New sufficient conditions for a center and global phase portraits for polynomial systems

Publicacions Matem`atiques, Vol 40 (1996), 351–372. NEW SUFFICIENT CONDITIONS FOR A CENTER AND GLOBAL PHASE PORTRAITS FOR POLYNOMIAL SYSTEMS H. Giacomini and M. Ndiaye Abstract In this paper we consider cubic polynomial systems of the form: ˙x=y+P(x, y), ˙y=−x+Q(x, y), where Pand Qare polynomials of degree 3 without linear part. If M(x, y) is an integrating factor of the system, we propose its reciprocal V(x, y)= 1 M(x,y)as a linear function of certain coefficients of the system. We find in this way several new sets of sufficient conditions for a center. The resulting integrating factors are of Darboux type and the first integrals are in the Liouville form. By induction, we have generalized these results for polynomials systems of arbitrary degree. Moreover, for the cubic case, we have constructed all the phase portraits for each new family with a center. Introduction In a previous paper [1], we considered the following system of differential equations (S) (1) ˙x=P(x, y), ˙y=Q(x, y), where P(x, y) and Q(x, y) are polynomials of degree nof the form (2) P(x, y)=y+ n  j=2 j  i=0 ai,j−ixiyj−i, Q(x, y)=−x− n  j=2 j  i=0 bi,j−ixiyj−i. 352 H. Giacomini, M. Ndiaye For this system, we found sufficient conditions for the existence of a center at the origin. This has been accomplished by proposing a constant of motion linear in some coefficients of the system. We recall that the problem of the center plays an important role in the second part of Hilbert’s 16th problem, which asks for the maximum number of limit cycles that a vector field given by polynomials of degree nshould have. There are well-known necessary and sufficient conditions for quadratic systems [2], [3], [4] and for systems in which P(x, y) and Q(x, y) are cubic polynomials without quadratic terms [5]. Sufficient conditions are also well-known for other classes of cubic systems [6]. In [7], the author classified the reversible cubic systems with center and recently, Sokulsky began the classification of Darboux integrals for cubic systems [8]. In this paper we continue the study of the problem of the center. In section 1 we consider the general cubic system (n= 3). Instead of looking for a constant of motion, we work with the reciprocal of an integrating factor of (S): V(x, y). In [9] it has been shown that V(x, y) has very interesting properties and that it is relevant in the analysis of the problem of limit cycles. By studying quadratic systems, we found that, while the known constants of motion are complicated functions, the reciprocal of the integrating factor V(x, y) turns out to be a simple polynomial. This has also been showed in [10]. We hoped to find the same result for the general cubic system, but unfortunately, this is not always the case. A counter-example has been given in [11], where V(x, y) is not a polynomial function. Nevertheless, we found particular cases of the general cubic system, for which the integrating factor 1 V(x,y)is of Darboux type (see [12], [13], [14]) and leads to an elementary first integral of the Liouville form [15] as we can see in the