This is a preprint of: “Phase portraits of a family of Kolmogorov systems with infinitely many singular points at infinty”, Erika Diz Pita, Jaume Llibre, M. Victoria Otero Espinar, Commun. Nonlinear Sci. Numer. Simul., vol. 104, 106038:1–16, 2022. DOI: [10.1016/j.cnsns.2021.106038] Phase portraits of a family of Kolmogorov systems with infinitely many singular points at infinty Érika Diz-Pitaa , Jaume Llibreb , M. Victoria Otero-Espinara a Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain b Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain Abstract We give the topological classification of the global phase portraits in the Poincaré disc of the Kolmogorov systems  ẋ = x a0 + c1 x + c2 z 2 + c3 z ,  ż = z c0 + c1 x + c2 z 2 + c3 z , which depend on five parameters and have infinitely many singular points at the infinity. We prove that these systems have 22 topologically distincs phase portraits. Keywords: Kolmogorov system, Phase portrait, Poincaré disc 1. Introduction and statement of the main results Kolmogorov systems are polynomial differential systems of the form ẋi = xi Pi (x1 , . . . , xn ) for i = 1, ..., n, where Pi are polynomials. These include, for instance, Lotka-Volterra or May-Leonard systems. Kolmogorov systems can be used for modelling problems from different sciences as the evolution of competing species [4, 9, 26, 27, 32], plasma physics [20], hydrodynamics [8], chemical reactions [18], the study of black holes [7], and economic [16, 17, 34] or social problems, as the evolution of the number of internet users [15]. Recently, some works on the global dynamics of these systems have been carried out. For example, for the May-Leonard systems, which have the form ẋ = x(1 − x − ay − bz), ẏ = y(1 − bx − y − az), ż = z(1 − ax − by − z), their global dynamics on the Poincaré sphere when a + b = 2 or a = b were studied in [5], and the case with a + b = −1 were studied in [6]. The global dynamics of some particular Lotka-Volterra systems on dimension three has also been described on the Poincaré sphere as in [1], where the authors give the global phase portraits of a system that appears in the study of black holes, or in [24] where the description of the global dynamics of a system previously proposed and studied in [21, 25, 33] is finally completed. Also the global study of some families depending on a small number of parameters has been done, as in [23] where the authors study a family depending on two parameters, or in [22] where the family depends on three parameters but with some restrictions as all of them must be positive. Email addresses: erikadiz.pita@usc.es (Érika Diz-Pita), jllibre@mat.uab.cat (Jaume Llibre), mvictoria.otero@usc.es (M. Victoria Otero-Espinar) Preprint submitted to Communications on Nonlinear Science and Numerical Simulation July 5, 2021 With the aim of studying the global dynamics of the general 3-dimensional Lotka-Volterra systems de- pending on 12 parameters, ẋ = x(a0 + a1 x + a2 y + a3 z), ẏ = y(b0 + b1 x + b2 y + b3 z), (1.1) ż = z(c0 + c1 x + c2 y + c3 z), in [11] and [12] the global dynamics of two families of Kolmogorov systems is studied. Those families are obtained from (1.1) assuming the existence of a rational first integral of degree two of the form xi y j z k , by applying the Darboux theory of integrability. For the obtained families, the condition that they have a Darboux invariant of the form x` y m est is required. Here we will focus on the family   ẋ = x a0 − µ(c1 x + c2 z 2 + c3 z) , ż = z c0 + c1 x + c2 z 2 + c3 z , (1.2) that depends on five parameters. This family is studied in [11] where the topological classification of the global phase portraits in the Poincaré disc is given for all the values of the parameters with µ 6= −1. The particular case with µ = −1, in which all the singular points at infinty are singular points was not studied, so here we carry out the study of this case, i.e. we study the systems   ẋ = x a0 + c1 x + c2 z 2 + c3 z , ż = z c0 + c1 x + c2 z 2 + c3 z , (1.3) under conditions  H = c2 6= 0, a0 ≥ 0, c1 ≥ 0, c3 ≥ 0, a0 6= c0 , a20 + c21 6= 0 . In Section 4 of [11] a set of conditions is given and it is proved that Kolmogorov systems (1.2) can be reduced to satisfy such conditions either using symmetries, or eliminating known phase portraits, or eliminating phase portraits with infinitely many finite singular points. For instance, it is not necessary to study the cases in which these systems can be reduced to Lotka-Volterra systems in dimension two, as the global qualitative dynamics of those systems has been completely studied in [31]. Asumming µ = −1 those conditions become the conditions H given above. Also from the mentioned results, if a0 = 0 we can consider c0 > 0, as for c0 < 0 the results can be obtained by symmetry. We give the topological classification of all global phase portraits of systems (1.3) on the Poincaré disc, and our main result is the following. Theorem 1.1. Kolmogorov systems (1.3) have 22 topologically distinct phase portraits in the Poincaré disc, given in Figure 1. In Section 2 we summarize the basic results and techniques that we need for proving Theorem 1.1. In Section 3 we give the classification of the local phase portraits of the finite singular points, in Section 4 we study the local phase portraits at the infinite singular points, and finally in Section 5 we prove Theorem 1.1. 