Codimension 4 singularities of reflectionally symmetryc planar vector fields

Publicacions Matem`atiques, Vol 43 (1999), 501–533. CODIMENSION 4 SINGULARITIES OF REFLECTIONALLY SYMMETRYC PLANAR VECTOR FIELDS F. Dumortier and S. Ib´ a˜ nez∗ Abstract The paper deals with the topological classification of singularities of vector fields on the plane which are invariant under reflection with respect to a line. As it has been proved in previous papers, such a classification is necessary to determine the different topological types of singularities of vector fields on R3whose linear part is invariant under rotations. To get the classification we use normal form theory and the blowing-up method. 1. Introduction Consider a C∞vector field Xon R2with X(0) = 0. The set of all C∞vector fields which coincide with Xin some neighbourhood of 0 is called the germ of Xin 0 and denoted by any of its elements. Let ¯ G2 be the set of all germs in 0 having a representative Xwhich is invariant under the symmetry R(x, y)=(−x, y). We say that Xis reflectionally symmetric. In the sequel we assume that each representative which is given for a germ in ¯ G2is reflectionally symmetric. Given X,Y∈¯ G2, they are said to be k-jet equivalent, with k∈N∪ {∞}, if their Taylor approximations up to order kare equal. The set of all germs which are k-jet equivalent to X∈¯ G2is called the kjet of Xat 0 and denoted by jkX(0). ¯ J2 kdenotes the set of k-jets in ¯ G2. ∗Partially supported by the D.G.E.S. Project PB95-1054. 502 F. Dumortier, S. Ib´ a˜ nez Let us notice that there exists a one to one correspondence between ¯ J2 k and the space of reflectionally symmetric vector fields Yon R2with Y(0) = 0 and whose components are polynomials of degree less or equal than k. Therefore we will consider ¯ J2 kas a vector space of finite dimension with the usual topology. Take the projections jk:¯ G2→¯ J2 kinduced by the k-jet equivalence. ¯ G2is considered to be endowed with the coarsest topology which makes jkcontinuous for all k∈N. A set A⊂¯ G2is said to be algebraic if A=j−1 k( A) with  A⊂¯ J2 kan algebraic set of ¯ J2 k. The codimension of Ais the codimension of  Ain ¯ J2 k. On the other hand X,Y∈¯ G2are said to be C0equivalent if for some (and hence for all) representatives  Xand  Yof Xand Y, respectively, there are open neighbourhoods Uand Vof 0 ∈R2and a homeomorphism h:U→Vwhich sends orbits of  Xto orbits of  Ypreserving the sense but not necessarily the parametrization. To give the topological classification of the singularities having a codimension at most four in ¯ G2, we will define a stratification: ¯ G2=V0⊃V1⊃V2⊃V3⊃V4⊃V5, where for each i=1,2,3,4,5, Viis a closed algebraic set of codimension i and Vi−1\Viis a regular manifold of codimension i−1 such that for all X∈Vi−1\Vithere is a neighbourhood Uof Xin ¯ G2for which Y∈(Vi−1\Vi)∩Uimplies that Xand Yare C0equivalent. Consider XR∈¯ G2given by (1) XR=f(x, y)∂ ∂x +g(x, y)∂ ∂y, where fand gare C∞functions. Since XRis reflectionally symmetric, it follows that f(−x, y)=−f(x, y) and g(−x, y)=g(x, y). As a consequence of the Malgrange Division Theorem [CH] we know that there exist C∞functions ˜ f,˜g:R2→Rsuch that f(x, y)=x˜ f(x2,y),g(x, y)=˜g(x2,y). Codimension 4singularities 503 Therefore, we can use the change of coordinates w=x2and z=yto write (2) XR=w˜ f(w,z)∂ ∂w +˜g(w,z)∂ ∂z. Let us now