expression (34) of section 3. V(x, y) is given by a fourth-degree polynomial that factorizes as a product of a polynomial of degree one and another of degree three. These two polynomials are directly related to particular algebraic integrals of the system [16]. The corresponding constant of motion associated to the integrating factor 1 V(x,y)is analytic in the neighbourhood of the origin and therefore the origin is a center. These results have been found by proposing a function V(x, y) which is linear in the coefficient a21 of the general cubic system. In fact, our first choice was the coefficient a11, because for quadratic systems, V(x, y) is linear in this coefficient, but the calculations rapidely became very involved. After that, we have proposed a first integral linear in the New sufficient conditions for a center 353 coefficient a21. This choice is natural and leads to new conditions for a center. By using the fact that V(x, y) satisfies the equation: (3) ∂V (x, y) ∂x P(x, y)+∂V (x, y) ∂y Q(x, y) −V(x, y)∂P(x, y) ∂x +∂Q(x, y) ∂y =0 we found in this way new sufficient conditions for which the origin is a center of the general cubic system. In section 2, we generalize these results to polynomial systems (S)of arbitrary degree n. This is done by proposing a function V(x, y) of the form: (4) V(x, y)=V1(x, y)an−1,1+V2(x, y) where V1(x, y) and V2(x, y) don’t depend on an−1,1. We found that, similarly to the case n=3,V(x, y) factorizes as a polynomial of degree 1 and another of degree n, associated to two particular algebraic integrals of the system. Finally, in section 3, we use the particular algebraic integrals for constructing global phase portraits of the system (S1) with n=3. 1. Sufficient conditions for a center in the case n=3 Here, we consider the general cubic system (S1) (5) ˙x=P(x, y), ˙y=Q(x, y), where (6) P(x, y)=y+ 3  j=2 j  i=0 ai,j−ixiyj−i, Q(x, y)=−x− 3  j=2 j  i=0 bi,j−ixiyj−i. In order to obtain sufficient conditions for the existence of a center, we propose the reciprocal of an integrating factor V(x, y) which is linear in the coefficient a21 of (S1): (7) V(x, y)=V1(x, y)a21 +V2(x, y), 354 H. Giacomini, M. Ndiaye where V1(x, y) and V2(x, y) are independent of a21. By replacing (7) in equation (3) and imposing the condition that (3) must be identically zero for arbitrary values of a21, we obtain three equations u1(x, y)=0,(8) u2(x, y)=0,(9) u3(x, y)=0,(10) where (11) u1(x, y)=∂V1(x, y) ∂x x5y−2V1(x, y)x4y, and u2(x, y) and u3(x, y) are complicated expressions in terms of V1(x, y), V2(x, y) and their partial derivatives. From (11), we have (12) V1(x, y)=h(y)x2. When we replace (12) in (9), we obtain (13) W1(x, y)+∂V2(x, y) ∂x x5y−2V2(x, y)x4y=0, where W1(x, y) is a function of x,yand h(y). From (13), we can calculate V2(x, y) (14) V2(x, y)=(W2(x, y)+h1(y))x2, where h1(y) is an arbitrary function of yand W2(x, y) is a function of x, yand h(y). If we replace (12) and (14) in (10), we obtain an equation of the form (15) W3(x, y)=0, where W3(x, y) is function of x, y, h(y),h 1(y),h (y),h (y) and h 1(y). With respect to the variable x,W3(x, y) is a polynomial in xand log |x| (of first degree in log |x|). As (15) must be identically zero for arbitrary values of x, the coefficients of log |x|and (log |x|)0in (15) must be identically zero. From this, we obtain two equations W4(x, y)=0,(16) W5(x, y)=0,(17) New sufficient conditions for a center 355 where W4(x, y) and W5(x, y) are polynomials in the variable xof degree 7 and 5 respectively. The coefficients of these polynomials are functions of yand must be identically zero. One of the coefficients of W4(x, y) gives the equation (18) (1+a20y+a03y2)(−(b11 +2b12)h(y)+(1+b11y+b12y2)h(y)) = 0. As the first factor cannot be zero, the second one must vanish. This condition enable us to determine (19) h(y)=k(1 + b11y+b12y2), where kis an arbitrary constant factor. By replacing (19) in the others coefficients of (16), we find that they are identically zero. By replacing (19) in (17), we obtain 6 equations: (20) r1(y)=0, r2(y)=0, . . . r6(y)=0. Each of these equations must be identically zero for arbitrary values of y. One of them is a polynomial of fourth degree in y. As each coefficient of this polynomial must vanish, we obtain the following system of five algebraic equations between the coefficients of (S1): (21) b02 =0, 3a02b02 +6b03 +2b02b11 =0, 2a03b02 +5a02b03 +a02b02b11 +4b03b11 =0, 4a03b03 +3a02b03b11 −a02b02b11 +2b03b12 =0, 2a03b03b11 −2a03b02b12 +a02b03b12 =0. It is easy to verify that the solution of this system is b02 =b03 = 0, the remainder coefficients being arbitrary. By replacing these results in the other equations of the system (20), we obtain the condition (22) b11b2 30 =0. If we take b11 = 0, by following the calculation, we find that b30 =0. Therefore, we must choose the solution b30 = 0 and b11 arbitrary. 356 H. Giacomini, M. Ndiaye Using this result, we obtain from equations (20), after a long calculation (23) a02 =−2a20,b 02 =b03 =b30 =0, a03 =a2 20,b 11 =−a20 +b21 b20 , a30 =a20b20,b 12 =−a20b21 b20 , a12 =−a20a11,h 1(y)=b12(1 + b11y+b12y2), where we supposed that b20 =0. The final expression that we obtain for V(x, y) is polynomial in xand y: V(x, y)=(b20 +b21y)(−a20b20(a20b20 +b21)x3 +(a21b20 −a20b21)(1 −a20y)x2 +a11b20(1 −a20y)2x+b20(1 −a20y)3)/b2 20. We wrote V(x, y) in this form in order to be able to generalize this expression for a system (S) of arbitrary degree. Since V(x, y) is the reciprocal of an integrating factor, we can evaluate the associated constant of motion I(x, y) from the relations: (24) ∂I(x, y) ∂x =−Q(x, y) V(x, y),∂I(x, y) ∂y =P(x, y) V(x, y). It can be easily verified that the behaviour of I(x, y) in a neighbourhood of the origin is equivalent to (x2+y2)/2. This proves that I(x, y)is analytic at the origin and the level curves in a neighbourhood of this point are circles surrounding the origin O. The explicit form of I(x, y) is given in the Proposition 2 of section 3. We conclude now with the following theorem: Theorem 1.1. Consider system (S1)with b20 =0. Under conditions (25) a02 =−2a20,b 02 =b03 =b30 =0, a03 =a2 20,b 11 =−a20 +b21 b20 , a30 =a20b20,b 12 =−a20 b21 b20 , a12 =−a20a11, New sufficient conditions for a center 357 the origin is a center and the associated reciprocal of an integrating factor is a polynomial form given by (26) V(x, y)=(b20 +b21y)(−a20b20(a20b20 +b21)x3 +(a21b20 −a20b21)(1 −a20y)x2 +a11b20(1 −a20y)2x+b20(1 −a20y)3)/b2 20. 