ax + c), with c 6= a2 /4, (LV.1) ẋ = x(α + r y + β x), ẏ = y(α + (r − q + β) y + q x), (LV.2) ẋ = x(p + q x + r y), ẏ = y(y + c), with c(c + 1) = 0. (LV.3) ẋ = −x(y + α(y + a x + c)), ẏ = y(a x + β(y + a x + c)), with a c 6= 0, (RP L) ẋ = x2 − 1, ẏ = y(α + β x + γ y), (DL) ẋ = x2 , ẏ = y(α + β x + γ y), (CP L) ẋ = x2 + 1, ẏ = y(α + β x + γ y), (p.1) ẋ = (β/2)(x2 + y 2 ) − β3 y 2 + x(α3 + γ3 y), ẏ = y(α3 + β3 x + γ3 y), (p.2) ẋ = −(x2 + y 2 ) + (β x − α y)(y + ax + c), ẏ = a(x2 + y 2 ) + (β y + α x)(y + ax + c), with c 6= 0, where a, c, A, p, q, r, α, β, α1 , α2 , β1 , β2 , γ1 and γ2 are the parameters of the system. Theorem D. The global phase portrait in the Poincaré disc of each quadratic differential system admitting a reducible invariant cubic f (x, y) = 0 and having a Darboux invariant of the form e−st f (x, y) is topologically equivalent to one of the phase portraits presented in Figures 2-7. Their normal forms according to Theorem C is labelled in the corresponding figure. Theorem E. Systems of type (CE), (E.1), (H.1), (H.5), (P.3) do not admit Darboux in- variants of the form e−st f (x, y). QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 5 Figure 2. Phase portraits of systems of type (E) and (H) when they have a Darboux invariant. 3. Preliminary and basic results The objective of this section is introduce some definitions and results which shall be used in next sections for the study of the Darboux invariants and to obtain the global phase portrait of the systems of Theorems B and C. 3.1. Invariants. A nonconstant C 1 function H : U = R, defined in the open and dense set U ⊂ R2 is a first integral of system (1) on U if H(x(t), y(t)) is constant for all of the values of t for which (x(t), y(t)) is a solution of system (1) contained in U . In other words H is a first integral of system (1) if and only if ∂H ∂H (3) P +Q = 0, ∂x ∂y for all (x, y) ∈ U . An invariant of system (1) on the open subset U of R2 is a nonconstant C 1 function I in the variables x, y and t such that I(x(t), y(t), t) is constant on all solution curves (x(t), y(t)) of system (1) contained in U , i.e. ∂I ∂I ∂I (4) P+ Q+ = 0, ∂x ∂y ∂t for all (x, y) ∈ U . In short, I is a first integral of system (1) depending on the time t. 6 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES Figure 3. Phase portraits of systems of type (P ) when they have a Dar- boux invariant. QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 7 Figure 4. Phase portraits of systems of type (P ) when they have a Dar- boux invariant. On the other hand given f ∈ C[x, y] we say that the curve f (x, y) = 0 is an invariant algebraic curve of system (1) if there exists K ∈ C[x, y] such that ∂f ∂f (5) P +Q = Kf. ∂x ∂y The polynomial K is called the cofactor of the invariant algebraic curve f = 0. When K = 0, f is a polynomial first integral. Note that if a real polynomial differential system has a complex invariant algebraic curve then it has also its conjugate. It is important to consider the complex invariant algebraic curves of the real systems because sometimes 8 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES Figure 5. Phase portraits of systems of type (LV ) when they have a Darboux invariant. QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 9 Figure 6. Phase portraits of systems of type (RP L) and (DL) when they have a Darboux invariant. these force the real integrability of the system, for more details see Chapter 8 of [9], or the subsection 3.2. Let f, g ∈ C[x, y] and assume that f and g are relatively prime in the ring C[x, y], or that g = 1. Then the function exp(f /g) is called a exponential factor of system (1) if for some polynomial L ∈ C[x, y] of degree at most m − 1 we have ∂ exp(f /g) ∂ exp(f /g) (6) P +Q = L exp(f /g). ∂x ∂y 10 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES Figure 7. Phase portraits of systems of type (CP L) and (p) when they have a Darboux invariant. As previously we say that L is the cofactor of the exponential factor exp (f /g). We observe that in the definition of exponential factor exp(f /g) if f, g ∈ C[x, y] then the exponential factor is a complex function. Again when we look for a complex exponential factor of a real polynomial system we are thinking the real polynomial system as a complex polynomial system. 3.2. Darboux invariants. An invariant I is called a Darboux invariant if it can be written into the form λ µ (7) I(x, y, t) = f1λ1 · · · fp p F1µ1 · · · Fq q es t , where fi = 0 are invariant algebraic curves of system (1) for i = 1, . . . p, and Fj are exponential factors of system (1) for j = 1, . . . , q, λi , µj ∈ C and s ∈ R \ {0}. Observe that if among the invariant algebraic curves a complex conjugate pair f = Re(f ) + Im(f )i = 0 and f¯ = Re(f ) − Im(f )i = 0 occurs, then the Darboux invariant has a factor of the form f λ f¯λ̄ , which is the real multi-valued function  Re(λ) (Re(f ))2 + (Im(f ))2 e−2Im(λ) arctan(Im(f )/Re(f )) . So if system (1) is real then the Darboux invariant is also real, independently of the fact of having complex invariant curves or complex exponential factors. The next result is proved in Proposition 8.4 of [9]. Proposition 1. Suppose that f ∈ C[x, y] and let f = f1n1 . . . frnr be its factorization into irreducible factors over C[x, y]. Then for a polynomial differential system (1), f = 0 is an QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 11 invariant algebraic curve with cofactor kf if and only if fi = 0 is an invariant algebraic curve for each i = 1, . . . , r with cofactor kfi . Moreover kf = n1 kf1 + . . . + nr kfr . The next result, proved in [6], explain how to obtain a Darboux invariant using the algebraic invariant curves of a polynomial differential system. Proposition 2. Suppose that a polynomial system (1) of degree m admits p invariant algebraic curves fi = 0 with cofactors ki for i = 1, ..., p, q exponential factors exp(gj /hj ) with cofactors Lj for j = 1, ..., q, then, if there exist λi and µj ∈ C not all zero such that p X q X (8) λi ki + µj Lj = −s, i=1 j=1 for some s ∈ R\{0}, then substituting fiλi by |fi |λi if λi ∈ R, the real (multi-valued) function   µ1   µq λ1 λp g1 gq f1 . . . fp exp . . . exp est h1 hq is a Darboux invariant of system (1). The search of first integrals is a classic tool in order to describe the phase portraits of a 2–dimensional differential system. As usual the phase portrait of a system is the decomposition of the domain of definition of this system as union of all its orbits. It is well known that the existence of a first integral or an a invariant for a planar differential system allow to draw its phase portrait. Here we investigate the existence of invariants of the form f (x, y)est , called Darboux invariants, see section 3.2 for details. Such invariants describe the asymptotic behavior of the solutions of the system. Indeed let φp (t) be the solution of system (1) passing through the point p ∈ R2 , defined on its maximal interval (αp , ωp ) such that φp (0) = p. If ωp = ∞ we define the ω-limit set of p as ω(p) = {q ∈ R2 : ∃{tn } with tn = ∞ and φp (tn ) = q when n = ∞}. In the same way, if αp = −∞ we define the α-limit set of p as α(p) = {q ∈ R2 : ∃{tn } with tn = −∞ and φp (tn ) = q when n = ∞}. For more details on the ω– and α–limit sets see for instance section 1.4 of [9]. The existence of a Darboux invariant of system (1) provides information about the ω– and α–limit sets of all orbits of system (1). More precisely, we have the following result, where the definition of Poincaré compactification and Poincaré disc is given in subsection 3.3. Its proof can be found in [13]. Proposition 3. Let I(x, y, t) = f (x, y)est be a Darboux invariant of system (1). Let p ∈ R2 and φp (t) the solution of system (1) with maximal interval (αp , ωp ) such that φp (0) = p. (1) If ωp = ∞ then ω(p)⊂ {f (x, y) = 0} ∪ S1 , (2) If αp = −∞ then α(p)⊂ {f (x, y) = 0} ∪ S1 . Here S1 denotes the infinity of the Poincaré disc. 12 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES ∂ ∂ 3.3. Poincaré compactification. LetX = P (x, y) ∂x + Q(x, y) ∂y be the planar polyno- mial vector field of degree m associated to the polynomial differential system (1). The Poincaré compactified vector field π(X ) corresponding to X is an analytic vector field induced on S2 as follows (for more details, see [9]). Let S2 = {y = (y1 , y2 , y3 ) ∈ R3 ; y12 + y22 + y32 = 1} and Ty S2 be the tangent plane to S2 at point y. We identify R2 with T(0,0,1) S2 and we consider the central projection f : T(0,0,1) S2 = S2 . The map f defines two copies of X on S2 , one in the southern hemisphere and the other in the northern hemisphere. Denote by X 0 the vector field D(f ◦ X ) defined on S2 \ S1 , where S1 = {y ∈ S2 ; y3 = 0} is identified with the infinity of R2 . For extending X 0 to a vector field on S2 , including S1 , X must satisfy convenient condi- tions. Since the degree of X is m, π(X ) is the unique analytic extension of y3m−1 X 0 to S2 . On S2 \ S1 there is two symmetric copies of X , and once we know the behavior of π(X ) near S1 , we know the behavior of X in a neighborhood of the infinity. The Poincaré com- pactification has the property that S1 is invariant under the flow of π(X ). The projection of the closed northern hemisphere of S2 on y3 = 0 under (y1 , y2 , y3 ) 7→ (y1 , y2 ) is called the Poincaré disc, and its boundary is S1 . 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 π(X ) into orbits of the flow induced by π(Y) preserving or not the orientation of all the orbits. As S2 is a differentiable manifold, in order to compute the explicit expression of π(X ), we consider six local charts Ui = {y ∈ S2 ; yi > 0} and Vi = {y ∈ S2 ; yi < 0}, where i = 1, 2, 3, and the diffeomorphisms Fi : Ui = R2 and Gi : Vi = R2 , for i = 1, 2, 3, which are the inverses of the central projections from the tangent planes at the points (1, 0, 0), (−1, 0, 0), (0, 1, 0), (0, −1, 0), (0, 0, 1) and (0, 0, −1), respectively. We denote by z = (u, v) the value of Fi (y) and Gi (y), for any i = 1, 2, 3, therefore z means different things depending on the local charts where we are working. So after some computations π(X ) is given by:        m 1 u 1 u 1 u (9) v ∆(z) Q , − uP , , −vP , in U1 , v v v v v v        m u 1 u 1 u 1 (10) v ∆(z) P , − uQ , , −vQ , in U2 , v v v v v v (11) ∆(z)(P (u, v), Q(u, v)) in U3 , where ∆(z) = (u2 + v 2 + 1)−(m−1)/2 . The expressions for Vi ’s are the same as that for Ui ’s but multiplied by the factor (−1)m−1 . In these coordinates v = 0 always denotes the points of the infinity S1 . 3.4. Irreducible invariant cubics. The next results characterize all irreducible cubics, their proofs can be found in [2]. Proposition 4 (Theorem 8.3 in [2]). A cubic is non–singular and irreducible and has a flex (a generalized inflection point) if and only if it can be transformed with affine trans- formations into either y 2 = x(x − 1)(x − r) with r > 1, QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 13 or y 2 = x(x2 + sx + 1) with −2 < s < 2. Proposition 5 (Theorem 8.4 in [2]). A cubic is singular and irreducible if and only if it can be transformed with affine transformations into one of the forms y 2 = x3 , y 2 = x2 (x + 1), y 2 = x2 (x − 1). Moreover in [2] it is proved that every non-singular and irreducible curve has a flex. So we have the complete characterization of the irreducible cubics. 