Affine invariant conditions for the topological distinction of quadratic systems with a critical point of the 4th multiplicity

Publicacions Matem`atiques, Vol 40 (1996), 431–441. AFFINE INVARIANT CONDITIONS FOR THE TOPOLOGICAL DISTINCTION OF QUADRATIC SYSTEMS WITH A CRITICAL POINT OF THE 4TH MULTIPLICITY Mark Voldman, Iu. T. Calin and N. I. Vulpe Abstract The affine invariant partition of the set of quadratic systems with one finite singular point of the 4th multiplicity with respect to different topological classes is accomplished. The conditions corresponding to this partition are semi-algebraic, i.e. they are expressed as equalities or inequalities between polynomials. Let us consider the system of differential equations (1) dxj dt =aj+aj αxα+aj αβxαxβ,(j, α, β =1,2) where aj αand aj αβ (j, α, β =1,2) are real numbers (the tensor aj αβ is symmetric in the lower indices, with respect to which the complete contraction was made). The topological classification of the quadratic system (1) in the case when it has a unique real critical point on its phase plane is done in [4] and [6]. The conditions for the classification into topological classes described in these papers are expressed using the parameters of the corresponding canonical forms. The topological classification of system (1) with a unique real simple critical point is done in [7]. In [9] the conditions for the topological classification of quadratic systems with a unique finite critical point of multiplicity 4 were gived. However, the critical point was situated at the origin of the coordinates of system (1) and the obtained conditions are center affine invariant. Thus 432 M. Voldman, Iu. T. Calin, N. I. Vulpe the conditions given in [9] are not valid for the full system (1) in the case when such a point does not coincide with the origin. Using the results of article [9] we shall find the corresponding affine invariant partition of coefficient space E12 of system (1). The affine invariant conditions corresponding to this partition are semi-algebraic, i.e. they are expressed as equalities or inequalities between polynomials. Preliminaires Let a∈E12 be an element of the space of the coefficients of system (1) and let us consider the group Qof nondegenerate real linear transformations of the phase plane. We denote by rqthe linear presentation of any element q∈Qinto the coefficient space E12 of system (1). Definition 1 [8].A polynomial K(a, x) of the coefficients of system (1) and the unknown variables x1and x2is called a comitant of system (1) in the group Q, if there exists a function λ(q) such that K(rq·a, q ·x)≡λ(q)K(a, x) for every q∈Q,a∈E12 and x=(x1,x 2). The function λ(q) is called a multiplicator. If λ(q)≡1, then the comitant K(a, x) is called absolute; otherwise it is called relative. It is known (see [8]), that λ(q)=∆ −x q, where ∆q= 0 is the determinant of the linear transformation matrix and the integer κis called the weight of the comitant. A comitant Kof system (1) in the group Q=GL(2,R) of linear homogeneous transformations of the phase plane of system (1) (which is also called a group of center-affine transformations) is called center affine. A comitant Kof system (1) in the group Q=Aff(2,R) of affine (linear non-homogenous) transformations is called affine. If the comitant Kdoes not depend explicitly on the variables x1and x2then it is called an invariant (center affine or affine, respectively). Remark 1. We say that the comitant of system (1) equals zero when all its coefficients vanish. The signs of the comitants which take part in some sequences of conditions should be calculated at one and the same point, where they do not vanish. We denote by T(2,R) the group of shift transformations and by rtthe linear presentation of any element t∈Tinto the coefficient space R12 of system (1). Conditions for topological distinction of QS 433 Definition 2 [3].A center affine comitant K(a, x) of system (1) is called a T-comitant if the relation K(rt·a, x)≡K(a, x) is valid for every t∈Tand a∈E12. We shall say that comitant Kis of the type (ρ, κ,d) if it is a homogeneous polynomial of degrees ρand din the coordinates of vector xand in the coefficients of system (1), respectively, and if its weight is equal to κ. Definition 3 [5].The polynomial (f,ϕ)(k)=(r−k)!