This is a preprint of: “Planar Kolmogorov systems with infinitely many singular points at infinity”, Erika Diz Pita, Jaume Llibre, M. Victoria Otero Espinar, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 32(5), 2250065, 2022. DOI: [10.1142/S0218127422500651] Planar Kolmogorov systems with infinitely many singular points at infinity Érika Diz-Pita Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, 15705, Santiago de Compostela, Spain erikadiz.pita@usc.es Jaume Llibre Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Spain jllibre@mat.uab.cat M. Victoria Otero-Espinar Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, 15705, Santiago de Compostela, Spain mvictoria.otero@usc.es Received (to be inserted by publisher) We classify the global dynamics of the five-parameter family of planar Kolmogorov systems ẏ = y (b0 + b1 yz + b2 y + b3 z) , ż = z (c0 + b1 yz + b2 y + b3 z) , which has been obtained from the Lotka-Volterra systems of dimension three. We give the topological classification of their phase portraits in the Poincaré disc, so we can describe the dynamics of these systems near infinity. We prove that these systems have 13 topologically distinct global phase portraits. Keywords: Kolmogorov system, Lotka–Volterra system, phase portrait, Poincaré disc. 1. Introduction Kolmogorov systems are differential systems of the form ẋi = xi Pi (x1 , . . . , xn ), i = 1, ..., n, where Pi are polynomials. Particular cases of these systems are, for example, Lotka-Volterra systems. All of them have been used for modelling problems from different sciences as the interaction between species [Arnoedo et al., 1980; Coste et al., 1979; Llibre & Xiao , 2014; Lois-Prados & Precup , 2020; Smale , 1976], plasma physics [Laval & Pellat , 1975], hydrodynamics [Busse , 1981], chemical reactions [Hering , 1990] or economic and social problems [Gandolfo , 2009, 2008; Wijeratne et al., 2009]. 1 2 For the Lotka-Volterra systems in dimension three the global dynamics has been described in some particular cases. In [?] the authors give the global phase portraits in the Poincaré disc of a system related with the study of black holes; in [Llibre & Martínez , 2020] the authors complete the description of the global dynamics of a system previously proposed and studied in [Leach & Miritzis , 2006; Llibre & Valls , 2011; Tudoran & Girban , 2012]. There are also some works about the global dynamics of certain Lotka- Volterra families depending on a small number of parameters. In [Llibre & Martínez , 2020] the authors study a family depending on two parameters, and in [Llibre & Martínez , 2020] the family studied depends on three parameters, but with some restrictions such as all of them must be positive. There are few works that study these kind of systems when they have a line consisting of singular points; an example can be found in [Schlomiuk & Vulpe , 2008]. In [Diz-Pita et al., 2021a] and [Diz-Pita et al., 2021b] the global dynamics of two Kolmogorov families in dimension two was studied. Those families are obtained from general 3-dimensional Lotka-Volterra systems depending on 12 parameters, ẋ = x ( a0 + a1 x + a2 y + a3 z ), ẏ = y ( b0 + b1 x + b2 y + b3 z ), ż = z ( c0 + c1 x + c2 y + c3 z ), with 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 est y λ1 z λ2 is required. In this work we focus on the systems studied in [Diz-Pita et al., 2021b] which are ẏ = y (b0 + b1 yz + b2 y + b3 z) , (1) ż = z (c0 − µ(b1 yz + b2 y + b3 z)) , and depend on six parameters. In [Diz-Pita et al., 2021b] the authors give the topological classification of the global phase portraits in the Poincaré disc for all the values of the parameters such that µ 6= −1. The particular case with µ = −1, in which there exists a line of singular points (more precisely, all the infinity consist on singular points) was not studied, so here we carry out the study of this case, i.e. we deal with the systems ẏ = y (b0 + b1 yz + b2 y + b3 z) , (2) ż = z (c0 + b1 yz + b2 y + b3 z) . In this paper we study the global dynamics