3.5. Reducible invariant cubics. Proposition 6. A real quadratic system having an invariant conic after an affine change of coordinates can be written in one of the following forms (real ellipse) ẋ = (A/2)(x2 + y 2 − 1) + 2y(p + q x + r y), ẏ = (B/2)(x2 + y 2 − 1) − 2x(p + q x + r y), (complex ellipse) ẋ = (A/2)(x2 + y 2 + 1) + 2y(p + q x + r y), ẏ = (B/2)(x2 + y 2 + 1) − 2x(p + q x + r y), (hyperbola) ẋ = (A/2)(x2 − y 2 − 1) − 2y(p + q x + r y), ẏ = −(B/2)(x2 − y 2 − 1) − 2x(p + q x + r y), (parabola) ẋ = A(y − x2 ) − (p + q x + r y), ẏ = B(y − x2 ) − 2x(p + q x + r y), (Lotka-Volterra) ẋ = x(p1 + q1 x + r1 y) ẏ = y(p2 + q2 x + r2 y), (two parallel real lines) ẋ = x2 − 1 ẏ = Q(x, y), (double line) ẋ = x2 ẏ = Q(x, y), (two parallel complex lines) ẋ = x2 + 1 ẏ = Q(x, y), (two non-parallel complex lines) ẋ = (A/2)(x2 + y 2 ) + (C/2)x + 2y(p + q x + r y), ẏ = (B/2)(x2 + y 2 ) + (C/2)y − 2x(p + q x + r y). Here Q(x, y) denotes an arbitrary polynomial of degree 2. The proof of the previous result can be found in [4], except to the normal form of the system with a parabola that is proved in [11]. The next result is due to Christopher, Llibre, Pantazi, Zhang and Zholadek, see [5, 7, 17]. An algebraic proof of it also can be found in [7]. Theorem 7. Let fi = 0 for i = 1, . . . , q be q irreducible algebraic curves in C2 , and let Xq k= degfi . We assume i=1 (i) there are no points at which fi and its first derivatives all vanish, (ii) the highest order terms of fi have no repeated factors, (iii) no more than two curves meet at any point in the finite plane and are not tangent at these points, 14 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES (iv) no two curves have a common factor in their highest order terms, then any poly- nomial vector field X of degree m tangent to all fi = 0 is of the form describe bellow. (a) If m > k − 1 then !   Yq Xq Yq (12) X=Y fi +  fj  Xfi , i=1 i=1 j=1,j6=i where Xfi = (−∂fi /∂y, ∂fi /∂x) is a Hamiltonian vector field, the hi are poly- nomials of degree ≤ m − k + 1 and Y is a polynomial vector field of degree ≤ m − k. (b) If m = k − 1 then   X q Yq (13) X= αi  fj  Xfi , i=1 j=1,j6=i where αi ∈ C. In this case a Darboux first integral exists. (c) If m < k − 1 then X ≡ 0. Theorem 8 (Lemma 7 of [7]). Assume that f = 0 and g = 0 are different irreducible invariant algebraic curves of system (1) of degree m, and that they satisfy conditions (i) and (iii) of Theorem 7. If gcd(fx , fy ) = 1 and gcd(gx , gy ) = 1, then system (1) has the normal form (14) ẋ = Af g − h1 fy g − h2 f gy ẏ = Bf g + h1 fx g + h2 f gx , where A, B and hj are polynomials, for i = 1, 2. 4. Proof of Theorems A and B Here we denote P (x, y) = a00 + a01 y + a02 y 2 + a10 x + a11 xy + a20 x2 , Q(x, y) = b00 + b01 y + b02 y 2 + b10 x + b11 xy + b20 x2 . 4.1. Proof of Theorem A. If a quadratic system (1) has a singular irreducible invariant cubic f (x, y) = 0 by Proposition 5 the function f can be written as f (x, y) = y 2 − x3 or f (x, y) = y 2 − x2 (x + 1) or f (x, y) = y 2 − x2 (x − 1). The curve f (x, y) = y 2 − x3 = 0 is an invariant cubic for system (1) if and only if equation (5) is satisfied. The solution of this equation in terms of the parameters of the system is a00 = a02 = b00 = b10 = 0, b01 = 3a10 /2, b02 = 3a11 /2, b11 = 3a20 /2, b20 = 3a01 /2. So the cofactor of f is K = 3(a10 + a20 x + a11 y). Doing a10 = a, a20 = b, a01 = c, a11 = d and a rescaling of the time we obtain system (i) of Theorem A. When f (x, y) = y 2 − x2 (x ± 1) we obtain the normal forms given in (ii) and (iii) of the theorem following similar steps. Now if a quadratic system (1) has an invariant non–singular irreducible cubic f (x, y) = 0 then by Proposition 4 we can write f (x, y) = y 2 − x(x − 1)(x − r) with r > 1 or f (x, y) = y 2 − x(x2 + sx + 1) with −2 < s < 2. In the first case solving equation (5) we obtain three solution but fixing r > 1 only one solution can hold a00 = a02 = a10 = a20 = b01 = b11 = 0, b00 = a01 r/2, b02 = 3a11 /2, b10 = −a01 (r + 1) − a11 r, b20 = (3a01 + a11 r + a11 )/2. It corresponds to system (iv) of Theorem A. QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 15 For f (x, y) = y 2 − x(x2 + sx + 1) we obtain only one solution corresponding to system (v) of the theorem. Using the normal forms described in Theorem A we investigate when these systems admit a Darboux invariant of the form est f (x, y). 4.2. Proof of Theorem B. First of all is easy to see that the cofactor K of f in systems (ii) − (v) of Theorem A has no constant terms. Then equation (8) becomes λK + s = 0 which never holds if s ∈ R \ {0} and λ ∈ C \ {0}. Therefore we conclude that systems (ii) − (v) do not admit a Darboux invariant of such form. Now considering system (i) of Theorem A we have f (x, y) = y 2 − x3 = 0 as invariant curve with cofactor K = 6(a + cx + dy). In this case the solution of equation (8) is given by {c = 0, d = 0, s = −6aλ}. Taking λ = −s/(6a) we obtain the system ẋ = 2(ax + by), ẏ = 3(ay + bx2 ), with Darboux invariant e−6at (y 2 − x3 ). The normal form described in Theorem B is obtained doing the following change of 2a2 2a3 1 coordinates and rescaling of the time x = 2 X, y = 3 Y, t = T. 3b 3b 2a Now it remains to study the phase portrait of system (2). This system has two singular points, namely z1 = (0, 0) yperbolic unstable node, and z2 = (3/2, −3/2) a hyperbolic saddle. Applying the Poincaré compactification in the local chart U1 and on the line v = 0 the compactified system has no singular points. However in the local chart U2 the origin (0, 0) is a nilpotent singularity. With the notation of Theorem 3.5 of [9] the compactified system has F (u) = −u5 − (3/2)u6 and G(u) = −4u2 − (7/2)u3 . Hence the origin of U2 is a nilpotent stable node. By the previous statements it follows that the phase portrait of system (2) is the one described in Figure 1. 5. Proof of Theorem C The proof is done according to the conic that appears in the expression of the reducible cubic. 5.1. Systems of type (E). If system (1) has an invariant cubic of the form f (x, y) = f1 (x, y)f2 (x, y) with f1 = x2 + y 2 − 1 and f2 = ax + by + c, then applying a rotation we can assume b = 1. Therefore it follows from Proposition 1 that fj is an invariant curve with cofactor kj = αj + βj x + γj y, j = 1, 2. Consider two cases: a = 0 and a 6= 0. If a = 0 then using equation (5) we have Q = k2 f2 and P = (k1 f1 − 2yk2 f2 )/(2x). As P is a polynomial the parameters of the system must satisfy on the of following conditions s1 = {c = −1, α1 = 0, γ1 = 2α2 , γ2 = α2 }, s2 = {c = 1, α1 = 0, γ1 = −2α2 , γ2 = −α2 }, s3 = {α1 = 0, γ1 = 0, γ2 = 0}. Moreover the solutions s1 and s2 provide equivalent systems, and we can summarize the solutions s1 and s3 writing the system ẋ = (β1 /2)(x2 + y 2 − 1) − y(β2 y − α2 x + cβ2 ), (15) ẏ = (y + c)(α2 y + β2 c x + α2 ), with α2 (c + 1) = 0. This is exactly system (E.1) of Theorem C. When a 6= 0 we check when the hypotheses of Theorem 7 are satisfied. Clearly f1 and f2 satisfies (i), (ii) and (iv). Condition (iii) is not satisfied when c2 = a2 + 1 because the line 16 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES f2 = 0 is tangent to the real ellipse f1 = 0. Indeed if the straight line f2 = y + ax + c = 0 is tangent to the real ellipse f1 = x2 + y 2 − 1 = 0 at the point (x0 , y0 ), then their gradients are parallel in such point, what means √ that x0 − ay0 = 0. Replacing√y0 = x0 /a in the ellipse we conclude that x0 = ±a/ a2 + 1. From f2 = 0 we get c = ∓ a2 + 1. Therefore the condition for the tangency is c2 = a2 + 1. In this case applying a rotation we can get f2 = y − 1. Again we are in system (15) with c = −1. Now assuming c2 6= a2 + 1 it follows from Theorem 7 that our system is given by (16) ẋ = −α2 (x2 + y 2 − 1) − 2α1 y(y + ax + c), ẏ = aα2 (x2 + y 2 − 1) + 2α1 x(y + ax + c), where α1 , α2 ∈ C and a, c ∈ R. As we are looking for a real system, then α1 , α2 ∈ R, and doing a rescaling of the time we can assume α2 = 1. Note that system (16) is exactly system (E.2) of Theorem C. 5.2. Systems of type (CE). In this case we can follow the same steps applied previously. If system (1) has an invariant cubic of the form f = f1 f2 with f1 = x2 + y 2 + 1 and f2 = ax + by + c we suppose, without loss of generality, b = 1. Since the coefficients a, b and c are real numbers the straight line f2 = 0 cannot be tangent to the complex ellipse f1 = 0. So we get (17) ẋ = −α2 (x2 + y 2 + 1) − 2α1 y(y + ax + c), ẏ = a α2 (x2 + y 2 + 1) + 2α1 x(y + ax + c), where α1 , α2 ∈ C and a, c ∈ R. Applying a rescaling we have α2 = 1 in (17), and we get the normal form for the systems of type (CE). 5.3. Systems of type (H). Let f1 = x2 −y 2 −1 and f2 = ax+by +c be two real algebraic invariant curves of system (1), so a2 + b2 6= 0. Proceeding as before if a = 0 then we can assume b = 1 and the system can be written in the form (18) ẋ = (β1 /2)(x2 − y 2 − 1) + β2 y(y + c), ẏ = β2 y(y + c), with β1 β2 6= 0. This is system (H.1) of Theorem C. If a 6= 0 and b = 0 we take a = 1 and system (1) satisfies P = k2 f2 and 2y Q = 2xP − k1 f1 , where kj = αj + βj x + γj y, for j = 1, 2. Since Q is a polynomial in the parameters of the system it must satisfy one of the following conditions s1 = {c = −1, α1 = 0, β1 = 2α2 , β2 = α2 }, s2 = {c = 1, α1 = 0, β1 = −2α2 , β2 = −α2 }, s3 = {α1 = 0, α2 = 0, β1 = 0, β2 = 0}. Applying the change of coordinates x = −X, y = Y we conclude that case s1 and s2 provide equivalent systems. Moreover we can summarize solutions s1 and s3 in the unique system (19) ẋ = (x + c)(α2 x + γ2 y + α2 ), ẏ = −(γ1 /2)(x2 − y 2 − 1) + x(γ2 x + α2 y + cγ2 ), with α2 (c + 1) = 0. System (19) corresponds to system (H.2) of Theorem C. If ab 6= 0 we assume b = 1 and consider three cases, according to the conditions of Theorem 7. Note that condition (i) of Theorem 7 holds because ∇f1 (x, y) = (2x, −2y) and ∇f2 (x, y) = (a, 1), where ∇ indicates the gradient. Condition (ii) also holds. However condition (iv) is not verified when a2 − 1 = 0. Indeed in this case f1 = (x + y)(x − y) − 1 and f2 = (y ± x) + c. Condition (iii) does not hold when c2 = a2 − 1 since the straight line f2 = y + ax + c = 0 is tangent to the hyperbola. The proof of this last statement can be done analogously as for the systems of type (E). Hence when a2 − 1 = 0 or c2 = a2 − 1 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 17 Theorem 7 does not hold and we split the study of systems of type (H) for ab 6= 0 in three cases: a2 − 1 = 0, c2 = a2 − 1 and (a2 − 1)(c2 − a2 + 1) 6= 0. For the first two cases we apply Propositions 1 and 6 to conclude that f1 is an algebraic invariant curve of a quadratic system (1) and it can be written as (20) ẋ = (A/2)(x2 −y 2 −1)−2y(p+q x+r y), ẏ = −(B/2)(x2 −y 2 −1)−2x(p+q x+r y), where A, B, p, q, r ∈ R. Fixing the cofactor of f2 = 0 as k2 = α + β x + γ y , where α, β, γ ∈ R and using system (20) we solve (5). First considering a = −1 (the case a = 1 is analogous except by a reflection) equation (5) has two possible solutions s1 = {B = −A, c = 0, p = α/2, q = β/2, r = γ/2}, s2 = {B = −A + 2cα, p = (αc − β)/2, q = (β − cα)/2, r = −(β + cα)/2, γ = −β}. Using the two above solutions we get the system ẋ = (A/2)(x2 − y 2 − 1) − y(α − cβ + x(β − cα) + y(γ − cα)), ẏ = (A/2)(x2 − y 2 − 1) − x(α − cβ + βx + y(γ − cα)) + cα(y 2 + 1), with c(γ + β) = 0. This is system (H.3) of Theorem C. Now considering c2 = a2 − 1 we investigate the conditions √ that must be satisfied by the parameters of system (20) in order that f2 =√y + ax ± a2 − 1 be an invariant curve. Without loss of generality we can assume c = a2 − 1. Equation (5) has one solution, namely √ √ √ √ B = aA − 2α d, p = (β d − aα)/2, r = −β/2, q = (α d − aβ)/2, γ = aβ − α d, where d = a2 − 1. Replacing it in (20) we get √ √ ẋ = (A/2)(x2 − y 2 − 1) + y(aα − β d√ + x(aβ − α d) + βy), √ (21) ẏ = (−Aa/2)(x2 − y 2 − 1) + x(aα − β d + aβx + βy) − α d(y 2 + 1), where d = a2 − 1, and this systems corresponds to system (H.4) of Theorem C. Finally if (a2 − 1)(c2 − a2 + 1) 6= 0 applying Theorem 7 we obtain the system ẋ = −α2 (x2 − y 2 − 1) + 2α1 y(y + ax + c), (22) ẏ = aα2 (x2 − y 2 − 1) + 2α1 x(y + ax + c), which is system (H.5) of Theorem C. 5.4. Systems of type (P ). Let f = (y − x2 )(ax + by + c) = 0 be an invariant cubic of system (1). When b = 0 we can assume f = x(y − x2 ). Indeed if b = 0 we take a = 1 and do the change of coordinates x = X − c, y = Y − 2cX + c2 . Using that f2 = x = 0 is an invariant straight line we have P = k2 f2 with k2 = α2 + β2 x + γ2 y, and a quadratic system (1) can be written as (23) ẋ = x(α2 + β2 x + γ2 y), ẏ = α1 (y − x2 ) + 2α2 x2 + 2y(β2 x + γ2 y). If b 6= 0 and a = 0 we can take b = 1 and proceed as in systems of type (H) and (E), then we get the system (24) ẋ = −β1 (y − x2 ) + y(β2 + γ2 x) + (α2 + γ2 c)x + cβ2 , ẏ = 2(y + c)(α2 + β2 x + γ2 y), with c α2 = 0. Observe that when c = 0 the invariant line is y = 0 and when α2 = 0 it is y + c = 0. If ab 6= 0 and f2 = y ± ax + a2 /4, f2 = 0 is tangent to the parabola. In this case we can assume f2 = y + ax + a2 /4 (the other case is a reflection). Applying the change of coordinates x = −X − a/2 and y = Y + aX + a2 /4 the cubic f = (y − x2 )(y + ax + a2 /4) 18 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES becomes f = (Y − X 2 )Y , which already has been studied above. Indeed it corresponds to system (23) with c = 0. Otherwise there is no tangency between the straight line and the parabola, and we apply Theorem 7 to get the differential system (25) ẋ = −(y − x2 ) − α(y + ax + c), ẏ = a(y − x2 ) − 2α x(y + ax + c). Systems (23), (24) and (25) correspond to systems (P.1), (P.2) and (P.3) of Theorem C, respectively. 