study expression (2) using again (x, y) instead of (w,z) and f,ginstead of ˜ f,˜g: (3) XR=xf(x, y)∂ ∂x +g(x, y)∂ ∂y. We only need to study the restriction of XRto the halfplane {(x, y)∈ R2:x≥0}. Consider the following developments of fand g: f(x, y)=a00 +a10y+a01x+a20y2+a11yx +a02x2+◦((x, y)2) g(x, y)=b10y+b01x+b20y2+b11yx +b02x2 +b30y3+b21y2x+b12yx2+b03x3+◦((x, y)3). If a00 = 0 and b10 = 0 the singularity is hyperbolic. Therefore, such conditions define the codimension 0 singularities and their corresponding phase-portraits are well known. The same happens with the semihyperbolic cases. They correspond to one of the following conditions a00 =0,b 10 =0 or a00 =0,b 10 =0. To determine the subsequent stratification we use the reduction to a 1-dimensional center manifold. To draw the different topological types one only needs to consider the classification of the semihyperbolic singularities on the plane up to codimension 5 (see [D]), bearing in mind that the y-axis is always invariant. Therefore we will find singularities which are topologically equivalent to either saddles or nodes or saddle-nodes. We will start the classification on a stratum W2⊂V2of codimension 2 given by the conditions a00 =0,b 10 =0. 504 F. Dumortier, S. Ib´ a˜ nez Hence (3) reduces to (4) XR=x(a10y+a01x+a20y2+a11yx +a02x2+◦((x, y)2)) ∂ ∂x +(b01x+b20y2+b11yx +b02x2 +b30y3+b21y2x+b12yx2+b03x3+◦((x, y)3)) ∂ ∂y. Moreover we will distinguish two cases: b01 = 0 (nilpotent case), b01 = 0 (quadratic case). They will be studied in sections 2 and 3, respectively. For each of them we will consider aditional restrictions. Although the study of the nilpotent case for codimension 2 is already present in the literature (see [T]) and also some singularities of codimension 3 in the quadratic case are considered in [KR], we include them here for the sake of completeness. Our main motivation for the study of the topological classification of reflectionally symmetric planar vector fields is that they play an essential role in the classification of singularities in R3. As it is shown in [T], any C∞vector field Xon R3, with X(0) = 0 and 1-jet linearly conjugate to the infinitesimal rotation λ(y∂ ∂x −x∂ ∂y), can be given in a normal form which is, up to flat terms, invariant under rotations. After normalizing the rotational component and introducing cylindrical coordinates (θ,r, z), Xcan be written as 2π∂ ∂θ +XR+Y, where j∞Y(θ,0,0) = 0 for all θ∈S1and XRis a reflectionally symmetric planar vector field, depending on the coordinates (r, z), such that XR∈W2.In[DI] it is shown that, at least up to codimension four, flat terms have no influence on the topological type of the singularity which, in fact, is given by the topological type of XR. Nevertheless, neither the classification of the reflectionally symmetric vector fields XRnor the topological types were described in [DI]. In this paper we provide such a complete classification. The pictures that we give for the phase portraits have been made with the computer package P4 (see [DH] and [H]). The models which are used will be indicated in each case. We would like to thank Chris Herssens and Peter de Maesschalck for having helped us in making these pictures. Codimension 4singularities 505 2. Nilpotent case (b01 =0) A linear coordinate change permits to take b01 = 1. Hence we can write the ∞-jet