of systems (2) and we give the topological classification of all their global phase portraits on the Poincaré disc. Our main result is the following. Theorem 1. Kolmogorov systems (2) have 13 topologically distinct phase portraits in the Poincaré disc, given in Figure 1. In order to give a detailed proof of Theorem 1, in Section 2 we give some definitions and results that will be useful. In Sections 3 and 4 we give, respectively, the classification of the local phase portraits of the finite and infinite singular points, and in Section 5 we study the global phase portraits on the Poincaré disc to prove Theorem 1. 2. Preliminaries We shall study systems (2) and it will suffice to do so under the conditions given in the following result. Proposition 1. To determine all global phase portraits of systems (2) it is sufficient to study those systems whose parameters satisfy the following conditions:  H = b1 6= 0, c0 − b0 6= 0, b0 ≥ 0, b2 ≥ 0, b3 ≥ 0, b23 + c20 6= 0, b22 + b20 6= 0 . If b2 b3 = 0 then it is enough to study the case with b1 > 0, and if b0 = 0 it is enough to consider c0 > 0. 3 (R1) (R2) (R3) (R4) (R5) (R6) (R7) (R8) (R9) (R10) (R11) (R12) (R13) Fig. 1. The topologically distinct phase portraits of systems (2) in the Poincaré disc. Proof. The proof follows from Propositions 1 and 2, and Corollary 4.1 of [Diz-Pita et al., 2021b]. In those results it is proved that Kolmogorov systems (1) can be reduced to satisfy conditions  H̃ = b1 6= 0, b0 µ + c0 6= 0, b0 ≥ 0, b2 ≥ 0, b3 ≥ 0, (µb3 )2 + c20 6= 0, b22 + b20 6= 0 , either using symmetries, or eliminating known phase portraits, or eliminating phase portraits with infinitely many finite singular points. Asumming µ = −1 the conditions H̃ become the conditions H given above. Also it was proved in the mentioned results of [Diz-Pita et al., 2021b] that if b2 b3 = 0 then it is enough to study the case with b1 > 0, as case b1 < 0 can be reduced to this one by symmetry. Similarly, in the case with b0 = 0 it is enough to consider c0 > 0.  As we want to study the global dynamics of systems (2) we must determine the behaviour of the orbits near the infinity. In order3 to 2do that we will use the Poincaré compactification. We call S = y ∈ R : y1 + y2 + y32 = 1 the Poincaré sphere, and we will consider a polynomial vector 2 2 field defined in its tangent plane at the point (0, 0, 1). Let that field be X = (P (x, y), Q(x, y)) and d the maximum of the degrees of the polynomials P and Q. We can obtain another vector field X on S2 \S1 by means of the differentials Df + and Df − of the central projections. Although X is not defined on the equator S1 , which corresponds with the points of the infinity of R2 , multiplying by y3d we can extended it analytically to another vector field ρ(X) defined on the closed Poincaré sphere. We say that ρ(X) is the Poincaré compactification of the vector field X on R2 . We will work with the expressions of the Poincaré compactification in the local charts (Ui , φi ) and (Vi , ψi ) where, for i = 1, 2, 3:   Ui = w ∈ S2 : wi > 0 , φi : Ui −→ R2 , Vi = w ∈ S2 : wi < 0 , ψi : Vi −→ R2 , and φi (w) = ψi (w) = (wm /wi , wn /wi ) for m < n and m, n 6= i. As the field ρ(X) on S2 is symmetric with respect to the origin of R3 , it will be enough to study the orbits on the closed northern hemisphere of S2 , which we will project onto the plane y3 = 0 by means of the orthogonal projection, so we will draw the global phase portraits in the so called Poincaré disc, denoted by D2 . 