5.5. Systems of type (LV ). In this case f = x y (a x + b y + c) = 0 is the invariant curve and except by a rotation we can assume b = 1. We consider different cases according to ac = 0 or ac 6= 0. Note that if c = 0 hypothesis (iii) of Theorem 7 is not valid, whereas a = 0 breaks the hypothesis (iv). Y √ When c = 0 and a 6= 0, doing the change of coordinates x = − √ 3 2, y = 3 aX the cubic a becomes F = X Y (Y − X). So using Proposition 6 the differential system can be written as (26) ẋ = x(p1 + q1 x + r1 y) ẏ = y(p2 + q2 x + r2 y). If (26) has f3 = y − x as an invariant curve with cofactor k = α + β x + γ y, then equation (5) must be satisfied. Solving it we get s1 = {p2 = α, r2 = β − q2 + r1 , q1 = β, p1 = α, γ = β − q2 + r1 }. Replacing in (26) and writing q = q2 , r = r1 we obtain system (LV.1) of Theorem C. Now if c = a = 0 then f2 = y = 0 is a double line, and it is not difficult to see that we can write the system as (27) ẋ = x(p + q x + r y), ẏ = y 2 . Finally, when a = 0 and c 6= 0, doing the change of coordinates x = X/c2 , y = cY − c the cubic f = 0 becomes F = X Y (Y − 1). So without loss of generality we can work with f3 = y − 1. Again the idea is to write the system as in (26), and see what are the conditions in order that f3 = 0 to be an invariant curve for such system. Solving equation (5) and replacing the solutions in (26) we get (28) ẋ = x(p + q x + r y), ẏ = y(y − 1). Systems (27) and (28) can be summarized as ẋ = x(p + q x + r y), ẏ = y(y + c), with c = 0 or c = −1. This is exactly system (LV.2) of Theorem C. In the last case, a c 6= 0 the invariant cubic is f = x y (y + a x + c) = 0 and by the geometry to the curves we can assume a < 0 and c < 0. Applying Theorem 7 we get the system ẋ = −α2 x(y + a x + c) − α3 x y, ẏ = α1 y(y + a x + c) + a α3 x y. Note that we can take α3 = 1. Doing α = α2 , β = α1 we obtain system (LV.3) of Theorem C. QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 19 5.6. Systems of type (RP L). Here the invariant cubic is f = f1 f2 f3 = 0 where f1 = x + 1, f2 = x − 1 and f3 = a x + b y + c. When b = 0 we apply Proposition 6 (case (RPL)), then it is easy to see that the corresponding normal form has one additional invariant curve f3 = 0 as invariant straight line if and only if it is a multiple of f1 or f2 . However we cannot consider any of these cases because if the system has f2 as an invariant double straight line for example, then there would be a change of coordinates so that the system would be written as ẋ = (x − 1)(x + 1)2 , ẏ = Q(x, y), then having degree 3 instead of 2. When b 6= 0 we can fix b = 1. In this case the cubic f = (x2 − 1)(y + ax + c) = 0 can be reduced to F = y(x2 − 1) by change of coordinates x = X, y = Y − a X − c. If the quadratic differential system (1) has the invariant curve f = y(x2 − 1) = 0, then f1 = 0 and f2 = 0 are invariant curves and by Proposition 6 such system can be written as (29) ẋ = x2 − 1, ẏ = Q(x, y), where Q(x, y) is an arbitrary polynomial of degree 2. Imposing that f3 = y = 0 is an additional invariant curve with cofactor k3 = α + β x + γ y, the above system must satisfy Q(x, y) = y(α + β x + γ y). This expression justify the normal form given in (RP L) of Theorem C. 5.7. Systems of type (DL). These systems have a double straight line as invariant curve which can be taken as f1 = x. We write f2 = a x + b y + c and use the normal form of a system having f = f1 f2 = 0 as an invariant cubic. For such normal form, if b = 0 then f2 = 0 is an invariant straight line if and only if c = 0 but in this case the system cannot have a triple invariant straight line. If b 6= 0 we can take b = 1 and f = x2 (y + a x + c). Doing the change x = X, y = Y − a X − c the function f can be written as F = X 2 Y . Hence it is enough to consider f2 = y. By Proposition 6 a quadratic system (1) can be written as ẋ = x2 , ẏ = Q(x, y), where Q(x, y) is an arbitrary polynomial of degree 2. Imposing that f2 = 0 is an additional invariant curve with cofactor k2 = α+β x+γ y, we conclude that Q(x, y) = y(α+β x+γ y). This expression justify the normal form given in (DL) of Theorem C. 5.8. Systems of type (CP L). The proof for this case is analogous to the case (DL) so we will omit some details. In short the cubic is given by f = f1 f2 f3 = 0 where f1 = x + i, f2 = x − i and f3 = ax + by + c. In order to f3 = 0 to be an invariant curve with b = 0 it is necessary that c = ±i. So b 6= 0 and we assume b = 1. This reduce f to the cubic F = y(x2 + 1) and then we get the normal form (CP L) described in Theorem C. 5.9. Systems of type (p). In this case the cubic is given by f = (x2 +y 2 )(a x+b y+c) = 0 and except by a rotation we can assume b = 1. When c = 0 the three curves intersect at the same point and the conditions of Theorem 7 are not satisfied. But if c = 0 doing the change of coordinates X aY aX Y x = −p +p , y= p +p , 3 (a2 + 1)2 3 (a2 + 1)2 3 (a2 + 1)2 3 (a2 + 1)2 the cubic f = (x2 + y 2 )(y + a x) = 0 is reduced to f = Y (X 2 + Y 2 ). Now using that system (1) has f3 = y = 0 as a third invariant curve it follows that Q(x, y) = k3 f3 where 20 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES k3 = α3 + β3 x + γ3 y is the cofactor of f3 . Moreover f1 f2 = 0 is also an invariant curve then we must have 2xP (x, y) + 2yQ(x, y) = k(x, y)(x2 + y 2 ), with k(x, y) = α + β x + γ y being the sum of the cofactors of f1 and f2 . So a quadratic system (1) can be written as ẋ = (β/2)(x2 + y 2 ) − β3 y 2 + x(α3 + γ3 y), ẏ = y(α3 + β3 x + γ3 y), which is exactly system (p.1) of Theorem C. When c 6= 0 we apply Theorem 7 and conclude that a quadratic system (1) can be written as ẋ = −α3 (x2 + y 2 ) − ((α2 + α1 )y − i(α2 − α1 )x)(y + ax + c), (30) ẏ = aα3 (x2 + y 2 ) + ((α2 + α1 )x − i(α2 − α1 )y)(y + ax + c), with α1 , α2 and α3 ∈ C. Writing αj = mj + i nj with mj , nj ∈ R and using that such system have real parameters we conclude that m2 = m1 , n2 = −n1 and n3 = 0. Replacing this conditions in (30) we get the system (31) ẋ = −m3 (x2 +y 2 )+2(n1 x−m1 y)(y+a x+c), ẏ = am3 (x2 +y 2 )+2(m1 x+n1 y)(y+a x+c). Note that if m3 = 0 then the system has a common factor, so we can take m3 = 2. Applying a rescaling of the time and writing α = m1 , β = n1 we obtain system (p.2) of Theorem C. It follows from the previous study the proof of Theorem C. 6. Proof of Theorems D and E In this section we investigate the conditions in order that a given quadratic system with an algebraic invariant cubic has a Darboux invariant. Moreover, using the obtained normal forms in Theorem C we study the phase portrait in the Poincaré disc of such systems. Proposition 9 (E). Each real planar quadratic differential system with a real ellipse and a straight line having a Darboux invariant can be written, after an affine change of coordinates, as system (E.2) with c = −1, α2 6= 0. Moreover, such system has the Darboux invariant − 1 I1 (t, x, y) = e−t (y − 1)1/α2 (x2 + y 2 − 1) 2α2 . and, these systems have only two non equivalent phase portraits, see phase portraits EL.2.1 and EL.2.2 of Figure 2. Proof. If follows from the reducible cubic classification that we can fix f1 = x2 + y 2 − 1 = 0 as the real ellipse and by Theorem C there are only two families of systems having f1 = 0 and a straight line as invariant curves (E.1) and (E.2). We shall prove later that (E.1) does not admit a Darboux invariant. Now we study system (E.2). By Proposition 2 system (E.2) has a Darboux invariant if there exist λ1 , λ2 ∈ R not both equal to zero such that (8) holds with s ∈ R \ {0} and k1 , k2 being the cofactors of f1 = 0 and f2 = y + c = 0, respectively. But for system (E.2) we must have α2 = 0 or c = −1. If α2 = 0 the cofactors are k1 = β1 x and k2 = β2 x and the equation λ1 k1 + λ2 k2 + s = 0 has no solution for s 6= 0. Hence if α2 = 0 system (E.2) has no Darboux invariant. If α2 6= 0 and c = −1 then the cofactors are k1 = β1 x + 2α2 y and k2 = α2 + β2 x + α2 y and the unique solution of (8), with s 6= 0 is (32) β1 = 2β2 , s = −α2 λ2 , λ1 = −λ2 /2. QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 21 Taking λ1 = 1/α2 and replacing (32) in system (E.2) we obtain the system (33) ẋ = β2 (y − 1) + x(β2 x + α2 y), ẏ = (y − 1)(α2 + β2 x + α2 y), which has the Darboux invariant − 2α1 I1 (t, x, y) = e−t (y − 1)1/α2 (x2 + y 2 − 1) 2 . In order to study the global phase portrait of system (E.2) we start considering its finite singularities. Note that (33) has at 2most  three finite singularities, namely z1 = (0, 1), 2β2 β2 −1 z2 = (−1/β2 , 1) and z3 = − β 2 +1 , β 2 +1 . The eigenvalues associated to z1 are 2 and 1, 2 2 if β2 6= 0, the eigenvalues associated to z2 are −1 and 1 and the eigenvalues of z3 are −1 and −2. So for β2 6= 0 z1 , z2 and z3 are an unstable node, a saddle and a stable node, respectively. When β2 = 0 we have only z1 and z3 as finite singularities. In the local chart U1 the compactified system is (34) u̇ = −v(β2 + β2 u2 − β2 uv + v), v̇ = −v(β2 + β2 uv + u − β2 v 2 ), so v = 0 is a common factor, this means that v = 0 is a line of singular points. Eliminating the common factor v, system (34) has no singular points if β2 6= 0. Otherwise u1 = (0, 0) is a singular point with eigenvalues −1 and 1 which implies that the origin is a hyperbolic saddle besides the line of singular points. In the local chart U2 the compactified system is written as u̇ = v(β2 + β2 u2 + uv − β2 v), v̇ = v(v − 1)(β2 u + v + 1). Eliminating the common factor v the origin is not a singular point of the compactified system. Note that if β2 = 0 there are an additional invariant straight line given by y + 1 = 0. From the study of the finite and infinite behavior of system (E.2) we conclude that such system has two non–equivalent phase portraits when c = −1: phase portrait EL.2.1, if β2 6= 0 and phase portrait EL.2.2, if β2 = 0. See Figure 2.  Proposition 10 (H). Each real planar quadratic differential system with a hyperbola and a straight line having a Darboux invariant can be written, after an affine change of coordinates, as (i) system (H.2) with α2 6= 0 and c = −1. Its Darboux invariant is I2 (t, x, y) = e−α2 t (x2 − y 2 − 1)−1/2 (x − 1). ii) system (H.3) with Aα 6= 0, c = 0 and β = −γ. Its Darboux invariant is I3 (t, x, y) = e−Aαt (x2 − y 2 − 1)γ (y − x)A . (iii) system (H.3) with α 6= 0 and β = γ = 0. Its Darboux invariant is I4 (t, x, y) = eαt (y − x + c)−1 . (iv) system (H.4) with α 6= 0 and A = 2β. Its Darboux invariant is p I5 (t, x, y) = e−αt (x2 − y 2 − 1)−1/2 (y + ax − a2 − 1). Moreover the are 13 non-equivalent phase portrait in the Poincaré disc of these systems. They are in Figure 2 HL.2.1–HL.2.3, HL.3.1–HL.3.13. 