Quadratic vector fields with a weak focus of third order

Publicacions Matem`atiques, Vol 41 (1997), 7–39. QUADRATIC VECTOR FIELDS WITH A WEAK FOCUS OF THIRD ORDER Joan C. Art´ es and Jaume Llibre Abstract We study phase portraits of quadratic vector fields with a weak focus of third order at the origin. We show numerically the existence of at least 20 different global phase portraits for such vector fields coming from exactly 16 different local phase portraits available for these vector fields. Among these 20 phase portraits, 17 have no limit cycles and three have at least one limit cycle. 1. Introduction A vector field X:R2→R2of the form X=(P,Q) where P= aijxiyjand Q=bijxiyj,0≤i+j≤n, is called a planar polynomial vector field of degree nif i+j=n(|aij|+|bij|)= 0. The M=(n+1)(n+2) real numbers aij,bij are called the coefficients of X. The space of these vector fields, endowed with the structure of affine RM-space in which X is identified with the M-tuple (a00,a 10,... ,a 0n,b 00,b 10,... ,b 0n)ofits coefficients, is denoted by Pn(R2). In particular, the polynomial vector fields of degree 2 are called quadratic vector fields. Aweak focus of a planar vector field is defined as a critical point such that the real part of its eigenvalues is zero. Several levels of weakness are defined for these points. In particular, for quadratic vector fields, we may have up to three levels of weakness. For more details see Section 3. The main open problem in the study of the planar polynomial vector fields is the number of limit cycles and their distribution, the so called Hilbert’s sixteen problem, see [Hi]. This problem remains open even for the quadratic vector fields, where it seems that the answer must be the following: A quadratic vector field has at most 4 limit cycles, and when The authors are partially supported by a DGICYT grant number PB93-0860. 8J. C. Art´ es, J. Llibre they exist, three are surrounding one focus and the other is surrounding a different focus. All the analytic examples realizing these four limit cycles are obtained perturbing a quadratic system having a weak focus of third order. This is the main reason which motivates our classification of the quadratic vector fields with a weak focus of third order. Prior to 1950, there was some interest in the qualitative theory of quadratic vector fields, but a real impetus was given to the development of this subject in the fifties. In fact, more than eight hundred papers have been published on this subject, see Reyn [R1]. The Poincar´e compactification of X∈P n(R2) is defined to be the unique analytic vector field p(X)∈P n(S2) tangent to the sphere S2= {(x, y, z)∈R3:x2+y2+z2=1}, whose restriction to the northern hemisphere S2 +={(x, y, z)∈S2:z>0}is given by zn−1Df+(X), where f+is the central projection from R2≡{(x, y, 1) ∈R3:(x, y)∈R2}to S2 +, defined by f+(x, y)=(x, y, 1)/(x2+y2+1)1/2. The projection of the closed northern hemisphere of S2on y3= 0 under (y1,y 2,y 3)→ (y1,y 2) is called the Poincar´e disc. See Section 2 for more details. Let p(X)∈P n(S2). We define the local phase portrait around the critical points as the union of the set of all critical points of p(X) (finite and infinite) together with their local phase portrait. We will say that two vector fields p(X1) and p(X2)∈P n(S2) have equivalent local phase portrait if there is a homeomorphism of S2leaving the equator S1invariant and such that restricted to some neighborhood of each finite or infinite critical points carries orbits of the flow induced by p(X1) onto orbits of the flow induced by p(X2). Two vector fields p(X1) and p(X2)∈P n(S2) having equivalent local phase portraits may have non-equivalent global phase portraits. This is due to the fact that the global behavior of their separatrices (different from critical points and limit cycles) is different, or to the fact that they can have different number or distribution of limit cycles. In this paper we get the local phase portrait around the critical points on the Poincar´e disc of all the quadratic polynomial vector fields with one weak focus of third order. More precisely our main result can be stated as follows. Theorem 1.1. A quadratic vector field with a weak focus of third order may have up to 16 different local phase portraits around the critical points. Two of these local phase portraits are realizable at least by three different global phase portraits. The remainder 14 different local phase portraits around the critical points are realized