2. Preliminaries 2.1. Poincaré compactification Our objective is to study the dynamics of system (1.3) in R2 adding the infinity and The Poincaré compactification allow us to study the orbits of a polynomial differential system near the infinity (see Chapter 5 of [14]). Consider a polynomialvector field X = (P (x, y), Q(x, y)) defined in R2 such that d = max {deg P, deg Q}. We call the sphere S2 = y ∈ R3 : y12 + y22 + y32 = 1 the Poincaré sphere. We will assume that our vector field is defined in the tangent plane to the Poincaré sphere at the point (0, 0, 1). We can define a vector field X on S2 \S1 by means of the differentials Df + and Df − of the central projections f + : R2 → S2 and f − : R2 → S2 . The obtained vector field is defined at all points in S1 except at those over the equator S1 , which correspond with the points at infinity of R2 , but we can extend it analytically 2 (R1) (R2) (R3) (R4) (R5) (R6) (R7) (R8) (R9) (R10) (R11) (R12) (R13) (R14) (R15) (R16) (R17) (R18) (R19) (R20) (R21) (R22) Figure 1: The topologically distincts phase portraits of systems (1.3) in the Poincaré disc. to a vector field ρ(X) defined on the closed Poincaré sphere multiplying by y3d . We say that ρ(X) is the Poincaré compactification of the vector field X on R2 . To obtain the expressions of  the vector field ρ(X) we use the local charts (Ui , φi ) and (Vi , ψi ), where Ui = y ∈ S2 : yi > 0 , Vi = y ∈ S2 : yi < 0 , φi : Ui −→ R2 and ψi : Vi −→ R2 for i = 1, 2, 3, with φi (y) = −ψi (y) = (ym /yi , yn /yi ) for m < n and m, n 6= i. The expression of ρ(X) in the local chart (U1 , φ1 ) is        1 u 1 u 1 u u̇ = v d −u P , +Q , , v̇ = −v d+1 P , , (2.1) v v v v v v and in the local chart (U2 , φ2 ) is        1 u 1 u 1 u u̇ = v d P , − uQ , , v̇ = −v d+1 P , . (2.2) v v v v v v We will not need the expression of ρ(X) in the local chart (U3 , φ3 ) and neither we need the explicit expressions of ρ(X) in the local charts (Vi , ψi ), with i = 1, 2, 3, as they can be obtained multiplying by (−1)d−1 the expressions in (Ui , φi ), so the behaviour will be the same as in those charts when d is even (the orientation should be reversed if d were odd). The singular points of ρ(X) which are over the equator are called the infinite singular points of X and if p ∈ S1 is an infinite singular point, then −p is also an infinite singular point with the same stability if d is odd and opposite stability if d is even. We will classify the global phase portraits of the Kolmogorov systems (1.3) in the Poincaré disc, which is the orthogonal projection of the closed northern hemisphere of S2 onto the plane y3 = 0. This is enough to 3 get the complete classification because the orbits of ρ(X) on S2 are symmetric with respect to the origin of R3 . 2.2. Topological equivalence and separatrix skeleton There are some important elements in a phase portrait that help us to carry out their topological clas- sification. Special attention should be paid to the separatrices, namely, the orbits at the infinity, the finite singular points, the orbits on the boundary of a hyperbolic sector at a singular point and the limit cycles. Each of the connected components resulting from removing all separatrices from D2 are called canonical regions. By the separatrix skeleton of π(ρ(X)) we mean the union of all the separatrices together with an orbit of each canonical region. In order to perform the topological classification, we recall the concept of topological equivalence. We say that two compactified polynomial differential systems in the Poincaré disc are topologically equivalent if there exists a homeomorphism that sends orbits to orbits preserving or reversing the orientation of all orbits. Similarly, we say that two separatrix skeletons Σ1 and Σ2 are topologically equivalent if there is a home- omorphism h : D2 → D2 such that h(Σ1 ) = Σ2 . To determine the topological classification of a polynomial differential system in the Poincaré disc it is enough to study the separatrix skeleton, as stated in the following result of Markus [28], Neumann [29] and Peixoto [30]. Theorem 2.1. The phase portraits in the Poincaré disc of two compactified polynomial vector fields π(ρ(X)) and π(ρ(Y )) with finitely many separatrices are topologically equivalent if and only if their separatrix skeletons are topologically equivalent. This result applies to vector fields with finitely many separatrices, so we could not apply it to systems 1.3 in the closed Poincaré disc as they have infinitely many singular points at the inifinity, and so infinitely many separatrices, but we can apply it in the open Poincaré disc. 2.3. Desingularization of singular points The first step to obtain the global phase portraits of our systems is to determine the local phase portraits at the singular points. For hyperbolic and semi-hyperbolic singular points we will use Theorem 2.15 and Theorem 2.19 in [14]. Whereas to study the singular points for which the Jacobian matrix is identically zero, called linearly zero points, we will need to go through a desingularization process, particularly we will use the blow up technique [2, 3]. Roughly speaking the blow up technique consist in explode the singular point to a line, and then study the new singular points that appear on that line. The process must be repeated if some of the new singular points are also linearly zero, but this is not a problem as it is proved by Dumortier in [13] that this iterative process is finite. Let P and Q be coprime polynomials, Pm and Qm homogeneous polynomials of degree m ∈ N and consider the differential system ẋ = P (x, y) = Pm (x, y) + o(m), (2.3) ẏ = Q(x, y) = Qm (x, y) + o(m), where o(m) represents terms of order higher than m. The origin of this system is a singular point as m > 0, and we say that it is nondicritical if the characteristic polynomial F(x, y) := xQm (x, y) − yPm (x, y) is not identically zero. We will only deal with nondicritical singular points on our study. If we consider a new variable z, we say that the mapping (x, y) → (x, z) = (x, y/x) is a homogeneous directional blow up in the vertical direction. This map transforms the origin of (2.3) into the line x = 0 and we call this line the exceptional exceptional divisor. After this blow up system (2.3) becomes Q(x, xz) − zP (x, xz) ẋ = P (x, xz), ż = . (2.4) x This system is always well-defined because we are assuming that the origin is a singular point. 