Phase portraits of a family of Kolmogorov systems depending on six parameters

PHASE PORTRAITS OF A FAMILY OF KOLMOGOROV SYSTEMS DEPENDING ON SIX PARAMETERS ÉRIKA DIZ-PITA1, JAUME LLIBRE2AND M. VICTORIA OTERO-ESPINAR1 Abstract. Consider a general 3-dimensional Lotka-Volterra system with a rational first integral of degree two of the form H=xiyjzk. The restriction of this Lotka-Volterra system to each surface H(x, y, z) = hvarying h∈R provide Kolmogorov systems. With the additional assumption that they have a Darboux invariant of the form xℓymest they reduce to the Kolmogorov systems ˙x=xa0−µ(c1x+c2z2+c3z), ˙z=zc0+c1x+c2z2+c3z. In this paper we classify the phase portraits in the Poincaré disc of all these Kolmogorov systems which depend on six parameters. 1. Introduction The Lotka-Volterra systems have been used for modelling many natural phenomena, such as the time evolution of conflicting species in biology [20], chemical reactions, plasma physics [15] or hydrodynamics [6], just as other problems from social science and economics. These systems, which are polynomial differential equations of degree two, were initially proposed, independently, by Alfred J. Lotka in 1925 and Vito Volterra in 1926, both in the context of competing species. Later on Lotka-Volterra systems were generalized and considered in arbitrary dimension, i.e. ˙xi=xi ai0+ n X j=1 aijxj , i = 1, ..., n. Consequently the applications of these systems started to multiply. Moreover Kolmogorov in [14] extended the Lotka-Volterra systems as follows ˙xi=xiPi(x1, . . . , xn), i = 1, ..., n, where Piare polynomials of degre at most m. These kind of systems are now known as Kolmogorov systems. They have in particular all the applications of the Lotka-Volterra systems as for instance in the study of the black holes in cosmology, see [1]. The global qualitative dynamics of the Lotka-Volterra systems in dimension two has been completely studied in [24], where all possible phase portraits on the Poincaré disc have been classified. 2010 Mathematics Subject Classification. Primary 34C05. Key words and phrases. Kolmogorov system, Lotka-Volterra system, Phase portrait, Poincaré disc. 1 2 E. DIZ-PITA, J. LLIBRE AND M.V. OTERO-ESPINAR There are few results about the global dynamics of the Lotka-Volterra systems in dimension three. Our objective is to study the phase portraits of the 3-dimensional Lotka-Volterra systems ˙x=x(a0+a1x+a2y+a3z), ˙y=y(b0+b1x+b2y+b3z), ˙z=z(c0+c1x+c2y+c3z), (1.1) which have a rational first integral of degree two of the form xiyjzk. We have used the Darboux theory of integrability to obtain a characterization of these systems. As a result, we have reduced the initial problem to a problem in dimension two, the study of the global dynamics of two families of Kolmogorov systems. In this paper we focus on the first family, which is ˙x=x(a0+a1x+a2z2+a3z), ˙z=z(c0+c1x+c2z2+c3z), (1.2) Kolmogorov systems (1.2) depend on eight parameters, this is a big number in order to classify all their distinct topological phase portraits. Then we require that Kolmogorov systems (1.2) have a Darboux invariant of the form xℓymest, then these systems are reduced to study the Kolmogorov systems ˙x=xa0−µ(c1x+c2z2+c3z), ˙z=zc0+c1x+c2z2+c3z, (1.3) which now depend on six parameters. For these Kolmogorov systems we give the topological classification of all their phase portraits in the Poincaré disc. Roughly speaking the Poincaré disc is the closed unit disc centered at the origin of R2. Its interior is identified with R2and the circle of its boundary is identified with the infinity of R2. In the plane R2we can go or come from the infinity in as many directions as points have the circle. The polynomial differential systems can be extended to the closed Poincaré disc, i.e. they can be extended to infinity and in this way we can study their dynamics in a neighborhood of infinity. This extension is called the Poincaré compactification, for more details see subsection 2.1. Thus our main result is the following. Theorem 1.1. Kolmogorov systems (1.3) have 102 topologically distinct phase portraits in the Poincaré disc under condition (H2)given in Figure 16. The condition (H2)is {c2= 0, a0≥0, c1≥0, c3≥0, a0+c0µ= 0, a0c1µ= 0, µ =−1}. We will see that we can assume condition (H2)because Kolmogorov systems (1.3) can be reduced to satisfy such condition either using symmetries, or eliminating known phase portraits, or eliminating phase portraits with infinitely many finite or infinite singular points. Other papers where some topological phase portraits have been classified in the Poincaré disc are, for instance, [4, 13, 17]. In Section 3 using the Darboux theory of integrability we explain the reduction from the Lotka-Volterra system (1.1) to the Kolmogorov systems (1.3). In Section 4 we give some properties of the system obtained. In Section 5 we study the local phase portrait of the finite singular points, and in Section 6 we do the same with A CLASS OF KOLMOGOROV SYSTEMS 3 the infinite singular points, applying the blow-up technique. Finally in Section 7 we prove Theorem 1.1. 2. Preliminaries 2.1. Poincaré compactification. In order to study the behavior of the trajectories of our polynomial differential systems near the infinity we will use the Poincaré compactification. We provide a short summary about this method, more details can be found in Chapter 5 of [9]. Let X= (P(x, y), Q(x, y)) be a polynomial vector field of degree ddefined in R2. Consider the Poincaré sphere S2=y∈R3:y2 1+y2 2+y2 3= 1and its tangent plane at the point (0,0,1) which is identified with R2. We consider the central projections f+:R2→S2and f−:R2→S2. By definition, f+(x)is the intersection of the straight line passing through the point xand the origin with the northern hemisphere of S2, and respectively for f−(x)with the southern hemisphere. The differential Df+and respectively Df−send the vector field Xinto a vector field Xon S2\S1. Note that the points at infinity of R2are in bijective correspondence with the points of the equator S1of S2. The vector field Xcan be extended analytically to a vector field on S2multiplying Xby yd 3. We denothe this vector field by ρ(X), and it is called the Poincaré compactification of the vector field Xon R2. For studying the dynamics of Xin the neighborhood of the infinity, we must study the dynamics of ρ(X)near S1. The sphere S2is a 2-dimensional manifold so we need to know the expressions of the