of XRas   i+j≥1 aijyixj x∂ ∂x + x+ i+j≥2 bijyixj ∂ ∂y. One can prove that, up to a C∞coordinate change which leaves {x=0} invariant, the ∞-jet of XRcan be written as: (5) x  i≥1 aiyi ∂ ∂x + x+ i≥2 biyi ∂ ∂y. Indeed, let m≥2 be an integer. Since x∂ ∂y,x j−1ym−j+1 ∂ ∂x=(m−j+1)xjym−j∂ ∂x −xj−1ym−j+1 ∂ ∂y, all the terms xjym−j∂ ∂x with j≥2 can be removed after a change of coordinates which does not affect neither the remaining terms of the m-jet of the ∂ ∂x-component nor the terms of the (m−1)-jet of the ∂ ∂y-component. Moreover, since x∂ ∂y,x j−1ym−j+1 ∂ ∂y=(m−j+1)xjym−j∂ ∂y, all the terms xjym−j∂ ∂y with j≥1 can be removed after a change of coordinates which does not affect the remaining terms of the m-jet of the vector field. All the changes of coordinates which are needed leave {x=0}invariant and it is important to observe that (5) is the simplest normal form which preserves such an invariance. 506 F. Dumortier, S. Ib´ a˜ nez We define the following stratification N2,1={X∈W2|b01 =0}, N3,1={X∈N2,1|b2=0}, N3,2={X∈N2,1|b2=0,a 1=0}, N3,3={X∈N2,1|b2=0,a 1=0,a 1=2b2}, N4,1={X∈N2,1|b2=0,a 1=0}, N4,2={X∈N2,1|b2=0,a 1=0,b 3=0}, N4,3={X∈N2,1|b2=0,a 1=0,a 2=0}, N5,1={X∈N2,1|b2=0,a 1=0,a 2=0}, N5,2={X∈N2,1|b2=0,a 1=0,a 2=0,b 3=0}, N5,3={X∈N2,1|b2=0,a 1=0,a 2=0,b 3=0,a 2=3b3}, N5,4={X∈N2,1|b2=0,a 1=0,b 3=0,b 4=0}, N5,5={X∈N2,1|b2=0,a 1=0,a 2=0,a 3=0}. N2,1 N3,1N3,2N3,3 N4,1N4,2N4,3 N5,1N5,2N5,3N5,4N5,5 Figure 2.1. Stratification in the nilpotent case. Note that each Ni,j with i=2,3,4,5 is a semi-algebraic set of codimension i. In Figure 2.1 we show how the previous strata are organized. Let us now consider the different equivalence classes which are found in each stratum. To do this we use the blow-up x=r2u, y =rv Codimension 4singularities 507 with u2+v2= 1 and r∈R+and as usual the calculations are made by means of directional blow-up’s. Since the exponent of rin the change y=rv is odd the vector field obtained by blowing-up in the y-direction with v=−1 is related with the one obtained by taking v=1 by means of the change (r, u)→(−r, u) and the rescaling t→−t.We keep this in mind in order to draw the different phase-portraits after blowing up. Further simplifications are possible in each case separately. Remark 2.1. Let us recall that in the previous observations and calculations made on expression (3) the coordinates (x, y) represent in fact (w,z) in expression (2). Going back to the original coordinates (x, y) in (1) we have to use w=x2and z=y. Now by means of blow-up we will, in many cases, prove the existence of invariant curves for vector fields like in (2) of the form w=αz2+o(z2) with αpositive. Hence, with respect to the original (x, y)-coordinates, we get invariant curves of the form x=√αy +o(y) for vector fields (1). However, in the cases drawn in Figure 2.9, the invariant curves in {x>0}are obtained by a blow-up (w=r3u, z =rv) and will be of the form x=γy3/2+o(y3/2) for some γ>0. 