Sufficient conditions for the existence of a center in polynomial systems of arbitrary degree

Publicacions Matem`atiques, Vol 40 (1996), 205–214. SUFFICIENT CONDITIONS FOR THE EXISTENCE OF A CENTER IN POLYNOMIAL SYSTEMS OF ARBITRARY DEGREE H. Giacomini and M. Ndiaye Abstract In this paper, we consider polynomial systems of the form ˙x= y+P(x, y), ˙y=−x+Q(x, y), where Pand Qare polynomials of degree nwihout linear part. For the case n= 3, we have found new sufficient conditions for a center at the origin, by proposing a first integral linear in certain coefficient of the system. The resulting first integral is in the general case of Darboux type. By induction, we have been able to generalize these results for polynomial systems of arbitrary degree. Introduction In this paper, we consider systems of differential equations (S) ˙x=P(x, y), ˙y=Q(x, y), in which P(x, y) and Q(x, y) are polynomials of the form P(x, y)=y+ n  j=2 j  i=0 aij−ixiyj−i, Q(x, y)=−x− n  j=2 j  i=0 bij−ixiyj−i, and we seek for sufficient conditions under which the origin is a center i.e., a critical point in a neighbourhood of which all orbits are closed. The problem of the center is an important one in the theory of plane polynomial vector fields. It plays an important role in the second part 206 H. Giacomini, M. Ndiaye 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. The necessary and sufficient conditions for a center are known for very few classes of systems. There are well-known conditions for quadratic systems [1], [2], [3] (and references therein) and for systems in which P(x, y) and Q(x, y) are cubic polynomials without quadratic terms [4], [5]. Conditions have been obtained for other classes of cubic systems. For instance, in [6] and [7], Kukles studied systems of the form: ˙x=y, ˙y=−x+a1x2+a2xy +a3y2+a4x3+a5x2y+a6xy2+a7y3, and claimed to have found necessary and sufficient conditions under which the origin is a center. However, Jin and Wang showed in [8] that Kukles’conditions are only sufficient but not necessary, by giving explicitly another condition which is not covered by the Kukles’ones (see also [9]). C. J. Christopher and N. G. Lloyd showed in [10] that Kukles’conditions are necessary only in the case where a7=0. More recently, important results have been obtained about the problem of the center for cubic polynomial vector fields [11], [12]. For polynomial systems of arbitrary degree, there are three classes for which sufficient conditions for the existence of the center are known: a family of homogeneous polynomial vector fields [13], reversible system [14], the Hamiltonian case. For this case, the constant of motion (the Hamiltonian) is linear in all the coefficients of the polynomials P(x, y) and Q(x, y). In the first section of this paper we analyse cubic systems. We propose a constant of motion which is linear in the coefficient of y2in the polynomial P(x, y). The reason of this choice comes from the analysis of the form of the constant of motions for the quadratic systems [1], [2]. For one of integrability conditions, the first integral is linear in the coefficient a02. For cubic system, by proposing a first integral linear in a02, we have found several new sets of sufficient conditions for a center. The corresponding first integrals are in general case of Darboux type [15], [16], [17], [18]. We