vector field ρ(X)in 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−→ R2for i= 1,2,3with ϕi(y) = −ψi(y)=(ym/yi, yn/yi)for m<n and m, n =i. In the local chart (U1, ϕ1)the expression of ρ(X)is (2.1) ˙u=vd−u P 1 v,u v+Q1 v,u v,˙v=−vd+1 P1 v,u v. In the local chart (U2, ϕ2)the expression of ρ(X)is (2.2) ˙u=vdP1 v,u v−uQ 1 v,u v,˙v=−vd+1 P1 v,u v, and in the local chart (U3, ϕ3)the expression of ρ(X)is (2.3) ˙u=P(u, v),˙v=Q(u, v). In the charts (Vi, ψi), with i= 1,2,3, the expression for ρ(X)is the same as in the charts (Ui, ϕi)multiplied by (−1)d−1. The equator S1is invariant by the vector field ρ(X)and all the singular points of ρ(X)which lie in this equator are called the infinite singular points of X. If y∈S1is an infinite singular point, then −yis also an infinite singular point and they have the same (respectively opposite) stability if the degree of vector field is odd (respectively even). The image of the closed northern hemisphere of S2onto the plane y3= 0 under the orthogonal projection πis called the Poincaré disc D2. Since the orbits of ρ(X) on S2are symmetric with respect to the origin of R3, we only need to consider the 4 E. DIZ-PITA, J. LLIBRE AND M.V. OTERO-ESPINAR flow of ρ(X)in the closed northern hemisphere, and we can project the phase portrait of ρ(X)on the northern hemisphere onto the Poincaré disc. We shall present the phase portraits of the polynomial differential systems (1.3) in the Poincaré disc. 2.2. Topological equivalence between two polynomial vector fields. Two polynomial vector fields X1and X2on R2are topologically equivalent if there exists a homeomorphism on the Poincaré disc which preserves the infinity S1and sends the trajectories of the flow of π(ρ(X1)) to the trajectories of the flow of π(ρ(X2)), preserving or reversing the orientation of all the orbits. Aseparatrix of the Poincaré compactification π(ρ(X)) is an orbit at the infinity S1, or a finite singular point, or a limit cycle, or an orbit on the boundary of a hyperbolic sector at a finite or an infinite singular point. The set of all separatrices of π(ρ(X)) is closed and we denote it by ΣX. An open connected component of D2\ΣXis a canonical region of π(ρ(X)). The separatrix configuration of π(ρ(X)) is the union of an orbit of each canonical region with the set ΣX, and it is denoted by Σ′ X. We denote by S (respectively R) the number of separatrices (respectively canonical regions) of a vector field π(ρ(X)). We say that two separatrix configurations Σ′ X1and Σ′ X2are topologically equivalent if there is a homeomorphism h:D2→D2such that h(Σ′ X1)=Σ′ X2. The following theorem of Markus [21], Neumann [22] and Peixoto [23] allows to investigate only the separatrix configuration of a polynomial differential system in order to determine its phase portrait in the Poincaré disc. Theorem 2.1. 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 configurations Σ′ X1and Σ′ X2are topologically equivalent. 2.3. Blow-up technique. There exist classification theorems for hyperbolic and semi-hyperbolic singular points, and also for nilpotent singular points which can be found in Chapter 2 and 3 of [9]. The centers are more difficult to study, see for instance Chapter 4 of [9]. Whereas to study a singular point for which the Jacobian matrix is identically zero, the only possibility is studying each singular point case by case. The main technique to perform the desingularization of a linearly zero singular point is the blow-up technique. We give a short summary about this method, more details can be found in [2]. Roughly speaking the idea behind the blow up technique is to explode, through a change of variables that is not a diffeomorphism, the singularity to a line. Then, for studying the original singular point, one studies the new singular points that appear on this line, and this is simpler. If some of these new singular points are linearly zero, the process is repeated. Dumortier proved that this iterative process of desingularization is finite, see [8]. Consider a real planar polynomial differential system of the form ˙x=P(x, y) = Pm(x, y) + . . . , ˙y=Q(x, y) = Qm(x, y) + . . . , (2.4) where Pand Qare coprime polynomials, Pmand Qmare homogeneous polynomials of degree m∈Nand the dots mean higher order terms in xand y. Note that we are assuming that the origin is a singular point because m > 0. We define the A CLASS OF KOLMOGOROV SYSTEMS 5 characteristic polynomial of (2.4) as (2.5) F(x, y) := xQm(x, y)−yPm(x, y), and we say that the origin is a nondicritical singular point if F ≡ 0and a dicritical singular point if F ≡ 0. In this last case Pm=xWm−1and Qm=yWm−1, where Wm−1≡ 0is a homogeneous polynomial of degree m−1. If y−vx is a factor of Wm−1and v= tan θ∗,θ∗∈[0,2π), then θ∗is a singular direction. The homogeneous directional blow up in the vertical direction is the mapping (x, y)→(x, z) = (x, y/x), where zis a new variable. This map transforms the origin of (2.4) into the line x= 0, which is called the exceptional divisor. The expression of system (2.4) after the blow up in the vertical direction is (2.6) ˙x=P(x, xz),˙z=Q(x, xz)−zP (x, xz) x, that is always well-defined since we are assuming that the origin is a singularity. After the blow up, we cancel an appearing common factor xm−1(xmif F ≡ 0). Moreover, the mapping swaps the second and the third quadrants in the vertical directional blow up. Propositions 2.1 and 2.2 of [2] provide the relationship between the original singular point of system (2.4) and the new singularities of system (2.6). For additional details see [3]. Finally, to study the behavior of the solutions around the origin of system (2.4), it is necessary to study the singular points of system (2.6) on the exceptional divisor. They correspond to either characteristic directions in the nondicritical case, or singular directions in the dicritical case. It may happen that some of these singular points are linearly zero, in which case we have to repeat the process. As we said before, it is proved in [8] that this chain of blow ups is finite. 2.4. Indices of planar singular points. Given an isolated singularity qof a vector field X, defined on an open subset of R2or S2, we define the index of qby means of the Poincaré Index Formula. We assume that qhas the finite sectorial decomposition property. Let e,hand pdenote the number of elliptic, hyperbolic and parabolic sectors of q, respectively, and suppose that e+h+p > 0. Then the index of qis iq= 1 + (e−h)/2, and it is always an integer. We recall that the Poincaré compactification of a vector field in R2introduced in Subsection 2.1 is a tangent vector field on the sphere S2, so the next result will be very useful in our study. Theorem 2.2 (Poincaré-Hopf Theorem).For every tangent vector field on S2with a finite number of singular points, the sum of their indices is 2. 