4 We can cover all the Poincaré disc with the charts U1 , U2 , V1 and V2 so it will be no necessary to study the expressions of the field on U3 and V3 . Also, as the expression for ρ(X) in the local charts (Vi , ψi ), with i = 1, 2, can be obtained multiplying by (−1)d−1 the expression in (Ui , φi ), it will be enough to study the Poincaré compactification in U1 and U2 . The expression of ρ(X) in the local chart (U1 , φ1 ) is        d 1 u 1 u d+1 1 u u̇ = v −u P , +Q , , v̇ = −v P , , (3) v v v v v v and in the local chart (U2 , φ2 ) is        d 1 u 1 u d+1 1 u u̇ = v P , − uQ , , v̇ = −v P , . (4) v v v v v v The Poincaré compactification will allow to study the infinite singular points of X, which are the singular points of ρ(X) over boundary of the Poincaré disc. Note that if we have a singular point p ∈ S1 then the opposite −p is also a singular point and it has the same stability if d is odd and opposite stability if d is even. We note that the points at the infinity in the local charts Ui and Vi for i = 1, 2 have coordinates (u, 0). For more details about the Poincaré compactification see Chapter 5 of [Dumortier et al., 2006]. To draw and classify the phase portraits on Poincaré disc, we have to pay special attention to the separatrices, i. e., the orbits at the infinity, the singular points (finite and infinite), the orbits on the boundary of a hyperbolic sector at a singular point and the limit cycles. We call canonical regions to each one of the connected components resulting from removing all the separatrices from D2 , and separatrix skeleton of π(ρ(X)) to the union of all the separatrices together with an orbit of each canonical region. We recall that two polynomial vector fields X1 and X2 are topologically equivalent if there exists a homeomorphism on the Poincaré disc that sends orbits of X1 to orbits of X2 , preserving or reversing the orientation of all the orbits, and it also preserves the infinity. The same definition is applicable to separatrix skeletons. The following result of Markus [Markus , 1954], Neumann [Neumann , 1975] and Peixoto [Peixoto , 1973] allows to study only the separatrix skeletons to determine the topological classification of a polynomial differential system in the Poincaré disc. Theorem 2. The phase portraits in the Poincaré disc of two compactified polynomial vector fields π(ρ(X1 )) and π(ρ(X2 )) with finitely many separatrices are topologically equivalent if and only if their separatrix skeletons are topologically equivalent. Although this result can be applied only to vector fields with finitely many separatrices, and this will not be the case of systems (2), we can apply it to those systems in the open Poincaré disc. If two phase portraits are topologically distinct in the open Poincaré disc, they will be distinct in the closed disc and if two phase portraits are topologically equivalent in the open Poincaré disc, they will be still equivalent if we add the boundary filled of singular points and consider the closed Poincaré disc. As we have already mentioned systems (2) will have an infinite number of singular points, namely all points at infinity. To study those singular points which form a continuum we will need the following result, which can be found in [Devaney , 1978; Hirsch , 1977]. Let ϕt be a smooth flow on a manifold M , and consider a submanifold C consisting entirely of singular points. The submanifold C is said 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 (respectively, expands) E s (respectively, E u ) exponentially and T C is the tangent bundle of C. Theorem 3. 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. 5 3. Local phase portraits of the finite singular points Asumming the condition µ = −1, from Section 5 in [Diz-Pita et al., 2021b] we know that the singular points of systems (2) are     c0 b0 P0 = (0, 0), P1 = 0, − if b3 6= 0 and P2 = − , 0 if b2 6= 0, b3 b2 and from Table 1 of [Diz-Pita et al., 2021b], we distinguish four cases depending on the existence of the singular points. These cases are given in Table 1. Table 1. The different cases for the finite singular points. Case Conditions Finite singular points 1 b3 6= 0, b2 6= 0. P0 , P1 , P2 . 2 b3 6= 0, b2 = 0, b0 6= 0. P0 , P1 . 3 b3 = 0, c0 6= 0, b2 =6 0. P0 , P2 . 4 b3 = 0, c0 6= 0, b2 = 0, b0 6= 0. P0 . Also from [Diz-Pita et al., 2021b], taking µ = −1 in Lemma 1 and Tables 2 to 5, we get the following local classification in 15 subcases for the finite singular points. Table 2. Classification of the local phase portraits of the finite singular points of case 1 of Table 1. Case 1: b3 6= 0, b2 6= 0. Sub. Conditions Classification 1.1 b0 > 0, c0 < 0, c0 − b0 < 0. P0 saddle, P1 unstable node, P2 stable node. 1.2 b0 > 0, c0 > 0, c0 − b0 < 0. P0 unstable node, P1 saddle, P2 stable node. 1.3 b0 > 0, c0 > 0, c0 − b0 > 0. P0 unstable node, P1 stable node, P2 saddle. 