have also proposed first integrals that are linear in the other coefficients of the system but we have not found new integrability conditions. Let us remark that the first integrals of Darboux type that we have found are not contains in the Sokulsky’s list [12]. We verify that this associated constant of motion is also linear in the coefficients of yand y3(by symmetry, analogous constants of motion exist that are linear in the coefficients of x,x2and x3of the polynomial Q(x, y)). It is natural to think that this result is valid for polynomial systems of Centers in polynomial systems 207 an arbitrary degree. In order to check this hypothesis, we perform the calculations for polynomials P(x, y) and Q(x, y) of degree 4 ≤n≤8. In all these cases, we find that the constant of motion is linear in the coefficients of yjof P(x, y), with 1 ≤j≤n. The analysis of the integrability conditions and the explicit expressions of constants of motion will enable us to conjecture a general result, valid for an arbitrary degree n. The conjectured expressions for the integrability conditions and constants of motion reduce to the correct results for 2 ≤n≤8. These results, valid for arbitrary n, are given in Section 2. 1. Case n=3 We consider the system: (S1) ˙x=y+ 3  j=2 j  i=0 aij−ixiyj−i, ˙y=−x− 3  j=2 j  i=0 bij−ixiyj−i. We propose a constant of motion f(x, y) linear in the coefficient a02: f(x, y)=f1(x, y)a02 +f2(x, y) where f1(x, y) and f2(x, y) are independent of a02. Taking the rate of change of f(x, y) along a trajectory of (S1), we obtain ˙ f(x, y)=∂f1(x, y) ∂x a02 +∂f2(x, y) ∂x P(x, y) +∂f1(x, y) ∂y a02 +∂f2(x, y) ∂y Q(x, y). If f(x, y) is a constant of motion, ˙ f(x, y) must be identically zero for arbitrary values of a02. In consequence, we obtain the following conditions for the functions f1(x, y) and f2(x, y): ∂f1(x, y) ∂x =0, Q(x, y)∂f1(x, y) ∂y +y2∂f2(x, y) ∂x =0, C(x, y)∂f2(x, y) ∂x +Q(x, y)∂f2(x, y) ∂y =0, 208 H. Giacomini, M. Ndiaye which imply f1(x, y)=f1(y), ∂f2(x, y) ∂x =−Q(x, y) y2 ∂f1(y) ∂y ,(α) ∂f2(x, y) ∂y =C(x, y) y2 ∂f1(y) ∂y ,(β) where C(x, y)=y+a20x2+a11xy +a30x3+a21x2y+a12xy2+a03y3. By imposing the condition ∂2f2(x, y) ∂x∂y =∂2f2(x, y) ∂y∂x . We obtain from (α) and (β): (−a11y3+(−a12 +b03)y4)∂f1(y) ∂y +(b02y4+b03y5)∂2f1(y) ∂y2 +((−2y+(−2a20 −b11)y2−2a21y3)∂f1(y) ∂y +(y2+b11y3+b12y4)∂2f1(y) ∂y2)x +((−2b20y+(−3a30 −b21)y2)∂f1(y) ∂y +(b20y2+b21y3)∂2f1(y) ∂y2)x2 +((−2y∂f1(y) ∂y +y2∂2f1(y) ∂y2)x3)b30 =0. As this condition must be satisfied for arbitrary values of xand y, and f1(y) is a function only of y, we have the following equations for the function f1(y): (S2) (1) (−a11 +(−a12 +b03)y)∂f1(y) ∂y +(b02y+b03y2)∂2f1(y) ∂y2=0, (2) (−2+(−2a20 −b11)y−2a21y2)∂f1(y) ∂y +(y+b11y2+b12y3)∂2f1(y) ∂y2=0, (3) (−2b20 +(−3a30 −b21)y)∂f1(y) ∂y +(b20y+b21y2)∂2f1(y) ∂y2=0, (4) −2∂f1(y) ∂y +y∂2f1(y) ∂y2=0,or (5) b30 =0. Centers in polynomial systems 209 Then we must consider two different cases: 1) Equations (1), (2), (3) and (4) are satisfied: These equations are compatible if and only if the following conditions holds −b21 +3a30 =−b11 +2a20 =−b12 +a21 =−2b02 +a11 =−3b03 +a12 =0. These conditions insure that System (S1)isHamiltonian. Therefore, in order to obtain new results we must consider condition (1), (2), (3) and (5). 