22 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES Proof. Fixing f1 = x2 − y 2 − 1 = 0, Proposition 2 says that system (H.2) has a Darboux invariant if equation (8) holds for λ1 , λ2 not both zero, where s ∈ R \ {0}, and k1 , k2 are cofactors of f1 = 0 and f2 = x + c = 0, respectively. Moreover c = −1 or α2 = 0 in system (H.2). For α2 = 0 we have k1 = γ1 y and k2 = γ2 y and the equation λ1 k1 + λ2 k2 + s = 0 has no solution with s 6= 0. So in this case system (H.2) has no Darboux invariant. If α 6= 0 and c = −1 then k1 = 2α2 x + γ1 y and k2 = α2 + α2 x + γ2 y and (8) has a unique solution s = −α2 λ2 , γ1 = 2γ2 , λ1 = −λ2 /2. The proof of (i) follows taking λ2 = 1 and replacing γ1 = 2γ2 in system (H.2), from that we have the system (35) ẋ = (x − 1)(α2 + α2 x + γ2 y), ẏ = −γ2 (x2 − y 2 − 1) + x(−γ2 + γ2 x + α2 y), having the Darboux invariant I1 (t, x, y) = e−α2 t (x2 − y 2 − 1)−1/2 (x − 1). To prove (ii) and (iii) we study system (H.3) where we consider two cases: c = 0 and β = −γ. It is easy to see that if c = 0 (H.3) has a Darboux invariant when α 6= 0 and β = −γ. In this case we have the differential system (36) ẋ = (A/2)(x2 − y 2 − 1) − y(α − γx + γy), ẏ = (A/2)(x2 − y 2 − 1) − x(α − γx + γy), having the Darboux invariant I1 (t, x, y) = e−Aαt (x2 − y 2 − 1)γ (y − x)A . If β = −γ system (H.3) has a Darboux invariant only when γ = 0 and α 6= 0. In this case the system is (37) ẋ = (A/2)(x2 −y 2 −1)−α y(1−cx−cy), ẏ = (A/2)(x2 −y 2 −1)−α x(1−c y)+c α(y 2 +1), and it has the Darboux invariant I1 (t, x, y) = eαt (y − x + c)−1 . √ The study of (iv) follows from system (H.4) where the invariant line is f2 = y + ax − a2 − 1 = 0. In this case the unique solution of equation (8) is (38) s = −αλ2 , A = 2β, λ1 = −λ2 /2. So taking λ2 = 1 we obtain the Darboux invariant p I2 (t, x, y) = e−αt (x2 − y 2 − 1)−1/2 (y + ax − a2 − 1). We start the study of the phase portraits of system (35). Since α2 6= 0 we can take α2 = 1 and the transformation x = X, y = −Y takes the system with parameter γ2 to the system with parameter −γ2 . So we may also assume γ2 ≥ 0. Considering the finite singularities, if γ2 6∈ {0, 1} system  2 (35) hasthree finite singu- γ +1 larities, namely z1 = (0, 1), z2 = (1, −1/γ2 ) and z3 = γ22 −1 , − γ2γ 2 2 . The eigenvalues 2 2 −1 associated to z1 are 2 and 1, if β2 6= 0, the eigenvalues associated to z2 are −1 and 1 and the eigenvalues of z3 are −1 and −2. So for γ2 6∈ {0, 1} z1 , z2 and z3 are respectively, an unstable node, a saddle and a stable node. When β2 = 0 we have only z1 and z3 as finite singularities. In the local chart U1 the compactified system is (39) u̇ = v(−γ2 + γ2 u2 + uv + γ2 v), v̇ = v(v − 1)(γ2 u + v + 1), QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 23 so v is a common factor, this means that v = 0 is a line of singular points. Eliminating the common factor v, system (39) has no singular points if γ2 6= 1. Otherwise u1 = (−1, 0) is a singular point with eigenvalues −2 and −1, which implies that u1 is a hyperbolic stable node. Moreover if γ2 = 0 there an additional invariant straight line given by x + 1 = 0. In the local chart U2 the compactified system is written as u̇ = −v(γ2 − γ2 u2 + γ2 uv + v), v̇ = −v(γ2 + γ2 v 2 − γ2 uv + u). So after eliminating the common factor v the origin is a singular point of the compactified system if and only if γ2 = 0. In this case (0, 0) is a hyperbolic saddle. It is easy to see that if γ2 ∈ (0, 1) the singulatities z1 and z3 are in distinct branches of the hyperbola, and if γ2 ∈ (1, +∞) they are in the same branch as shows Figure 8. Figure 8. Possible phase portraits of sytem (35) when γ2 6∈ {0, 1}. From Theorem 1.43 of [9] (Markus-Neumann-Peixoto Theorem) we conclude that these two phase portraits are topologically equivalent. By continuity and the study done previ- ously we conclude that system of type (H.2) having a Darboux invariant can have three non-equivalent phase portrait. The case γ2 6= 0, 1 corresponds to HL.2.1 in Figure 2 and when γ2 = 1 or γ2 = 0 we have the phase portraits HL.2.2 and HL.2.3 of Figure 2, respectivelly. Now we study the global phase portrait of system (H.3). Remember that the parameters of (H.3) must satisfies c(γ+β) = 0. We start considering c = 0, then the differential system is (40) ẋ = (A/2)(x2 − y 2 − 1) − y(α − γx + γy), ẏ = (A/2)(x2 − y 2 − 1) − x(α − γx + γy), that has f1 = x2 − y 2 − 1 = 0 and f2 = y − x = 0 as invariant algebraic curves. Since α 6= 0 we can take α = 1 and the transformation x = −X, y = −Y allows to assume  A > 0. If γ 6= 0 then z1 = (−A/2, −A/2) and z2 = (γ 2 + 1)/(2γ), (γ 2 − 1)/(2γ) are the two finite singular points. If γ = 0 exists only one finite singular point. The eigenvalues associated to z1 are −1 and 1 so z1 is a saddle. The eigenvalues associated to z2 are A/γ and −1, so z2 is a stable node if γ < 0, and a saddle if γ > 0. Moreover z1 is on the straight line and z2 is on the hyperbola. In the local chart U2 we have the system u̇ = (1/2)(u − 1)(Av 2 − (A + 2γ)u2 + 2uv + 2v + A + 2γ), v̇ = (1/2)v(Av 2 − (A + 2γ)u2 + 2γu + 2uv + A), and the origin is a singular point only when A + 2γ = 0 but in this case the line v = 0 is filled up of singular points. In the local chart U1 we have system u̇ = (1/2)(u − 1)((A + 2γ)u2 + Av 2 + 2uv + 2v − A − 2γ), v̇ = (1/2)v((A + 2γ)u2 + Av 2 + 2uv − 2γu − A), 24 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES which has the infinity filled up by singularities when A + 2γ = 0, otherwise, there are two singularities u1 = (−1, 0) and u2 = (1, 0). Assuming A + 2γ 6= 0. The point u1 has eigenvalues 2γ and 2(A + 2γ), and u2 is linearly zero because the Jacobian matrix of the linear part of the system evaluated in u2 is null. To decide the local behavior of u2 we must do blow up. From now on we fix l1 = γ, l2 = A + 2γ. After translate the singular point u2 to the origin, making the change of coordinates u = U, v = U W and rescaling the common factor U we get the differential system U̇ = (1/2)U (AU W 2 + (A + 2γ)U + 2U W + 4W + 2A + 4γ), Ẇ = −W (W + γ). Note that such system have two singularities when l1 l2 6= 0, namely, U1 = (0, 0) and U2 = (0, −γ); one singular point when l1 = 0 and l2 6= 0, namely U1 = U2 . The eigenvalues of U1 are −γ and A+2γ, whereas the eigenvalues of U2 are A and γ. From the combination of the signs of l1 and l2 , as described in Figure 9, we get the possible local behavior of U1 and U2 . Figure 9. The possible combination of signs of l1 and l2 describe the cases to be considered for system (H.3) when c = 0. Applying the blow down we get all possible phase portraits for system (H.3) when c = 0. Note that each one is realizable, indeed, the phase portrait HL.3.2 corresponds to subcase (1.1) which is realizable with A = 4 and γ = −1; HL.3.3 corresponds to subcase (1.2) which is realizable with A = 1 and γ = −1. Notice that if γ 6= 0 there is a third invariant straight line, given by f3 = γ(x − y) − 1 = 0 so HL.3.3 is the only possible phase portrait for subcase (1.2). The phase portraits HL.3.4 and HL.3.5 correspond, respectively, to subcases (2.1) and (3.1). The phase portrait HL.3.4 is realizable with A = 1 and γ = 1, and HL.3.5 is realizable with A = 1 and γ = 0. It remains to consider the case l2 = 0. With this condition the infinity is filled up of singular points. After eliminating the common factor v we have only one singular point at the local chart U1 . The eigenvalues associated to this point are 2 and 1, so this is a unstable node. By continuity the only possible phase portrait in this case is HL.3.1 of Figure 2, which is realizable with A = 2 and γ = −1. Now considering system (H.3) with β + γ = 0 we have seen above that the system has a Darboux invariant when β = γ = 0 and α 6= 0. Under these conditions the differential system is ẋ = (A/2)(x2 − y 2 − 1) − α y(1 − cx − cy), (41) ẏ = (A/2)(x2 − y 2 − 1) + cα(y 2 + 1) − α x(1 − cy). Such system has f1 = x2 − y 2 − 1 = 0 and f2 = y − x + c = 0 as algebraic invariant curves. If c = 0 then we get system (40) when γ = 0, so we can take c 6= 0 here. Moreover, doing the transformation x = −X, y = −Y in the algebraic cubic we can assume c > 0. Finally, since α is different from zero we can take α = 1 in (41). System (41) has two finite singular points, namely z1 = ((2c − A)/2, −A/2) and z2 = ((c2 + 1)/(2c), (1 − c2 )/(2c)). Defining l1 = c2 − Ac − 1, l2 = A − c and l3 = A − 2c, we QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 25 Figure 10. On the left the local phase portrait after blow up. Here they are indexed according to the signs of l1 and l2 . On the right the local behavior at origin after the Blow down for system (H.3). have z1 coalesces with z2 if and only if l1 = 0. Moreover the eigenvalues associated to z1 are l1 and 1, and the eigenvalues associated to z2 are −l1 and 1. So we conclude that z1 is a unstable node and z2 is a saddle if l1 > 0; z1 is a saddle and z2 , an unstable node, if l1 < 0 and, if l1 = 0, z1 = z2 is a saddle-node. In the local chart U1 the system becomes (42) u̇ = (1/2)((A − 2c)u3 − Au2 + Auv 2 − Au − Av 2 + 2cu + 2cv 2 + 2u2 v − 2v + A), v̇ = (1/2)v((A − 2c)u2 + Av 2 − 2cu + 2uv − A), A which has three singularities u1 = (−1, 0) and u2 = (1, 0) and u3 = ( A−2c , 0), if A 6= 2c. Note that when l3 = 0 the point u3 does not exist and u1 = u3 when l2 = 0. The eigenvalues associated to u1 are 2l2 and 0, the point u2 has both eigenvalues equal to −2c, and u3 has eigenvalues 0 and 2cl2 /l3 . It is not difficult to see that when l2 6= 0, u1 and u3 26 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES are saddle–nodes. In the local chart U2 the origin (0, 0) is a singular point if and only if l3 = 0. Assuming l1 l2 6= 0 and considering all possible combinations of the sign of l1 , l2 and l3 we observe that there are some impossible combinations, for instance when l2 < 0 we have l3 < 0. In Figure 11 we describe the possible combinations and introduce a label for each one. Figure 11. The possible combinations of signs of l1 , l2 and l3 for system (H.3) when c 6= 0. The case (2.2.1) presents a unique phase portrait, HL.3.6 of Figure 2 and it is realizable with A = 1/2 and c = 1. In case (2.1.1) we have three possibilities for the finite saddle separatrix ω-limit set: we can have a connection of separatrix as in HL.3.7; the separatrix can go to the stable node, generating a phase portrait equivalent to HL.3.6, or the separatrix can go to the parabolic sector of the saddle node u3 which corresponds to HL.3.8. Moreover HL.3.8 is realizable with A = 2 and c = 1/2, and as we see above, HL.3.6 is realizable with A = 1/2 and c = 1. Since HL.3.6 and HL.3.8 are realizable then by continuity of the parameters we conclude that HL.3.7 is also realizable. The analysis of case (2.1.2) can be done as the case (2.1.1) and it has the phase portraits equivalent to them. The possible phase portraits of (2.1.3) are also equivalent to the phase portraits of (2.1.1). Also the case (1.1.1) has a phase portrait equivalent to (2.2.1). When l2 = 0 it follows that l1 , l3 < 0 and in the local chart U1 the singular point u1 = u3 is non–elementary. After translate this singular point to the origin, making the change of coordinates u = U, v = U W and rescaling the common factor U we get U̇ = (U/2)(AU W 2 − AU + 2U W − 4W + 2A), Ẇ = W (W − A). This system has two singularities U1 = (0, 0) and U2 = (0, A) being both saddles. Figure 12 shows the blow down. Figure 12. The local phase portrait of system (42) when l2 = 0. On the left the local phase portrait after blow up. On the right the local behavior at the origin after blow down. QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 27 To obtain the phase portrait for system (41) with l2 = 0 we note that there is more two invariant straight lines, given by f3 = x + y = 0 and f4 = A x + A y − 1. The finite saddle z1 is on f3 = 0 and the finite node is on the intersection of f2 = 0 and f4 = 0 so by continuity there is only one phase portrait, which is topologically equivalent to HL.3.3. Finally it remains to study the case l1 = 0. Here l2 < 0 and l3 < 0 so the only possibility is the phase portrait HL.3.9 of Figure 2, which is realizable with A = 0 and c = 1. To conclude the proof of Proposition 10 it remains to study the global phase portrait of system (H.4) when A = 2β and α 6= 0. In this case we assume α = 1, so (H.4) is written as √ √ ẋ = βx2 + √ (aβ − a2 − 1)xy + (a − √ a2 − 1β)y − β, √ ẏ = (aβ − a2 − 1)y 2 + βxy + (a − a2 − 1β)x + (aβ − a2 − 1). √ √ Denoting δ = aβ− a2 − 1 and η = a− a2 − 1β there are at most three finite singularities z1 = (−δ/η, β/η), z2 = ((δη − β)/(β 2 − δ 2 ), (δη − β)/(β 2 − δ 2 )) and z3 = ((β + δη)/(β 2 − δ 2 ), −(β + δη)/(β 2 − δ 2 )). We observe that such singular points never coalesce but if η = 0, z1 does not exist and if β 2 − δ 2 = 0 the same happens with z2 or z3 . With respect to the localization of these points, z3 is the intersection of the hyperbola and the straight line, z1 is on the straight line and z2 is on the hyperbola. Moreover is not difficult to check that z1 , z2 and z3 are hyperbolic points, being z1 a saddle, z2 a stable node and z3 an unstable node. Concerning to the behavior at infinity, in the local chart U1 the compactified system is given by u̇ = v(η − ηu2 + βuv + δv), v̇ = −v(β − βv 2 + ηuv + δu), so v is a common factor what means that v = 0 is a line of singular points. Eliminating this common line it remains singularities if and only if η = 0 or β 2 − δ 2 = 0. When η = 0 the point u1 = (−a, 0) is a saddle. When δ = β the point u2 = (−1, 0) is a node with eigenvalues η and 2η. Finally if δ = −β then the point u3 = (1, 0) has eigenvalues −η and −2η so it is a node. In the local chart U2 the system becomes u̇ = −v(−η + βv + ηu2 + δuv), v̇ = v(δ + βu + ηuv + δv 2 ). So eliminating the common factor v the origin is not a singular point. By the previous study and continuity of the solutions we conclude that there exist three possible phase portraits and they are topologically equivalent to the ones obtained from system (H.2) and described in Figure 2. Indeed when η, β 2 − δ 2 6= 0 we have the phase portrait HL.2.1, when β 2 − δ 2 = 0 we have HL.2.2, and the case η = 0 corresponds to phase portrait HL.2.3.  