1.4 c0 = 0, b0 > 0. P0 ≡ P1 saddle-node, P2 stable node. 1.5 b0 = 0, c0 > 0. P0 ≡ P2 saddle-node, P1 stable node. Table 3. Classification of the local phase portraits of the finite singular points of case 2 of Table 1. Case 2: b3 6= 0, b2 = 0, b0 6= 0. Sub. Conditions Classification 2.1 b0 > 0, c0 < 0, c0 − b0 < 0. P0 saddle, P1 unstable node. 2.2 b0 > 0, c0 > 0, c0 − b0 < 0. P0 unstable node, P1 saddle. 2.3 b0 > 0, c0 > 0, c0 − b0 > 0. P0 unstable node, P1 stable node. 2.4 c0 = 0, b0 > 0. P0 ≡ P1 saddle-node. Table 4. Classification of the local phase portraits of the finite singular points of case 3 of Table 1. Case 3: b3 = 0, c0 6= 0, b2 6= 0. Sub. Conditions Classification 3.1 b0 > 0, c0 < 0, c0 − b0 < 0. P0 saddle, P2 stable node. 3.2 b0 > 0, c0 > 0, c0 − b0 > 0. P0 unstable node, P2 saddle. 3.3 b0 > 0, c0 > 0, c0 − b0 < 0. P0 unstable node, P2 stable node. 3.4 b0 = 0, c0 > 0. P0 ≡ P2 saddle-node. 6 Table 5. Classification of the local phase portraits of the finite singular points of case 4 of Table 1. Case 4: b3 = 0, c0 6= 0, b2 = 0, b0 6= 0. Sub. Conditions Classification 4.1 b0 > 0, c0 < 0. P0 saddle. 4.2 b0 > 0, c0 > 0. P0 unstable node. 4. Local phase portraits at the infinite singular points Here we study the local phase portrait at the infinite singular points, and as it was said previously, we work under the hypothesis H. The expression of the Poincaré compactification of systems (2) in the local chart U1 according to equations (3) is u̇ = (c0 − b0 )uv 2 , (5) v̇ = −b3 uv 2 − b0 v 3 − b1 uv − b2 v 2 . In the chart U2 according to equations (4) the expression is u̇ = (b0 − c0 )uv 2 , (6) v̇ = −b2 uv 2 − c0 v 3 − b1 uv + b3 v 2 . We want to study all the points at the infinity, which correspond with the line v = 0 of these systems. To do that it is enough to study the singular points over v = 0 in the chart U1 and the origin of the chart U2 . We easily check in system (5) that all points over the line v = 0 are singular points. The eigenvalues of the Jacobian matrix at these singular points are both zero at the origin and at any other point (u0 , 0) the eigenvalues are zero and −b1 u0 . If b1 > 0 (respectively, b1 < 0), the nonzero eigenvalue is positive (respectively, negative) at the points on the negative u-axis, which correspond with the infinite singular points at the second and fourth quadrants of the Poincaré disc; the nonzero eigenvalue is negative for the infinite points at the first and third quadrants on the Poincaré disc (respectively, positive). Then, by Theorem 3 we get the following result: Lemma 1. For all the infinite singular point of systems (2) distinct from the origin of the charts U1 and U2 the following statements hold. • If b1 > 0, to the points on the first and third quadrants arrives exactly one orbit from outside the infinity, and from the points on the second and fourth quadrants leaves exactly one orbit outside the infinity. • If b1 < 0, from the points on the first and third quadrants leaves exactly one orbit outside the infinity, and to the points on the second and fourth quadrants arrives exactly one orbit from outside the infinity. As the two eigenvalues at the origin of systems (5) are zero, we must eliminate a common factor v from these systems and then study the singular points over the line v = 0. We do that on Subsection 4.1 and there we prove Theorem 4. The same occurs with the origin of the chart U2 , as the origin of systems (6) is a singular point and the eigenvalues of the Jacobian matrix at that point are both zero. We study this point in Subsection 4.2 proving Theorem 5. Note that Theorem (4) and Theorem (5) determine the local phase portrait at the origin of the charts U1 and U2 in the Poincaré disc, but also at the origins of charts V1 and V2 . Theorem 4. The origin of systems (5) is a singular point and it has 3 topologically distinct local phase portraits, which taking into account the position of the sectors and orientation of the orbits give raise to the 8 phase portraits described in Figure 2. 