2) Equations (1), (2), (3) and (5) are satisfied. The compatibility relation between (1) and (2) gives: (−2b02 +a11)y+((−b11 −2a20)b02 −3b03 +a12 +a11b11)y2 +(−2b11b03 −2a20b03 −2a21b02 +a12b11 +a11b12)y3 +(−2a21b03 −b03b12 +a12b12)y4=0 for arbitrary values of y. The compatibility condition between (1)-(2) and (2)-(3) gives, respectively ((−2b02 +a11)b20)y+(−b21b02 −3a30b02 −3b20b03 +b20a12 +b21a11)y2 +(−2b21b03 −3a30b03 +b21a12)y3=0, (3a30−2a20b20+b11b20−b21)y2+(3a30b11−2a21b20+2b12b20−2a20b21)y3 +(3a30b12 −2a21b21 +b12b21)y4=0. As these conditions must be satisfied for arbitrary values of y, we have b30 =0, −2b02 +a11 =0, −2b11b03 −2a20b03 −2a21b02 +a12b11 +a11b12 =0, −b11b02 −2a20b02 −3b03 +a12 +a11b11 =0, −2a21b03 −b03b12 +a12b12 =0, −b21b02 −3a30b02 −3b20b03 +b20a12 +b21a11 =0, −2b21b03 −3a30b03 +b21a12 =0, 3a30 −2a20b20 +b11b20 −b21 =0, 3a30b11 −2a21b20 +2b12b20 −2a20b21 =0, 3a30b12 −2a21b21 +b12b21 =0. 210 H. Giacomini, M. Ndiaye If we assume that (3a30 +b21)(3a30 +2b21)=0,b 20 =0, the general solution of the above system of equations is b11 =2a21b21 3a30 +b21 +b21 b20 ,b 02 =a12b20 3a30 +2b21 ,b 12 =2a21b21 3a30 +b21 , b03 =a12b21 3a30 +2b21 ,a 20 =a21b20 3a30 +b21 +3a30 2b20 ,a 11 =2a12b20 3a30 +2b21 , b30 =0. In this way, System (S2) is compatible. We can now determine the function f1(y), in order to obtain an explicit expression for the constant of motion f(x, y). From Equation (3), and assuming that a30(3a30 +b21)(3a30 +2b21)=0,b 21 =0, we obtain f1(y)=2y2(b20 +b21y) 3a30 b21 3a30 −2(b20 +b21y) 3a30+b21 b21 3a30(3a30 +b21) +2(b20 +b21y) 3a30+2b21 b21 3a30(3a30 +b21)(3a30 +2b21). Now, from Equations (α) and (β), and assuming that a30(3a30 +b21)(3a30 +2b21)(a30 +b21)=0,b 21 =0, we determine f2(x, y): f2(x, y)=2x2 2+b20 3x3+b11 2x2y+b02xy2+b21 3x3y+b12 2x2y2+b03xy3 +2y(b20 +b21y) 3a30 −(b20 +b21y)2 3a30(3a30 +b21) +2a03 y3(b20 +b21y) 3a30 −3y2(b20 +b21y)2 3a30(3a30 +b21) +6y(b20 +b21y)3 3a30(3a30 +b21)(3a30 +2b21) −6(b20 +b21y)4 3a30(3a30 +b21)(3a30 +2b21)(3a30 +3b21)(b20 +b21y) 3a30 −b21 b21 then, we conclude with the following theorem: Centers in polynomial systems 211 Theorem 1. Under conditions b11 =2a21b20 3a30 +b21 +b21 b20 ,b 02 =a12b20 3a30 +2b21 ,b 12 =2a21b21 3a30 +b21 b03 =a12b21 3a30 +2b21 ,a 20 =a21b20 3a30 +b21 +3a30 2b20 ,a 11 =2a12b20 3a30 +2b21 , b30 =0, with b20 =0,b21 =0,a30(3a30 +b21)(3a30 +2b21)(a30 +b21)=0, the origin is a center of (S1)and the constant of motion is given by f(x, y)=2y(b20 +b21y) 3a30 −(b20 +b21y)2 3a30(3a30 +b21) +2a02 y2(b20 +b21y) 3a30 −2y(b20 +b21y)2 3a30(3a30 +b21) +2(b20 +b21y)3 3a30(3a30 +b21)(3a30 +2b21) +2a03 y3(b20 +b21y) 3a30 −3y2(b20 +b21y)2 3a30(3a30 +b21) +6y(b20 +b21y)3 3a30(3a30 +b21)(3a30 +2b21) −6(b20 +b21y)4 3a30(3a30 +b21)(3a30 +2b21)(3a30 +3b21) +x2+2 3b20x3+b11x2y+2b02xy2 +2 3b21x3y+b12x2y2+2b03xy3(b20 +b21y) 3a30−b21 b21 . We have written the constant of motion in such a way that it will be possible later on, to guess its general form for polynomial systems of arbitrary degree. Let us remark that all the constants of motion that we have found in this section are also linear in the coefficient a03. New integrability cases and new constants of motion can be obtained from the above results by interchanging xand y,aij and bji. This is due to the invariance of (S) under the transformation x↔y,aij ↔bji, t→−t. 2. General case Using the results obtained in Section 1, we remark that when we seek for a constant