4 Similarly, we say that the mapping (x, y) → (z, y) = (x/y, y) is a homogeneous directional blow up in the horizontal direction. In this case the exceptional divisor is the line y = 0 and the expression of the system is P (yz, y) − zQ(yz, y) ż = , ẋ = P (yz, y). (2.5) y After the blow up in the vertical or horizontal direction, we cancel an appearing common factor xm−1 . The mapping swaps the second and the third quadrants in the vertical directional blow up and the third and the fourth quadrants in the horizontal blow up.The relationship between the behaviour at the origin of system (2.3) and the behaviour at the new singular points of system (2.4) is provided by Propositions 2.1 and 2.2 in [2], so we must study the singular points of system (2.4) on the exceptional divisor and then apply these results and undo the blow ups to determine the behaviour of the orbits around the origin of system (2.3). We recall that if some of these singular points on the exceptional divisor is linearly zero we have to repeat the process. 2.4. Normally hyperbolic submanifolds Here we summarize a result that allows us to study systems which have a submanifold consisting of singular points, for more details see [10, 19]. If we have a smooth flow ϕt on a manifold M and C is a submanifold of M consisting entirely of singular points, we say that C is normally hyperbolic if the tangent bundle to M over C splits into three subbundles T C, E s and E u invariant under the flow and satisfying that dϕt contracts E s exponentially, dϕt expands E u exponentially and T C is the tangent bundle of C. For these submanifolds the following result holds: Theorem 2.2. Let C be a normally hyperbolic submanifold consisting of singular points for a flow ϕt . Then there exist smooth stable and unstable manifolds tangent along C to E s ⊕ T C and E u ⊕ T C respectively. Furthermore, both C and the stable and unstable manifolds are permanent under small perturbations of the flow. 3. Local study of the finite singular points From Section 5 in [11], if we consider the condition µ = −1, we get that the singular points of systems (1.3) are • P0 = (0, 0),     Rc − c3 Rc + c3 • P1 = 0, and P2 = 0, − if c23 > 4c0 c2 ; 2c2 2c2   c3 • P3 = 0, − if c23 = 4c0 c2 ; 2c2   a0 • P4 = − , 0 if c1 6= 0. c1 From Table 1 in [11], we get six cases depending on the coexistence of finite singular points, given in Table 1. Case Conditions Finite singular points 1 c23 > 4c0 c2 , c1 6= 0. P0 , P1 , P2 , P4 . 2 c23 > 4c0 c2 , c1 = 0, a0 6= 0. P0 , P1 , P2 . 3 c23 = 4c0 c2 , c1 6= 0. P0 , P3 , P4 . 4 c23 = 4c0 c2 , c1 = 0, a0 =6 0. P0 , P3 . 5 c23 < 4c0 c2 , c1 6= 0. P0 , P4 . 6 c23 < 4c0 c2 , c1 = 0, a0 =6 0. P0 . Table 1: The different cases for the finite singular points. 5 From Lemma 5.1 and Tables 2 to 7 in [11], asumming the condition µ = −1, we get the following classification for the local phase portraits of the finite singular points, with 34 subcases. Case 1: c23 > 4c0 c2 , c1 6= 0. Sub. Conditions Classification 1.1 a0 > 0, c0 = 0, c2 < 0. P0 ≡ P1 saddle-node, P2 saddle, P4 stable node. 1.2 a0 > 0, c0 = 0, c2 > 0. P0 ≡ P1 saddle-node, P2 unstable node, P4 stable node. 1.3 a0 = 0, c0 > 0, Rc − c3 < 0, c2 > 0. P0 ≡ P4 saddle-node, P1 stable node, P2 saddle. 1.4 a0 = 0, c0 > 0, c2 < 0, Rc − c3 > 0. P0 ≡ P4 saddle-node, P1 stable node, P2 stable node. 1.5 a0 > 0, c0 < 0, c2 < 0, a0 − c0 > 0, P0 saddle, P1 unstable node, P2 saddle, P4 stable (Rc − c3 ) < 0. node. 1.6 a0 > 0, c0 < 0, a0 − c0 > 0, c2 > 0, P0 saddle, P1 unstable node, P2 unstable node, P4 Rc − c3 > 0. stable node. 1.7 a0 > 0, c0 > 0, a0 − c0 > 0, P0 unstable node, P1 saddle, c2 < 0, Rc − c3 > 0. P2 saddle, P4 stable node. 1.8 a0 > 0, c0 > 0, Rc −c3 < 0, a0 −c0 > 0, P0 unstable node, P1 saddle, c2 > 0. P2 unstable node, P4 stable node. 1.9 a0 > 0, c0 > 0, a0 −c0 < 0, Rc −c3 < 0, P0 unstable node, P1 stable node, P2 saddle, P4 c2 > 0. saddle. 1.10 a0 > 0, c0 > 0, a0 − c0 < 0, c2 < 0, P0 unstable node, P1 stable node, P2 stable node, Rc − c3 > 0. P4 saddle. Table 2: Classification in case 1 of Table 1 according with the local phase portraits of finite singular points. Case 2: c23 > 4c0 c2 , c1 = 0, a0 > 0. Sub. Conditions Classification 2.1 c0 < 0, a0 −c0 > 0, Rc −c3 < 0, c2 < 0. P0 saddle, P1 unstable node, P2 saddle. 2.2 c0 < 0, a0 −c0 > 0, c2 > 0, Rc −c3 > 0. P0 saddle, P1 unstable node, P2 unstable node. 2.3 c0 > 0, c2 (a0 − c0 ) < 0, Rc − c3 > 0. P0 unstable node, P1 saddle, P2 saddle. 2.4 c0 > 0, Rc −c3 < 0, a0 −c0 > 0, c2 > 0. P0 unstable node, P1 saddle, P2 unstable node. 2.5 c0 > 0, a0 −c0 < 0, Rc −c3 < 0, c2 > 0. P0 unstable node, P1 stable node, P2 saddle. 