7 v v v v u u u u L11 L12 L13 L14 v v v v u u u u L15 L16 L17 L18 Fig. 2. Local phase portraits at the infinite singular point O1 . Remark 4.1. Note that phase portraits L11 to L14 correspond to the first equivalence class, L15 and L16 to the second class, and L17 and L18 to the third class. Theorem 5. The origin of the chart U2 is an infinite singular point of systems (2) and it has 3 topologically distinct local phase portraits, which taking into account the position of the sectors and orientation of the orbits give raise to the 10 phase portraits described in Figure 3. v v v v u u u u L21 L22 L23 L24 v v v v v v u u u u u u L25 L26 L27 L28 L29 L210 Fig. 3. Local phase portraits at the infinite singular point O2 . Remark 4.2. Note that phase portraits L11 to L14 correspond to the first equivalence class, L15 to L18 to the second class, and L19 and L110 to the third class. 4.1. Study of the origin of the chart U1 To study the origin of the chart U1 , first we eliminate a common factor v from systems (5) obtaining: u̇ = (c0 − b0 )uv, (7) v̇ = −b3 uv − b0 v 2 − b1 u − b2 v. The only singular point of these systems over v = 0 is the origin, and the eigenvalues of the Jacobian matrix at that point are zero and −b2 . Then this singular point can be semi-hyperbolic or nilpotent. Semi-hyperbolic case. If b2 6= 0 then the origin of systems (7) is semi-hyperbolic so its phase portrait can be determined by Theorem 2.19 in [Dumortier et al., 2006], concluding that it is always a saddle-node. 8 In order to determine its local phase portrait it will be necessary to know the position of the different sectors and the orientation of the orbits in the saddle-node, so we must determine these depending on the parameters. If b1 > 0, c0 − b0 > 0 and b0 = 0, then by the information given by the theorem and the sense of the flow in the different regions, the position of the sectors of the saddle-node and the orientation of the orbits for systems (7) is the one given in Figure 4(a). To obtain the local phase portrait of the origin of the chart U1 we must multiply by v, so that all the points over the v-axis become singular points and the orbits on the third and fourth quadrants reverse their orientation. Thus we obtain the phase portrait of Figure 4(b), which is also L11 of Figure 2. v v u u (a) (b) Fig. 4. Local phase portraits of the origins of systems (7) and (5) with b1 > 0 c0 − b0 > 0 and b0 = 0. If b1 > 0, c0 − b0 > 0 and b0 > 0, the fact that the parameter b0 is nonzero makes that systems (7) have a singular point on the negative v-axis, so that affects the phase portrait but not in a neighbourhood of the origin. Then we obtain the same phase portrait for O1 as in the previous case, L11 . Similarly we determine the position of the sectors and the orientation of the orbits in the remaining cases. If b1 > 0, c0 − b0 < 0 and b0 > 0 we obtain the phase portrait L12 of Figure 2. If b1 < 0, c0 − b0 > 0 and b0 ≥ 0 we obtain the phase portrait L13 , and if b1 < 0, c0 − b0 < 0 and b0 > 0 the phase portrait is L14 . Nilpotent case. If b2 = 0 then the origin of systems (7) is nilpotent so its phase portrait can be determined by Theorem 3.5 in [Dumortier et al., 2006], which concludes that in this case the singular point is either a saddle or it has a local phase portrait consisting of a hyperbolic sector and an elliptic sector, depending on the parameters. It is also necessary to determine the position of the sectors and the orientation of the orbits, and in order to do that we must take into account the information given by the theorem and also analyze the sense of the flow in the different regions depending on the parameters. Once we have determined the local phase portrait for systems (7) we must multiply by v so all the points over