2.5. Invariants and Application of the Darboux Theory. The Darboux Theory of Integrability provides a link between the integrability of polynomial vector fields and the number of invariant algebraic curves that they have. The basic results on dimension two can be found in Chapter 8 of [9], and these results have been extended to Rnand Cnin [16, 18, 19]. We consider a real polynomial differential system in dimension three, that is a system of the form dx/dt = ˙x=P(x, y, z), dy/dt = ˙y=Q(x, y, z), dz/dt = ˙z=R(x, y, z), (2.7) 6 E. DIZ-PITA, J. LLIBRE AND M.V. OTERO-ESPINAR where P, Q and Rare polynomials in the variables x, y and z. We denote by m= max{deg P, deg Q, deg R}the degree of the polynomial system, and we always assume that the polynomials P, Q and Rare relatively prime in the ring of the real polynomials in the variables x, y and z. Theorem 2.3 (Darboux Integrability Theorem).Suppose that a polynomial system (2.7) of degree madmits pirreducible invariant algebraic surfaces fi= 0 with cofactors Kifor i= 1, . . . , p. Then the next statements hold. (a) There exist λi∈Cnot all zero such that Pp i=1 λiKi= 0 if and only if the function fλ1 1. . . fλp pis a first integral of system (2.7). (b) There exist λi∈Cnot all zero such that Pp i=1 λiKi=−sfor some s∈R\ {0} if and only if the function fλ1 1. . . fλp pexp(st)is a Darboux invariant of system (2.7). 3. Reduction of the Lotka-Volterra systems in R3to the Kolmogorov systems in R2 As we said our objective is to study the global dynamics of the Lotka-Volterra systems (1.1) in dimension three, which have a rational first integral of degree two of the form xiyjzk. The Darboux theory of integrability allow us to obtain a characterization of these systems. We consider the irreducible invariant algebraic surfaces f1(x, y, z) = x= 0, f2(x, y, z) = y= 0 and f3(x, y, z) = z= 0 of system (1.1), with cofactors K1,K2 and K3, respectively. As Kiis the cofactor of fiwe have that Xfi=P∂fi ∂x +Q∂fi ∂y +R∂fi ∂z =Kifi. Then for the invariant algebraic surfaces considered we get the cofactors K1= a0+a1x+a2y+a3z,K2=b0+b1x+b2y+b3zand K3=c0+c1x+c2y+c3z, respectively. Applying Theorem 2.3, since we assume that xλ1yλ2zλ3is a first integral of system (1.1), we get that there exist λi∈C, with i∈ {1,2,3}, not all zero, such that P3 i=1 λiKi= 0.Apart from the trivial solution {λ1= 0, λ2= 0, λ3= 0}, there are the following three solutions of this equation: S1={c0= 0, c1= 0, c2= 0, c3= 0, λ2= 0, λ1= 0}, S2=b0=−c0λ3 λ2 , b1=−c1λ3 λ2 , b2=−c2λ3 λ2 , b3=−c3λ3 λ2 , λ1= 0,and S3=a0=−b0λ2−c0λ3 λ1 , a1=−b1λ2−c1λ3 λ1 , a2=−b2λ2−c2λ3 λ1 , a3=−b3λ2−c3λ3 λ1, which give rise to three families of Lotka-Volterra polynomial differential systems of degree two in R3, with a first integral of the form xλ1yλ2zλ3. If we consider the family given by solution S1, as the parameters ci, i = 0, ..., 3, are zero, we have that ˙z= 0 and the Lotka-Volterra system is reduced to: ˙x=x(a0+a1x+a2y+a3z), ˙y=y(b0+b1x+b2y+b3z), ˙z= 0. A CLASS OF KOLMOGOROV SYSTEMS 7 As ˙z= 0,zis constant and this system has H=zas a first integral. Note that if we consider the first integral H=xλ1yλ2zλ3, and apply the conditions given by S1, it is λ1=λ2= 0, we obtain H=zλ3, with λ3= 2 for getting the degree two, but in this case we will consider the simplest first integral. In each invariant plane with z constant, we have a Lotka-Volterra polinomial differential system in R2. The phase portrait of these systems has been studied in [24], so we are not going to deal with this case. In this paper we study the family given by the solution S2. This solution provides the values of parameters bias a function of the parameters λ2,λ3and ci, with i= 0, .., 3, so we can replace them in the expression of ˙yobtaining ˙y=y−c0λ3 λ2 −c1λ3 λ2 x−c2λ3 λ2 y−c3λ3 λ2 z. If we denote λ=−λ3/λ2, then the original Lotka-Volterra system becomes ˙x=x(a0+a1x+a2y+a3z), ˙y=λy(c0+c1x+c2y+c3z), ˙z=z(c0+c1x+c2y+c3z). Given that λ1= 0, the first integral H=xλ1yλ2zλ3is reduced to H=yλ2zλ3, but if this is a first integral, also H=yλ2zλ3−1 λ2=y−1z−λ3 λ2=y−1zλ=zλ/y is a first integral. If we want Hto be rational of degree two, we must take λ= 2. In each level H= 1/h, with h= 0, we will have 1/h =z2/y, so y=hz2and then, for each h, the initial Lotka-Volterra system on dimension three reduces to the system on dimension two ˙x=x(a0+a1x+a2hz2+a3z), ˙z=z(c0+c1x+c2hz2+c3z). We must study the phase portrait of the systems of this family, but it is equivalent to study the phase portraits of the family of Kolmogorov systems in dimension two ˙x=x(a0+a1x+a2z2+a3z), ˙z=z(c0+c1x+c2z2+c3z). (3.1) In the particular cases in which His zero or infinity, the differential system on dimension three is reduced to a Lotka-Volterra system on dimension two, having in each case z= 0 and y= 0, respectively. We recall that these systems had already been studied in [24]. Systems (3.1) depend on eight parameters and the classification of all their distinct topological phase protraits is huge. For this reason we study the subclass of them having a Darboux invariant of the form xλ1zλ2est. By statement (b) of Theorem 2.3 the expression λ1Kx+λ2Kz+smust be zero, where Kxand Kyare the cofactors of the invariant planes x= 0 and z= 0, respectively. Note that sand λ2 1+λ2 2cannot be zero. We obtain the cofactors Kx=a0+a1x+a2z2+a3zand Kz=c0+c1x+c2z2+c3zand then, solving the equation λ1Kx+λ2Kz+s= 0, we get the following two non-trivial solutions ˜ S1=s=−a0λ1−c0λ2, a1=−c1λ2 λ1 , a2=−c2λ2 λ1 , a3=−c3λ2 λ1,and ˜ S2={s=−c0λ2, c1= 0, c2= 0, c3= 0, λ1= 0}. 8 E. DIZ-PITA, J. LLIBRE AND M.V. OTERO-ESPINAR So we have two subsystems from the initial system (3.1). According to the conditions given by solution ˜ S1the first subsystem is ˙x=xa0−c1λ2 λ1 x−c2λ2 λ1 z2−c3λ2 λ1 z, ˙z=zc0+c1x+c2z2+c3z. If we denote λ2/λ1=µand λ1=λ, then this subsystem becomes ˙x=xa0−µ(c1x+c2z2+c3z), ˙z=zc0+c1x+c2z2+c3z, (3.2) and its Darboux invariant is xλzλµe−tλ(a0+c0µ). But if this is a Daboux invariant, also it is xzµe−t(a0+c0µ). Note that in order that we have a Darboux invariant a0+c0µcannot be zero. If we consider now the solution ˜ S2we get the subsystem ˙x=xa0+a1x+a2z2+a3z, ˙z=c0z, which is equivalent to the previous one, taking µ= 0 and interchanging the variables xand z, so it is sufficient to study the Kolmogorov systems (3.2) depending on six parameters. 