2.6 c0 > 0, a0 −c0 < 0, c2 < 0, Rc −c3 > 0. P0 unstable node, P1 stable node, P2 stable node. 2.7 c0 = 0, a0 > 0, c2 < 0. P0 ≡ P1 saddle-node, P2 saddle. 2.8 c0 = 0, a0 > 0, c2 > 0. P0 ≡ P1 saddle-node, P2 unstable node. Table 3: Classification in case 2 of Table 1 according with the local phase portraits of finite singular points. Case 3: c23 = 4c0 c2 , c1 6= 0. Sub. Conditions Classification 3.1 a0 > 0, c0 < 0, a0 − c0 > 0. P0 saddle, P3 saddle-node, P4 stable node. 3.2 a0 > 0, c0 > 0, a0 − c0 < 0. P0 unstable node, P3 saddle-node , P4 saddle. 3.3 a0 > 0, c0 > 0, a0 − c0 > 0. P0 unstable node, P3 saddle-node , P4 stable node. 3.4 a0 = 0, c0 > 0. P0 ≡ P4 saddle-node, P3 saddle-node. 3.5 c0 = 0, a0 > 0, c2 < 0. P0 ≡ P3 topological saddle, P4 stable node. 3.6 c0 = 0, a0 > 0, c2 > 0. P0 ≡ P3 topological unstable node, P4 stable node. Table 4: Classification in case 3 of Table 1 according with the local phase portraits of finite singular points. 6 Case 4: c23 = 4c0 c2 , c1 = 0, a0 > 0. Sub. Conditions Classification 4.1 c0 < 0. P0 saddle, P3 saddle-node. 4.2 c0 > 0. P0 unstable node, P3 saddle-node. 4.3 c0 = 0, c2 < 0. P0 ≡ P3 topological saddle. 4.4 c0 = 0, c2 > 0. P0 ≡ P3 topological unstable node. Table 5: Classification in case 4 of Table 1 according with the local phase portraits of finite singular points. Case 5: c23 < 4c0 c2 , c1 6= 0. Sub. Conditions Classification 5.1 a0 = 0. P0 ≡ P4 saddle-node. 5.2 a0 > 0, c0 < 0, a0 − c0 > 0. P0 saddle, P4 stable node. 5.3 a0 > 0, c0 > 0, a0 − c0 < 0. P0 unstable node, P4 saddle. 5.4 a0 > 0, c0 > 0, a0 − c0 > 0. P0 unstable node, P4 stable node. Table 6: Classification in case 5 of Table 1 according with the local phase portraits of finite singular points. Case 6: c23 < 4c0 c2 , c1 = 0, a0 > 0. Sub. Conditions Classification 6.1 c0 < 0. Problemas de espacio P0 saddle. 6.2 c0 > 0.Problemas de espacio P0 unstable node. Table 7: Classification in case 6 of Table 1 according with the local phase portraits of finite singular points. 4. Local study at the infinite singular points In order to study the behaviour of the trajectories of systems (1.3) near infinity we consider the Poincaré compactification. We assume the hypothesis H. According to equations (2.1) and (2.2), we get the compact- ification in the local charts U1 and U2 respectively. To study all the infinite singular points, it is enough to study the singular points over v = 0 in the chart U1 and the origin of the chart U2 . We start with the study of the origin of the chart U2 , which is simpler. The systems in this chart have the expression u̇ = (a0 − c0 )uv 2 , v̇ = −c1 uv 2 − c0 v 3 − c3 v 2 − c2 v. (4.1) Over the line v = 0 we get u̇ |v=0 = v̇ |v=0 = 0, then all points at infinity are singular points, including the origin which is the only we must study in this chart. At the origin of this chart, one of the eigenvalues of the Jacobian matrix is zero and the other is −c2 . Applying Theorem 2.2 given in Subsection 2.4, we can conclude that if c2 > 0 there is exactly one orbit outside the infinity that goes to the origin of U2 and if c2 < 0 there is exactly one orbit from outside the infinity that leaves the origin of U2 . Now we address the study of the infinite singular points in the local chart U1 , where the expression of the systems is u̇ = (c0 − a0 )uv 2 , v̇ = −c2 u2 v − c3 uv 2 − a0 v 3 − c1 v 2 . (4.2) Taking v = 0 we get again that all points at infinity in this chart are singular points. At the origin, the eigenvalues of the Jacobian matrix are both zero. At a point (u0 , 0) with u0 6= 0 the eigenvalues are one zero and the other −c2 u20 so, if c2 > 0, then the nonzero eigenvalue is negative and exactly one orbit outside the infinity arrives at each infinite singular point on the chart U1 distinct from the origin. If c2 < 0 the nonzero eigenvalue is positive so from each inifinite singular point on the chart U1 distinct from the origin leaves exactly one orbit from outside the infinity. 7 From the previous reasonings we can state the following result. Lemma 4.1. For any infinite singular point of systems (1.3) distinct from the origin of the chart U1 the following statements hold: • If c2 > 0 exactly one orbit outside the infinity arrives to the singular point. • If c2 < 0 exactly one orbit from outside the infinity leaves the singular point. Now we must study insightfully the origin of the chart U1 , which we will name O1 . For this singular point we will prove the following result. Lemma 4.2. The origin of the chart U1 is an infinite singular point of systems (1.3) and it has 12 distinct local phase portraits described in Figure 2. v v v v v v u u u u u u L1 L2 L3 L4 L5 L6 v v v v v v u u u u u u L7 L8 L9 L10 L11 L12 Figure 2: Local phase portraits at the infinite singular point O1 . To prove Lemma 4.2, at first, we must eliminate a common factor v from systems (4.2) obtaining: u̇ = (c0 − a0 )uv, v̇ = −c2 u2 − c3 uv − a0 v 2 − c1 v. (4.3) Now we must study the origin of this system, which we name Õ1 , and which is now the only singular point over v = 0. The eigenvalues of the Jacobian matrix at Õ1 are zero and −c1 , so if c1 6= 0 the singular point of system (4.3) is semi-hyperbolic and we can study it applying Theorem 2.19 of [14]. If c1 = 0 we must study this singular point with the blow up technique. 