at least by one global phase portrait. Quadratic systems with a weak focus of third order 9 After this we characterize numerically the global phase portraits from these local phase portrait by finding realizable cases from each one of them. Numerical result. There are at least 20 phase portraits for the quadratic vector fields having a weak focus of third order. All these phase portraits are shown in Figure 1.1. The study is divided in four different parametric normal forms. In the study of these parametric families we will find several phase portraits having a center instead of a weak focus. The different phase portraits for such systems having a center are given in Figure 1.2 and are denoted by V#. We use the classification of Vulpe [V]. This paper is an improved version of the Master Ph. D. of the first author under the direction of the second author [A1]. 2. Basic definitions and results In this section we introduce some notation and definitions that we will need later. Given the differential system (2.1) x=P(x0,y 0), y=Q(x0,y 0), where Pand Qare two real analytic functions in the variables xand y,we say that (x0,y 0)isacritical or singular point if P(x0,y 0)=Q(x0,y 0)= 0. In order to classify it we denote by ∆(x0,y 0) and by ρ(x0,y 0), the determinant and the trace of the linear part of system (2.1) at the critical point (x0,y 0) respectively. Set also δ(x0,y 0)=ρ(x0,y 0)2−4∆(x0,y 0). If ∆(x0,y 0)= 0 then we say that the critical point is non-degenerated and elementary; and we can apply the results of [ALGM] (see Chapter IV) in order to determine its local behavior. If ∆(x0,y 0)=0butρ(x0,y 0)= 0 then we say that the critical point is degenerated and elementary; and we apply Theorem 65 of [ALGM] in order to determine its local behavior. If ∆(x0,y 0) = 0 and ρ(x0,y 0) = 0 but the matrix of the linear part of (2.1) at (x0,y 0) is not identically zero, then we say that the critical point is degenerated and nilpotent; and we apply the result of [An] in order to determine its local behavior. If the matrix of the linear part of (2.1) at (x0,y 0) is identically zero, then we say that the critical point is degenerated with linear part equal to zero, and we need to make a particular study around this point by using blow up’s in order to determine its local behavior [ALGM]. 10 J. C. Art´ es, J. Llibre W1 W5 W9 W13 W17 W18 W14 W10 W6 W2W3 W7 W11 W15 W19 W4 W8 W12 W16 W20 Figure 1.1. Phase portraits Wifor quadratic vector fields with a weak focus of third order. Quadratic systems with a weak focus of third order 11 V1V8V9 V10 V12 V16 V19 V20 V21 V23 V24 V26 V32 Figure 1.2. Phase portraits Vifor quadratic vector fields with a center which appear during the study of systems with a weak focus of third order. 12 J. C. Art´ es, J. Llibre For X∈P n(R2) the Poincar´e compactified vector field p(X)corresponding to Xis a vector field induced on S2as follows (see, for instance [G], [S] and [ALGM]). Let S2={y=(y1,y 2,y 3)∈R3:y2 1+y2 2+y2 3=1} (the Poincar´e sphere) and TyS2the tangent space to S2at point y. Consider the central projections f+:T(0,0,1)S2→S2 +={y∈S2:y3>0} and f−:T(0,0,1)S2→S2 −={y∈S2:y3<0}. These maps define two copies of X, one in the northern hemisphere and the other in the southern hemisphere. Denote by Xthe vector field defined on S2except on its equator S1={y∈S2:y3=0}by Df+◦Xand Df−◦X. Clearly S1 is identified to the infinity of R2. In order to extend Xto an analytic vector field on S2(including S1) it is necessary that Xsatisfies suitable hypotheses. In the case that X∈P n(R2), the Poincar´e compactification p(X)ofXis the only analytic extension of yn−1 3Xto S2. The set of all compactified vector fields p(X) with X∈P n(R2) is denoted by Pn(S2). For the flow of the compactified vector field p(X), the equator S1is invariant. On S2\S1there are two symmetric copies of X, and knowing the behavior of p(X) around S1, we know the behavior of Xat infinity. The projection of the closed northern hemisphere of S2on y3= 0 under (y1,y 2,y 3)→ (y1,y 2) is called the Poincar´e disc, and it is denoted by D2. As S2is a differenciable manifold for computing the expression of p(X), we can consider the following six local charts Ui={y∈S2: yi>0}, and Vi={y∈S2:yi<0}where i=1,2,3, and the diffeomorphisms Fi:Ui→R2and Gi:Vi→R2which are the inverses of the central projections from the vertical planes tangents at points (1,0,0),(−1,0,0),(0,1,0),(0,−1,0),(0,0,1),(0,0,−1) respectively. We denote by z=(z1,z 2) the value of Fi(y)orGi(y) for any