2.1. Singularities of codimension 2. Let X∈N2,1\(N3,1∪N3,2∪N3,3). In this case we have b2=0,a1=0 and a1=2b2. By means of the change of variables x=¯x/b2,y=¯y/b2, we get from (5) the vector field (6)            ¯x=a1 b2 ¯x¯y+¯x i≥2 ai bi 2 ¯yi ¯y=¯x+¯y2+ j≥3 bj bj−1 2 ¯yj. Blowing-up in the ¯y-direction with v= 1, we get      r=ru +r+o(r) u=a1 b2−2u−2u2+u(O(r)). This vector field restricted to {r=0}has exactly two hyperbolic singularities at (r, u)=(0,0) and (r, u)=(0,(a1/b2−2)/2). Blowing-up in the ¯x-direction, no singularity is found at (r, v)=(0,0). The study of the singularities leads to the phase-portraits given in Figure 2.2 (a) for a neighbourhood of S1×{0}. In Figure 2.2 (b) we represent the different 508 F. Dumortier, S. Ib´ a˜ nez topological types corresponding to XRas in (1). Note that, because of the change of variables used to get (6), we must take into account the sign of b2. (a) (b) b2>0 b2<0 a1 b2<0 0 <a1 b2<2 2 <a1 b2 (a1,b 2)=(−1,1) (a1,b 2)=(1,1) (a1,b 2)=(3,1) (a1,b 2)=(1,−1) a1 b2<0 (a1,b 2)=(−1,−1) 0<a1 b2<2 (a1,b 2)=(−3,−1) 2<a1 b2 Figure 2.2. Topological types for vector fields in N2,1\(N3,1∪N3,2∪N3,3). We use the model a1 2xy ∂ ∂x +(x2+b2y2)∂ ∂y. Codimension 4singularities 509 Before continuing with the study of the singularities of higher codimension let us quickly describe the dynamic degeneracies that will occur in each of the next strata. For that we look at the singularities which appear after the previous directional blowing-up. For vector fields in N3,1 the singularity at (0,0) has a vanishing radial eigenvalue because of the coalescence with an extra hyperbolic singular point. Along the branches N3,1,N4,2and N5,4new singular points coalesce with (0,0) leading to more degenerate singularities, however staying semi-hyperbolic in all cases. A similar phenomenon happens at (0,(a1/b2−2)/2) for the branches N3,2,N4,3and N5,5. For vector fields in N3,3both singularities on the blown-up circle {r=0}coalesce at (0,0) which hence becomes a semi-hyperbolic singular point whose center manifold is contained in {r=0}; further degeneracies are not possible. Something similar happens for N4,1but in this case the singularity at (0,0) has a double zero eigenvalue and an extra blowing-up should be used. We will find again two singularities on the new blown-up circle. Then the strata N5,j with j=1,2,3 will have the same dynamical meaning, with respect to these new singularities, as N3,j has with respect to those obtained after the first blowing-up. In any case, as we will see later, it is better to modify the original blowing-up instead of making an extra one. On the other hand, looking at the quadratic case as limit of the nilpotent case, when b01 = 0, and applying the same blowing-up, we would find degenerate singularities on the second blown-up circle. This is the reason why we use, after further simplifications, a different blowing-up. If we take into account the influence of b01, the singularity (0,(a1/b2−2)/2) will tend to infinity when b01 tends to 0, leading to the coalescence of two singular points on the blown-up circle. Such degeneracy is considered in the quadratic case. 2.2. Singularities of codimension 3. We have three different cases: 1. X∈N3,3. In this case b2=0,a1= 0 and a1=2b2, 2. X∈N3,2\N4,3. In this case b2=0,a1= 0 and a2=0, 3. X∈N3,1\(N4,1∪N4,2). In this case b2=0,a1= 0 and b3=0, which are studied separately. Case 1: Since b2= 0 we use (6) which is given, taking into account that a1=2b2,by          ¯x=2¯x¯y+¯x i≥2 ai bi 2 ¯yi ¯y=¯x+¯y2+ j≥3 bj bj−1 2 ¯yj. 516 F. Dumortier, S. Ib´ a˜ nez On the other hand, when a1b4<0 we use the change of variables and the rescaling given by (9)                    x=−a2 1 b4 ¯x t=−−b4 a13 21 b4 ¯ t y=−a1 b41 2 ¯y to reduce (5) to ¯x=¯x¯y+¯x(o(¯y)) ¯y=¯x−¯y4+o(¯y4). In this case, by blowing-up in the ¯y-direction with v= 1, we get r=ru −r3+o(r3) u=u−2u2+u(O(r)). As in the previous cases the above vector field gives us all the relevant information. We obtain the phase-portrait given in Figure 2.8 (a) and the topological types given in Figure 2.8 (b) taking into account the signs of a1and b4in (9). (a) (b) (a1,b 4)=(−1,1) a1<0, b4>0 (a1,b 4)=(1,−1) a1>0, b4<0 Figure 2.8. Topological types for vector fields in N4,2\N5,4with a1b4<0. We use the model a1 2xy ∂ ∂x +(x2+b4y4)∂ ∂y. Case 3: Now we use a different blowing-up given by ¯x=r3u, ¯y=rv. Codimension 4singularities 517 Blowing-up in the ¯y-direction with v= 1 we obtain, after division by r2, r=ru +b3r+o(r) u=a2u−3u(u+b3)+O(r). b3>0 a2>0 b3a2<0 b3<0 a2<0 (a2,b 3)=(4,1) (a2,b 3)=(1,1) (a2,b 3)=(1,−1) (a2,b 3)=(−1,1) (a2,b 3)=(−1,−1) a2−3b3>0 (a2,b 3)=(−4,−1) a2−3b3<0 Figure 2.9. Topological types for vector fields in N4,1\(N5,1∪N5,2∪N5,3). We use the model a2 2xy2∂ ∂x +(x2+b3y3)∂ ∂y. 518 F. Dumortier, S. Ib´ a˜ nez We find exactly two hyperbolic singularities at (r, u)=(0,0) and (r, u)= (0,(a2−3b3)/3). The blowing-up in the y-direction with v=−1 is related to the one obtained by taking v= 1 by means of the change (r, u)→ (−r, −u). All the relevant information is given by the previous vector field and glueing the different charts together we get the phase-portraits given in Figure 2.9 for the blowing-up near S1×{0}as well as for the different topological types corresponding to XR. 3. Quadratic case (b01 =0) Under certain conditions we will consider further simplifications of the expresion of XRgiven in (4). These simplifications are obtained by means of linear changes of coordinates and they are such that the 2-jet of (4) with respect to the new coordinates presents some specific invariant lines. Let us recall that the 2-jet of XRis given by (10) x(a10y+a01x)∂ ∂x +(b20y2+b11yx +b02x2)∂ ∂y. Simple calculations permit to show that if Ais such that (11) (a10 −b20)A2+(a01 −b11)A−b02 =0, then the line y=Ax is invariant for the vector field given in (10). Let ∆=(a01 −b11)2+4b02(a10 −b20) be the discriminant of equation (11). If ∆ <0 then a10 −b20 = 0 and hence (11) is a quadratic equation which has no real solutions. In this case no further simplifications are done and we merely define the following strata in W2: C3,1={X∈W2|b01 =0,∆<0}, C4,1={X∈C3,1|b20 =0}, C5,1={X∈C3,1|b20 =0,b 30 =0}. Codimension 4singularities 519 If ∆ >0 and a10 −b20 = 0 then (11) is a quadratic equation with two different real solutions A1and A2for which condition A2>A 1is assumed. The linear change of coordinates (12) x=¯x y=A1¯x+(A2−A1)¯y sends the lines {y=A1x}and {y=A2x}to {¯y=0}and {¯y=¯x}, respectively. Hence, with respect to the (¯x, ¯y)-coordinates, XRstill maintains {¯x=0}as an invariant line and presents the following 3-jet: (13) ¯x(¯a10 ¯y+¯a01 ¯x+¯a20 ¯y2+¯a11 ¯y¯x+¯a02 ¯x2)∂ ∂¯x +( ¯ b20 ¯y2+¯ b11 ¯y¯x+¯ b30 ¯y3+¯ b21 ¯y2¯x+¯ b12 ¯y¯x2+¯ b03 ¯x3)∂ ∂¯y with ¯ b20 +¯ b11 =¯a01 +¯a10 since ¯y=¯xis an invariant line for the 2-jet. Under these conditions we define the following stratification: C3,2={X∈W2|b01 =0,∆>0,a 10 −b20 =0}, C4,2={X∈C3,2|¯ b20 =0}, C4,3={X∈C3,2|¯ b20 =0,¯a01 =0}, C4,4={X∈C3,2|¯ b20 =0,¯a01 =0,¯a01 +¯a10 =0}, C5,2={X∈C3,2|¯ b20 =0,¯a01 =0}, C5,3={X∈C3,2|¯ b20 =0,¯a01 =0,¯a01 +¯a10 =0}, C5,4={X∈C3,2|¯ b20 =0,¯a01 =0,¯a01 +¯a10 =0,¯ b30 =0}, C5,5={X∈C3,2|¯ b20 =0,¯a01 =0,¯a10 =0}, C5,6={X∈C3,2|¯ b20 =0,¯a01 =0,¯a10 =0,¯a10¯ b03 +¯a02(¯ b20 −¯a10)=0}, C5,7={X∈C3,2|¯ b20 =0,¯a01 =0,¯a01 +¯a10 =0,¯ b20(¯a20 +¯a11 +¯a02) −¯a10(¯ b30 +¯ b21 +¯ b12 +¯ b03)=0}. 