4.1. Case with Õ1 semi-hyperbolic In order to apply the theorem for semi-hyperbolic singular points in [14], we must change the sign of the flow, as the result requires the nonzero eigenvalue to be positive. We obtain systems u̇ = (a0 − c0 )uv, v̇ = c2 u2 + c3 uv + a0 v 2 + c1 v, (4.4) and applying the mentioned result we get that the origin of these systems is a saddle point if c2 (a0 − c0 ) > 0 and a topological unstable node if c2 (a0 − c0 ) < 0. Then, reversing the orientations, the singular point Õ1 is a saddle if c2 (a0 − c0 ) > 0 and a topological stable node if c2 (a0 − c0 ) < 0. Before obtaining the corresponding phase portraits for O1 , we will distinguish four cases according to the sign of c2 and a0 − c0 . (a) If c2 > 0 and a0 − c0 > 0 then Õ1 is a saddle as in Figure 3(a). If we multiply by v to go back to systems (4.2), then all the points at the u-axis become singular points and the orientation of the orbits in the third and fourth quadrants is reversed. Thus, for O1 we obtain the local phase portrait in Figure 3(b), which is L1 of Figure 2. 8 v v u u (a) Origin of systems (4.3) (b) Origin of systems (4.2) Figure 3: Local phase portraits of Õ1 and O1 with c2 > 0 and a0 − c0 > 0. (b) If c2 < 0 and a0 − c0 < 0 then Õ1 is also a saddle, but with the sectors in a different position as in the previous case. Going back to systems (4.2) we obtain phase portrait L2 of Figure 2. (c) If c2 > 0 and a0 − c0 < 0 then Õ1 is a stable topological node as represented in Figure 4(a). If we multiply by v to go back to systems (4.2), then all the points of the u-axis become singular points and the orientation of the orbits in the third and fourth quadrants is reversed. We obtain the local phase portrait in Figure 4(b), which is L3 of Figure 2. v v u u (a) Origin of systems (4.3) (b) Origin of systems (4.2) Figure 4: Local phase portraits of Õ1 and O1 with c2 > 0 and a0 − c0 < 0. (d) If c2 < 0 and a0 − c0 > 0 then Õ1 is also a stable topological node, but with the sectors in a different position as in the previous case. Going back to systems (4.2) we obtain phase portrait L4 of Figure 2. 4.2. Case with Õ1 linearly zero In this subsection we study the local phase portrait of the origin of systems (4.3) assuming c1 = 0. In order to do that we use the blow up technique. First we note that the characteristic polynomial is F = −c2 u3 − c3 u2 v − c0 uv 2 , which can not be identically zero because c2 6= 0, so the singular point Õ1 is nondicritical. Now we introduce the variable w1 by means of the variable change uw1 = v, and we obtain the systems: u̇ = (c0 − a0 )u2 w1 , w˙1 = −c0 uw12 − c3 uw1 − c2 u. (4.5) We eliminate a common factor u so we get u̇ = (c0 − a0 )uw1 , w˙1 = −c0 w12 − c3 w1 − c2 . (4.6) We must study the singular points of these systems over the line u = 0, which are the points with the first coordinate zero and the second one a solution of the equation −c0 w12 − c3 w1 − c2 = 0. In the following we will distinguish several subcases. (A) If c0 = 0 and c3 = 0, then there are no singular points over the line u = 0, as c2 6= 0 by hypothesis. Subcase (A.1). If c2 > 0 we have for systems (4.6) the phase portrait given in Figure 5(a). If we multiply by u, then all the points over the w1 -axis become singular points, and the orbits on the second and third 9 quadrants reverse their orientation, so we get phase portrait in Figure 5(b). If we undo the blow up, contracting the exceptional divisor to the origin, and swapping the second and third quadrants, we obtain for systems (4.3) the phase portrait on Figure 5(c). Lastly, if we multiply by v we get the local phase portrait for O1 , the origin of systems (4.2), which has a line consisting of singular points, the u-axis, and it is the phase portrait L5 given in Figure 2. w1 w1 v u u u (a) (b) (c) Figure 5: Desingularization of the origin of systems (4.2) with c0 = c1 = c3 = 0 and c2 > 0. Subcase (A.2). If c2 < 0 the vertical blow up does not determine the phase portrait, it only give us the information that over the u-axis the flow is vertical and it goes in the positive sense. We must do a horizontal blow up to determine the phase portrait. In systems (4.3) we introduce the variable w2 by means of the change vw2 = u, and with the hypothesis of this case we get the systems w˙2 = c2 w23 v, v̇ = −c2 w22 v − a0 v 2 . (4.7) Eliminating a common factor v we obtain w˙2 = c2 w23 , v̇ = −c2 w22 − a0 v, (4.8) and for these systems the only singular point over the line v = 0 is the origin, which is semi-hyperbolic. Applying Theorem 2.19 in [14] we obtain that it is a stable topological node. The phase portrait around the origin for systems (4.8) is the one in Figure 6(a), and multiplying by v, the phase portrait for systems (4.7) is the one in Figure 6(b). Undoing the blow up, contracting the exceptional divisor into the origin and swapping the third and fourth quadrants, we get that in the first and second quadrants the orbits arrive to the origin tangent to the v-axis and in the third and fourth qudrants the orbits leave the origin tangent to the v-axis, this together with the information from the vertical blow up leads to the phase portrait in Figure 6(c). At last, if we multiply again by v we obtain the local phase portrait for O1 which is L6 of Figure 2. v v v w1 w 2 u (a) (b) (c) Figure 6: Desingularization of the origin of systems (4.2) with c0 = c1 = c3 = 0 and c2 < 0. (B) If c0 = 0 and c3 > 0 then we have the hyperbolic singular point Q1 = (0, −c2 /c3 ). At this point the eigenvalues of the Jacobian matrix are a0 c2 /c3 and −c3 so we have two subcases. Subcase (B.1). If c2 > 0 the vertical blow up does not determine the behaviour of the orbits around the v-axis in the second and