i=1,2,3. So zrepresents different things according to the local chart under consideration. Some easy computations give for p(X) the following expressions (2.2) zn 2·∆(z)Q1 z2 ,z1 z2−z1P1 z2 ,z1 z2,−z2P1 z2 ,z1 z2, (2.3) zn 2·∆(z)Pz1 z2 ,1 z2−z1Qz1 z2 ,1 z2,−z2Qz1 z2 ,1 z2, ∆(z)[P(z1,z 2),Q(z1,z 2)] , in the local charts U1,U2and U3respectively. Here ∆(z)=(z12+z22+ 1)−n−1 2. The expression for Viis the same as that for Uiexcept for a multiplicative factor (−1)n−1. In these coordinates and in the local charts Uiand Vifor i=1,2, z2= 0 denotes always the points of S1.In what follows we omit the factor ∆(z) by rescaling the vector field p(X). Quadratic systems with a weak focus of third order 13 So, in each local chart we obtain a polynomial vector field. We denote by Pn(S2) all the polynomial vector fields p(X)onS2defined as above and endowed with the coefficient topology. A critical point qof X∈P n(R2) is called infinite (respectively finite) if it is a critical point of p(X)inS1(respectively in S2\S1). Then to compute the infinite critical points of Xwe take the nonzero terms of the expressions (2.2) and (2.3) with z2= 0 and we obtain F(z1)=Qn(1,z 1)−z1Pn(1,z 1),(2.4) G(z1)=Pn(z1,1) −z1Qn(z1,1),(2.5) respectively, where Pnand Qnare the homogeneous part of degree nof Pand Q. Thus, the infinite critical points of Xare the points (z1,0) satisfying F(z1)=0 if(z1,0) ∈U1, G(z1)=0 if(z1,0) ∈U2. We remember that the equation F(z1) = 0 gives us all the infinite critical points of Xexcept, perhaps, the origin of coordinates of U2. In order to study the local behavior at the infinite critical points (z1,0) we need its linear part given by the Jacobian matrix (2.6) dF(z1) dz1∗ 0−Pn(1,z 1)in U1, and (2.7) dG(z1) dz1∗ 0−Qn(z1,1) in U2. 3. Weak focus of third order We will start from the expression given by Ye Yanqian (see [Y1]or [Y2]) of the quadratic vector fields which have a weak focus at the origin: x=−y+lx2+mxy +ny2, y=x(1 + ax +by). From the papers of Bautin [B], we know how to classify a weak focus in four types, but here we use the notation of Li [Li1]. More precisely, given the constants: 14 J. C. Art´ es, J. Llibre L1=m(l+n)−a(b+2l), L2=ma(5a−m)[(l+n)2(n+b)−a2(b+2l+n)], L3=ma2[2a2+n(l+2n)][(l+n)2(n+b)−a2(b+2l+n)], we say that the origin of the above system is (1) a weak focus of first order if L1=0, (2) a weak focus of second order if L1= 0 and L2=0, (3) a weak focus of third order if L1=L2= 0 and L3=0, (4) a center if L1=L2=L3=0. A center is a singular point such that all the orbits in a neighborhood of it are periodic. We also know that if the origin is a focus of order k (with k=1,2,3), its stability is given by the sign of Lk; so we have that the origin is a stable weak focus of order k(respectively unstable) if and only if Lk<0 (respectively >0). If a quadratic system has a center, then it is integrable, see [C] and [LS]. As we want to study the quadratic vector fields with a weak focus of third order, we must force certain conditions to the coefficients l,m,n,a and b. In concrete, L1=L2= 0, and L3= 0. So we have that: m=0, a=0, 2a2+n(l+2n)=0, (l+n)2(n+b)−a2(b+2l+n)=0. This, with the fact that L2= 0, implies that m=5a. Substituting in L1= 0 we get that 5a(l+n)−a(b+2l)=0. Asa= 0 we have b=3l+5n. In short, a quadratic vector field with a weak focus of third order at the origin may be written in the form x=−y+lx2+5axy +ny2, y=x+ax2+(3l+5n)xy, with L3=5a3[2a2+n(l+2n)][(l+n)2(6n+3l)−a2(5l+6n)] = 0. This vector field depends on three parameters; we can simplify it to two. If n= 0, then with the change x=n−1X,y=n−1Y, and naming l=ln−1and a=an−1, we get (3.1) X=−Y+lX2+5aXY +Y2, Y=X+aX2+(3l+5)XY. Quadratic systems with a weak focus of third order 15 If n= 0 and a= 0, then with the change x=a−1X,y=a−1Y, and naming l=la−1,weget (3.2) X=−Y+lX2+5XY, Y=X+X2+3lXY. If n=a= 0, then L3= 0, and the origin is a center. If l= 0, then with the change x=l−1X,y=l−1Y, we get (3.3) X=−Y+X2, Y=X+3XY. Finally, if l= 0, then (3.4) X=−Y, Y=X. These two last vector fields are integrable and so we know their phase portraits corresponding to phase portraits V12 and V1respectively. So we only have to study vector fields (3.1) and (3.2). In order to do that, we will split the plane of parameters (a,l) of the vector field (3.1), or the line of parameters (l) of the vector field (3.2) in convenient regions (from now on we omit the primes and change X, Y by x, y). These regions are limited by curves which mean when critical points, finite or infinite, appear or disappear, when any of the eigenvalues of the Jacobian matrix at a critical point is zero, or when the vector field is integrable. With this we can control the local behavior at critical points of all these vector fields. In order to obtain the global phase portraits we need to study the existence of limit cycles and the different αand ω-limit sets for the separatrices of the systems. 