Before to study the systems of type (P), we present two lemmas that will help to show the realization or not of the phase portraits that follow. Lemma 11. On any straight line which is not composed of orbits the total number of contact points is at most two for any quadratic system. If there are two such points p1 and p2 , then the orbits intersecting the segment ∞p1 cross in the same sense as the orbits intersecting p2 ∞, and the opposite sense to the path intersecting p1 p2 . Lemma 12. The straight line connecting one finite singular point and a pair of infinite singular points in a quadratic system is either formed by trajectories or a line with exactly one contact point. If this contact point is the finite singular point, the flow goes in different directions on each half straight line. 28 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES The proof of Lemma 11 can be founded in [8]. Lemma 12 in the case that the pair of infinite singular points are saddles was proved in [16]. When such a pair are saddle-nodes, the proof appeared in [1]. Proposition 13 (P). Each real planar quadratic differential system with a parabola and a straight line having a Darboux invariant can be written, after an affine change of coor- dinates, as (i) (P.1) with α1 − 2α2 6= 0 and Darboux invariant I6 (t, x, y) = e(α1 −2α2 )t (y − x2 )−1 x2 . (ii) (P.2) with α2 (β1 − β2 ) 6= 0, c = 0 and Darboux invariant I7 (t, x, y) = e2α2 (β1 −β2 )t (y − x2 )β2 y −β1 , (iii) (P.2) with c γ2 6= 0, β1 = β2 , α2 = 0 and Darboux invariant I8 (t, x, y) = e−2cγ2 t (y − x2 )(y + c)−1 , Moreover there are 41 non-equivalent phase portrait in the Poincaré disc for these systems. They are in Figures 3 and 4. Proof. We fix the invariant parabola as f1 = y − x2 = 0. Here we describe in details the proof of the existence of a Darboux invariant for system (P.2), the other cases are analogous. System (P.2) is given by ẋ = −β1 (y − x2 ) + y(β2 + γ2 x) + (α2 + γ2 c)x + cβ2 , ẏ = 2(y + c)(α2 + β2 x + γ2 y), where c α2 = 0. If c = 0 then the additional invariant line is written as f2 = y = 0 and if α2 = 0, such line is f2 = y + c = 0. System (P.2) has a Darboux invariant if there exist λ1 , λ2 not all zero satisfying equation (8) with s ∈ R \ {0}, and k1 , k2 being the cofactors of f1 = 0 and f2 = 0, respectively. For c = 0, k1 = 2(α2 + β1 x + γ2 y) and k2 = 2(α2 + β2 x + γ2 y). Equation (8), with s 6= 0 has the solution (43) s = −2α2 (λ1 + λ2 ), β2 = −β1 λ1 /λ2 , γ2 = 0, Taking λ1 = β2 and λ2 = −β1 the solution can be rewritten as (44) s = −2α2 (β2 − β1 ), λ1 = β2 , λ2 = −β1 , γ2 = 0, and the Darboux invariant is I1 (t, x, y) = e2α2 (β1 −β2 )t (y − x2 )β2 y −β1 . In this case we assume β2 − β1 6= 0 otherwise system (P.2) has a common factor. Moreover if α2 = c = 0 (P.2) does not admit a Darboux invariant. When α2 = 0 then f2 = y + c and the cofactors of f1 = 0 and f2 = 0 are, respectively, k1 = 2(cγ2 + β1 x + γ2 y) and k2 = 2(β2 x + γ2 y). In this case equation (8) has only one solution s = −2cγ2 λ1 , β2 = β1 , λ2 = −λ1 . So taking λ1 = 1 we get the Darboux invariant I1 (t, x, y) = e−2cγ2 t (y − x2 )(y + c)−1 .  QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 29 From now on we study the possible global phase portraits for systems (P) when they have a Darboux invariant. We start studying system (P.1). Remember that such system is given by ẋ = x(α2 + β2 x + γ2 y), ẏ = α1 (y − x2 ) + 2α2 x2 + 2y(β2 x + γ2 y). We consider two cases: γ2 6= 0 and γ2 = 0. If γ2 6= 0 we assume γ2 = 1. In this last case system (P.1) have at most four singular points, given by z1 =  (0, 0), z = (0, −α1 /2), p2 p  z3 = −(β2 + β22 − 4α2 )/2, (β22 − 2α2 + β2 β22 − 4α2 )/2 ,  p p  z4 = −(β2 − β22 − 4α2 )/2, (β22 − 2α2 − β2 β22 − 4α2 )/2 . Observe that unless of the p change x = −X, y = Y we can assume β2 ≥ 0. Let l1 = α1 , l2 = α2 , l3 = β22 − 4α2 − β2 β22 − 4α2 and l4 = α1 − 2α2 be. It follows from Proposition 13 (i) l4 6= 0. Moreover • z1 has eigenvalues l1 and l2 ; • z2 has eigenvalues −l1 and −l4 ; p • z3 has eigenvalues l4 and (β22 − 4α2 + β2 β22 − 4α2 )/2; • z4 has eigenvalues l3 and l4 , so l12 + l22 6= 0 and the topological type of the finite singular points can be studied using the Hartman-Grobman Theorem and Theorem 2.19 of [9]. With respect to the position of the finite singularities, z1 is on the intersection of the parabola and the straight line, z2 is on the straight line, and z3 , z4 are on the parabola. In the local chart U1 system (P.1) is written as u̇ = u2 + β2 u + (α1 − α2 )uv + 2α2 − α1 , v̇ = −v(α2 v + u + β2 ), which has at most two singular points when v = 0, namely q q u1 = (−β2 − β22 + 4(α1 − 2α2 )/2, 0), u2 = (−β2 + β22 + 4(α1 − 2α2 )/2, 0). p p 2 −(β2 − β22 + 4l4 )/2 while the The eigenvalues associated to up 1 are − β2 + 4l4 andp eigenvalues associated to u2 are β22 + 4l4 and −(β2 + β22 + 4l4 )/2. Since we are assuming β2 ≥ 0 it follows that when β22 + 4l4 > 0 the point u2 is a saddle and it is not difficult to see that if l4 > 0, then u1 is a saddle, and if l4 < 0, u1 is a stable node. When β22 + 4l4 = 0 u1 and u2 coalesce and we conclude that this point is a saddle-node, using Theorem 2.19 [9]. When β22 + 4l4 < 0 there is no infinite points in the local chart U1 . In the local chart U2 the origin (0, 0) is a stable node. Observe that l1 , l2 , l3 , l4 , β22 − 4α2 and β22 + 4l4 are bifurcation surfaces, i.e. where topological changes in the global phase portrait of (P.1) can happen. To draw all non- equivalent phase portraits of system (P.1) we split the study in three cases: β22 − 4α2 > 0, β22 − 4α2 = 0 and β22 − 4α2 < 0. Choosing a representative of each region defined by such surfaces we have a configuration of finite and infinite points. Considering the behavior of the separatrices of these systems we obtain all possible phase portraits when β22 − 4α2 > 0, thus we obtain the 40 phase portraits described in Figures 13 and 14 and the phase portraits 41 − 50 of Figure 18. We study all these cases bellow. Among the phase portraits 1 − 18 of Figure 13, we claim that 1 and 3, as well as 7 to 18, are not realizable. Indeed these 18 phase portraits, 1−3 present the possible combinations 30 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES Figure 13. Phase portraits of system (P.1) when γ2 = 1 and β22 − 4α2 > 0. when the singular points in the local chart U1 are both saddles. In the finite part we have z1 and z3 unstable nodes, z2 is a stable node and z4 is a saddle. So we have l1 , l2 , l4 > 0 and l3 < 0. In phase portrait 1 of Figure 13, consider the straight line joining the finite singular point z3 to the infinity singular point u1 as shows Figure 15. We can see that near the singular point z3 but on opposite sides, the vector field has the same direction, which contradicts Lemma 12. So the phase portrait 1 of Figure 13 is not realizable. With the same argument the portrait 3 of Figure 13 is also not realizable. So phase portrait 2 of Figure 13 is the only realizable and corresponds to phase portrait PL.1.1 of Figure 3. QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 31 Figure 14. Phase portraits of system (P.1) when γ2 = 1 and β22 − 4α2 > 0. Figure 15. The straight line joining the finite singular point z3 to the infinity singular point u1 in phase portrait 1 of Figure 13. 32 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES Considering the phase portraits 4 − 18 of Figure 13 we shall prove that 7 − 18 are not realizable. First consider the phase portrait 7 and the straight line joining the middle point between the infinity singular points u1 and u2 and the middle point between the finite singular points z3 and z4 as shows Figure 16. By Lemma 11 this line should have at most two points of contact with the vector field, which does not occur. In Figure 16 we can see at least four contact points, represented by the smaller points that are not singularities of the system. This fact guarantees that the ω−limit set of u2 is the finite point z4 on the parabola. So phase portraits 7 − 18 are not realizable using similar arguments. So among the phase portraits 4 − 18 only 4, 5 and 6 are realizable, which correspond, respectively to phase portraits PL.1.2, PL.1.3 and PL.1.4 of Figure 3. The values of the parameters that realize these systems can be found in Table 2. Figure 16. The straight line joining the middle point between the infinity singular points u1 and u2 and the middle point between the finite singular points z3 and z4 in phase portrait 7 of Figure 13. The phase portraits 19 − 20 in Figure 13 and 21 − 26 in Figure 14 are topologically equivalent to one of the phase portraits 1−18 in Figure 13 so they can be realizable or not, depends on their configuration. In Table 1 we present the relation among the equivalent phase portraits of system (P.1) when c 6= 0. In the case where they are topologically equivalent to a realizable phase portrait, we need not consider the study again. However if they are topologically equivalent to a phase portrait which was not realizable, we need to study it. Considering the same straight line used to prove the non-realization of phase portraits 7 − 18 of Figure 13 we apply Lemma 11 to conclude that 21, 22, 25 and 26 of Figure 14 are not realizable. The phase portraits 27 − 31 in Figure 14 present all the possibilities when there are four finite singular points and one infinite singular point on the local chart U1 . Phase portraits 27, 28 and 29 are realizable and correspond to phase portraits PL.1.5, PL.1.6 and PL.1.7 of Figure 3. The values of the parameters that realize these systems can be found in Table 2. Moreover 30 and 31 are topologically equivalent to one of these three phase portraits. Finally if there are four finite singular points and the local chart U1 has no singular point we get the phase portraits 32 − 36 in Figure 14. For phase portraits 32 and 33 of Figure 14 we consider the straight line x = z41 where the finite singualarity z4 is z4 = (z41 , z42 ), and apply Lemma 12 to see that they are not realizable (see Figure 17). Moreover the phase portraits 35 and 36 are topologically equivalent to the phase portrait 34 which is the only realizable phase portrait for this case and it is represented by PL.1.8 in Figure 3. The values of the parameters that realize this system can be found in Table 2. For β22 − 4α2 > 0 we consider the cases with three finite singular points. When z1 = z2 the origin is a saddle-node and there are ten possible phase portraits, namely 37 − 40 in Figure 14 and 41 − 46 in Figure 18. But since the nodal sector of the saddle node QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 33 Phase portrait Topologicaly equivalent to 19 2 20 6 21 12 22 9 23 2 24 6 25 12 26 9 30 29 31 29 35 34 36 34 60 50 Table 1. Table of relations among all the possible phase portraits of sys- tem (P.1) when c 6= 0. p Figure 17. The straight line x = z41 = −(β2 − β22 − 4α2 )/2 in phase portrait 32 of Figure 14. must have its orbits tangent to its separatrix, the phase portraits 37 and 38 in Figure 14 are not realizable. In other words the separatrices of the saddle-node z1 must be on the invariant parabola. With the same argument the phase portraits 41, 42, 45 and 46 of Figure 18 also are not realizable. So when z1 = z2 the realizable phase portraits are 39, 40, 43 and 44 of Figure 18, corresponding to PL.1.9, PL.1.10, PL.1.11 and PL.1.12, in Figure 3, respectively. The values of the parameters that realize these systems can be found in Table 2. When there are three finite singularities with z1 = z4 then by continuity we have the phase portraits 47, 48, 49 and 50 of Figure 18. All these for phase portraits are realizable and correspond, to PL.1.13, PL.1.14, PL.1.15 and PL.1.16 in Figure 3, respectively. The values of the parameters that realize these systems can be found in Table 2 For β22 − 4α2 = 0 there is another case with three finite singularities that correspond to the case z3 = z4 . Here we can have ten phase portraits, given by 51 − 60 in Figure 18. The phase portraits 51, 52 and 55 are realizable and corresponds, respectively, to PL.1.17, PL.1.18 and PL.1.19 in Figure 3. The values of the parameters that realize these systems can be found in Table 2. The phase portraits 53 and 54 are not realizable. The ideia again is to use Lemma 12 with the straight line joining the origin of the local chart U3 to the singular point u2 of the local chart U1 . By Figure 19 and this lemma the phase portraits 53 and 54 are not realizable. 