fourth quadrants. Then we do a horizontal blow up introducing the variable 10 vw2 = u in systems (4.3): w˙2 = c2 w23 v + c3 w22 v, v̇ = −c2 w22 v + c3 w2 v 2 − a0 v 2 , (4.9) and we eliminate a common factor v: w˙2 = c2 w23 − c3 w22 , v̇ = −c2 w22 v − c3 w2 v − a0 v. (4.10) For systems (4.10) there are two singular points over the line v = 0, the origin which is a semi-hyperbolic saddle-node and the point (−c3 /c2 , 0) which is a saddle. The phase portrait around the w2 -axis for systems (4.10) is the one in Figure 7(a), and multiplying by v, the phase portrait for systems (4.9) is the one in Figure 7(b). Undoing the blow up we get the phase portrait in Figure 7(c). Finally, if we multiply again by v we obtain the local phase portrait for O1 which is L7 of Figure 2. v v v w1 w 1 u (a) (b) (c) Figure 7: Desingularization of the origin of systems (4.2) with c0 = c1 = 0, c3 > 0 and c2 < 0. Subcase (B.2). If c2 < 0 then Q1 is a stable node. We must do again a horizontal blow up to determine the local phase portrait, and thus we obtain again the phase portrait L6 of Figure 2. (C) If c0 6= 0, c3 = 0 and c0 c2 > 0, there are no singular points over u = 0. Subcase (C.1). If c0 and c2 are positive, we obtain again the phase portrait L5 of Figure 2 by undoing the blow up. Subcase (C.2). If c0 and c2 are negative, we obtain again the phase portrait L6 of Figure 2, but in this case it is necessary to do a horizontal blow up to conclude. (D) If cp 0 6= 0, c3 = 0 and c0 c2

0 and a0 − c0 > 0 then Q2 is a stable node and Q3 is an unstable node. Undoing the blow up we obtain phase portrait L6 of Figure 2. Subcase (D.2). If c0 > 0 and a0 − c0 < 0 then Q2 and Q3 are saddle points with the orientation of the hyperbolic orbits given in Figure 8(a). If we multiply by u and then we undo the blow up we obtain, respectively, the phase portraits in Figure 8(b) and (c). Multiplying by u again we obtain that the local phase portrait for O1 is L8 of Figure 2. w1 w1 v u u u (a) (b) (c) Figure 8: Desingularization of the origin of systems (4.2) with c3 = 0, c0 > 0, c2 < 0 and a0 − c0 < 0. 11 Subcase (D.3). If c0 < 0 and a0 − c0 > 0 then Q2 and Q3 are saddle points with a different orientation than in the previous case. Now the vertical blow up does not determine the behaviour of the orbits around the v-axis, so we must do a horizontal blow up introducing the variable w2 such that vw2 = u: w˙2 = c2 w23 v + c0 w2 v, v̇ = −c2 w22 v − a0 v 2 , (4.11) and we eliminate a common factor v: w˙2 = c2 w23 + c0 w2 , v̇ = −c2 w22 − a0 v. (4.12) For systems (4.12) there are p three singular points over the line v = 0, the origin which is a stable node, and two saddle points (± −c0 /c2 , 0). The phase portrait around the w2 -axis for systems (4.12) is the one in Figure 9(a). If we multiply by v we get the phase portrait in Figure 9(b) for systems (4.11). Blowing down we obtain the phase portrait in Figure 9(c). If we multiply again by v we obtain the local phase portrait for O1 which is L9 of Figure 2. v v v w1 w1 u (a) (b) (c) Figure 9: Desingularization of the origin of systems (4.2) with c3 = 0, c0 < 0, c2 > 0 and a0 − c0 > 0. (E) If c0 6= 0, c3 6= 0 and c23 − 4c0 c2 < 0 then there are no singular points over u = 0. Subcase (E.1). If c2 > 0 we obtain phase portrait L5 of Figure 2. Subcase (E.2). If c2 > 0 it is necessary to do a horizontal blow up and thus we obtain the phase portrait L6 of Figure 2. (F) If c0 6= 0, c3 6= 0pand c23 − 4c0 c2 > 0 then there are two hyperbolic p singular points over u = 0, namely, Q4 = (0, −(c3 + c23 − 4c0 c2 )/(2c0 )) and Q5 = (0, −(c3 − c23 − 4c0 c2 )/(2c0 )). Subcase (F.1). If a0 − c0 > 0, c0 > 0 and Rc − c3 > 0 then Q4 is an unstable node and Q5 a stable node. Blowing down we obtain phase portrait L6 of Figure 2. Subcase (F.2). If a0 − c0 > 0, c0 > 0 and Rc − c3 < 0 then Q4 is an unstable node and Q5 a saddle. Blowing down we obtain phase portrait L7 of Figure 2. Subcase (F.3). If a0 − c0 > 0, c0 < 0 and Rc − c3 > 0 then Q4 and Q5 are saddle points, but the vertical blow up does not determine the behaviour of the orbits around the v-axis. Doing a horizontal blow up we obtain phase portrait L9 of Figure 2. Subcase (F.4). If a0 − c0 > 0, c0 < 0 and Rc − c3 < 0 then Q4 is a saddle and Q5 is a stable node. Again the vertical blow up is not enough to determine the phase portrait. With a horizontal blow up we obtain phase portrait L10 of Figure 2. Subcase (F.5). If a0 − c0 < 0, c0 > 0 and Rc − c3 > 0 then Q4 and Q5 are saddle points. Blowing down we obtain phase portrait L8 of Figure 2. Subcase (F.6). If a0 − c0 < 0, c0 > 0 and Rc − c3 < 0 then Q4 is a saddle and Q5 is a stable node. Blowing down we obtain phase portrait L11 of Figure 2. 12 (G) If c0 6= 0, c3 6= 0 and c23 − 4c0 c2 = 0 then we have the singular point Q6 = (0, −c3 /(2c0 )), which is a semi-hyperbolic saddle-node. Attending to the position of the different sectors we have the following cases. Subcase (G.1). If c0 (a0 − c0 ) > 0 and c0 > 0, blowing down we obtain phase portrait L7 of Figure 2. Subcase (G.2). If c0 (a0 − c0 ) < 0 and c0 > 0, blowing down we obtain phase portrait L11 of Figure 2. Subcase (G.3). If c0 (a0 − c0 ) < 0 and c0 < 0 the vertical blow up does not determine the phase portrait of O1 . Doing a horizontal blow up we get the phase portrait L12 of Figure 2. 