4. Study of the normal form 3.1 The phase portraits of Figure 1.1 represent all the local phase portraits for normal form (3.1) changing aand l(except phase portraits W8and W9which are only realizable under normal form (3.2)). These local phase portraits have been included in a global phase portrait which is provided from it and that we have checked numerically its existence. The plane R2with parameters aand lhas been divided in a collection of regions of dimensions 0, 1 and 2, so that for all the points (a, l) belonging to the same region, the local phase portrait is the same. 22 J. C. Art´ es, J. Llibre (vi) If we are in the conditions of Lemma 4.2 (vi), then the critical point Sis an unstable node if l(3l+5)2−3a2(5l+8) >0and l>0; a stable node if l(3l+5) 2−3a2(5l+8)<0and l<−5/3, and a saddle in the rest of the cases; and the point Ris an unstable node if l(3l+5) 2−3a2(5l+8) >0and −8/5>l>−2;or l(3l+5) 2−3a2(5l+8)<0and −5/3>l>−2, and a saddle in the rest of the cases. Proof: (i) It comes immediately from Section 2. (ii) When the point is elementary the computations are easy. For the degenerate cases it is easy to apply the corresponding results mentioned in Section 2. It is easy to see that next curves are important: ∆(0,1) = −(3l+6)=0andδ(0,1) =25a2+ 12(l+2)=0. In order to prove (iii) and (iv) we must carry out a long computations to reduce our equations to a single parameter, but they do not add any theoretical complexity. Moreover, as we are in a curve, we may get the same results studying the adjacent regions and applying continuity criteria. The case (v) is easier because the conditions are linear. To demonstrate (vi) we must carry out again a long computations to get next results for S: ∆(S)=2(9a2−3l(l+ 2))(3l+5)+6a(2l+3) 9a2−3l(l+2) l(3l+5) 2−3a2(5l+8) , ρ(S)=5 10al +3al2+3a3+9a+(3l2+8l+5−a2)9a2−3l(l+2) l(3l+5) 2−3a2(5l+8) . Then it is easy to decide when we have a saddle. In the cases in which we have a focus or a node we must decide about its stability. In order to decide if we have a node or a focus we must find δ(S). δ(S)=(2a(−75a4+ 510a2l2+ 1466a2l+ 1014a2+9l4+ 306l3+ 1370l2 + 2150l+ 1125)9a2−3l(l−2) + 450a6+ 2265a4l2+ 8334a4l + 7740a4−324a2l4−1008a2l3+ 2014a2l2+ 9240a2l+ 7650a2 −27l6−414l5−2370l4−6300l3−7875l2−3750l)/ (l(3l+5) 2−3a2(5l+ 8))2. Quadratic systems with a weak focus of third order 23 If we make this equal to zero and eliminate the square root we get: −4860000a10l2−15552000a10l−12441600a10 + 5394375a8l4 +19808100a8l3+ 7978140a8l2−38256192a8l−35852544a8 −1282500a6l6−1077480a6l5+ 27567516a6l4+ 88464936a6l3 +81585216a6l2−12083904a6l−36288000a6−89910a4l8 −2411856a4l7−13276044a4l6−21397488a4l5+ 25879082a4l4 +123421440a4l3+ 134411400a4l2+ 36072000a4l−12960000a4 −18468a2l10 −390744a2l9−3813372a2l8−20058552a2l7 −59800380a2l6−98862600a2l5−73522500a2l4+ 14055000a2l3 +58500000a2l2+ 27000000a2l−729l12 −22356l11 −299376l10 −2302560l9−11258550l8−36585000l7−80122500l6 −117000000l5−109265625l4−59062500l3−14062500l2=0. From this expression we have not managed to get any analytical conclusion but we know some numerical behaviors. So, for values (a, l) for which δ(S) is defined, we have seen that δ(S)>0 always, so we conjecture that if Sis not a saddle, then it must be a node. Similar functions and arguments work for the point R. Lemma 4.4. The normal form (3.1) has at infinity six critical points if we have that (a, l)∈{(a, l)∈R2: 125a4+a2(25l2+170l+262)+(2l+ 5)3>0}, four if we have (a, l)∈{(a, l)∈R2: 125a4+a2(25l2+ 170l+ 262) + (2l+5) 3=0,a=0}and two otherwise. Proof: From Section 2 we have that the point (0,1,0) of S2is never critical and so all the critical points will come from the equation: F(z)=z3+5az2−(2l+5)z−a=0. The discriminant of this polynomial is −(125a4+a2(25l2+ 170l+262)+(2l+5) 3), and the rest is immediate. We denote by K1,K2and K3with K1>K 2>K 3the three real roots of F(z) = 0. In studying the equation F(z) = 0 we see that if two roots disappear, the remainder one corresponds to the highest one, i.e. K1. We suppose that if we have two roots they will be K1and K2. 