34 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES Figure 18. Phase portraits 41 − 50 corresponds to phase portraits of sys- tem (P.1) when γ2 = 1 and β22 − 4α2 > 0; Phase portraits 51−60 corresponds to phase portraits of system (P.1) when γ2 = 1 and β22 − 4α2 = 0. Considering the phase portraits 56 and 57 we will show that they are not realizable. Take the straight line passing through the origin of the local chart U1 and the infinite singular point u1 = u2 (see Figure 20). The contact points on this straight line contradicts Lemma 12 so the phase portraits 56 and 57 are not realizable. About the phase portraits 58 and 59, considering the straight line passing through the points z1 and z3 we have Figure 21 QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 35 Figure 19. The straight line connecting the origin of the local chart U3 with the singular poin u2 of the local chart U1 in phase portrait 53 of Figure 18. that is a contradiction with Lemma 11. So they are not realizable. The phase portrait 60 is topologically equivalent to 50 of Figure 18. Figure 20. The straight line connecting the origin of the local chart U3 with the singular poin u1 = u2 of the local chart U1 in phase portrait 56 of Figure 18. Figure 21. The straight line passing through the points z1 and z3 in phase portrait 58 of Figure 18. If z3 = z4 and z1 = z2 we have the phase portraits 61, 62 and 63 of Figure 22. But using the straight line joining z1 and z3 as done in Figure 21 and applying Lemma 11 we see that 61 and 62 are not realizable. The phase portrait 63 is realizable and corresponds to PL.1.20 in Figure 3. The values of the parameters that realize this system can be found in Table 2. For β22 − 4α2 < 0 the points z3 and z4 are complex. The possible phase portraits are described by 64 − 72 of Figure 22. The phase portraits 64, 65, 68 and 71 are realizable and corresponds, respectively, to PL.1.21, PL.1.22, PL.1.23 and PL.1.24 of Figure 3. The values of the parameters that realize these systems can be found in Table 2. To prove that the phase portraits 66, 67, 69 and 70 are not realizable, it is enough to consider the straight line passing through the origin of the local chart U3 and the infinity singularity u1 = u2 of the local chart U1 (see Figure 23). This straight line generates a contradition with Lemma 12 so the phase portraits 66, 67, 69 and 70 are not realizable. 36 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES Figure 22. Phase portraits 61 − 63 corresponds to phase portraits of sys- tem (P.1) when γ2 = 1 and β22 − 4α2 = 0; Phase portraits 64−72 corresponds to phase portraits of system (P.1) when γ2 = 1 and β22 − 4α2 < 0. Figure 23. The straight line connecting the origin of the local chart U3 with the singular poin u1 = u2 in the local chart U1 in phase portrait 66 of Figure 22. To end the case γ2 = 1 we consider the case where there is only one finite singular point. Using Theorem 2.19 of [9] we can see that the point is a saddle, which generates phase portrait 72 of Figure 22 which corresponds to phase portrait PL.1.25 of Figure 4. The values of the parameters that realize this system can be found in Table 2. Now we consider the case γ2 = 0. The system is (45) ẋ = x(α2 + β2 x), ẏ = α1 (y − x2 ) + 2x(α2 x + β2 y). When α1 = 0 such system has a common factor so assume α1 = 1. By the change x = −X, y = Y it is enough to consider the case β2 ≥ 0. QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 37  Assuming β2 > 0. In the finite part the points z1 = (0, 0) and z2 = −α2 /β2 , (α2 /β2 )2 are the singular points and the system has an additional invariant straight line, given by f3 = x + α2 /β2 = 0. Defining l1 = α2 and l2 = 1 − 2α2 the eigenvalues associated to z1 are 1 and l1 , while the eigenvalues associated to z2 are −l1 and l2 . We assume l2 6= 0 (otherwise such system has a common factor and it is equivalent to a linear system). In the local chart U1 the unique singular point is u1 = (l2 /β2 , 0) and it is a saddle. In the local chart U2 the compactified system is u̇ = u((1 − 2α2 )u2 + (α2 − 1)v − β2 u), v̇ = v((1 − 2α2 )u2 − 2β2 u − v). The origin (0, 0) is a linearly zero singularity. Doing the blow up u = U V, v = V and rescaling by V we get the system U̇ = U (α2 + β2 U ), V̇ = V ((1 − 2α2 U 2 V ) − 2β2 U − 1). When V = 0 the singularities are u1 = (0, 0) and u2 = (−α2 /β2 , 0). The eigenvalues associated to u1 are −1 e l1 while the eigenvalues of u2 are −l1 e −l2 . The blowing down process is described in Figure 24 (1)-(4) according to the signs of l1 and l2 . When β2 = 0 the point z1 is the unique finite singular point, being a saddle or an unstable node depending on the sign of l1 . In the local chart U1 there is no singular point and the origin (0, 0) of U2 is linearly zero. To study such point we apply blow ups, in Figure 24 is described the blowing down (5) and (6). Summarizing the study done previously we get the local behaviour at origin of U2 : (1) β2 > 0, l1 > 0 and l2 > 0: the origin of U2 has two elliptic sectors; (2) β2 > 0, l1 > 0 and l2 < 0: the origin of U2 has two hyperbolic sectors; (3) β2 > 0, l1 < 0 and l2 > 0: the origin of U2 has two elliptic sectors; (4) β2 > 0, l1 = 0 and l2 > 0: the origin of U2 has two elliptic sectors. (5) β2 = 0, l1 > 0: the origin of U2 has two hyperbolic sectors; (6) β2 = 0, l1 < 0: the origin of U2 has two elliptic sectors; By continuity and the above analysis we conclude that the case (3) is topologically equivalent to case (1) and the cases (1), (2), (4), (5) and (6) correspond, respectively, to the phase portraits PL.1.26, PL.1.27, PL.1.28, PL.1.29 e PL.1.30 of Figure 4. Table 4 has the values of the parameters that realizes the phase portraits of system (P.1) System (P.2) with c 6= 0 has a Darboux invariant if γ2 6= 0, and it can be written as ẋ = β1 (x2 + c) + γ2 x(y + c), ẏ = 2(y + c)(β1 x + γ2 y). Note that if β1 = 0 such system has a common factor so we can assume β1 = 1. Applying the change of coordinates x = −X, y = Y and rescaling the time we can assume√γ2 > 0. 2 √ system has three finite singular points z1 = (−1/γ2 , 1/γ2 ), z2 = (− −c, −c) If c < 0 the and z3 = ( −c, −c). Otherwise, only z1 . Defining l1 = c 6= 0 and l2 = 1 + c γ22 the √eigenvalues associated √ to z1 are 2γ2 l1 and l2 /γ2 , the eigenvalues√associated to z2 are√−2 −c and −2(γ2 c + −c); the eigenvalues associated to z3 are 2 −c and −2(γ2 c − −c). So when c < 0 the point z3 exists and it is an unstable node. In the local chart U1 we have two singular points u1 = (0, 0) being a hyperbolic saddle and u2 = (−1/γ2 , 0) being a saddle-node. In the local chart U2 the origin is a stable node. When l2 = 0 then z1 = z3 is a semi-hyperbolic node and the infinity part does not change. Note that z1 is a saddle-node in this case. So by continuity and the reasoning above, if c > 0 we have phase portrait PL.2.1 of Figure 4 which is realizable with c = γ2 = 1. When c < 0 and l2 6= 0 the system has two possible phase portraits, also described 38 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES Figure 24. Blow down of system (P.1) when γ2 = 0. QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 39 γ2 β2 α2 α1 PL.1.1 1 1 1/8 1 PL.1.2 1 1 1/16 1/16 PL.1.3 by continuity PL.1.4 1 1 1/16 1/150 PL.1.5 1 1/2 3/64 1/32 PL.1.6 by continuity PL.1.7 1 1 -3/8 -1 PL.1.8 1 1 3/16 1/16 PL.1.9 1 1 1/16 0 PL.1.10 1 1 -1 0 PL.1.11 1 1 1/18 0 PL.1.12 1 1 3/16 0 PL.1.13 1 1 0 1 PL.1.14 1 1 0 -1/8 PL.1.15 1 1 0 -1/4 PL.1.16 1 1 0 -1 PL.1.17 1 1 1/4 1 PL.1.18 1 1 1/4 3/8 PL.1.19 1 1 1/4 1/4 PL.1.20 1 1 1/4 0 PL.1.21 1 1 2 5 PL.1.22 1 3 4 6 PL.1.23 1 1 9/8 2 PL.1.24 1 1 2 13/4 PL.1.25 1 1 2 0 PL.1.26 0 1 1/4 1 PL.1.27 0 1 3/2 1 PL.1.28 0 1 0 1 PL.1.29 0 0 1 1 PL.1.30 0 0 -1 1 Table 2. Table of values for the parameters of system (P.1). Figure 25 40 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES in Figure 4: PL.2.2 (realizable with c = −1/2 and γ2 = 1) and PL.2.3 (realizable with c = −2 and γ2 = 1). Finally if c < 0 and l2 = 0, we see that the line y + c = 0 is one of the separatix of the saddle-node. So the only possible phase picture is PL.2.4 (realizable with c = −1 and γ2 = 1). Phase portraits of systems (P.2) when c = 0 and they have a Darboux invari- ant. The differential system is ẋ = −β1 (y − x2 ) + β2 y + α2 x, ẏ = 2y(β2 x + α2 ). Since α2 6= 0 we take α2 = 1. Moreover doing the change of coordinates x = −X, y = Y we can assume β2 ≥ 0. The system has at most three finite singular points, namely, z1 = (0, 0) and z2 = (−1/β1 , 0) and z3 = (−1/β2 , 1/β22 ). The point z1 has eigenvalues 2 and 1, so it is an unstable node. On the other hand the topological type of z2 and z3 . . depends on the numbers l1 = β1 and l2 = β1 − β2 6= 0. Indeed the point z2 has eigenvalues −1 and 2l2 /l1 and z3 has the eigenvalues −1 and −2l2 /l1 . In the local chart U1 the system has u1 = (0, 0) as a singularity with eigenvalues −l1 . and −l3 , where l3 = β1 − 2β2 . In the local chart U2 the compactified system has the origin as a nilpotent singularity. This mean that the linear part of the system, evaluated in (0, 0), is not null but their eigenvalues are both equal to zero. To classify this type of singular point we use Theorem 3.5 of [9]. This result use two functions, F (u) = aM uM +o(uM ) and G(u) = bN uN +o(uN ), defined from the differential system. In short the caracterization is done using aM , bN and the natural numbers M, N . For the compactified system in the local chart U2 these functions are 2(β2 − 3β2 ) 5l3 2β2 l3 3 2l32 4 G(u) = − u + 2 u2 , F (u) = u + 3 u . l2 l2 l22 l2 So when l3 > 0 the origin (0, 0) is a saddle as in (b) of Figure 25. If l3 < 0 the origins consists of one hyperbolic and one elliptic sector as in (a) of Figure 25. By continuity, when l1 > 0 and l3 > 0 we have the phase portrait PL.2.5 of Figure 4 (realizable with β1 = 4 and β2 = 1). If l3 < 0 we have the phase portraits PL.2.6 (realizable with β1 = 3/2 and β2 = 1) and PL.2.7 (realizable with β1 = 1/2 and β2 = 1) of Figure 4. Now if l1 < 0 the only possibility is l3 < 0 and we have the phase portrait PL.2.8 (realizable with β1 = −1 and β2 = 1) of Figure 4. If l1 = 0 the point z2 goes to the infinity and collide with u1 becoming a saddle-node. Moreover l1 = 0 implies l3 < 0, so the origin of U2 has a hyperbolic and one elliptic sector. This case corresponds to phase portrait PL.2.9 of Figure 4, realizable with β1 = 0 and β2 = 1. If β2 = 0 the point z3 goes to the infinity and collide with the origin of U2 becoming (0, 0) a nilpotente saddle-node as (c) or (d) in Figure 25. Moreover the only possible phase portrait is given by PL.2.10 of Figure 4, realizable with β1 = 1 and β2 = 0). Finally when l3 = 0 then the infinity if filled of singular