1.1. Particular case b20 =0. By using the same procedure that above, we find the following conditions for a center: (27) a02 =−2a20 b30 =b02 =b03 =b21 =a30 =0, a03 =a2 20 b12 =−(a2 20 +a20b11), a12 =−a20a11, and the reciprocal of an integrating factor is for this case (28) V(x, y)=(1+(a20 +b11)y)(1 −a20y)((1 −a20y)2 +a11(1 −a20y)x+(a21 −a2 20 −a20b11)x2). The associated constant of motion is analytic at the origin. 2. General case We also carried out of calculations of section 1 for the cases n=4, n= 5 and n=6. We proposed the following form for V(x, y): V(x, y)=V1(x, y)an−1,1+V2(x, y), where V1(x, y) and V2(x, y) are independent of an−1,1. The regular form taken by these results has allowed us to propose a general compact expression for the integrability conditions and the reciprocal of an integrating factor, for arbitrary values of n. For 2 ≤n≤6, we found that the difference between the number of coefficients of the system (dimension): (n−1)(n+ 4) and the number of integrability conditions unhas an arithmetic sequence progress: (n−1)(n+4)−un=3+2(n−2). Then we have un=n2+n−3. The general expressions for integrability conditions, V(x, y) and un that we induced from the results obtained for n≤6, is verified for n=7 and n=8. So, we conjecture the following theorem: 358 H. Giacomini, M. Ndiaye Theorem 2.1. Consider the system (S), with narbitrary, and assums that bn−1,0=0. Under conditions ai,j =(−1)j−1(n−i)! j!(n−(i+j))!(j−1)a2 20bi−1,0 +(n−i−1)! (j−1)!(n−(i+j))!ai,1aj−1 20 ,j=0, ai,0=a20bi−1,0, ai,j =(−1)j−1−(n−i−1)! j!(n−(i+j))!a20 +(n−i−1)! (j−1)!(n−(i+j+ 1))! bn−1,1 bn−1,0bi,0aj−1 20 ,i+j < n, j =0, (29) bi,j =(−1)j−1aj−1 20 bi,0 bn−1,1 bn−1,0 ,i+j=n, j =0, bn,0=0, O(0,0) is a center and the reciprocal of an integrating factor is given by (30) V(x, y)=(bn−1,0+bn−1,1y) n  k=0 ((n−k−1)a2 20bn−1,0bk−1,0 +ak,1bn−1,0−a20bk−1,0bn−1,1)(1 −a20y)n−kxk, with bi,j =ai,j =0if i<0,j<0or i+j>n. Particular case. Suppose that bn−1,0= 0. Under conditions (31) ai,j =(−1)j−1(n−i)! j!(n−(i+j))!(j−1)a2 20bi−1,0 +(n−i−1)! (j−1)!(n−(i+j))!ai,1aj−1 20 ,j=0, ai,0=a20bi−1,0, bi,j =(−1)j−1 (n−i)! j!(n−(i+j))!(j−1) +(n−i−1)! (j−1)!(n−(i+j))!(i−1)a20 +(n−i−1)! (j−1)!(n−(i+j))!b11bi,0aj−1 20 ,j=0, bn−1,1=an,0=0, New sufficient conditions for a center 359 the origin is a center and the function V(x, y) is given by (32) V(x, y)=(1+((n−2)a20 +b11)y) n−1  k=0 Ωk(y)xk, where Ωk(y)=(−(k−1)a20bk−1,0−a20b11bk−1,0+ak,1)(1 −a20y)n−k. 3. Phase portrait of the system (S1) Our purpose in this section is to demonstrate the following theorem Theorem 3.1. If the conditions (33) a20 >0,b 20 >0,b 21 >0, ∆<0, r1=a21 −a11b20 +a2 20 +b2 20 =0, and the conditions (25) are satisfied, the phase portrait of (S1)is homeomorphic to one of the configurations shown in Figures 1,... ,4. F2∞F1∞ F0∞ F3∞ F2∞ F2∞ F2∞ F1∞ F1∞ F1∞ F0∞F0∞ F0∞ F3∞ F3∞ F3∞ Figure 1 Figure 2 Figure 3 Figure 4 The phase portraits of the system (S1)associated with conditions (25) and (33). 