the line v = 0 become singular points, and the orientation of the orbits on the third and fourth quadrants is reversed. Thus we obtain for O1 the 8 phase portraits in Figure 3 under the following conditions: If b2 = 0, b0 > 0, b1 > 0, c0 − b0 6= 2b0 b1 and c0 − b0 > 0 the phase portrait at O1 is L15 . We obtain the same phase portrait if b2 = 0, b0 > 0, b1 > 0 and c0 − b0 = 2b0 b1 . If b2 = 0, b0 > 0, b1 > 0, c0 − b0 6= 2b0 b1 and c0 − b0 < 0 the phase portrait at O1 is L17 . If b2 = 0, b0 > 0, b1 < 0, c0 − b0 6= 2b0 b1 and c0 − b0 > 0 the phase portrait at O1 is L16 . The same result is obtained for b2 = 0, b0 > 0, b1 < 0 and b0 − c0 = 2b0 b1 . If b2 = 0, b0 > 0, b1 < 0, c0 − b0 6= 2b0 b1 and c0 − b0 < 0 the phase portrait at O1 is L18 . 4.2. Study of the origin of the chart U2 As in the previous section to determine the phase portrait at the singular point O2 we eliminate a common factor v from systems (6). Then we study the singular points over the line v = 0 of systems u̇ = (b0 − c0 )uv, (8) v̇ = −b2 uv − c0 v 2 − b1 u + b3 v. The only singular point is the origin, and it presents a similar behaviour than in the previous case: it is semi-hyperbolic if b3 6= 0 and it is nilpotent if b3 = 0. In the semi-hyperbolic case the singular point is always a saddle-node, and attending to the information given by Theorem 2.19 in [Dumortier et al., 2006] and to the sense of the flow, we get four possibilities for the position and orientation of the sectors in the 9 saddle-node, which are associated with their corresponding conditions in Table 6. In the nilpotent case the singular point can be a saddle or have a hyperbolic and an elliptic sector. In the first case we found two possibilities for the position of the saddle, and in the second case we found four different cases attending to the position and orientation of the two sectors. The results are given in Table 6. Table 6. Conditions for each local phase portrait of O2 . Conditions Phase portrait O2 b3 6= 0, b1 > 0, b0 − c0 > 0. L21 b3 6= 0, b1 > 0, b0 − c0 < 0, c0 > 0. L22 b3 6= 0, b1 < 0, b0 − c0 > 0. L23 b3 6= 0, b1 < 0, b0 − c0 < 0, c0 > 0. L24 b3 6= 0, c0 6= 0, b1 > 0, b0 − c0 6= 2b1 c0 , b0 − c0 < 0, c0 > 0. L25 b3 6= 0, c0 6= 0, b1 > 0, b0 − c0 6= 2b1 c0 , b0 − c0 > 0, c0 < 0. L26 6 2b1 c0 , b0 − c0 > 0, c0 > 0. 6 0, c0 6= 0, b1 > 0, b0 − c0 = b3 = L29 b3 = 6 0, b1 > 0, b0 − c0 = 2b1 c0 , c0 > 0. 6 0, c0 = 6 2b1 c0 , b0 − c0 < 0, c0 > 0. 6 0, c0 6= 0, b1 < 0, b0 − c0 = b3 = L210 b3 = 6 0, b1 < 0, b0 − c0 = 2b1 c0 , c0 > 0. 6 0, c0 = b3 6= 0, c0 6= 0, b1 < 0, b0 − c0 6= 2b1 c0 , b0 − c0 > 0, c0 > 0. L27 b3 6= 0, c0 6= 0, b1 < 0, b0 − c0 6= 2b1 c0 , b0 − c0 < 0, c0 < 0. L28 5. Global phase portraits In this section we prove Theorem 1 by obtaining all the possible global phase portraits from the local information obtained in Sections 3 and 4. In each case of Tables 2 to 5 we must consider two subcases by setting the sign of b1 , and once this sign is fixed the local phase portrait at the infinite singular points is determined by Lemma 1 and Theorems 4 and 5. There is an exception to this which is case 4.2 in Table 5, as in this case we must consider four subcases fixing also the sign of c0 − b0 . Thus we have 32 cases. According to Theorem 2 we have to draw the separatrix skeleton in each case. We recall that the separatrices are the finite and infinite singular points, the limit cycles and the separatrices of the hyperbolic sectors. Systems (2) do not have any limit cycles as if they had a limit cycle it must surround a finite singular point, but all the finite singular points are over invariant lines, particularly over the axes, so there are no limit cycles. Then we have to draw the local phase portraits of the singular points and the separatrices of the hyperbolic sectors for which we have to determine their α and ω-limits. In 30 of the 32 cases the place where born and die the separatrices is determined in a unique way, so we obtain the corresponding global phase portrait by drawing them and one orbit in each canonical