5. Global phase portraits In this section we bring together the local information obtained in Sections 3 and 4 to prove Theorem 1.1. In each case of Tables 2 to 7, the conditions determine the phase portrait of O1 , i.e., only one of the local phase portraits L1 to L12 in Figure 2 can appear in each case, except in case 4.2. In this case 4.2 we must distinguish two cases determined by the sign of a0 − c0 . Once determined the local phase portrait at the singular points, in most cases the place where born and die the separatrices is determined in a unique way. We draw these separatrices, one orbit in each canonical region which does not have an inifinite number of singular points in the boundary, and three orbits (representing the infinite number of them existing) in each canonical region with an infinite number of singular points in the boundary. Thus we obtain the global phase portraits in Figure 15. In Table 8 we indicate which is the global phase portrait obtained in each case from the ones included in Figure 15. Now we focus on the cases in which the separatrices can be connected in three different ways, namely 1.8, 1.9, 2.4, 2.5, 3.2 and 3.3. In the following we explain how it can be determined which of the three options is realizable. Case 1.8. By Theorem 4.8(1) in [11] we know that on any straight line z = cte 6= 0 there exists exactly one contact point. Two of the three global phase portraits obtained by connecting the separatrices contradict this result. In both cases, if we take the line z = (Rc − c3 )/2c2 we can find two contact points on it, one is the singular point P1 and the other is a point on the third quadrant, as shown in Figure 10. We can ensure the existence of this contact point as if we choose, for example, the point indicated with a square in Figure 10, the orbit passing through this point has as α-limit the origin and as ω-limit the singular point P4 , i. e. it is an orbit as the one drawn with dashed line, and that allow to prove the existence of the contact point indicated with a cross. The same occurs in the two phase portraits included in Figure 10, then we can conclude that these two phase portraits are no realizable, and there is only one global phase portrait in case 1.8, and it is the G8 as it is indicated in Table 8. Figure 10: Global phase portraits appearing on case 1.8 that are not realizable. Case 1.9. Here we can apply the same result than in the previous case to dismiss two global phase portraits. If we take a straight line z = z0 with −(Rc + c3 )/(2c2 ) < z0 < (Rc − c3 )/(2c2 ), then over this line there are two contact points, one in the third quadrant and other in the fourth quadrant, as shown in Figure 11. Thus the only global phase portrait in case 1.9 is the G9. 13 Figure 11: Global phase portraits appearing on case 1.9 that are not realizable. Case 2.4. In this case we obtain first three possible global phase portraits by applying Corollary 4.3 in [11], because it ensures that the phase portraits must be symmetric with respect to the z-axis. Otherwise we would have more possible phase portraits. Then, by Theorem 4.8(2) in [11], we know that there must exist two invariant straight lines z = (−c3 ± Rc )/(2c2 ), i.e., the lines z = cte passing through the singular points P1 and P2 . This is only possible in one of the three global phase portraits obtained by connecting the separatrices, namely in the G14 of Figure 15. In the other two global phase portraits these two invariant lines does not exist, as can be seen in Figure 12. Figure 12: Global phase portraits appearing on case 2.4 that are not realizable. Case 2.5. Here we can use the same arguments than in the previous case to prove that the only realizable phase portrait is the G15, so we will not give more details. Case 3.2. We must apply again Theorem 4.8(1) in [11]. In two of the three global phase portraits obtained connecting the separatrices, if we take the line z = −c3 /(2c2 ) we can find two contact points on it, one is the singular point P3 and the other is a point on the third quadrant, as shown in Figure 13. Thus, this two phase portraits are no realizable, and the only global phase portrait in case 3.2 is the G20, as indicated in Table 8. Figure 13: Global phase portraits appearing on case 3.2 that are not realizable. Case 3.3. The same arguments that in the previous case are valid here. In Figure 14 the line with two contact points is represented in the two global phase portraits that are indeed not realizable. The only possible phase portrait in this case is the G21 of Figure 15. 14 Figure 14: Global phase portraits appearing on case 3.3 that are not realizable. Case Subcase O1 Global 1.1 L4 G1 1.2 L1 G2 1.3 L3 G3 1.4 L2 G4 1.5 L4 G5 1.6 L1 G6 1.7 L4 G7 1.8 L1 G8 1.9 L3 G9 1.10 L2 G10 2.1 L10 G11 2.2 L9 G12 2.3 L6 G13 2.4 L7 G14 2.5 L11 G15 2.6 L8 G16 2.7 L6 G17 2.8 L7 G18 3.1 L4 G19 3.2 L3 G20 3.3 L1 G21 3.4 L3 G22 3.5 L4 G23 3.6 L1 G24 4.1 L12 G25 a0 − c0 > 0 L7 G26 4.2 a0 − c0 < 0 L11 G27 4.3 L6 G28 4.4 L5 G29 5.1 L3 G30 5.2 L4 G23 5.3 L3 G31 5.4 L1 G24 6.1 L6 G28 6.2 L5 G29 Table 8: Classification of global phase portraits of systems 1.3 15 (G1) (G2) (G3) (G4) (G5) (G6) (G7) (G8) (G9) (G10) (G11) (G12) (G13) (G14) (G15) (G16) (G17) (G18) (G19) (G20) (G21) (G22) (G23) (G24) (G25) (G26) (G27) (G28) (G29) (G30) (G31) Figure 15: Global phase portraits of systems (1.3) in the Poincaré disc. We have obtained 34 global phase portraits given in Figure 15, but some of them are topologically equivalent so we must study these equivalences. Note that in order to apply Theorem 2.1 and determine which of them are topologically equivalent by studying the separatrix skeleton we consider only the open Poincaré disc. We will distinguish classes of equivalence according to the following invariants: the number of finite singular points and the sum of the indices at the finite singular points, denoted by indF . This classification is given in Table 9. Then within each class we prove which of the phase portraits are topologically equivalent. 