The null divergence factor

Publicacions Matem`atiques, Vol 41 (1997), 41–56. THE NULL DIVERGENCE FACTOR J. Chavarriga∗, H. Giacomini and J. Gin´ e∗ Abstract Let (P, Q)beaC1vector field defined in a open subset U⊂R2. We call a null divergence factor a C1solution V(x, y)of the equation P∂V ∂x +Q∂V ∂y =∂P ∂x +∂Q ∂y V. In previous works it has been shown that this function plays a fundamental role in the problem of the center and in the determination of the limit cycles. In this paper we show how to construct systems with a given null divergence factor. The method presented in this paper is a generalization of the classical Darboux method to generate integrable systems. 1. Introduction We consider in this paper two-dimensional autonomous systems of differential equations of the form (1) ˙x=P(x, y),˙y=Q(x, y),·=d dt, with P(x, y), Q(x, y)∈C1(E) and where Eis an open subset of R2. The two fundamental problems of the qualitative theory of system (1) are the problem of the center and the determination of the number of limit cycles and their location in phase space. In recent works it has been shown that a unified method can be used to study these problems [1], [2], [3], [4], [5], [9], [10], [11], and [12]. The method is based on the determination of a function V(x, y)∈ C1(E) that satisfies the equation (2) P∂V ∂x +Q∂V ∂y =∂P ∂x +∂Q ∂y V. ∗Research partially supported by a University of Lleida Project 93-3. 42 J. Chavarriga, H. Giacomini, J. Gin´ e Let us consider first the center problem for polynomial systems of the form: (3) ˙x=−y+X(x, y),˙y=x+Y(x, y), where X(x, y) and Y(x, y) are polynomials without independent and linear terms. The problem of the center consists in giving necessary and sufficient conditions on the coefficients of X(x, y) and Y(x, y) in order to have a continuous family of periodic orbits in a certain neighbourhood of the origin. If for system (3) we can find a solution V(x, y)∈C1(E) of (2) that is not zero at the origin, then we can obtain a first integral of (3) well defined in a neighbourhood of the origin, because M(x, y)= 1 V(x,y) is an integrating factor of the system. In that case, the origin will be a center of (3). For many systems of type (3) having a center at the origin, it has been shown in [1], [2], [4], [5] and [11] that the function V(x, y) has very simple properties, being very often a polynomial. By contrary, the first integral is, in general, a complicated expression that can not be written in terms of elementary functions. In particular, when in system (3) Xand Yare both quadratic or cubic homogeneous polynomials the function V(x, y) is a polynomial for all center cases (see [1]). In the general case, i.e. for system (1), it has been shown in [3], [9], [10] and [12] that any solution of (2) plays a fundamental role in the determination of the limit cycles of the system. Esentially, V(x, y) must vanish on all limit cycles of (1) (for a precise formulation of these results see [9]). In this paper we present a method which enables us to generate (or construct) systems of type (1) with a known function V(x, y). In this way, for all systems generated with this method, we know at once all limit cycles and all centers. These results are presented in sections 2 and 3. 2. Construction of systems with a known function V(x, y) We generalise in this section the classical Darboux method for constructing integrable systems [8]. Based in the Darboux method the next result follows easily from Christopher results (see [6] and [13]). The null divergence factor 43 The vector field defined by: (4) P= n  i=1 ai    n  j=1 j=i fj(x, y)    ∂fi(x, y) ∂y , Q=− n  i=1 ai    n  j=1 j=i fj(x, y)    ∂fi(x, y) ∂x , where fj(x, y) (with j=1,... ,n) are arbitrary C2functions, n∈Nand the aiare arbitrary real parameters, has a first integral given by (5) I(x, y)= n  i=1 fai i, and an integrating factor (6) M(x, y)= n  i=1 f−1 i. As it is well known, a system with a Darboux type first integral (5) can not have limit cycles in the domain of definition of the Darboux first integral. In particular if aiare rational numbers system (4) can not have limit cycles. Instead of giving a vector field with a known first integral, we construct a system with a known function V(x, y) as follows: Proposition 1. Let (Pi,Q i), with i=1,... ,n,beC1vector fields defined in an open subset U⊂R2, which have C2null divergence factors Vi(x, y), i.e. (7) Pi ∂Vi ∂x +Qi ∂Vi ∂y =∂Pi ∂x +∂Qi ∂y Vi, with i=1,... ,n. 44 J. Chavarriga, H. Giacomini, J. Gin´ e Then, the vector field (8) P=λ0 ∂(n i=1 Vi(x, y)) ∂y + n  i=1 λi    n  j=1 j=i Vj(x, y)    Pi(x, y), Q=−λ0 ∂(n i=1 Vi(x, y)) ∂x + n  i=1 λi    n  j=1 j=i