region which does not have an infinite 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. The two remaining cases are 1.2 and 1.3 in Table 2, with b1 > 0. In these cases the α and ω-limits are not determined in a unique way, and we can connect the sepatrices in three different ways. In case 1.2, if we fix b1 > 0, we obtain the phase portraits G3, G4 and G5 of Figure 6, depending on how we connect the separatrices on the third quadrant. We know from the local information that there is a separatrix which ω-limit is the origin of the chart V1 and a separatrix which α-limit the saddle P1 in the negative z-axis. Studying their possible α and ω-limits, respectively, we obtain the configuration given in Figure 5. Note that in the second case, the two separatrices are connected and so there is actually only one separatrix on the quadrant. The global phase portrait in the Poincaré disc for the values of the parameters b0 = 2, b1 = b2 = b3 = 1 and c0 = 1/2 is the G3 as the configuration in the third quadrant is the one in Figure 5(a). For the values of the parameters b0 = 2, b1 = b2 = b3 = 1 and c0 = 3/2 we obtain the configuration in Figure 5(c) and then phase portrait G5. For continuity in the parameter space, as we have only modified the value of c0 to get from G3 to G5, there must exists a value of c0 for which the two separatrices in the third quadrant are connected as in Figure 5(b) and so the phase portrait G4 gets realized. 10 Similarly, if we fix b1 > 0 in case 1.3, we obtain three phase portraits, G7, G8 and G9 in Figure 6. For the values b0 = 1/2, b1 = b2 = b3 = 1 and c0 = 2 the phase portrait is the G7, and for the values b0 = 3/2, b1 = b2 = b3 = 1 and c0 = 2 the phase portrait is G9. Now we have only modified the value of b0 to get from G7 to G9 and then there must exists a value of b0 for which the phase portrait G8 is realizable. (a) (b) (c) Fig. 5. Possible configurations on the third quadrant on case 1.2 with b1 > 0. We include all global phase portraits obtained in Figure 6 and in Table 7 we indicate which of them are obtained in each case. Table 7. Classification of the global phase portraits of systems (2). Case Subcase O1 O2 Global b1 > 0 L12 L21 G1 1.1 b1 < 0 L14 L23 G2 b1 > 0 L12 L21 G3, G4 or G5 1.2 b1 < 0 L14 L23 G6 b1 > 0 L11 L22 G7, G8 or G9 1.3 b1 < 0 L13 L24 G10 b1 > 0 L12 L21 G11 1.4 b1 < 0 L14 L23 G12 b1 > 0 L11 L22 G13 1.5 b1 < 0 L13 L24 G14 b1 > 0 L17 L21 G15 2.1 b1 < 0 L18 L23 G16 b1 > 0 L17 L21 G17 2.2 b1 < 0 L18 L23 G18 b1 > 0 L15 L22 G19 2.3 b1 < 0 L16 L24 G20 b1 > 0 L17 L21 G21 2.4 b1 < 0 L18 L23 G22 b1 > 0 L12 L26 G23 3.1 b1 < 0 L14 L28 G24 b1 > 0 L11 L25 G25 3.2 b1 < 0 L13 L27 G26 b1 > 0 L12 L29 G27 3.3 b1 < 0 L14 L210 G28 b1 > 0 L11 L25 G29 3.4 b1 < 0 L13 L27 G30 b1 > 0 L17 L26 G31 4.1 b1 < 0 L18 L28 G32 b1 > 0, c0 − b0 >0 L15 L25 G33 b1 > 0, c0 − b0 <0 L17 L29 G34 4.2 b1 < 0, c0 − b0 >0 L16 L27 G35 b1 < 0, c0 − b0 <0 L18 L210 G36 11 (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) (G32) (G33) (G34) (G35) Fig. 6. Global phase portraits of systems (2) in the Poincaré disc. 12 We have obtained the 36 global phase portraits given in Figure 6 and now we study which of them are topologically equivalent. As Theorem 2 only works in regions with a finite number of singular points, we will consider the equivalences on the open Poicaré disc, but this does not affect the result as if two separatrix skeletons are topologically equivalent, they will be still equivalents if we add the boundary of the disc because the boundary is filled of singular points,and if they are not topologically equivalent they will not be equivalent by adding the boundary of the disc. We will consider 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 . We give this first classification in Table 8 and then within each class we prove which of the phase portraits are topologically equivalent. Table 8. Classes of equivalence according to the number of finite singular points and to the indF . Class No finite singular points indF Global phase portraits 1 3 1 G1, G2, G4, G4, G5, G6, G7, G8, G9, G10. 