4. Properties of system (1.3) In this section we state some results that will be used on the classification in order to reduce the number of phase portraits appearing. Note that if c2= 0, then the system (1.3) is a Lotka-Volterra system in dimension 2. A global topological classification of these systems has been completed in [24], so we limit our study to the case c2= 0. We recall that for obtaining system (1.3) we have supposed that system (3.1) has the Darboux invariant I=xzµe−t(a0+c0µ), so it is required that a0+c0µ= 0. Proposition 4.1. Consider system (1.3) and suppose that (˜x(t),˜z(t)) is a solution of this system. If we change c1by −c1(respectively c3by −c3), then (−˜x(t),˜z(t)) (respectively (˜x(t),−˜z(t))) is other solution of the obtained system. Remark 4.2. By Proposition 4.1 we can limit our study to Kolmogorov systems (1.3) with c1and c3non-negatives. In the cases with these parameters negatives, we will obtain phase portraits symmetric to the ones obtained in the positive cases, with respect to the z-axis when we change the sign of c1, and with respect to the x-axis when we change the sign of c3. Corollary 4.3. Consider system (1.3) and suppose (˜x(t),˜z(t)) is a solution. If c1= 0 (respectively c3= 0), then (−˜x(t),˜z(t)) (respectively (˜x(t),−˜z(t))) is also a solution. Remark 4.4. Corollary 4.3 simplifies the study of the cases with c1= 0 or c3= 0, because it proves that the phase portraits have to be symmetric with respect to the z-axis and x-axis respectively, and this fact will be useful in obtaining the global phase portraits from the local results. Proposition 4.5. Let (˜x(t),˜z(t)) be a solution of system (1.3). In the next cases we obtain another system with solution (−˜x(−t),−˜z(−t)). A CLASS OF KOLMOGOROV SYSTEMS 9 (1) If a0,c0and c2are not zero, and we change the sign of all of them. (2) If a0= 0 and we change the sign of c0and c2, which are not zero. (3) If c0= 0 and we change the sign of a0and c2, which are not zero. Remark 4.6. In order to classify all the phase portraits of the Kolmogorov systems (1.3), according with the previous results, it is sufficient to consider a0≥0. And when a0= 0 we will consider also c0>0. Remark 4.7. In short, according with the previous results and considerations, from now on it will be sufficient to study the Kolmogorov systems (1.3) with the their parameters satisfying (H) = {c2= 0, a0≥0, c1≥0, c3≥0, a0+c0µ= 0}. Theorem 4.8. For system (1.3) the next statements hold. (1) If c1= 0, then on any straight line z=cte = 0, there exists only one contact point. (2) If c1= 0, then there exist two invariant straight lines z= (pc2 3−4c0c2− c3)/(2c2)and z=−(pc2 3−4c0c2+c3)/(2c2)if c2 3>4c0c2, and one invariant straight line z=−c3/(2c2)if c2 3= 4c0c2. There are not contact points on any other straight line z=cte = 0. Proof. First we suppose c1= 0 and consider a straight line z=z0= 0. Then the contact points on this straight line are those on which ˙z= 0 and, as z0= 0, the only possible contact point is the one that satisfies c0+c1x+c2z2 0+c3z0= 0, i.e. the point such that its first coordinate is x=−(c2z2 0+c3z0+c0)/c1. We consider now the case with c1= 0. Then looking for the points on the straight line z=z0= 0 satisfying ˙z= 0, we obtain that they must verify the condition c0+c2z2 0+c3z0= 0, and solving this equation we get that either there are no contact points, or a full straight line of contact points , or two straight line of contact points, depending on the solutions z0of that equation. □ 5. Local study of finite singular points System (1.3) has the following finite singularities: •P0= (0,0), •P1=0,Rc−c3 2c2and P2=0,−Rc+c3 2c2if c2 3>4c0c2, •P3=0,−c3 2c2if c2 3= 4c0c2, •P4=a0 c1µ,0if c1µ= 0. We use the notation Rc=pc2 3−4c0c2in order to simplify the expressions which will appear. Moreover if a0= 0 and c1µ= 0, all the points on the z-axis are singular points, and the system can be reduced to a Lotka-Volterra system in dimension 2. Therefore from now on we will consider the hypothesis (H1) = c2= 0, a0≥0, c1≥0, c3≥0, a0+c0µ= 0, a2 0+ (c1µ)2= 0. 16 E. DIZ-PITA, J. LLIBRE AND M.V. OTERO-ESPINAR which is proved in Subsections 6.1 and 6.2. From now on we include the condition µ=−1in our hypothesis, so we will work under the conditions (H2) = {c2= 0, a0≥0, c1≥0, c3≥0, a0+c0µ= 0, a0c1µ= 0, µ =−1}. Lemma 6.1. Asumming hypothesis (H2)the origin of the chart U1is an infinite singular point of system (1.3), and it has 47 distinct local phase portraits described in Figure 1. u v (L1) u v (L2) u v (L3) u v (L4) u v (L5) u v (L6) u v (L7) u v (L8) u v (L9) u v (L10) u v (L11) u v (L12) u v (L13) u v (L14) u v (L15) u v (L16) u v (L17) u v (L18) u v (L19) u v (L20) u v (L21) u v (L22) u v (L23) u v (L24) u v (L25) A CLASS OF KOLMOGOROV SYSTEMS 17 u v (L26) u v (L27) u v (L28) u v (L29) u v (L30) u v (L31) u v (L32) u v (L33) u v (L34) u v (L35) u v (L36) u v (L37) u v (L38) u v (L39) u v (L40) u v (L41) u v (L42) u v (L43) u v (L44) u v (L45) u v (L46) u v (L47) Figure 1. Local phase portraits of the infinite singular point O1. For system (6.1), if c1= 0 the characteristic polynomial is F=−c1uv2≡ 0, so the origin is a nondicrital singular point. If c1= 0 the characteristic polynomial is F=−c3u2v−c2u3v−c0uv3, which cannot be identically zero because c2= 0. We will study this two cases separately. 