366 H. Giacomini, M. Ndiaye If βi=−a20 b21 −1 b20 then we have Pβi,−b20 b21 =−(a21 −a11b20 +a2 20 +b2 20)(a20b20 +b21)2 b20b3 21 =0, and the point βi,−b20 b21 cannot be a critical point. In consequence we have λ1(Fi)=0(i=1,2,3). If λ2(Fi) = 0, then the abscisse βiof the critical point Fimust be a zero of λi(Fi), then the remainder of the euclidian division of Pwith λ2(Fi), calculated at the point βi,−b20 b21 must be a quantity independent from βi. But a simple calculation shows that it is of the form r2=m1+m2βi, where m2=−2a20 9b20 −2a2 21b20 9a20b2 21 −2a11a20b2 20 3b2 21 +4a21 9b21 −2a11b20 3b21 . In order to have a remainder independent of βi, we must impose the condition m2= 0, but it is easy to show that this condition is incompatible with ∆ <0. In consequence, we have λ2(Fi)=0(i=1,2,3). Signs of λ1(Fi)and λ2(Fi). Let us define the functions k1(x) and k2(x) as follows k1(x)=−x(a20b20 +b21 +b20b21x) b20 , k2(x)=−(a11b20(a20b20 +b21)+2b21(a21b20 −a20b21)x −3a20b20b2 21x2)/b2 21. It is evident that k1(βi)=λ1(Fi) and k2(βi)=λ2(Fi). In the following, the identity k2(x)=∂P ∂x x, −b20 b21 will be very useful for us. Let x1and x2be the zeros of ∂P ∂x x, −b20 b21 . We must consider two different cases. 1) Case where r1<0. For this case, we give the distribution of signs of the quantities that are relevant for the determination of the signs of λ1(Fi) and λ2(Fi)in New sufficient conditions for a center 367 Table 2. x30 ✁ ✁ ✁☛ ❆❆ ❆ ✁ ✁ ✁☛ ❆❆ ❆ x β1x1β2x2β3 ∂P ∂x x, −b20 b21 + + − − + + Px, −b20 b21 −+ + − − + k2(x) + + − − + + k1(x)− ± ± − + + ± ± − − Table 2 For the signs of k1(x), we have already seen that O∈]α2,α 3[soO∈ ]β2,β 3[. To be precise, O∈]β2,x 2[ifa11 ≥0 and O∈]x2,β 3[ifa11 <0. Obviously, the position of Oin the intervalle ]β2,β 3[ is not important for the determination of the signs of k1(x) and k2(x)atβi(i=1,2,3), as can be seen from Table 2. The other zero of k1(x)isx3=−a20 b21 −1 b20 <−1 b20 . As −1 b20 ∈]α1,α 2[, then −1 b20 ∈]β1,β 3[. Furthermore Px3,−b20 b21 =−r1(a20b20+b21 )2 b20b3 21 >0, which implies that x3∈]β1,β 2[. From these results, Table 2 follows. We see that for the three points Fi, the signs of the two eigenvalues associated to each point are distinct. Then we conclude that F1,F2and F3are saddle points. 2) Case where r1>0. We have P0,−b20 b21 =−b20(a20b20 +b21 )2 b3 21 <0 and Px3,−b20 b21 <0. So we must consider three possibilities: a) If x3<β 1<0. From Table 3, we conclude that F1is an unstable node, while F2and F3are saddle points. Let us remark that −1 b20 cannot be contained in the intervalle ]β2,0[. 368 H. Giacomini, M. Ndiaye Indeed we have that −1 b20 ∈]x3,β 1[. 0 ✁ ✁ ✁☛ ❆❆ ❆ x x3β1x1β2x2β3 ∂P ∂x x, −b20 b21 + + + − − + + Px, −b20 b21 − − + + − − − k2(x) + + + − − + + k1(x)−+ + + + ± ± − − Table 3 b) If β2<x 3<0. For this case we have Table 4, from which we conclude that F2is a stable node while F1and F3are saddle points. 