points, without special singu- larities and the corresponding phase portrait is PL.2.11 of Figure 4 (realizable with β1 = 2 and β2 = 1). Proposition 14 (LV). Each real polynomial differential system having two real lines that intersect at a single point and a third straight line having a Darboux invariant can be written, after an affine change of coordinates, as QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 41 (i) (LV.1) with α(q − β) 6= and Darboux invariant I9 (t, x, y) = eα(q−β) t y β xβ−q+r (y − x)−(β+r) , (ii) (LV.2) with c = q = 0, p 6= 0 and Darboux invariant I10 (t, x, y) = e−pt x y −r , (iii) (LV.2) with c = −1 and Darboux invariant I11 (t, x, y) = et y (y − 1)−1 , (iv) (LV.3) with α = −(β + 1), c β 6= 0 and Darboux invariant I12 (t, x, y) = e−c β t y (y + a x + c)−1 . Moreover there are 27 non-equivalent phase portraits in the Poincaré disc. They are in Figure 5. Proof of Proposition 14 (LV). Let f1 = x = 0, f2 = y = 0 be the two real straight lines intersecting in a point. Considering system (LV.1) the third line is f3 = y − x and the cofactors associated to f1 , f2 and f3 are, respectivelly, k1 = α + ry + βx, k2 = α + y(β − q + r) + qx and k3 = α + y(β − q + r) + βx. One solution for equation λ1 k1 + λ2 k2 + s = 0 is βλ1 (β + r)λ1 α(q − β)λ1 λ2 = , λ3 = − ,s= , β−q+r β−q+r β−q+r Taking λ1 = β − q + r we obtain the Darboux invariant I1 (t, x, y) = eα(q−β) t y β xβ−q+r (y − x)−(β+r) . Now we analize system (LV.2) that has f3 = y+c as the third invariant straigh line(remember that c = 0 or c = −1). Here the cofactors are k1 = p + qx + ry, k2 = y + c and k3 = y. If c = 0 then equation (8) has only one the solution q = 0, λ3 = −rλ1 − λ2 , s = −pλ1 . Taking λ1 = 1 we get the Darboux invariant I1 (t, x, y) = e−pt xy −r . Otherwise if c = −1 then the more general solution is λ1 = 0, λ3 = −λ2 , s = λ2 . Taking λ2 = 1 we obtain the Darboux invariant I1 (t, x, y) = et y(y − 1)−1 . The last case to be considered is system (LV.3) that has f3 = y + a x + c = 0 as the third straight line. The cofactors are k1 = −α(y + a x + c) − y, k2 = β(y + a x + c) + a x and k3 = β y − a α x. Solving equation (8) we get the solution α = −(β + 1), λ2 = −λ1 − λ2 , s = −c(λ1 + β(λ1 + λ2 )). Taking λ1 = 0 and λ2 = 1 then we obtain the Darboux invariant I1 (t, x, y) = e−c β t y (y + a x + c)−1 .  42 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES Phase portrait of systems (LV.1) when they have a Darboux invariant. Re- member that if system (LV.1) has a Darboux invariant then β − q 6= 0 and α 6= 0 so we can take α = 1 getting (46) ẋ = x(1 + β x + r y), ẏ = y(1 + q x + (β − q + r) y). Define l1 = (β − q)/(β − q + r), l2 = (β − q)/β and l3 = (β − q)/(β + r). The finite part presents at most four singularities • z1 = (0, 0) with eigenvalues both equal to 1; • z2 = (0, −1/(β − q + r)) with eigenvalues −1 and l1 ; • z3 = (−1/β, 0) with eigenvalues −1 and l2 ; • z4 = (−1/(β + r), −1/(β + r)) with eigenvalues −1 and −l3 . . In the local chart U1 the compactified system has two singular points, being u1 = (0, 0) with eigenvalues −β and −(β − q) and u2 = (1, 0) with eigenvalues β − q and −(β + r). Moreover in the local chart U2 the origin (0, 0) is a singular point with eigenvalues −(β −q) and −(β − q + r). Thus when one of the finite singularities goes to infinity, it collides with u1 , u2 , or the origin of the local chart U2 . When l1 , l2 and l3 are non-zero, the combinations between their signs generate the possible phase portraits of system (46). There are exactally three possible phase portraits, all of them described in Figure 5: LVL.1.1, realizable for β = 1, q = r = 0; LVL.1.2, realizable for β = 1, q = r = −2; LVL.1.3, realizable for β = 1, q = −r = 3/4. Now we consider the case β = −r 6= 0. Here only the point z4 goes to the infinity and collides with u2 making it a semi hyperbolic saddle-node. There are two possible phase portraits, given by LVL.1.4 of Figure 5 (realizable with β = 1, q = r = −1) and LVL.1.5 of Figure 5 (realizable with β = 2, q = 1, r = −2). The cases where z2 or z3 goes to the infinity generate phase portraits equivalent to the previous ones. Finally when two finite singular points go to the infinity (for example when β = −r and q = 0), then there is only one phase portrait, given by LVL.1.6 of Figure 5. This last phase portrait is realizable for β = 1, q = 0 and r = −1. Phase portraits of systems (LV.2) when they have a Darboux invariant. First we consider the case c = −1, when the system is given by ẋ = x(p + q x + r y), ẏ = y(y − 1). If q 6= 0 unless of the change x = X/q we can assume q = 1. Considering q = 1 and defining l1 = p, l2 = −(p + r) and l3 = r − 1 the system has at most four finite singular points, namely • z1 = (0, 0) with eigenvalues −1 and l1 ; • z2 = (0, 1) with eigenvalues 1 and −l2 ; • z3 = (−p, 0) with eigenvalues −1 and −l1 ; • z4 = (−p − r, 1) with eigenvalues 1 and l2 . In the local chart U2 the origin (0, 0) is a singularity with eigenvalues −1 and l3 . In the local chart U1 the sytem has two singularities if l3 6= 0: u1 = (0, 0) being a hyperbolic unstable node and u2 = (1/l3 , 0) with eigenvalues 1 and 1/l3 . Hence if l3 = 0 the point u2 collides with the origin of U2 making it a semi-hyperbolic singularity of type saddle node. By continuity and using all the possible combinations of the signs of l1 , l2 and l3 when q = 1 and l3 6= 0 we obtain the phase portraits LVL.2.1- LVL.2.7 of Figure 5. When l3 = 0, i.e., r = 1 has three possible phase portraits: LVL.2.8, LVL.2.9 and LVL.2.10 of Figure 5. The values of the parameters that realize these systems can be found in Table 3. QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 43 Now it remains to study the case q = 0. Note that since the system cannot have commom factors it follows that l1 and l2 are different from zero. When q = 0 both the finite part and the analyzes in the local chart U2 remain almost the same. The only difference in the finite part is that the singularities z3 and z4 go to infinity. However in the local chart U1 the compactified system is u̇ = −u((r − 1)u + (p + 1)v), v̇ = −v(pv + ru). So the origin is a linearly zero singular point if l3 6= 0 and we apply the blow up doing the change of coordinates u = U, v = U W . The new system is U̇ = −U 2 ((p + 1)W + r − 1), Ẇ = U W (W − 1). After eliminating the common factor U it remains two singular points on U = 0: u1 = (0, 0) with eigenvalues −1 and −l3 , and u2 = (0, 1) with eigenvalues 1 and l2 . Hence they are hyperbolic points and doing the blow down the origin of U2 has (for l3 6= 0) • two elliptic sectors if u1 is a saddle and u2 is a unstable node. This case corresponds to phase portrait LVL.2.11 of Figure 5; • two elliptic sectors if u1 is a stable node and u2 is a saddle. This case corresponds to phase portrait LVL.2.12 of Figure 5; • two parabolic sectors if u1 and u2 are both saddles and there is a saddle and a node as singular finite points. This case corresponds to phase portrait LVL.2.13 of Figure 5; • two parabolic sectors if u1 and u2 are both saddles and there are two nodes as singular finite points. This case corresponds to phase portrait LVL.2.14 of Figure 5; • six parabolic sectors if u1 and u2 are both saddles and there are two nodes as singular finite points. This case corresponds to phase portrait LVL.2.15 of Figure 5. The last possibility when c = −1 is q = 0 and l3 = 0. But when this happens the system has the infinity line v = 0 filled up of singular points. After eliminating the common factor v, in the local chart U1 the point u1 = (0, 0) is a singular point, with eigenvalues −l1 and l2 . In the local chart U2 , After eliminating the common factor v, the origin is a singularity. By continuity the possible phase portraits are LVL.16 and LVL.2.17 of Figure 5. In Table 3 we put the values of the parameters that realizes each one of the phase portraits described in Figure 5. Finally when c = 0 we get the differential system (47) ẋ = x2 , ẏ = y(p + r x), with p 6= 0. So we can take p = 1 and the system becomes a particular case of system (DL) of Theorem C. The global phase portraits of this system will be done in the proof of Proposition 16 and the correponding phase portraits of system (47) are described by DL.1, DL.2 and DL.3 of Figure 6. Phase portraits of systems (LV.3) when they have a Darboux invariant. When (LV.3) has a Darboux invariant the parameter α must be equal to −(β + 1) so the differ- ential system is ẋ = x(a x + β(y + a x + c) + c), ẏ = y(a x + β(y + a x + c)). In the finite part there are three singular points, namely z1 = (0, 0), z2 = (0, −c) and z3 = (−c/a, 0) (remember that a c 6= 0). Defining l1 = c β 6= 0 and l2 = c (β + 1) 6= 0, 44 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES q r p LVL.2.1 1 -1 1/2 LVL.2.2 1 2 1 LVL.2.3 1 -1 2 LVL.2.4 1 -1 1 LVL.2.5 1 2 -2 LVL.2.6 1 1/2 -1/2 LVL.2.7 1 0 0 LVL.2.8 1 1 1 LVL.2.9 1 1 -1/2 LVL.2.10 1 1 -1 LVL.2.11 0 -2 1 LVL.2.12 0 2 1 LVL.2.13 0 0 1 LVL.2.14 0 2 -1 LVL.2.15 0 3/4 -1/4 LVL.2.16 0 1 1 LVL.2.17 0 1 -1/2 Table 3. Table of values for the parameters of system (LV.2) when c = −1. then the eigenvalues of the z1 are l1 and l2 ; the eigenvalues of z2 are c and −l1 , and the eigenvalues associated to z3 are −c and −l2 . In the local chart U1 the compactified system becomes u̇ = −c u v, v̇ = −v(c v + β(u + c v + a) + a). Hence the line v = 0 is filled of singular points after eliminating the common factor v there are no singular points. The same happens in the local chart U2 . So by continuity the only possible phase portrait is LVL.3.1 of Figure 5, which is realizable for β = 1 and a = c = −1. Proposition 15 (RPL). Each real planar quadratic differential system with two parallel real straight lines and a third straight line having a Darboux invariant can be written, after an affine change of coordinates, as system (RP L) and it has the Darboux invariant I13 (t, x, y) = e2t (x + 1)(x − 1)−1 . Moreover there are 17 non-equivalent phase portraits in the Poincaré disc for this system. They are described in Figure 6. Proof. Let f1 = x + 1 = 0, f2 = x − 1 = 0 and f3 = y = 0 be the three invariant straight lines. The cofactors of f1 , f2 and f3 are, respectivelly, k1 = x − 1, k2 = x + 1, k3 = α + β x + γ y. With these cofactors equation (8) with s ∈ R \ {0} has two solutions, namely s1 = {γ = 0, s = 2λ1 + (β − α)λ3 , λ2 = −(λ1 + βλ3 )} s2 = {s = 2λ1 , λ2 = −λ1 , λ3 = 0}. Since the second solution s2 is more general we conclude that every quadratic system that has two real parallel straight lines and a third real straight line as invariant straight lines also has a Darboux invariant. Taking λ1 = 1 we get the invariant I16 (t, x, y) = e2t (x + 1)(x − 1)−1 . QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 45  Phase portraits of systems (RP L). Remember that the system is ẋ = x2 − 1, ẏ = y(α + β x + γ y). When γ 6= 0 we can take γ = 1 (indeed, just do the change x = X, y = Y /γ). So the system can present at most four finite singularities, namely, z1 = (−1, 0), z2 = (−1, β − α), z3 = (1, 0) and z4 = (1, −β − α). Define l1 = α − β and l2 = α + β. The eigenvalues associated to z1 are −2 and l1 while the eigenvalues associated to z2 are −2 and −l1 . Moreover z1 = z2 when l1 = 0. Analogously the eigenvalues of z3 are 2 and l2 , while the eigenvalues associated to z4 are 2 and −l2 , with z3 = z4 when l2 = 0. So in the finite part the system can have two, three or four singulaties, depending on the values of l1 and l2 . In the local chart U1 the compactified system has at most two singularities on the infinity line: u1 = (0, 0) and u2 = (1 − β, 0). Defining l3 = β − 1 we see that u1 = u2 when l3 = 0 and the topological type of these singularities depends on the sign of l3 . Indeed the eigenvalues associated to u1 are −1 and l3 while the associated to u2 are −1 and −l3 . In the local chart U2 we just need to check if the origin (0, 0) is a singularity, which is true. It is a node, with the two eigenvalues equal to −1. So considering γ 6= 0 and combining all the possibilities of the signs of l1 , l2 and l3 we obtain the phase portraits RPL.1–RPL.10 of Figure 6. In Table 3 we put the values of the parameters that realizes each one of the phase portraits described in Figure 6. If γ = 0 then z2 and z4 goes to the infinity and the compactified system in the local chart U2 becomes u̇ = (1 − β)u2 − αuv − v 2 , v̇ = −v(βu + αv). Note that when l3 = 0(β = 1) the line v = 0 is filled up of singular points, and when l3 6= 0 the origin (0, 0) is a linearly zero singularity. Considering this case first and applying the blow up u = U, v = U W and dividing by U we get the system (48) U̇ = −U (β + W 2 + αW − 1), Ẇ = W (W − 1)(W + 1). When U = 0 the singularities of (48) are u1 = (0, −1) with eigenvalues 2 and l1 , u2 = (0, 0) with eigenvalues −1 and −l3 , and u3 = (0, 1) with eigenvalues 2 and −l2 . After blow-down we get the local phase portraits of the origin of U2 which depend on the signs of l1 , l2 and l3 . Doing all the combinations the origin of U2 consists of: • two elliptic sectors and parabolic sectors, see phase portraits RPL.11 and RPL.12 of Figure 6; • two hyperbolic sectors and parabolic sectors, see phase portraits RPL.13 and RPL.14 of Figure 6; • six hyperbolic sectors, see phase portrait RPL.15 of Figure 6. Finally if we consider β = 1 and after eliminating the common factor v the origin of the local chart U2 is either a hyperbolic node or a hyperbolic saddle, described respectively by the phase portraits RPL.16 and RPL.17 of Figure 6. The Table 4 has the values of the parameters that realizes the phase portraits of Figure 6. Proposition 16 (DL). Each real planar quadratic differential system with a double real straight line and a third straight line having a Darboux invariant can be written, after an affine change of coordinates, as system (DL), with γ = 0 and α 6= 0, and the Darboux invariant is I14 (t, x, y) = e−α t y x−β . 