(ρ−k)! r!ρ! k  h=0 (−1)hCh k ∂kf ∂(x1)k−h∂(x2)h ∂kϕ ∂(x1)k∂(x2)k−h is called a transvectant of index kof two forms fand ϕ. The degree of these forms in the coordinates of vector x=(x1,x 2) are equal to rand ρ, respectively and k≤min(r, ρ). Proposition 1 [3].The transvectant (f,ϕ)(k)of two T-comitants f and ϕof the types (r, κ1,d 1)and (ρ, κ2,d 2)respectively will be also a T-comitant of the type (r+ρ−2k,κ1+κ2+k,d1+d2). Let us write system (1) as dx1 dt =P0+P1+P2, dx2 dt =Q0+Q1+Q2, where Pi(i=0,1,2) are homogeneous polynomials of degree i, and consider the following center affine invariants and comitants, which are 434 M. Voldman, Iu. T. Calin, N. I. Vulpe constructed directly through the right-side parts of the given system: (2) J1=    ∂P1/∂x1∂P1/∂x2 ∂Q1/∂x1∂Q1/∂x2    =aα paβ qεαβεpq, C1=    ∂P1/∂x1∂P2/∂x2 ∂Q1/∂x1∂Q2/∂x2    −    ∂P1/∂x2∂P2/∂x1 ∂Q1/∂x2∂Q2/∂x1    =xαaβ qaγ pαεβγεpq, C2=    P0P1 Q0Q1     =xαaβaγ αεβγ, C3=1 4    ∂P2/∂x1∂P2/∂x2 ∂Q2/∂x1∂Q2/∂x2    =1 2xαxβaγ pαaδ qβεγδεpq, C4=    P0P2 Q0Q2     =xαxβaγaδ αβεγδ, C5=    P1P2 Q1Q2     =xαxβxγaδ αaµ βγεδµ, C6=x2P2−x1Q2=xαxβxγaδ αβεδγ, (where ε11 =ε22 =ε11 =ε22 =0,ε12 =ε12 =−ε21 =−ε21 = 1). We also introduce the following T-comitants: C7=xqxsxv[2apar αγ −ap αar γ]au βκεpqεrsεuvεαβεγκ, C8=xαxβ[am san βak pr −2ak ran βam ps +ak pam ran sβ −4amak pran sβ]al qαεklεmnεpqεrs , and transvectants: µ1=(C3,C 3)(2),H 1=(C3,C 1)(1), G1=(C1,C 5)(1),G 2=(C5,C 5)(2),G 3=(C3,C 4)(1), η1= (((C6,C 6)(2),C 6)(1),C 6)(3),M 1=(C6,C 6)(2), D1= (((C6,C 7)(2),C 6)(1),C 6)(3), D2=((C3,C 7)(1),C 6)(3), D3=(C3,C 8)(2), D4= (((C7,C 7)(2),C 7)(1),C 7)(3), which by Proposition 1 are also T-comitants. Conditions for topological distinction of QS 435 We introduce the following notation (3) µ=−2µ1,H=2H1,2G=4G1−3G2+8G3, F=J1C5+2C1C4+4C2C3,V =C2 4−C2C5, 2η=27η1,2M=9M1,L=C6,D=−D4, D1=D1−2D2,D 2=4D2+D3,D 3=D1(21D1−26D2+4D3). As it was shown in [2] the comitants µ,H,G,Fand Vare responsible for the number and multiplicities of the finite singular points of quadratic system (1). Definition 4. We shall say that the real finite singular point M0of quadratic system a∈E12 has multiplicity k(or, in other words, from the singular point M0bifurcate ksingular points, as the coefficients of system a∈E12 are varied), if the following conditions are satisfied: (i) there exist a positive ε0>0 and δ0>0 such that in the neighbourhood U(a, δ0) of the point a, there are no points, which correspond to a system (1) having more than ksingular points in the neighbourhood U(M0,ε 0) of the singular point M0; (ii) for every positive δ<δ 0and ε<ε 0, there is a point b∈ U(a, δ) which corresponds to a system (1) with ksingular points in U(M0,ε). According to [2] we shall construct the following T-comitants: (4) P=G2−6FH +12µV, R = 4(3H2−2µG), T=2µ[2G3+9µ(3F2−8GV )−18FGH + 108H2V]−PR. Proposition 2 [2].Quadratic system (1) has a finite singular point of multiplicity 4if, and only if, the following conditions hold: (5) µ=0,D=0,P=R=T=0. 436 M. Voldman, Iu. T. Calin, N. I. Vulpe Main result Theorem. A phase portrait of quadratic system (1) with a finite point of multiplicity 4(µ=0,D=P=R=T=0) in the Poincare disk is up to a homeomorphism given by: Figure 1 if η>0,D1<0,D2>0; Figure 2 if η>0,D1<0,D2<0or η>0,D1≥0; Figure 3 if η=0,M=0,D1<0,D2>0,D3≥0; Figure 4 if η=0,M=0,D1<0,D2>0,D3<0; Figure 5 if η=0,M=0,D1<0,D2<0or η=0,M=0,D1≥0; Figure 6 if η<0, or η=M=0,L=0. Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Proof: Let us assume that conditions (5) are satisfied, i.e. according to (6) there is a singular point of multiplicity 4 arbitrarily situated on the phase plane of system (1). By applying a shift transformation we can move the origin of the coordinates to this point. Thus we obtain the system (6) dxj dt =aj αxα+aj αβxαxβ(j, α, β =1,2) for which, from (2) and (3), we have that V= 0. Therefore, by virtue of conditions µ=0,V= 0 and in accordance with (4) the conditions (5) Conditions for topological distinction of QS 437 imply the following relations among the comitants µ,H,Gand F: (7) G2−6FH =0,3H2−2µG =0,2G3+27µF2−18FGH =0. We intend to show that for system (6) conditions (7) imply (8) H=F=G=0. Indeed, multiplying the first relation of (7) by −2Gand summing it with the third relation we obtain (9) F(9µF −2GH)=0. If F= 0, then from (7) it follows that H=G= 0. Otherwise, taking into account (7) we have 4H(G2−6FH)+9F(3H2−2µG)+2G(2G3+27µF2 −18FGH)=3FH2=0. Therefore, we have obtained that H= 0 and from (7) it again follows that F= 0. Thus, conditions (8) are valid for system (6) if its singular point (0,0) is of the fourth multiplicity and vice versa. Let us denote by λ1and λ2the eigenvalues corresponding to the singular point (0,0) of system (6). Case I. λ2 1+λ2 2= 0. According to [1] by applying a linear transformation and rescaling, system (6) can be written as follows (10) dx dt =gx2+2hxy +ky2, dy dt =y+lx2+2mxy +ny2, for which we have F=0,G=g(gx2+2hxy +ky2). By condition (8) we receive g= 0 and calculating the values of Hand µwe obtain H=hl(2hx +ky)=0,µ=l(4h2m−4hkn +k2l)=0. Thus, we have h=0,kl = 0 and by scaling of the parameters, system (10) can be put in the form (11) dx dt =y2, dy dt =y+x2+2mxy +ny2, for which the conditions for distinguishing the topological classes through the parameters mand nare found in [9]. Namely, it occurs in the following: 438 M. Voldman, Iu. T. Calin, N. I. Vulpe Proposition 3 [9].A phase portrait of quadratic system (11) in the Poincare disk is up to a homeomorphism given by: Figure 1 if η>0,X<0,Y>0; Figure 2 if η>0,X<0,Y≥0or η>0,X≥0; Figure 3 if η=0,R=0,X<0,Y<0,Z≥0; Figure 4 if η=0,R=0,X<0,Y<0,Z<0; Figure 5 if η=0,R=0,X<0,Y>0or η=0,R=0,X≥0; Figure 6 if η<0, or η=R=0,L=0, where (12) η=−4m3+4m2n2−36mn +32n3−27, R=(3m−4n2)x2−(2mn +9)xy −(m2+6n)y2, L=−x3+2nx2y−mxy2+y3,X=2n3, Y=mn, Z = (32n3+4m2n2+ 27)n2(m2+6n). It is easy to prove the following assertion: Proposition 4. The following statements hold. 1) The consequences of conditions a) n<0,mn < 0;b)n≥0or n<0,mn ≥0 are equivalent to the consequences of conditions a) m>0,mn < 0;b)mn ≥0or mn < 0,m<0, respectively. 2) If η=0then the consequences of conditions a) n<0,mn < 0,(32n3+4m2n2+ 27)(m2+6n)≥0; b) n<0,mn < 0,(32n3+4m2n2+ 27)(m2+6n)<0; are equivalent to the consequences of conditions a) m>0,mn < 0,(2m3+18mn + 27)m(m2+6n)≥0; b) m>0,mn < 0,(2m3+18mn + 27)m(m2+6n)<0, respectively. Indeed, the truth of the first statement of Proposition 4 follows directly from the expressions of the corresponding conditions. To prove the second one we assume that η= 0. According to (12) we have −4m3+4m2n2−36mn +32n3−27 = 0. Therefore, we have that 4m2n2+32n3+ 27 = 2(2m3+18mn + 27) which proves Proposition 4. Conditions for topological distinction of QS 439 For system (11) the following comitants can be calculated: (13) η=−4m3+4m2n2−36mn +32n3−27, M=(3m−4n2)x2−(2mn +9)xy −(m2+6n)y2, L=−x3+2nx2y−mxy2+y3,D 1=7 3mn, D2=2 27m3,D 3=14 243(2m3+18mn + 27)m(m2+6n). Taking into account Proposition 4 and relations (12) and (13), we deduce that the conditions of Theorem for the realization of each of the phase portraits of system (11) are equivalent to the corresponding conditions of Proposition 3. Thus, the Theorem is valid for system (1) with one non-zero eigenvalue of the singular point of multiplicity 4. Case II. λ1=λ2= 0. According to [1] by applying a linear transformation and rescaling (6) we have (14) dx dt =y+gx2+2hxy +ky2, dy dt =lx2+2mxy +ny2, for which have F=0,G=l(lx2+2mxy +ny2). By virtue of conditions (8) we have l= 0 and by calculating the values of Hand µwe obtain H=gm(2mx +ny)=0,µ=g(gn2−4hmn +4km2)=0. Thus, we have m=0,gn = 0 and by scaling of the parameters, system (14) becomes dx dt =y+gx2+2hxy +ky2, dy dt =y2, after the following substitution, x=x1−hy1,y=gy1,t1=gt we have (15) dx1 dt1 =y1+x2 1+ky2 1, dy1 dt1 =y2 1.