of motion which is linear in a coefficient of ymin P(x, y), 212 H. Giacomini, M. Ndiaye with 1 ≤m≤3, we obtain one which is linear in all coefficients of ym.It is natural to think that this result will be valid for polynomial systems of an arbitrary degree n. In order to verify this hypothesis, we have carried out similar calculations as in Section 1, for polynomial systems up to degree 8. We have found analogous results as for the case n=3: in all the cases, the constant of motion is linear in all coefficients of ym in P(x, y). In addition, it is always possible to find explicit expressions of the constant of motion. By throughly inspecting expressions for integrability conditions and constants of motion up to degree 8, we have been able to induce analogous results for polynomial systems of an arbitrary degree. Let us remark that, up to n= 8, we have found that for the System (S), which has (n−1)(n+4) coefficients, the number of integrability conditions is given by (n−1)(n+4) 2for the principal case, where n k=0(nan,0+kbn−1,1)=0, bn−1,0= 0 and bn−1,1=0. The above remarks lead us to conjecture the following theorem: Theorem 2.1. Consider the System (S). Assuming that n  r=0 (nan,0+rbn−1,1)=0,b n−1,0=0,b n−1,1=0, and aij =0if i+j>nor j<0, and under conditions a20 =nan,0 2bn−1,0 + m∈∗ (−1)m+1m!a2,mbm n−1,0 m r=1 (nan,0+rbn−1,1), b11 =bn−1,1 bn−1,0 +2 m∈∗ (−1)m+1m!a2,mbm n−1,0 m r=1 (nan,0+rbn−1,1), a11 = m∈∗ (−1)m+1(m+ 1)!a1,1+mbm n−1,0 m r=1(nan,0+(r+1)bn−1,1), b02 =1 2 m∈∗ (−1)m+1(m+ 1)!a1,1+mbm n−1,0 m r=1(nan,0+(r+1)bn−1,1), bi,j =(i+1)ai+1,j−1bn−1,1 nan,0+(j−1)bn−1,1 +(nan,0−bn−1,1) m∈∗ (−1)m−1 (j+m−1)! j! ai+1,m+j−1bm n−1,0 m+j r=j(nan,0+(r−1)bn−1,1), 2≤i+j≤n, (i, j)=(0,2),(i, j)=(1,1), Centers in polynomial systems 213 O(0,0) is a center and the associated constant of motion is f(x, y)= n  m=0 a0,m m  l=0 (−1)lm! (m−l)! ym+1−l(bn−1,0+bn−1,1y) nan,0+lbn−1,1 bn−1,1 l k=0(nan,0+kbn−1,1)+h(x, y), with h(x, y)=  n  j=0 j  i=0 bi,j−i xi+1 i+1yj−i (bn−1,0+bn−1,1y) nan,0−bn−1,1 bn−1,1, and a0,0=b0,0=a1,0=b0,1=0,a0,1=b1,0=1. We recall that this conjectured theorem has been verified for 2 ≤n≤ 8. Let us remark that we have also considered all the particular cases by using the same method. We have found nine particular cases, but due to the size of the calculations we have not included these cases in the present paper. We have also seek constants of motion which are linear in the remaining coefficients of (S1), but the resulting differential equations for the functions f1(x, y) and f2(x, y) are in all cases incompatible. Much of calculations performed in this work have been made with Mathematica computer algebra system. References 1. D. Schlomiuk, J. Guckenheimer and R. Rand, Integrability of plane quadratic vector field, Exp. Math 8(1990), 3–25. 2. D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc. 338 (1993), p. 799. 3. H. ˙ Zoladek, Quadratic systems with center and their perturbations, J. Differential Equations 109 (1994), 223–273. 4. V. A. Lunkevich and S. K. Sibirski, On the condition for a center, Differential’nye Uravneniyad 1(1965), 53–66. 5. J. Chavarriga, Integrable systems in the plane with center type linear part, Applicationes Mathematicae 22 (1994), 285–309. 6. I. S. Kukles, Necessary and sufficient conditions for the existence of a center, Dokl. Akad. Nauk. 42 (1944), 160–163.