520 F. Dumortier, S. Ib´ a˜ nez Finally, we have to consider the cases {∆>0,a 10 −b20 =0}and {∆=0,a 10 −b20 =0}. The condition in the first case implies that a01 −b11 = 0 and therefore (11) is a linear equation. In the second case (11) is a second order equation with a unique real solution. Let A1 be the unique solution of (11) under any of the previous conditions. In such a case, with respect to the (¯x, ¯y)-coordinates given by the linear transformation (14) x=¯x y=A1¯x+¯y, the vector field XRstill maintains {¯x=0}as an invariant line and its 3-jet is like in (13). That is, the 2-jet presents {¯y=0}as the unique invariant line of the type ¯y=A¯x. Thus, we define the following strata of codimension 4 and 5: C4,5={X∈W2|b01 =0,∆>0,a 10 −b20 =0}, C5,8={X∈C4,5|¯ b20 =0}, C5,9={X∈C4,5|¯ b20 =0,¯a01 =0} and C4,6={X∈W2|b01 =0,∆=0,a 10 −b20 =0}, C5,10 ={X∈C4,6|¯ b20 =0}, C5,11 ={X∈C4,6|¯ b20 =0,¯a01 =0}. C3,1C3,2 C4,1C4,2C4,3C4,4C4,5C4,6 C5,1C5,2C5,3C5,4C5,5C5,6C5,7C5,8C5,9C5,10 C5,11 C5,12 Figure 3.1. Stratification in the quadratic case. Codimension 4singularities 521 We also define the stratum C5,12 ={X∈W2|b01 =0,∆=0, a10 −b20 =0}. Note that each Ci,j with i=3,4,5 is a semi-algebraic set of codimension i. In Figure 3.1 we show how the different strata are organized. Remark 3.1. The aij and bij refer to coefficients in (4) while ¯aij and ¯ bij to coefficients in (13). Nevertheless, it is important to observe that after the change (12) (resp. (14)) ¯a10 =a10(A2−A1) and ¯ b20 =b20(A2−A1) (resp. ¯a10 =a10 and ¯ b20 =b20). Therefore, in both cases ¯a10 −¯ b20 and a10 −b20 have the same sign. Since ¯y= 0 is the unique invariant line of the type ¯y=A¯xfor the 2-jet of (13) if X∈C4,5, it follows that A= 0 is the unique solution of (¯a01 −¯ b11)A= 0 and hence ¯a01 −¯ b11 = 0. Similarly, if X∈C4,6then A= 0 is the unique solution of (¯a10 −¯ b20)A2+(¯a01 −¯ b11)A= 0 and hence ¯a01 =¯ b11. Remark 3.2. Putting together the different strata which we have just characterized with the ones given for the nilpotent case in section 2, we finally obtain the following stratification of W2: W22 =N2,1 W23 =∪3 j=1N3,j∪∪2 j=1C3,j, W24 =∪3 j=1N4,j∪∪6 j=1C4,j, W25 =∪5 j=1N5,j∪∪12 j=1C5,j. Note that each W2iwith i=2,3,4,5 is a semi-algebraic set of codimension i. We will obtain the different equivalence classes which are found in each stratum Ci,j \Ci,j+1. To do this we use the homogeneous blow-up given by x=ru, y =rv. Since the exponent of rin the change y=rv is odd the vector field obtained by blowing-up in the y-direction with v=−1 is related to the one obtained by taking v= 1 by means of the change (r, u)→(−r, −u) and t→−t. We will keep this in mind in the sequel. 