24 J. C. Art´ es, J. Llibre Lemma 4.5. The following statements hold. (i) If l(3l+5) 2−3a2(5l+8)>0and l>0, then K1and K3are stable nodes and K2is a saddle. (ii) If l(3l+5) 2−3a2(5l+8)=0and l>0, then K1and K3are stable nodes and K2is an elementary saddle-node, SN1, with the nodal behavior in the semiplane z2>0of the chart U1. (iii) If a=l=0, then K1and K3are stable nodes and K2is an elementary degenerated saddle. (iv) If l(3l+5) 2−3a2(5l+8)<0and l>−5/3, then K1and K3are stable nodes and K2is an unstable node. (v) If l(3l+5) 2−3a2(5l+8)=0and −8/5>l>−5/3, then K1is a stable node, K2is an unstable node and K3is an elementary saddle-node, SN1, with the hyperbolic component in the semiplane z2>0of the chart U1. (vi) If l(3l+5) 2−3a2(5l+8)>0and l<−8/5, then K1is a stable node, K2is an unstable node and K3is a saddle. (vii) If a=0and l=−5/3,K2is an unstable node, then K1is an elementary saddle-node, SN1, with the nodal component in the semiplane z2>0of the chart U1, and K3is an elementary saddlenode, SN1, with the hyperbolic component in the semiplane z2>0 of the chart U1. (viii) If l(3l+5) 2−3a2(5l+8)=0and l<−5/3, then K1is an elementary saddle-node, SN1, with the nodal component in the semiplane z2>0of the chart U1,K2is an unstable node, and K3is a saddle. (ix) If l(3l+5) 2−3a2(5l+8)<0,125a4+a2(25l2+ 170l+ 262) + (2l+5) 3>0and l<−5/3, then K1and K3are saddles and K2 is an unstable node. (x) If 125a4+a2(25l2+ 170l+ 262) + (2l+5) 3=0and a=0, then K1is a saddle and K2is an elementary saddle-node, SN2, with the nodal component in the semiplane z1>0of the chart U1. (xi) If 125a4+a2(25l2+ 170l+ 262) + (2l+5) 3<0, then K1is a saddle and its infinite separatrix is stable. (xii) If a=0and l=−5/2, then K1is an elementary degenerated saddle and its infinite separatrix is stable. Note that all the results of the Lemma 4.5 are refereed to the local chart U1. With the Poincar´e s compactification we may get the results for the chart V1. Quadratic systems with a weak focus of third order 25 Proof: The proof of the lemma implies to compute the roots of the third degree equation in an algebraic way and substituting them in the jacobian of the vector field written in the corresponding form of the chart U1. This means a long and tedious calculus. In other way, we may find out the same if we compute the curves of the degenerated points, that is, if we solve the systems: F(z)=0 F(z)=0,and F(z)=0 P2(1,z)=0. Isolating and substituting zwe get respectively 125a4+a2(25l2+ 170l+ 262) + (2l+5) 3= 0, and l(3l+5) 2−3a2(5l+ 8) = 0. These two curves and a= 0 limit some convenient regions. We now do a particular study of each region and we may generalize the results to the whole region. Now we have found the curves and points of bifurcation for the normal form (3.1) refereed to the number and nature of all critical points, finite or infinite. These curves limit a certain well-defined regions of dimension 2. We also see that these curves intersect among them in certain concrete points. These points differentiate the pieces of the curves such that different points in a same piece will imply different phase portraits. We must separate these curves in homeomorphic sets to the interval (0,1) such that they do not intersect each other. It remains the unexplained curve (C14) which we will comment later on. Now we see that these curves cannot intersect in a point different from the origin. The curve C1does not cross C2because C1increases with slope √3/3 and C2increases with slope 5/3. The curve C2does not cross C3because even though they increase with the same slope, C3begins in a slower position. The curve C4does not cross C5because C4is asymptotic to l=−5/6 and C5is asymptotic to l=−8/5. The curve C9does not cross C10 for the same reason that C2does not cross C3. The curve C10 does not cross C11 for the same reason that C1does not cross C2. The curve C11 does not cross C12 because C11 increases as a straight line and C12 increases as a parabola of coefficient −2. The curve C12 does not cross C13 because C13 increases as a parabola of coefficient −25/12 <−2. We note that with respect to the possible intersection between C13 and C15 the arguments of growing are not sufficient. In [QSC]itisproved 26 J. C. Art´ es, J. Llibre for our vector fields that, if we have an only infinite critical point, then the point (0,1) is always a focus. So, from Lemmas 4.3 and 4.4 C13 can not cross C15. We separate the points (a, l)∈R2in several regions so that we can assure that if two vector fields are in the same region, then they