6.1. Case c1non-zero. Consider c1= 0. We introduce the new variable w1by means of the variable change uw1=v, and get the system ˙u= (c0−a0)u3w2 1+c3(µ+ 1)u3w1+c2(µ+ 1)u3+c1(µ+ 1)u2w1, ˙w1=−c0u2w3 1−c3u2w2 1−c2u2w1−c1uw2 1. (6.2) 18 E. DIZ-PITA, J. LLIBRE AND M.V. OTERO-ESPINAR Now we cancel the common factor u, getting the system ˙u= (c0−a0)u2w2 1+c3(µ+ 1)u2w1+c2(µ+ 1)u2+c1(µ+ 1)uw1, ˙w1=−c0uw3 1−c3uw2 1−c2uw1−c1w2 1, (6.3) for which the only singular point on the exceptional divisor is the origin, and it is linearly zero, so we have to repeat the process. Now the characteristic polynomial is F=−c2(µ+ 2)u2w1−c1(µ+ 2)uw2 1, so the origin is a nondicritical singular point if µ=−2, and it is dicritical if µ=−2. In both cases we introduce the new variable w2doing the change uw2=w1, obtaining the system ˙u= (c0−a0)u4w2 2+c3(µ+ 1)u3w2+c2(µ+ 1)u2+c1(µ+ 1)u2w2, ˙w2= (a0−2c0)u3w3 2−c3(µ+ 2)u2w2 2−c1(µ+ 2)uw2 2−c2(µ+ 2)uw2. (6.4) In the nondicritical case we have to cancel the common factor uobtaining ˙u= (c0−a0)u3w2 2+c3(µ+ 1)u2w2+c2(µ+ 1)u+c1(µ+ 1)uw2, ˙w2= (a0−2c0)u2w3 2−c3(µ+ 2)uw2 2−c1(µ+ 2)w2 2−c2(µ+ 2)w2. (6.5) But in the dicritical case, when µ=−2, we must cancel the common factor u2from system (6.4), and we obtain the system ˙u= (c0−a0)u2w2 2−c3uw2−c2−c1w2, ˙w2= (a0−2c0)uw3 2. (6.6) 6.1.1. Nondicritical case. In this case it is necessary to study the singular points of system (6.5) on the exceptional divisor. The origin is always a singular point, and we denote it by Q0. As µ+ 2 = 0 there is another singular point, Q1= (0,−c2/c1) and we determine their local phase portraits. The origin Q0is always hyperbolic. It is a saddle if µ∈(−∞,−2) ∪(−1,+∞), a stable node if c2>0and µ∈(−2,−1), and an unstable node if c2<0and µ∈(−2,−1). The singular point Q1is semi-hyperbolic. If c2(a0+c0µ)>0then Q1is a topological saddle, if c2(µ+2) >0and (µ+2)(a0+c0µ)<0then it is a topological unstable node, and if c2(µ+ 2) <0and (µ+ 2)(a0+c0µ)>0it is a topological stable node. These conditions come together in the next 7 subcases. (1) If µ∈(−∞,−2)∪(−1,+∞)and c2(a0+c0µ)>0, then Q0is a saddle and Q1a topological saddle. In order to determine the phase portrait around the w2-axis for system (6.5), we must fix the sign of c2, which determines the position of the singular point Q1, and also the sign of µ+ 1, which determines the sense of the flow along the x-axis. Thus we deal with the following subcases. Subcase (1.1). Let µ < −2(so µ+1 <0) and c2>0. Then the singular point Q1is on the negative part of the w2-axis and the expression ˙u|w2=0=c2(µ+1)u determines the sense of the flow, so the phase portrait is the one in Figure 2(a). To return to system (6.4) we multiply by u, thus the orbits in the second and third quadrants change their orientation. Moreover all the points on the w2-axis become singular points. The resultant phase portrait is given in Figure 2(b). When going back to the (u, w1)-plane the second and the third quadrants swap from the (u, w2)-plane, and the exceptional divisor shrinks to a point, and hence the orbits are slightly modified. Attending to the expresions of A CLASS OF KOLMOGOROV SYSTEMS 19 ˙u|w1=0=c2(µ+1)u2and ˙w1|u=0=−c1w2 1, we know the sense of the flow along the axes. Following the results mentioned in Subsection 2.3, the separatrix of the singular point Q1= (0,−c2/c1)in the (u, w2)-plane, becomes the separatrix with slope −c2/c1in the (u, w1)-plane. We get the phase portrait given in Figure 2(c), and multiplying again by u, the one given in Figure 2(d). w 2 u (a) w 2 u (b) (c) (d) Figure 2. Desingularization of the origin of system (6.1) with c1= 0. Nondicritical case (1.1). Finally we must go back to the (u, v)-plane, swapping the second and the third quadrants and contracting the exceptional divisor to the origin. The orbits tending to the origin in forward or backward time, became orbits tending to the origin in forward or backward time with slope zero, i.e. tangent to the u-axis. According to the expressions ˙u|v=0=c1µv2−a0v3and ˙u|v=0=c2(µ+ 1)u3, which determine the sense of the flow along the axes, we get the local phase portrait at the origin for system (6.1) given in Figure 1(L1). Subcase (1.2). If we maintain µ < −2but take c2<0, the reasoning is essentially similar to the one we have given in the previous case, and we obtain the phase portrait (L2) of Figure 1. Subcase (1.3). Let µ > −1and c2>0. This determines the position of the singular point Q1and the sense of the flow along the axes, so around the w2-axis we obtain the phase portrait given in Figure 3(a). As in the previous subcase we multiply by uobtaining the phase portrait given in Figure 3(b), as the orbits in the second and third quadrants change their orientation and all the point in the w2-axis become singular points. In order to undo the variable change we analyze the sense of the flow along the axes according to the expression ˙u|w1=0=c2(µ+ 1)u2, which determines that the flow goes in the positive sense of the u-axis, and ˙w1|u=0=−c1w2 1which determines that the flow goes in the negative sense of the w1-axis. Moreover we swap the second and third quadrants, and press the exceptional divisor into the origin, modifying the orbits. We obtain the phase portrait given in Figure 3(c). Multiplying again by uwe obtain the phase portrait 3(d). Now we have to undo de second variable change. We note that ˙u|v=0= c2(µ+1)u3, so the flow gets away from the origin along the u-axis, nevertheless the sense of the flow along the v-axis is not determined by ˙v|u=0=c1µv2−a0v3, it depends on the constant µ. If µ > 0the flow goes in the positive sense of the v-axis, if µ < 0in the opposite sense and, if µ= 0 the flow goes to the origin. Thus we must distinguish three subcases and in each of them, modifying 20 E. DIZ-PITA, J. LLIBRE AND M.V. OTERO-ESPINAR the orbits properly, we obtain, respectively, the phase portraits given in Figure 1(L3), (L4) and (L5). Subcase (1.4). Let µ > −1and c2<0. By a similar reasoning to the previous one, we obtain the phase portraits (L6) and (L7) of Figure 1. w 2 u (a) w 2 u (b) w 1 u (c) w 1 u (d) Figure 3. Desingularization of the origin of system (6.1) with c1= 0. Nondicritical case (1.3). (2) If µ∈(−∞,−2) ∪(−1,+∞),c2(µ+ 2) >0and (µ+ 2)(a0+c0µ)<0, then Q0 is a saddle and Q1a topological unstable node. We must distinguish two cases according with the sign of c2. Subcase (2.1). We consider c2<0so µ < −2and a0+c0µ > 0. The singular point Q1is on the positive w2-axis and it is an unstable node, so the sense of the flow along the axes is determined, and we obtain the phase portrait given in Figure 4(a). Multiplying by uwe obtain Figure 4(b). We see that for system (6.3) the flow goes in the negative sense along the w1-axis and in the positive sense along the u-axis, according to the expressions ˙u|w1=0=c2(µ+ 1)u2and ˙w1|u=0=−c1w2 1. We undo the variable change modifying the orbits properly, and we note that it must exist an hyperbolic or elliptic sector in both first and third quadrants, thus it can appear the configuration given in Figure 4(c) or the one given in Figure 5(a). From the first of them multiplying by uwe obtain 4(d), and if we undo the variable change in a similarly way than in the previous cases, we get the phase portrait in Figure 1(L8). If we consider hyperbolic sectors we continue undoing the blow up from Figure 5(a), obtaining