16 Class No finite singular points indF Global phase portraits 1 0 G5,G7,G9. 4 2 2 G6,G8,G10. 3 -1 G11,G13. 4 0 G1, G3, G19, G20. 5 3 1 G12,G14,G15. 6 2 G2,G4,G21. 7 3 G16. 8 -1 G17, G25. 9 0 G22, G23, G31. 2 10 1 G18, G26, G27. 11 2 G24. 12 -1 G28. 13 1 0 G30. 14 1 G29. Table 9: Classes of equivalence according to the number of finite singular points and to the indF . Class 1. Global phase portrait G5 is topologically equivalent to G7 because if we move the unstable node in G5 to the origin we obtain G7. G5 is topologically distinct to G9 as in G9 all the orbits arriving to the stable node come from the unstable node or from a unique infinite singular point while in G5 there are orbits that arrive to the stable node from infinitely many infinite singular points. Then in this class there are two topologically different phase portraits represented by G5 and G9. Class 2. G6 is topologically equivalent to G10 by moving the stable node in G6 to the origin, doing a symmetry with respect to the z-axis and doing a change of the time variable t by −t. G6 is different from G8 because in G6 there are four separatrices that connect inifinite singular points with finite singular points and in G8 there are five separatrices of this kind. In this class there are two topologically different phase portraits represented by G6 and G8. Class 3. G11 is topologically equivalent to G12 by moving the node in G11 to the origin. Then in this class there is only one topologically different phase portrait. Class 4. G1 is different from G3 as to the node in G1 arrive orbits from ininitely many infinite singular points but in G3 only from one infinite singular point. G1 is topologically equivalent to G19 by moving the saddle in G1 to the origin and doing a symmetry with respect to the x-axis. G1 is topologically equivalent to G20: we must move the saddle in G20 to the origin and the unstable node to the positive x-axis, and then the saddle-node to the origin and the saddle to the positive z-axis. Then we must do a symmetry with respect to the z- axis and a change of the time variable t by −t. In this class there are two topologically different phase portraits represented by G1 and G3. Class 5. G12 is topologically equivalent to G14 by moving the saddle in G14 to the origin. G14 is different from G15 because in both global phase portraits there are two nodes, but in G14 they have the same stability and in G15 one is stable and the other is unstable. In this class there are two topologically different phase portraits represented by G12 and G15. Class 6. G2 is different from G4 because in both phase portraits there are two nodes, but in G2 they have different stability and in G4 the same. G21 is different from G2 and G4 because in G2 and G4 there are 9 separatrices in the open Poincaré disc and in G21 there are 10 separatrices. Then in this class all the phase portraits are topologically different. Class 8. G17 is topologically equivalent to G25 by moving the saddle-node in G17 to the origin and doing a symmetry with respect to the x-axis. Then in this class there is only one topologically different phase portrait. Class 9. G22 is different from G23 because they have different kind of finite singular points. G23 is topologically equivalent to G31 by moving the node in G23 to the origin, doing a symmetry with respect to the z-axis and a change of the time variable t by −t. In this class there are two topologically different phase portraits represented by G22 and G23. 17 Class 10. G18 is topologically equivalent to G26 by moving the node in G18 to the origin and doing a symmetry with respect to the x-axis. G26 is different from G27 because in G26 there are a separatrix that connects two finite singular points and in G27 there is not a such separatrix. In this class there are two topologically different phase portraits represented by G18 and G27. Note that classes 7,11,12,13 and 14 have only one global phase portrait each of them. In summary, we have obtained 22 different phase portraits in the Poincaré disc for systems (1.3), so we have proved Theorem 1.1. This 22 phase portraits are described in Figure 1, where we include a representative of each one of the topological equivalence classes. These representatives correspond with the phase portraits in Figure 15 as follows: Rep. Phase portraits Rep. Phase portraits Rep. Phase portraits R1 G1, G19, G20. R9 G11, G13. R17 G23, G31. R2 G2. R10 G12, G14. R18 G24. R3 G3. R11 G15. R19 G27. R4 G4. R12 G16. R20 G28. R5 G5, G7. R13 G17, G25. R21 G29. R6 G6, G10. R14 G18, G26. R22 G30. R7 G8. R15 G21. R8 G9. R16 G22. Acknowledgements The first and third authors are partially supported by the Ministerio de Economía, Industria y Com- petitividad, Agencia Estatal de Investigación (Spain), grant MTM2016-79661-P (European FEDER support included, UE) and the Consellería de Educación, Universidade e Formación Profesional (Xunta de Gali- cia), grant ED431C 2019/10 with FEDER funds. The first author is also supported by the Ministerio de Educacion, Cultura y Deporte de España, contract FPU17/02125. 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