have the same number of critical points, either finite or infinite, with the same local phase portrait. In order to determine if two vector fields in one of these regions have topologically equivalent phase portraits, we need to know their limit cycles and the global behavior of their separatrices. The determination of the limit cycles for a polynomial vector field of degree nis an open problem. In fact, to know the maximum number of limit cycles in function of nis still open. In particular, for the case n=2 it seems that this maximum number is 4. An example with at least 4 limit cycles can be obtained perturbing the phase portraits corresponding to the region R15 of Figure 4.1. Such phase portraits have a limit cycle around the strong focus and three small limit cycles can bifurcate from the third order weak focus at the origin under small perturbations. As a result of the classification we are given, we obtain the regions where there exist this limit cycle around the strong focus. We only know that there must exist at least one limit cycle, but we cannot prove that it is unique. In order to prove this we have made a numerical study of the region R15. If we cut the region R15 with an horizontal line l= constant with l<−5/2, we find three interesting points: the point (0,l), the point where the line cuts the curve C15, and the point where the line cuts the curve C13. The point (0,l) corresponds to an integrable system where the only pair of saddles at infinity connect their separatrices, and the two finite critical points are centers. If we move to the right along our line, we have that the point (0,1) becomes a strong focus and a limit cycle appears around it. It appears when the unstable separatrix of the infinity disconnects from the stable one and tries to go to the point (0,1), forcing the appearance of a limit cycle because the point (0,1) is an unstable focus. We know that when we arrive to the curve C13 this limit cycle has disappeared because we cannot have a limit cycle around a node. As the point (0,1) does not change from unstable to stable anywhere close to where we are now, the limit cycle cannot die at this point and, so, it has to die in an infinite graphic in the same way that it has appeared. As usual a graphic of p(X)∈P n(S2) denotes a closed simple curve which is the union of a finite number of critical points and separatrices, and at least in one of the two sides of the curve is defined a return or Poincar´e map. Quadratic systems with a weak focus of third order 27 When we cross the curve C15 there appears a new infinite critical point which is an elementary saddle-node with nodal sectors in the two sides of the infinity; that is, it is of type SN2. The separatrix of the infinite saddle which rotated around the point (0,1) now goes to the nodal sector of the saddle-node which we have at U2, and the separatrix of this saddlenode is the one which goes to the point (0,1), but since this separatrix is stable, the limit cycle can disappear. More concretely, the previous limit cycle can dissapear at infinity when the saddle-node appears; or perhaps it can remain and a new limit cycle appears simultaneously with the saddle-node at infinity. The numerical results detect only the first possibility. For example, now we do the global study for a concrete region. For the systems of region R15 we know that we have two separatrices which come from the infinity, a stable one and an unstable one. The Poincar´eBendixson Theorem says that the separatrices of any system may only end at a critical point, a periodic orbit or a graphic. So, we could have that the stable separatrix of the infinity comes from the critical points (0,1), or (0,0), or from any periodic orbit that there may exist, or that it connects with the unstable separatrix at the infinity. In order to analyze these different possibilities we will use the following results. [Yu]: If a quadratic vector field has a limit cycle, then it surrounds one and only one critical point, and this point is a focus. Therefore the limit cycles cannot be anywhere in the plane. They may only be around the points (0,0) and (0,1). [Li2]: There are no limit cycles around a weak focus of third order. Then only around the point (0,1) we may have limit cycles. Moreover, the unstable separatrix of the infinity may not turn around the critical point (0,0) because this is an unstable weak focus, and it should have to turn around a limit cycle which we know it can not exist. [SP]: If in a quadratic vector field the separatrix of an infinite saddle connects with the symmetric one, then they form an invariant straight line. Consequently, we can see that the two separatrices do not connect. Studying the normal form (3.1) we see that we cannot have an invariant straight line of the form x=cbecause the north pole is not a critical point. It should be of the form y=cx+d, which would imply y=cxon the points of this straight line. So, it should have the following equalities: a+(3l+5)c=c(l+5ac +c2), 1+(3l+5)d=c(−c+5ad +2cd), 0=c(−d+d2). 