522 F. Dumortier, S. Ib´ a˜ nez We can first give a survey of the different dynamics which are observed in each stratum as well as of the different transitions. We first study the singularities obtained after one blow-up. Only the side u≥0 on the blown-up circle needs to be considered. In all cases we will find two singularities at (u, v)=(0,±1), corresponding to the invariant axes. For C3,1there are no more singularities and further degeneracies in C4,1 and C5,1are given by the coalescence of an extra hyperbolic singular point with (0,±1). For C3,2we also find two singularities p1and p2 on the blown-up circle with u>0. Further degeneracies along the different strata contained in C3,2are given by the coalescence of an extra hyperbolic singular point with (0,±1), p1or p2.C4,5can be seen as the transition inside C3,2from a10 −b20 positive to negative. As a consequence, one of the singularities, either p1or p2, coalesce with (0,±1) in a saddle-node bifurcation. Finally C4,6shows the transition from C3,2 to C3,1, in which p1and p2coalesce in a saddle-node bifurcation. Remark 3.3. Let us recall that in the previous calculations on (3) the coordinates (x, y) represent in fact (w,z) in expression (2). Going back to the original coordinates (x, y) in (1) we have to use w=x2and z=y. By means of blow-up we prove the existence of invariant curves for vector fields like in (3) of the form z=αw+o(w). Hence, with respect to (x, y)-coordinates we get invariant curves of type y=αx2/2+o(x2). This must be kept in mind when drawing, as we do in this case, the different topological types for vector fields (1). 3.1. Singularities of codimension 3. We will distinguish two cases: 1. X∈C3,1\C4,1. In this case b01 =0,∆<0 and b20 =0, 2. X∈C3,2\(C4,2∪C4,3∪C4,4). In this case b01 =0,∆>0, a10 −b20 =0,¯ b20 =0,¯a01 = 0 and ¯a01 +¯a10 =0, which are studied separately. Case 1: We consider expression (4) for XR. The change of coordinates (x, y)→(x, −y) permits us to put b20 >0. Hence, we assume that b20 is positive. Blowing-up in the y-direction with v= 1 we get r=b20r+r(O((r, u))) u=u(a10 −b20 +(a01 −b11)u−b02u2)+u(O(r)). Codimension 4singularities 523 Since ∆ <0, on the invariant line {r=0}we find exactly one hyperbolic singularity at (r, u)=(0,0). Recall that ∆ <0 implies that (a10 −b20)= 0. One can check that blowing-up in the x-direction no singularity is found at (r, v)=(0,0). The study of the singularity leads to the phase-portraits given in Figure 3.2 for the blowing-up as well as for the corresponding topological types of XR. (a10,b 20,b 02)=(2,1,−1) a10 −b20 >0 (a10,b 20,b 02)=(−1,1,1) a10 −b20 <0 Figure 3.2. Topological types for vector fields in C3,1\C4,1. We use the model 1 2x(a10y+a01x2)∂ ∂x +(b20y2+b11yx2+ b02x4)∂ ∂y. Case 2: In this case we consider the expression of XRwhich is obtained after the change (12) and hence we assume that its 3-jet is given by (13). As it is explained in Remark 3.1, ¯a10 −¯ b20 = 0. Moreover we use that ¯ b20 +¯ b11 =¯a01 +¯a10 and assume, as in the previous case, that ¯ b20 >0. Blowing-up in the ¯x-direction with u= 1 we get r=(¯a10v+¯a01)r+o(r) v=( ¯ b20 −¯a10)v(v−1) + O(r). On the invariant line {r=0}we find exactly two hyperbolic singularities at (r, u)=(0,0) and (r, v)=(0,1). Blowing-up in the ¯y-direction with v= 1 we get an extra hyperbolic singularity at (r, u)=(0,0). Studying the singularities and glueing the different charts together we obtain the phase-portraits given in Figure 3.3. 