successively the phase portraits 5(b) and 5(c). However in our study we have proved, by means of the index theory, that only the case with elliptic sectors is feasible in the global phase portraits obtained. More detailed explanations will be given in Section 7 but, roughly speaking we know that the index of the vector field on the sphere must be 2, and this index is the sum of the indices of all singularities, which depend on the sectors that they have, so if the index is 2 considering two elliptic sectors in a particular singular point, it cannot be 2 if we change those sectors for hyperbolic ones. In conclusion the only phase portrait that will appear in this case is (L8) of Figure 1. A CLASS OF KOLMOGOROV SYSTEMS 21 w 2 u (a) w 2 u (b) w 1 u (c) w 1 u (d) Figure 4. Desingularization of the origin of system (6.1) with c1= 0. Nondicritical case (2.1). w 1 u (c) w 1 u (d) (b) Figure 5. Desingularization of the origin of system (6.1) with c1= 0. Alternative to nondicritical case (2.1). From now on we will omit the reasonings about how to undo the variable changes for obtaining the final phase portrait, because they are similar to the ones of the previous cases. The results obtained are the following. Subcase (2.2). Let c2>0so µ > −1and a0+c0µ < 0. We obtain the phase portraits (L9) and (L10) of Figure 1. In (L9) it is possible to consider hyperbolic sectors instead of the elliptic ones, but applying index theory to the global phase portraits obtained in our study, we note that only the phase portrait with elliptic sectors is feasible. (3) If µ∈(−∞,−2) ∪(−1,+∞),c2(µ+ 2) <0and (µ+ 2)(a0+c0µ)>0, then Q0is a saddle and Q1a topological stable node. If c2>0we obtain the phase portrait (L11) of Figure 1, and if c2<0we obtain the phase portraits (L12), (L13) and (L14) of Figure 1. In (L11) and (L12) it would be possible that the elliptic sectors appearing were hyperbolic sectors, but again we have proved that only the elliptic option is feasible according to the index theory. (4) If c2>0,µ∈(−2,−1) and a0+c0µ > 0, then Q0is a stable node and Q1a topological saddle. We obtain the phase portrait (L15) of Figure 1. (5) If c2>0,µ∈(−2,−1) and a0+c0µ < 0, then Q0is a stable node and Q1a topological unstable node. We obtain the phase portrait (L11) of Figure 1. (6) If c2<0,µ∈(−2,−1) and a0+c0µ < 0, then Q0is an unstable node and Q1 a topological saddle. We obtain the phase portrait (L16) of Figure 1. (7) If c2<0,µ∈(−2,−1) and a0+c0µ > 0, then Q0is an unstable node and Q1 a topological stable node. We obtain the phase portrait (L8) of Figure 1. 22 E. DIZ-PITA, J. LLIBRE AND M.V. OTERO-ESPINAR 6.1.2. Dicritical case. Now we must study the singular points on the exceptional divisor of system (6.6). In this case there is only one singular point, R= (0,−c2/c1) which is non-degenerated. We shall distinguish several subcases. •If c2 3<−4c2(a0−2c0)and c2c3<0, then Pis a stable focus. We shall distinguish two cases depending on the sign of the parameter c2, because it determines if the singular point is on the positive or negative u-axis. We consider c2>0. In Figure 6 the blowing-down process is represented. The phase portrait around the u-axis is the one given in Figure 6(a), multiplying by u2we obtain (b), undoing the second variable change we obtain (c), multiplying by uwe get (d) and finally, undoing the first variable change we get the phase portrait (L11) of Figure 1. Taking c2<0and by the same method we obtain the phase portrait (L8) of Figure 1. w 2 u (a) w 2 u (b) w 1 u (c) u w 1 (d) Figure 6. Desingularization of the origin of system (6.1) with c1= 0. Dicritical case (1), c2>0. From now on we omit explanations in cases in which similar arguments are valid, and same results are obtained. In order to simplify the notation we define β=pc2 3+ 4c2(a0−2c0). •If c2 3<−4c2(a0−2c0)and c2c3>0, then Pis an unstable focus. The reasoning is analogous to the one of the previous case and we obtain the same phase portraits: (L11) if c2>0, and (L8) if c2<0. •If c2 3=−4c2(a0−2c0)and c2c3<0or if c2 3>−4c2(a0−2c0),c2(c3−β)<0and c2(c3+β)<0, then Pis a stable node. If c2 3=−4c2(a0−2c0)and c2c3>0or if c2 3>−4c2(a0−2c0),c2(c3−β)>0and c2(c3+β)>0, then Pis an unstable node. In both cases the phase portrait obtained is again (L11) if c2>0, and (L8) if c2<0. •If c3= 0 and c2(a0−2c0)<0then Pis a linear center. In this case the singular point Pcould be a center or a focus, but the final phase portrait obtained when Pis a center is the same as the one we obtained previously for the case with a focus, so the result is (L11) if c2>0, and (L8) if c2<0. •If c2 3>−4c2(a0−2c0)and (c3−β)(c3+β)<0, or if c3= 0 and c2(a0−2c0)>0, then Pis a saddle. The blowing-down considering c2>0is represented in Figure 7. The final result is (L17) of Figure 1. If we take c2<0we obtain the phase portrait (L18). A CLASS OF KOLMOGOROV SYSTEMS 23 w 2 u System (6.6) w 2 u System (6.4) w 1 u System (6.3) w 1 u System (6.2) Figure 7. Desingularization of the origin of system (6.1) with c1= 0. Dicritical case (5), c2>0. 6.2. Case c1zero. We consider system (6.1) and do the same variable change that we did in the case with c1= 0, the result is obviously system (6.2) but taking c1= 0, i.e. ˙u= (c0−a0)u3w2 1+c3(µ+ 1)u3w1+c2(µ+ 1)u3, ˙w1=−c0u2w3 1−c3u2w2 1−c2u2w1. (6.7) In this case we can cancel a common factor u2getting the system ˙u= (c0−a0)uw2 1+c3(µ+ 1)uw1+c2(µ+ 1)u, ˙w1=−c0w3 1−c3w2 1−c2w1, (6.8) for which we must study the singular points on the exceptional divisor, i.e. on the straight line u= 0. The origin S0= (0,0) is always a singular point. The other singular points on this line are those for which w1is a solution of c0w2 1+c3w1+c2= 0. If c0= 0 and c2 3>4c0c2then S1= (0,−(Rc+c3)/(2c0)) and S2= (0,(Rc−c3)/(2c0)) are singular points. If c0= 0 and c2 3= 4c0c2, then S3= (0,−c3/2c0)is a singular point, and finally, if c0and c3are non-zero, S4= (0,−c2/c3)is a singular point. Case Conditions Singular points Ac0= 0,c3= 0.S0. Bc0= 0,c3= 0.S0,S4. Cc0= 0,c2 3<4c0c2.S0. Dc0= 0,c2 3= 4c0c2.S0,S3. Ec0= 0,c2 3>4c0c2.S0,S1,S2. Table 8. Cases with the singular points on the exceptional divisor of system (6.8). In summary we shall study the five cases given in Table 8. For doing this we study separately the local phase portrait of each singular point assuming in each case the necessary hypothesis for its existence. The singular points S0,S1and S2are hyperbolic. S0is a saddle if µ > −1, a stable node if c2>0and µ < −1, and an