28 J. C. Art´ es, J. Llibre This implies either c=0,ord=0,ord= 1. But d=0ord=1 imply that the line crosses the point (0,0) or (0,1) respectively, which is impossible because these points are focus. The possibility c= 0 tells us that a= 0 and the region R15 has no contact with this line. So, the only possible phase portrait for the systems of region R15 is W20 as it is shown in Figure 1.1 with an unknown number of limit cycles around the point (0,1). We may say that we will have at least one limit cycle because the point is unstable and the separatrix which turns around it is also unstable. A similar study should be made for each region so we limit the number of possible phase portraits in each one, but in this paper we will limit ourselves to give the local behavior and a phase portrait numerically found for each region, and will leave this study for latter. Only for the regions where we have a center, we determine analytically the global phase portrait (see the appendix). We also need the next two lemmas. Lemma 4.6. In all the regions where the point (0,1) is not a focus (i.e., except in the regions R13,C14,R15,C15,C16 and Q2) we cannot have any limit cycle. The proof is immediate from the results [Li2], [Yu] and from the Lemma 4.3. Lemma 4.7. For all (a, l)∈R15,R14 or C16, there exists at least one limit cycle around the focus (0,1). Proof: The lemma was proved for the systems defined by the region R15. We study now the systems of the region R14. Since the curve C14 is not defined yet, we may suppose that the region R13 includes for the moment, the regions R14 and C14. The same study we have done for the region R15 may be done now for R13. There are three possible ωlimits for the separatrix which comes from the infinite saddle of the local chart U1which are: The two nodes of the local chart U2or a limit cycle around the point (0,1) (it cannot be the point because it is unstable). For points close to P15, in the semiplane a>0 and in R13, it has been proved [QSD], that the separatrix of the infinite saddle turns around a limit cycle which has the critical point (0,1) in its interior. For many other points far from here, it is numerically clear that the ω-limit of the separatrix is the node in local chart U2which lays upwards of the saddle there. For continuity, there exists a saddle-to-saddle connection in a curve that we will denote by C14. Quadratic systems with a weak focus of third order 29 Numerically we have seen that the region R14 inside R13 and separated of this by the curve C14 (that its corresponding systems have at least one limit cycle) is quite bigger than the estimated analytically in [QSD], and it is of the form showed in Figure 4.2. We have not found the analytic expression of this curve, but numerically we may conjecture that it begins in the point (0,−2) and ends in the curve defined by the equation 125a4+a2(25l2+ 170l+ 262) + (2l+5) 3= 0, in a point of approximated coordinates (0.054684523,−2.4085) which we will denote by Q2, which produces that the latter curve splits in the arcs C15 and C16. Even though we do not know exactly the expressions of C14 and Q2we may guess the phase portraits of their corresponding systems by using continuity criteria. So, we conjecture that quadratic vector fields having a weak focus of third order can not have limit cycles anywhere except the corresponding ones to the regions R14,C16 and R15, and that those only have one limit cycle. 