0 ✁ ✁ ✁☛ ❆❆ ❆ x β1x1β2x3x2β3 ∂P ∂x x, −b20 b21 + + − − − + + Px, −b20 b21 −+ + − − − + k2(x) + + − − − + + k1(x)− − − − +± ± − − Table 4 c) If 0 <β 1<β 2<β 3. For this case, the table of signs is given in Table 5. We see that F2is a stable node, while F1and F3are saddle points. x x30β1x1β2x2β3 ∂P ∂x x, −b20 b21 + + + + − − + + Px, −b20 b21 − − − + + − − + k2(x) + + + + − − + + k1(x)−+− − − − − − Table 5 New sufficient conditions for a center 369 For the study of the critical points at infinity, we apply the same procedure that for the points F1,F2and F3, but we treat with the transformed system (S3). Is can be easily verified that λ1(F i∞) and λ2(F i∞) are both non-zero. In order to study their signs, it is convenient to introduce the polynomials: k 1(w)=(−a11a20b20 +2(a21b20 −a20b21)w+3b20(a20b20 +b21)w2)/b20, k 2(w)=b21w(−a20 +b20w) b20 . It is obvious that k 1(ξi)=λ1(F i∞) and k 2(ξi)=λ2(F i∞). Furthermore, a direct calculation shows that k 1(w)=∂P2 ∂w (w,0),P 2(0,0) = a2 20 >0 and P2a20 b20 ,0=a2 20r1 b2 20 . We use the same method as above. We must consider two cases: r1<0 and r1>0 and obtain the following conclusion. 1) Case where r1<0. F i∞(i=1,2,3) are unstable, stable and unstable nodes, respectively. 2) Case where r1>0. For this case we must consider three possibilities: a) If x4>γ 1,F 2∞and F 3∞are stable and unstable nodes, while F 1∞is a saddle point. b) If 0 <x 4<γ 2,F 1∞,F 3∞are both unstable nodes while F 2∞is a saddle. c) If 0 >γ 1>γ 2>γ 3,F 1∞,F 3∞are both unstable nodes and F 2∞ is a saddle. The results that have been obtained above for the points F i∞are also valid for the points Fi∞(i=1,2,3). We can note that the Tables 6, 7, 8 and 9 correspond respectively to Tables 2, 3, 4 and 5, respectively. A direct calculation gives the eigenvalues associated to the critical point M2:λ1(M2)=√r1 b20 and λ2(M2)=√r1 b20 . Therefore, if r1>0 then M2is a saddle and if r1<0, then M2is a center or a focus. 370 H. Giacomini, M. Ndiaye Proposition 3.5. If r1<0then M2is a center. Proof: If in the system (S1), we translate the origin at the point M2 we obtain the system (S4) ˙x=P3(x, y)=(a20b20x+(a21 −a11b20 +b2 20)y−2a20b2 20x2 +b20(a11b20 −2a21)xy +a20b20(a11 −2b20)y2 +a20b3 20x3+a21b2 20x2y−a11a20b2 20xy2+a2 20b2 20y3)/b2 20,(36) ˙y=Q3(x, y)=−(−1+b20x)(b20x−a20y)(b20 +b21y) b2 20 . As can be easily verified, the system (S4) admits the reciprocal of an integrating factor 1 V3, where V3(x, y) is given by V3(x, y)=(b20 +b21y)(−b20(a21 −a11b20 +a2 20 +b2 20) +b20(3a2 20b20 +2a21b20 −a11b2 20 +a20b21)x +a20(a21b20 −2a11b2 20 +3b3 20 −a20b21)y −b2 20(3a2 20b20 +a21b20 +2a20b21)x2 +2a20b20(−a21b20 +a11b2 20 +a20b21)xy +a2 20b2 20(a11 −3b20)y2+a20b3 20(a20b20 +b21)x3 +a20b2 20(a21b20 −a20b21)x2y−a11a2 20b3 20xy2+a3 20b3 20y3). Furthermore, the associated constant of motion I(x, y) defined by (29) is analytic in the neighbourhood of the origin M2:I(x, y)∼b3 20 2r1(b20x2− 2a20xy +(−a21 +a11b20 −b2 20)y2). 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I, Gauthier-Villars, Paris, 1951, pp. 3–84. 372 H. Giacomini, M. Ndiaye 18. L. Perko,“Differential equations and dynamical systems,” Springer-Verlag, New York, 1991. 19. A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier,“Qualitative theory of second-order dynamical systems,” John Wiley & Sons, New York, 1973. Laboratoire de mod`eles de Physique Math´ematique Facult´e des Sciences et Techniques Universit´e de Tours 37200 Tours FRANCE Primera versi´o rebuda el 23 d’Octubre de 1995, darrera versi´o rebuda el 10 de Novembre de 1995