46 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES α β γ RPL.1 -5/4 1/4 1 RPL.2 0 -1 1 RPL.3 -3 2 1 RPL.4 -2 1 1 RPL.5 0 1 1 RPL.6 -1/2 1/2 1 RPL.7 1/2 -1/2 1 RPL.8 -2 2 1 RPL.9 -1 1 1 RPL.10 0 0 1 RPL.11 -3 2 0 RPL.12 0 -1 0 RPL.13 -1 0 0 RPL.14 -1 2 0 RPL.15 -1/4 3/4 0 RPL.16 -2 1 0 RPL.17 0 1 0 Table 4. Table of values for the parameters of system (RP L). Moreover there are 3 non-equivalent phase portraits in the Poincaré disc for this systems. They are described in Figure 6. Proof. Let f1 = x = 0 be the double real invariant straight line. By the proof of Proposi- tion 6 we know that the second invariant straight line is f2 = y = 0. The cofactors of f1 and f2 are, respectivelly, k1 = x, k2 = α + β x + γ y. Equation (8) with s ∈ R \ {0} has only one solution γ = 0, s = −αλ2 , λ1 = −βλ2 . Taking λ2 = 1 and using this solution we get ẋ = x2 , ẏ = y(α + β x), with Darboux invariant I2 (t, x, y) = e−α t y x−β .  Phase portraits of systems (DL) when γ = 0 and α 6= 0. Since α 6= 0 we can take α = 1. The origin of the system is the only finite singularity, which is a saddle-node. For the infinity singularities we assume first that β − 1 6= 0. In the local chart U1 the origin is a saddle if β − 1 > 0, and a stable node if β − 1 < 0. In the chart U2 the system becomes u̇ = −u((β − 1)u + v), v̇ − v(βu + v), and the origin is a linearly zero singularity. Applying the blow up u = U, v = U W we get the system U̇ = −U 2 (β − 1 + W ), Ẇ = −U W, which after eliminating the common factor U has the origin as only singular point. If β − 1 > 0 the origin is a hyperbolic stable node and if β − 1 < 0 the origin is a saddle. After blow down we get the local phase portraits of the origin of U2 which depend on β. When β − 1 > 0 the origin has two elliptic sectors and parabolic sectors, see phase portrait DL.1 of Figure 6. If β − 1 < 0 then there are two hyperbolic sectors and parabolic ones, see phase portrait DL.2 of Figure 6. QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 47 When β = 1 the infinity is filled up of singular points and the origin in the local chart U2 is a hyperbolic stable node. The phase portrait of this case can be found and it is described by DL.3 of Figure 6. Proposition 17 (CPL). Each real planar quadratic differential system with two parallel complex straight line and a third straight line having a Darboux invariant can be written, after an affine change of coordinates, as system (CP L). A Darboux invariant is given by I15 (t, x, y) = et earctan(1/x) Moreover there are 6 non-equivalent phase portraits in the Poincaré disc for this system. They are described in Figure 7. Proof. Let f1 = x + i = 0, f2 = x − i = 0 be the two complex parallel straight lines. By the proof of Proposition 6 we know that the third invariant straight line is f3 = y = 0. The cofactors of f1 , f2 and f3 are, respectivelly, k1 = x − i, k2 = x + i, k3 = α + β x + γ y. The equation (8) with s ∈ R \ {0} has two solutions, namely s1 = {γ = 0, s = i(2λ1 + (β + iα)λ3 ), λ2 = −βλ3 − λ1 } s2 = {s = 2iλ1 , λ2 = −λ1 , λ3 = 0}. Using s2 (which is more general) we conclude that all systems with two parallel com- plex straight lines and a real straight line as invariants curves have a Darboux invariant. Moreover taking λ1 = −i/2 we get I2 (t, x, y) = et (x − i)i/2 (x + i)−i/2 . Using the polar form of the complex numbers it follows that (x − i)i/2 (x + i)−i/2 = earctan(1/x) so the Darboux invariant is I2 (t, x, y) = earctan(1/x)+t .  Phase portraits of systems (CP L). In [10] the authors already study the quadratic systems with f = x2 + 1 = 0 as invariant curve, given by ẋ = x2 + 1, ẏ = Q(x, y), with Q an arbitrary polynomial of degree 2. In our case we have Q(x, y) = y(α+β x+γ y). So the system studied here is a subcase of systems (V I) of the article [10]. In that article the study of those systems is divided in six cases and since we have the invariant straigh line y = 0 there are seven possible phase portraits. The case (V I.1) provides the phase portraits 1 and 2 of [10](Fig. 1), i.e. the phase portraits CPL.1 and CPL.2 of Figure 7; the case (V I.2) gives the phase portrait 6 of [10](Fig. 1), i.e. the phase portrait CPL.3 of Figure 7; the case (V I.4) generates the phase portraits 16 and 17 of [10](Fig. 1), i.e. the phase portraits CPL.4 and CPL.5 of Figure 7; the case (V I.5) gives the phase portrait 20 of [10](Fig. 1), i.e. the phase portrait CPL.6 of Figure 7. Finally the case (V I.6) provides the phase portrait 21 of [10](Fig. 1), i.e. the phase portrait CPL.7 of Figure 7. Proposition 18 (p). Each real planar quadratic differential system with two complex straight lines that intersects in a real point and a third straight line having a Darboux can be written, after an affine change of coordinates, as (i) (p.1) with α3 (β − 2β3 ) 6= 0 and Darboux invariant I16 (t, x, y) = eα3 (β−2β3 ) t e−2γ3 arctan(y/x) (x2 + y 2 )β3 y −β . (ii) (p.2) with c 6= 0, α = −1 and Darboux invariant I17 (t, x, y) = e− arctan(y/x)−c t 48 JAUME LLIBRE, REGILENE OLIVEIRA AND CAMILA RODRIGUES Moreover there are 5 non-equivalent phase portraits in the Poincaré disc for this system. They are described in Figure 7. Proof. Let f1 = x + iy = 0 and f2 = x − iy = 0 be the two complex straight lines that intersect at a real point. We have two systems, (p.1), with f3 = y, and (p.2) with f3 = y + a x + c. We shall do the calculations for (p.1), and for system (p.2) the computations are analogous. Consider system (p.1) the cofactors of f1 , f2 and f3 are, respecively, k1 = (1/2)(β x + 2γ3 y + 2α3 − i(β − 2β3 )y), (49) k2 = (1/2)(β x + 2γ3 y + 2α3 + i(β − 2β3 )y), k3 = α3 + β3 x + γ3 y. Solving equation (8) the most general solution is λ1 = β3 + iγ3 , λ2 = β3 − iγ3 , λ3 = −β, s = α3 (β − 2β3 ). Hence assuming α3 (β − 2β3 ) 6= 0 system (p.1) of Theorem C has the Darboux invariant (50) I2 (t, x, y) = eα3 (β−2β3 ) t y −β (x − iy)β3 −iγ3 (x + iy)β3 +iγ3 . Using the polar form of the complex numbers it follows that (x − i)i/2 (x − iy)β3 −iγ3 (x + iy)β3 +iγ3 = e−2γ3 arctan(y/x) (x2 + y 2 )β3 and we get the Darboux invariant I16 (t, x, y) = eα3 (β−2β3 ) t e−2γ3 arctan(y/x) (x2 + y 2 )β3 y −β For system (p.2) the third invariant straight line is f3 = y + a x + c with c 6= 0. In this case the system has a Darboux invariant if and only if α = −1, and with the same reasoning applied above we get the invariant I17 (t, x, y) = e− arctan(y/x)−c t . Phase portraits of systems (p.1) when they have a Darboux invariant. Since α3 6= 0 we can take α3 = 1. Systems (p.1) have at most two finite singularities, namely z1 = (0, 0) and z2 = (−2/β, 0). When β = 0 the point z2 goes to infinity. The point z1 is an unstable node and the eigenvalues associated to z2 are −1 and (β − 2β3 )/β. So the point z2 is either a stable node or a saddle. In the local chart U2 the origin is not a singularity for the compactfied system. In the local chart U1 the system compactified has only one infinity singularity u1 = (0, 0) with eigenvalues −β/2 and −(β − 2β3 )/2. Then if β(β −2β3 ) > 0, z2 is a saddle and u1 is a stable node and the only phase portrait is p.1.1 of Figure 7, realizable for β = 1, γ3 = 1 and β3 = −1/2. If β(β − 2β3 ) < 0, z2 is a stable node and u1 is a saddle and the corresponding phase portrait of this case is p.1.2 of Figure 7, realizable for β = 1, γ3 = 1 and β3 = 3/2. Finally if β = 0 then z2 goes to the infinity and u2 becomes a semi hyperbolic saddle-node generating the phase portrait p.1.3 of Figure 7, which is realizable for β = 0, γ3 = 1 and β3 = 2. Phase portraits of systems (p.2) when they have a Darboux invariant. In the local chart U1 system (p.2) becomes u̇ = −cv(u2 + 1), v̇ = −v(aβ + au + cuv + βcv + βu − 1). So the line v = 0 is filled up of singular points. The same happens in the local chart U2 . In the finite part the point (0, 0) is the only singularity, with complex eigenvalues. So the origin can be a node or a center. Both cases are described, respectively, by the QUADRATIC SYSTEMS AND DARBOUX INVARIANTS 49 phase portraits p.2.1 ,realizable with a = β = 1 and c = 2, and p.2.2, realizable with a = 1, β = 0 and c = 2, of Figure 7.  By the end we prove Theorem E. This result is about the differential systems having an invariant cubic but that do not have a Darboux invariant. Proof of Theorem E. First we consider systems of type (CE), i.e, the ones which has an invariant cubic of the form f = f1 f2 = 0 where f1 = x2 + y 2 + 1 and f2 = a x + b y + c. By Theorem C these systems can be written as ẋ = −(x2 + y 2 + 1) − 2α1 y(y + ax + c), ẏ = a(x2 + y 2 + 1) + 2α1 x(y + ax + c), with f1 = x2 +y 2 +1 and f2 = y+ax+c. The cofactors of f1 and f2 are k1 (x, y) = 2(a y+x) and k2 (x, y) = −2α1 (a y − x), respectively. So the cofactors have no constant terms, i.e, k1 (0, 0) = k2 (0, 0) = 0. The consequence of this is that equation (8) has no solution considering s 6= 0. Hence these systems do not have a Darboux invariant of the form es t f1λ1 f2λ2 . The proofs for other systems are very similar. In fact it suffices to observe that the cofactors of the invariant curves never have a constant term.  Acknowledgements The first author is partially supported by the Ministerio de Economı́a, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER) and grant MDM-2014-0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017- 777911. The second author is partially supported by Projeto Tem atico FAPESP number 2014/00304-2 and FAPESP grant number 2017/20854-5. The third author is partially supported by a Ph.D. CNPq Fellowship number 140292/2017-9 References [1] J.C. Artés, PhD thesis, Universitat Autónoma de Barcelona, Barcelona, 1990. Advisor J. Llibre. [2] R. Bix, Conics and Cubics. A concrete introduction of algebraic curves. UTM, Springer, 2006. [3] C. Bujac and N. Vulpe, Cubic systems with invariant straight lines of total multiplicity eight and with three distinct infinite singularities, Qual. Theory Dyn. Syst. 14 (2015), 109–137. [4] L. Cairó and J. 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Sotomayor and R. Paterlini, Quadratic Vector Fields with Finitely Many Periodic Orbits. IMPA. International Symposium on Dynamical Systems, 1983. [17] H. Zholadek On algebraic solutions of algebraic Pfaff equations. Studia Math. 114 (1995), 117–126. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, Avenida Trabalhador São-carlense, 400, 13.560-970, São Carlos, SP, Brazil. Email address: regilene@icmc.usp.br, camilaap@usp.br