unstable node if c2<0and µ < −1.S1is a saddle if c0(a0+c0µ)<0, a stable node if c0>0and (a0+c0µ)>0, and an unstable 24 E. DIZ-PITA, J. LLIBRE AND M.V. OTERO-ESPINAR node if c0<0and (a0+c0µ)<0.S2is a saddle if c0(a0+c0µ)(Rc−c3)<0, a stable node if c0(Rc−c3)>0and (a0+c0µ)>0, and an unstable node if c0(Rc−c3)<0 and (a0+c0µ)<0. The singular point S3is a semi-hyperbolic saddle-node and finally, S4is a hyperbolic saddle if c2>0, and a hyperbolic stable node if c2<0. Using these informations we study the next cases from the five given in Table 8. First of all we study case (A) in which the only singular point is the origin, so we have the next three possibilities. If c0=c3= 0 and µ > −1, then S0is a saddle. In order to determine the phase portrait around the w1-axis for system 6.8, we must fix the sign of c2, which determines the sense of the flow along the axes. Considering c2>0we get the phase portrait given in Figure (1)(L19), and with c2<0we obtain the phase portrait (L20). If c0=c3= 0,µ < −1and c2>0, then S0is a stable node, and we obtain the phase portrait (L21) of Figure (1). If c0=c3= 0,µ < −1and c2<0, then S0is an unstable node, and we get the phase portrait (L22) of Figure (1). In case (B), fixed the phase portrait of S4, only two phase portraits will be possible for the origin, as the sign of c2is determined, and so we get the four following cases. If c0= 0,c3= 0,µ > −1and c2>0, then S0and S4are both saddle points and from the blowing-down we obtain the phase portrait (L23) of Figure (1). If c0= 0,c3= 0,µ > −1and c2<0, then S0is a saddle and S4a stable node. We obtain the phase portrait (L24) of Figure (1). If c0= 0,c3= 0,µ < −1and c2>0, then S0is a stable node and S4a saddle. We obtain the phase portrait (L25) of Figure (1). If c0= 0,c3= 0,µ < −1and c2<0, then S0is an unstable node and S4a stable node. We obtain the phase portrait (L26) of Figure (1). Again in case (C) the only singular point is the origin so we distinguish three cases, and obtain the same local phase portrait that in case (A), but under different conditions. If c0= 0,c2 3<4c0c2and µ > −1, then S0is a saddle. Attending to the sign of c2, which determines the sense of the flow on the axes, we consider the following cases: if c2>0we obtain the phase portrait (L19) of Figure (1), and if c2<0we obtain the phase portrait (L20) of Figure (1). If c0= 0,c2 3<4c0c2,µ < −1and c2>0, then S0is a stable node. We obtain the phase portrait (L21) of Figure (1). If c0= 0,c2 3<4c0c2,µ < −1and c2<0, then S0is an unstable node. We obtain the phase portrait (L22) of Figure (1). In case (D) apart from the origin, there exists the singular point S3, which is always a saddle node, so again we get only three cases. If c0= 0,c2 3= 4c0c2and µ > −1, then S0is a saddle and S3a saddle-node. We must distinguish four subcases according to the signs of c0and a0+c0µ, which determine the position of the saddle-node S3and its sectors. If c0>0and a0+c0µ > 0we obtain the phase portrait (L27) of Figure (1), if c0>0and a0+c0µ < 0we get the phase portrait (L28) of Figure (1), if c0<0 A CLASS OF KOLMOGOROV SYSTEMS 25 and a0+c0µ > 0we have the phase portrait (L29), and if c0<0and a0+c0µ < 0 the phase portrait (L30). If c0= 0,c2 3= 4c0c2,µ < −1and c2>0, then S0is a stable node and S3 a saddle-node. We distinguish two subcases setting the sign of a0+c0µwhich determines the position of the sectors of the saddle-node S3. If a0+c0µ > 0we obtain the phase portrait (L31) of Figure (1), and if a0+c0µ < 0we obtain the phase portrait (L32) of Figure (1). If c0= 0,c2 3= 4c0c2,µ < −1and c2<0, then S0is an unstable node and S3 a saddle-node. The only possibility is that a0+c0µ > 0, and we obtain the phase portrait (L33) of Figure (1). In case (E) there exist three singular points, with three possible phase portraits for each of them, however, many of the combinations are not possible, and only 13 cases will be feasible. First, due to the conditions which define the local phase portrait in each singular point, it is obvious that if S1is a stable node, then S2cannot be an unstable node, and if S1is an unstable node, S2cannot be a stable node, due to the sign of a0+c0µ. If S0and S2were stable nodes and S1a saddle, the conditions c2>0,Rc−c3<0, and c0<0will hold. Squaring both terms in the condition Rc< c3we obtain c2 3−4c0c2< c2 3, and then c0c2>0, which is a contradiction. The same reasoning is valid in the next two cases. If S0and S2are unstable nodes and S1a saddle, then the conditions c2<0, Rc−c3<0and c0>0hold, and if S0is an unstable node, S1a stable node and S2a saddle, then the same three conditions hold. If S0,S1and S2are stable nodes, the conditions c2>0,Rc−c3>0and c0>0hold. Now we take condition Rc< c3and squaring both terms we obtain c2 3−4c0c2< c2 3, and then c0c2>0, which is a contradiction. If S0is a stable node and S1an unstable node, the conditions µ < −1,c0<0 and a0+c0µ < 0hold. Then according to the signs of c0and µwhich are fixed, a0<−c0µ < 0which contradicts the hypothesis (H2). The same reasoning is valid if S0and S1are unstable nodes, because the same conditions hold. Now we will study the feasible cases. (E1) If c0= 0,c2 3>4c0c2,µ > −1,c0(a0+c0µ)(2c0c2−c2 3−c3Rc)>0and c0(a0+c0µ)(Rc−c3)(2c0c2−c2 3+c3Rc)>0, then S0,S1and S2are saddles. We must distinguish two subcases depending on the position of the singular points S1and S2on the w1-axis. First if S1is on the negative w1-axis and S2on the positive w1-axis, that corresponds with conditions c0>0,Rc−c3>0and c2<0, we obtain the phase portrait (L34) of Figure 1. Note that if Rc−c3>0, the singular points S1and S2are one on the positive part of the axis and the other on the negative part, but in any case the absolute value of the second coordinate of S1is greater or equal than the absolute value of the second coordinate of S2, and this determines the relation between the slopes of orbits in the phase portraits. Conversely if we have S1on the positive w1-axis and S2on the negative one, i.e. under the conditions c0<0,Rc−c3>0and c2>0, we obtain the phase portrait (L35) of Figure (1). (E2) If c0= 0,c2 3>4c0c2,µ > −1,c0(a0+c0µ)(2c0c2−c2 3−c3Rc)>0, c0(Rc−c3)>0and (a0+c0µ)(2c0c2−c2 3+c3Rc)<0, then S0and S1 32 E. DIZ-PITA, J. LLIBRE AND M.V. OTERO-ESPINAR Subcase 1 Subcase 2 Subcase 3 Figure 15. Possible global phase portraits in case 4.2 with c1= 0, µ < −1and a0+c0µ < 0. The same methods that we have used in the previous cases for determine which of the global phase portraits are