2 1 G11, G12, G13, G14. 3 2 0 G15, G16, G17, G18, G23, G24, G25, G26. 4 2 G19, G20, G27, G28. 5 0 G21, G22, G29, G30. 6 1 -1 G31, G32. 7 1 G33, G34, G35, G36. Class 1. First we can distiguish two subclasses depending on the number of separatrices in the open Poincaré disc. There are 11 separatrices in phase portraits G1, G2, G4 and G8, and 12 separatrices in the phase portraits G3, G5, G6, G7, G9 and G10. In the first subclass, G1 is topologically equivalent to G2 by doing a symmetry with respect to the line z = −x and a change of the time variable t by −t. G1 is different from G4 as in G1 there are two separatrices that start in the unstable node and in G4 there are three. G1 is also different from G8 as in G1 there are two separatrices of the saddle that connect with the infinity and in G8 there are three. At last G4 is topologically equivalent to G8 by doing a 90o rotation of G8 and then a symmetry with respect to the z-axis. In the second subclass, G3 is different from G5 as in G3 the saddle has two separatrices that connect with the infinity and in G5 it has three. By doing a symmetry with respect to the line y = z we transform G3 intro G7, G5 into G9 and G6 into G10. G7 is different from G10 as in G7 there are three separatrices that start in the unstable node and in G10 there are four. G9 is different from G10 as in G10 there are a separatrix that connects two infinite singular points but in G9 there is not a such separatrix. Class 2. G11 is different from G12 as in G11 the saddle-node has three separatrices that connect with infinite singular points and in G12 it has four. G11 is topologically equivalent to G13 and G12 to G14 by doing a symmetry with respect to the line x = z. Class 3. G15 is topologically equivalent to G16 and G17 to G18 by doing a symmetry with respect to the z-axis. G15 is different from G17 as in G15 there are two separatrices that start at the node and in G17 there are four. G15 is topologically equivalent to G23 by doing a rotation of 90o in G15 and a change of the time variable t by −t. We also can transform G25 into G18 by a rotation of 90o . Lastly we can transform G23 into G24 and G25 into G26 with a symmetry with respect to the x-axis. Class 4. G19 is topologically equivalent to G20 by a symmetry with respect to the z− axis, G19 to G27 by a symmetry with respect to the line z = x and G27 to G28 by a symmetry with respect to x-axis. Class 5. G21 is topologically equivalent to G22 by a symmetry with respect to the z− axis, G21 to G29 by a symmetry with respect to the line z = x and G29 to G30 by a symmetry with respect to x-axis. Class 6. G31 is topologically equivalent to G32 by a symmetry with respect to the z-axis. Class 7. G33 is topologically equivalent to G34 with a symmetry with respect to the line z = x, and 13 by a symmetry with respect to the z-axis G33 is topologically equivalen to G35 and G34 to G36. In summary, among these seven clases, we have found 13 topologically different phase portraits in the Poincaré disc for systems (2), so we have proved Theorem 1. This 13 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 6 as follows: Rep. Phase portraits Rep. Phase portraits Rep. Phase portraits R1 G1, G2. R6 G11, G13. R10 G19, G20, G27, G28. R2 G3, G7. R7 G12, G14. R11 G21, G22, G29, G30. R3 G4, G8. R8 G15,G16, G23, G24. R12 G31, G32. R4 G5, G9. R9 G17, G18, G25, G26. R13 G33, G34, G35, G36. R5 G6, G10 Acknowledgements The first and third authors are partially supported by the Ministerio de Ciencia e Innovación, Agencia Estatal de Investigación (Spain), grant PID2020-115155GB-I00 and the Consellería de Educación, Uni- versidade e Formación Profesional (Xunta de Galicia), 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. The second author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grant PID2019-104658GB-I00, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017- 777911. References Alavez-Ramírez, J., Blé, G., Castellanos V. & Llibre, J. 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