5. Study of the normal form (3.2) The phase portraits that we study now represent all the local behaviors we may get from the normal form (3.2) in changing l. These local behaviors have been drawn as a part of a global phase portrait which is provided from it but that we have checked numerically its existence. The line Rwith parameter lhas been divided in a series of regions of dimensions 0 and 1, so that for all the points lbelonging to a same region the local behaviors are the same. We denote by Band S, with the corresponding subindex to certain regions of dimension 1 and 0, respectively. In Lemma 5.1 we show that it is sufficient to study the half-line l≥0. From Lemmas 5.2, 5.3, 5.4 and 5.5, the points that separate regions will be: √3, 5/3 and 0. Table 5.1 shows all the regions in which we have divided the halfline l≥0, taking into account the number and the different local phase portraits of the finite and infinite critical points, following the same rules as in Section 4. It also gives the corresponding phase portrait for each region. No one of these cases may have limit cycles as we will prove in Lemma 5.4. The examples of the global phase portraits for each one of these regions are included in Figure 1.1 and 1.2. We have obtained them with the same 30 J. C. Art´ es, J. Llibre method as in Section 4. (√3,∞)B1→W8fdr, S, NA, SN1A. √3S1→W7fdr, snr, S, NA, SN1A. (5/3,√3) B2→W9fdr, s, S, NA, SN1A. 5/3S2→V16 c, s, SN1R, NA, SN1A. (0,5/3) B3→W4fda, s, s, NR, NA, SN1A. 0S3→V19 c, s, NR, NA, SS. Table 5.1 We must study the system (5.1) X=−y+lx2+5xy, Y=x+x2+3lxy. Lemma 5.1. Given the phase portrait of a vector field in the normal form (5.1) with l≥0, in order to get the phase portrait of (5.1) with −l we must make a symmetry respect to the y-axis and change the sense of all the orbits. The proof of this lemma is immediate. Now we study the number of finite critical points of system (5.1) in function of l. Lemma 5.2. For system (5.1) the following statements hold. (i) The point (0,0) is always critical. (ii) If l=0we have only another critical point, namely the point (−1,0). (iii) If l=5/3we have only another finite critical point namely the point with coordinates (1/4,−5/3/4). (iv) If l=√3we have only another finite critical point namely the point with coordinates (1/2,−1/2√3). (v) If l>√3the only critical point is the point (0,0). (vi) If 0<l<√3and l=5/3we have only two other critical points, which are R=2−√9−3l2 3l2−5,3(1 −l2)+√9−3l2 3(3l2−5)l, S=2+√9−3l2 3l2−5,3(1 −l2)−√9−3l2 3(3l2−5)l. Quadratic systems with a weak focus of third order 31 Proof: (i) Trivial. From (5.1) any critical point different from (0,0) must be such that its second coordinate must satisfy next relation: y2(9l3−15l)+(6l2−6)y+l= 0. Then if l= 0 and l=5/3 we may have only one critical point different from (0,0) and we obtain (ii) and (iii). If we are not in the latter cases, we have a second degree equation with discriminant ∆ = 36 −12l2from which we easily have (iv), (v) and (vi). Now we study the local phase portrait of the finite critical points of system (5.1). Lemma 5.3. For system (5.1) the following statements hold. (i) The critical point (0,0) is an unstable weak focus of third order if l>5/3, a linear center if l=5/3or if l=0, and a stable weak focus of third order if 0<l<5/3. (ii) If l=0, then the critical point (−1,0) is a saddle. (iii) If l=5/3, then the critical point (1/4,−5/3/4) is a saddle. (iv) If l=√3, then the critical point (1/2,−1/2√3) is an elementary saddle-node. (v) If 0<l<√3and l=5/3, then the critical point Ris a saddle. (vi) If 5/3<l<√3, then the critical point Sis an unstable node. (vii) If 0<l<5/3, then the critical point Sis a saddle. The proof is easy just substituting the critical points in their corresponding jacobians. Now we analyze the infinite critical points of system (5.1). Lemma 5.4. System (5.1) has always three critical points at infinity which are: (K1,0),(K2,0) and (K3,0) in the local chart U2where K1= −l+√l2+5,K2=0, and K3=−l−√l2+5. Proof: From Section 2 we have that the point (0,0) of the local chart U1is never critical. So all the critical points come from the equation: −z3−2lz2+5z= 0, and they will be in the local chart U2. It is clear that this equation has the three mentioned roots. Next we study the local phase portraits at the infinite critical points of system (5.1). 38 J. C. Art´ es, J. Llibre [A3] J. C. 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[Y2] Ye Yanqian and others,“Theory of limit cycles,” Trans. Math. Monographs [66], Amer. Math. Soc., 1986. Quadratic systems with a weak focus of third order 39 [Yu] S. X. Yu, On limit cycles of quadratic systems, Acta Math. Sinica 20 (1977), 193–205. Keywords. Quadratic Vector Field, Weak Focus. 1991 Mathematics subject classifications: 34C05, 58F14. Departament de Matem`atiques Universitat Aut`onoma de Barcelona 08193 Bellaterra (Barcelona) SPAIN e-mail: [email protected] e-mail: [email protected] Rebut el 30 de Novembre de 1996