524 F. Dumortier, S. Ib´ a˜ nez ¯a01 >0 ¯a10 +¯a01 >0 ¯a01 >0 ¯a10 +¯a01 <0 ¯a01 <0 ¯a10 +¯a01 >0 ¯a01 <0 <0 (¯a10,¯a01,¯ b20)=(2,1,1) (¯a10,¯a01,¯ b20)=(2,−1,1) (¯a10,¯a01,¯ b20)=(−1,2,1) (¯a10,¯a01,¯ b20)=(−2,1,1) (¯a10,¯a01,¯ b20)=(1/2,−1/4,1) (¯a10,¯a01,¯ b20)=(2,−3,1) ¯a10 −¯ b20 >0 (¯a10,¯a01,¯ b20)=(−1,−1,1) ¯a10 −¯ b20 <0 ¯a10 +¯a01 Figure 3.3. Topological types for vector fields in C3,2\(C4,2∪C4,3∪C4,4). We use the model 1 2x(¯a10y+¯a01x2)∂ ∂x +(¯ b20y2+(¯a01 +¯a10− ¯ b20)yx2)∂ ∂y. Codimension 4singularities 525 3.2. Singularities of codimension 4. We have six cases: 1. X∈C4,1\C5,1. In this case b01 =0,∆<0, b20 = 0 and b30 =0, 2. X∈C4,2\(C5,2∪C5,3∪C5,4). In this case b01 =0,∆>0, a10 −b20 =0,¯ b20 =0,¯a01 =0,¯a01 +¯a10 = 0 and b30 =0, 3. X∈C4,3\(C5,5∪C5,6). In this case b01 =0,∆>0, a10 −b20 =0, ¯ b20 =0,¯a01 =0,¯a10 = 0 and ¯a10¯ b03 +¯a02(¯ b20 −¯a10)=0, 4. X∈C4,4\C5,7. In this case b01 =0,∆>0, a10 −b20 =0, ¯ b20 =0,¯a01 =0,¯a01 +¯a10 = 0 and ¯ b20(¯a20 +¯a11 +¯a02)−¯a10(¯ b30 + ¯ b21 +¯ b12 +¯ b03)=0, 5. X∈C4,5\(C5,8∪C5,9). In this case b01 =0,∆>0, a10 −b20 =0, ¯ b20 = 0 and ¯a01 =0, 6. X∈C4,6\(C5,10∪C5,11). In this case b01 =0,∆=0,a10−b20 =0, ¯ b20 = 0 and ¯a01 =0, which are studied separately. Case 1: In this case the change of coordinates (¯x, ¯y)→(¯x, −¯y) permit us to put a10 >0 and hence we assume that such a condition is satisfied. Blowing-up (4) in the y-direction with v= 1 we get r=r(b11u+b02u2)+r2(b30 +b21u+b12u2+b03u3)+o(r2) u=u(a10 +(a01 −b11)u−b02u2)+u(O(r)). (a10,b 02,b 30)=(1,−1,1) b30 >0 (a10,b 02,b 30)=(1,−1,−1) b30 <0 Figure 3.4. Topological types for vector fields in C4,1\C5,1. We use the model 1 2x(a10y+a01x2)∂ ∂x +(b11yx2+b02x4+ b30y3)∂ ∂y. 532 F. Dumortier, S. Ib´ a˜ nez Again (r, v)=(0,0) is the unique singularity on {r=0}and the linear part is given by ¯a01r∂ ∂r +¯ b03r∂ ∂v. Therefore {r=0}is a center manifold and the restriction of the vector field is given by (¯ b20 −¯a10)v2∂ ∂v. Glueing the different charts together we get the phase-portraits depicted in Figure 3.9 where the different topological types corresponding to XRare also shown. ¯a10 −¯ b20 >0 ¯a10 −¯ b20 <0 (¯a10,¯a01,¯ b20)=(2,1,1) ¯a01 >0 (¯a10,¯a01,¯ b20)=(0,1,1) (¯a10,¯a01,¯ b20)=(2,−1,1) ¯a01 <0 (¯a10,¯a01,¯ b20)=(0,−1,1) Figure 3.9. Topological types for vector fields in C4,6\(C5,10 ∪C5,11). We use the model 1 2x(¯a10y+¯a01x2)∂ ∂x +(¯ b20y2+¯a01yx2)∂ ∂y. References [CH] S. -N. Chow and J. Hale,“Methods of bifurcation theory,” Springer-Verlag, New York, Berlin, Heidelberg, 1982. [D] F. Dumortier, Singularities of vector fields on the plane, J. Differential Equations 23 (1977), 53–106. [DI] F. Dumortier and S. Ib´ a˜ nez, Singularities of vector fields on R3,Nonlinearity 11 (1998), 1037–1047. Codimension 4singularities 533 [DH] F. Dumortier and C. Herssens, Tracing phase portraits of planar polynomial vector fields with detailed analysis of the singularities, Preprint (1998). [H] C. Herssens, Drawing and classifying portraits of planar polynomial vector fields, Thesis, Limburgs Universitair Centrum (1998). [KR] B. Krauskopf and C. Rousseau, Codimension-three unfoldings of reflectionally symmetric planar vector fields, Nonlinearity 10 (1997), 1115–1150. [T] F. Takens, Singularities of vector fields, Inst. Hautes ´ Etudes Sci. Publ. Math. 43 (1974), 47–100. F. Dumortier: Limburgs Universitair Centrum Universitaire Campus B-3590 Diepenbeek BELGIUM e-mail: [email protected]e S. Ib´a˜nez: Departamento de Matem´aticas Universidad de Oviedo 33007 Oviedo SPAIN e-mail: [email protected]vi.es Primera versi´o rebuda el 25 de juny de 1998, darrera versi´o rebuda el 27 de gener de 1999