realizable, must be used in some other cases, namely, 1.17, 1.18 with µ≥ −2, 1.19, 2.7, 3.3, 3.4 with µ≥ −2, 3.5 with µ > 0, and finally 4.1 with c1= 0,µ > −1and a0+c0µ < 0. A CLASS OF KOLMOGOROV SYSTEMS 33 Case Conditions O1O2Global 1.1 L12 Unstable node G1 1.2 L3 Stable node G2 1.3 µ < −1L8 Saddle G3 µ∈(−1,0) L13 Unstable node G4 1.4 µ < −2L1 Saddle G5 µ∈(−1,0) L4 Saddle G7 µ∈(−2,−1) L15 Saddle G7 µ=−2L17 Saddle G8 1.5 L12 Unstable node G9 1.6 L3 Stable node G10 1.7 µ∈(−1,0) L10 Stable node G11 µ < −1L11 Saddle G12 1.8 µ < −2L2 Saddle G13 µ∈(−1,0) L7 Unstable node G14 µ∈(−2,−1) L16 Saddle G15 µ=−2L18 Saddle G16 1.9 L9 Stable node G19 1.10 L6 Unstable node G20 1.11 L12 Unstable node G17 1.12 µ < −1L8 Saddle G21 µ∈(−1,0) L13 Unstable node G22 1.13 L3 Stable node G18 1.14 µ < −2L1 Saddle G23 µ∈(−1,0) L4 Stable node G24 µ∈(−2,−1) L15 Saddle G25 µ=−2L17 Saddle G26 1.15 L12 Unstable node G27 1.16 µ < −1L8 Saddle G35 µ∈(−1,0) L13 Unstable node G36 1.17 L3 Stable node G28 1.18 µ < −2L1 Saddle G37 µ∈(−1,0) L4 Stable node G38 µ∈(−2,−1) L15 Saddle G39 µ=−2L17 Saddle G40 1.19 µ∈(−1,0) L10 Saddle G29 µ < −1L11 Saddle G30 1.20 µ < −2L2 Saddle G31 µ∈(−1,0) L7 Unstable node G32 µ∈(−2,−1) L16 Saddle G33 µ=−2L18 Saddle G34 2.1 L46 Stable node G41 2.2 L47 Unstable node G42 34 E. DIZ-PITA, J. LLIBRE AND M.V. OTERO-ESPINAR Case Conditions O1O2Global 2.3 µ= 0 L14 Unstable node G43 c1= 0,µ > −1L36 c1= 0,µ < −1L43 Saddle G44 2.4 µ= 0 L5 Stable node G45 c1= 0,µ > −1L35 Stable node G46 c1= 0,µ < −1L39 Saddle G47 2.5 µ= 0 L14 Unstable node G48 c1= 0,µ > −1L45 c1= 0,µ < −1L44 Saddle G49 2.6 µ= 0 L5 Stable node G50 c1= 0,µ > −1L38 Stable node G51 c1= 0,µ < −1L41 Saddle G52 2.7 µ > −1L37 Stable node G53 µ < −1L40 Saddle G54 2.8 µ > −1L34 Unstable node G55 µ < −1L42 Saddle G56 2.9 µ= 0 L14 Unstable node G57 c1= 0,µ > −1L24 c1= 0,µ < −1L26 Saddle G58 2.10 µ= 0 L5 Stable node G59 c1= 0,µ > −1L23 Stable node G60 c1= 0,µ < −1L25 Saddle G61 3.1 L12 Unstable node G62 3.2 µ > 0L6 Unstable node G63 µ < −1L8 Stable node G64 µ∈(−1,0) L13 Unstable node G65 3.3 µ > 0L3 Stable node G66 µ < −1L11 Stable node G68 µ∈(−1,0) L10 Stable node G67 3.4 µ < −2L1 Saddle G69 µ∈(−1,0) L4 Stable node G70 µ∈(−2,−1) L15 Saddle G71 µ=−2L17 Saddle G72 3.5 µ > 0L3 Stable node G7 µ∈(−1,0) L10 Stable node G74 µ < −1L11 Saddle G75 3.6 L12 Unstable node G76 3.7 µ < −1L8 Saddle G77 µ∈(−1,0) L13 Unstable node G78 3.8 L3 Stable node G79 3.9 µ < −2L1 Saddle G80 µ∈(−1,0) L4 Stable node G81 µ∈(−2,−1) L15 Saddle G82 µ=−2L17 Saddle G83 A CLASS OF KOLMOGOROV SYSTEMS 35 Case Conditions O1O2Global 4.1 µ= 0 L14 Unstable node G84 c1= 0,µ > −1,a0+c0µ > 0L29 c1= 0,µ > −1,a0+c0µ < 0L30 Unstable node G85 c1= 0,µ < −1L33 Saddle G86 4.2 µ= 0 L5 Stable node G87 c1= 0,µ > −1,a0+c0µ > 0L27 Stable node G88 c1= 0,µ > −1,a0+c0µ < 0L28 Stable node G89 c1= 0,µ < −1,a0+c0µ > 0L31 Saddle G90 c1= 0,µ < −1,a0+c0µ < 0L32 Saddle G91 4.3 µ= 0 L14 Unstable node G92 c1= 0,µ > −1L20 c1= 0,µ < −1L22 Saddle G93 4.4 µ= 0 L5 Stable node G94 c1= 0,µ > −1L19 Stable node G95 c1= 0,µ < −1L21 Saddle G96 5.1 µ > 0L3 Stable node G97 µ∈(−1,0) L10 Stable node G98 µ < −1L11 Saddle G99 5.2 L12 Unstable node G76 5.3 µ > 0L6 Unstable node G100 µ < −1L8 Saddle G77 µ∈(−1,0) L13 Unstable node G78 5.4 µ > 0L3 Stable node G79 µ∈(−1,0) L10 Stable node G101 µ < −1L11 Saddle G102 5.5 µ < −2L1 Saddle G80 µ∈(−1,0) L4 Stable node G81 µ∈(−2,−1) L15 Saddle G82 µ=−2L17 Saddle G83 6.1 µ= 0 L14 Unstable node G92 c1= 0,µ > −1L20 c1= 0,µ < −1L22 Saddle G93 6.2 µ= 0 L5 Stable node G94 c1= 0,µ > −1L19 Stable node G95 c1= 0,µ < −1L21 Saddle G96 Table 9. Classification of global phase portrais of system (1.3). 36 E. DIZ-PITA, J. LLIBRE AND M.V. OTERO-ESPINAR (G1) [R=8, S=21] (G2) [R=6, S=19] (G3) [R=5, S=18] (G4) [R=6, S=19] (G5) [R=5, S=18] (G6) [R=4, S=17] (G7) [R=6, S=19] (G8) [R=7, S=20] (G9) [R=7, S=20] (G10) [R=6, S=19] (G11) [R=6, S=19] (G12) [R=5, S=18] (G13) [R=5, S=18] (G14) [R=4, S=17] (G15) [R=6, S=19] (G16) [R=7, S=20] (G17) [R=8, S=23] (G18) [R=6, S=21] (G19) [R=7, S=22] (G20) [R=6, S=21] (G21) [R=5, S=20] (G22) [R=6, S=21] (G23) [R=5, S=20] (G24) [R=4, S=19] (G25) [R=6, S=21] (G26) [R=7, S=22] (G27) [R=9, S=24] (G28) [R=7, S=22] (G29) [R=6, S=21] (G30) [R=6, S=21] (G31) [R=5, S=20] (G32) [R=4, S=19] (G33) [R=6, S=21] (G34) [R=7, S=22] (G35) [R=5, S=20] A CLASS OF KOLMOGOROV SYSTEMS 37 (G36) [R=6, S=21] (G37) [R=5, S=20] (G38) [R=5, S=20] (G39) [R=6, S=21] (G40) [R=7, S=22] (G41) [R=8, S=21] (G42) [R=7, S=20] (G43) [R=6, S=19] (G44) [R=6, S=19] (G45) [R=4, S=17] (G46) [R=6, S=19] (G47) [R=8, S=21] (G48) [R=6, S=19] (G49) [R=6, S=19] (G50) [R=5, S=18] (G51) [R=6, S=19] (G52) [R=4, S=17] (G53) [R=6, S=19] (G54)[R=7, S=20] (G55) [R=4, S=17] (G56) [R=6, S=19] (G57) [R=6, S=17] (G58) [R=6, S=17] (G59) [R=4, S=15] (G60) [R=4, S=15] (G61) [R=6, S=17] (G62) [R=8, S=21] (G63) [R=6, S=19] (G64) [R=5, S=18] (G65) [R=6, S=19] (G66) [R=7, S=20] (G67) [R=6, S=19] (G68) [R=6, S=19] (G69) [R=5, S=18] (G70) [R=5, S=18] 38 E. DIZ-PITA, J. LLIBRE AND M.V. OTERO-ESPINAR (G71) [R=6, S=19] (G72) [R=7, S=20] (G73) [R=6, S=17] (G74) [R=6, S=17] (G75) [R=5, S=16] (G76) [R=6, S=17] (G77) [R=3, S=14] (G78) [R=4, S=15] (G79) [R=5, S=16] (G80) [R=3, S=14] (G81) [R=3, S=14] (G82) [R=4, S=15] (G83) [R=5, S=16] (G84) [R=6, S=17] (G85) [R=5, S=16] (G86) [R=6, S=17] (G87) [R=5, S=16] (G88) [R=4, S=15] (G89) [R=4, S=15] (G90) [R=4, S=15] (G91) [R=5, S=16] (G92) [R=4, S=13] (G93) [R=4, S=13] (G94) [R=3, S=12] (G95) [R=2, S=11] (G96) [R=2, S=11] (G97) [R=5, S=14] (G98) [R=4, S=13] (G99) [R=3, S=12] (G100) [R=5, S=16] (G101) [R=4, S=15] (G102) [R=5, S=16] Figure 16. Global phase portraits of system (1.3) in the Poincaré disc. A CLASS OF KOLMOGOROV SYSTEMS 39 Remark 7.1. Global phase portraits (G43), (G48), (G57), (G84) and (G92) can appear both under condition c1= 0 or under c1= 0 so, as we are interested in the topological classification we represent only the non-symmetric case respect to z-axis, but also the symmetric is possible. The same situation occurs with the global phase portraits (G9), (G13)–(G16), (G18), (G19), (G23)–(G27), (G31)–(G36), (G41), (G45)–(G49), (G55), (G56), (G76)–(G83), (G92)–(G102), which can appear under the condition c3= 0 or c3= 0 so they can present the symmetric or the nonsymmetric form respect to x-axis, altought we only represent one of them because we are only interested on the topological classification. Acknowledgements. 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