Vj(x, y)    Qi(x, y), has a null divergence factor V(x, y)given by (9) V(x, y)= n  i=1 Vi(x, y). Proof: The proof is straight-forward. The first integral of (8) can be calculated from the integrating factor M(x, y)= 1 V(x,y). In general, this first integral will not be defined in the whole domain of the definition of the differential system and it is possible for system (8) to have limit cycles. It is clear that (4) is a particular case of (8), with Vi(x, y)=fi(x, y), Pi(x, y)=∂fi ∂y ,Qi(x, y)=−∂fi ∂x ,λi=aiand λ0=0. In (4) all vector fields (Pi,Q i) used to generate the system (P, Q) are Hamiltonian, while in (8) they are arbitrary. This is the key point of our generalization of the classical Darboux method. It is interesting to note that well-known systems can be constructed from (8) by using linear systems and Hamiltonian systems (Pi,Q i). Let us consider several examples: Example 1. In [14], a quartic system with one center and one limit cycle has been studied. The system is: (10) P=−2y(x2+y2)(x−2)+(x−y)(x2+2y2−1)(x−2), Q=x(x2+y2)(x−2)+(x+y)(x2+2y2−1)(x−2) −7 10(x2+2y2−1)(x2+y2). The null divergence factor 45 The null divergence factor of this system is: (11) V(x, y)=(x−2)(x2+y2)(x2+2y2−1). The limit cycle of (10) is the ellipse x2+2y2−1 = 0 and the center is located at the point (3,1). This system can be generated by using (8), as follows: (12) P=P1V2V3+P2V1V3+P3V1V2, Q=Q1V2V3+Q2V1V3+Q3V1V2, where (P1,Q 1)=(−2y, x),with V1(x, y)=(x2+2y2−1), (P2,Q 2)=0,−7 10,with V2(x, y)=(x−2) and(13) (P3,Q 3)=(x−y, x +y),with V3(x, y)=(x2+y2). For this case we have n=3,λ0=0,λ1=λ2=λ3= 1. Systems (P1,Q 1) and (P2,Q 2) are Hamiltonian. System (P3,Q 3) is a linear non-Hamiltonian vector field. Let us recall that the null divergence factor of a linear system (14) P=ax +by, Q=cx +dy, is given by (15) V(x, y)=cx2+(d−a)xy −by2. For a Hamiltonian vector field P=∂H(x,y) ∂y ,Q=−∂H(x,y) ∂x , the null divergence factor is V(x, y)=f(H(x, y)) where fis an arbitrary function. Example 2. The cubic system (16) P=y+a20 x2+a11 xy −2a20 y2+a20 b20 x3+a21 x2y −a11 a20 xy2+a2 20 y3, Q=−x−b20 x2−b21 b20 −a20xy −b21 x2y+a20b21 b20 xy2, 46 J. Chavarriga, H. Giacomini, J. Gin´ e with b20 = 0 and a20b20 +b21 = 0, has been studied in [11]. It has a center at the origin and the null divergence factor is: (17) V(x, y)=(b20 +b21 y)(−a20 b20(a20 b20 +b21)x3 +(a21 b20 −a20 b21)x2(1 −a20 y) +a11 b20x(1 −a20 y)2+b20 (1 −a20 y)3). This system can be expressed as the composition of two systems (P1,Q 1) and (P2,Q 2), as follows: (18) P=P1V2+P2V1, Q=Q1V2+Q2V1, where (P1,Q 1)=−1 a20 b20 +b21 ,0,with V1=b20 +b21 yand (19) (P2,Q 2)=((a20 b20 +b21)−1(1 + a11 x+(a2 20 +a21)x2−2a20 y −a11 a20 xy +a2 20 y2),x b20 (−1−b20 x+a20 y)), with V2=−a20 b20(a20 b20 +b21)x3+(a21 b20 −a20 b21)x2(1 −a20 y) +a11 b20 x(1 −a20 y)2+b20(1 −a20 y)3. System (P1,Q 1) is a constant Hamiltonian vector field and (P1,Q 1)is an integrable quadratic system. Example 3. The cubic system (20) P=y, Q=−x+k(1 −l)x2+k2lx3+a2xy −a2kx2y+kly2−k2l2 1+lxy2, with l+1= 0, has been studied in [7]. It has a center at the origin and its null divergence factor is The null divergence factor 47 (21) V=1+2klx+k2l2x2−a2y−a2klxy−k2l3 1+ly2. This system can be expressed as the composition of two systems, when kl(l+1)= 0, as follows: (22) P=P1V2+P2V1, Q=Q1V2+Q2V1, with (23) (P1,Q 1)=y, (1 + l) kl 2(1 + klx−a2y),with V1=Vand (P2,Q 2)=0,(−1−l+klx) kl 2,with V2=1. Example 4. The quadratic system (24) P=−y−bx2−cxy −dy2, Q=x+ax2+Axy −ay2, has a center at the origin if and only if one of the following conditions is satisfied. (25) (i) A−2b=c+2a=0, (ii) c=a=0, (iii) b+d=0, (iv) c+2a=A+3b+5d=a2+bd +2d2=0. In all the cases, system (24) can be descomposed in terms of more simple systems. 48 J. Chavarriga, H. Giacomini, J. Gin´ e The case (i) corresponds to a Hamiltonian system and is a particular case of (8), with n=1,λ0= 1 and V1(x, y)=H(x, y), where H(x, y)is the Hamiltonian of the system. For the case (ii), with (A+b)(A+2b)= 0, the system can be written as (26) P=∂(V1V2) ∂y +P1V2+P2V1, Q=−∂(V1V2) ∂x +Q1V2+Q2V1, where (P1,Q 1)=−(1 + A3+3A2b+2Ab2) (A+b)(A+2b),0,with V1=1+Ay, and (27) (P2,Q 2)=(1+2A2b+6Ab2+4b3) (A+b)(A+2b)(−A−b+d−Ady −2bdy), (1+2A2b+6Ab2+4b3)x, with V2=−A−b+d+b(A+b)(A+2b)x2+2b(A+b−d)y+bd(A+2b)y2. For the case (iii), without loss of generality, we can take a= 0. In this case, system (24) can be decomposed as follows: (28) P=∂(V1V2) ∂y +P1V2+P2V1, Q=−∂(V1V2) ∂x +Q1V2+Q2V1, where A+b= 0 and The null divergence factor 49 (P1,Q 1)=1−A2−Ab A+b,0,with V1=1+Ay and (29) (P2,Q 2) = ((2Ab +2b2−1+(Acb +cb2−c)x+(b−2Ab2−2b3)y) (A+b)−1,c+x−2Abx −2b2x−cby), with V2=(1−by)2+c(1 −by)x−b(A+b)x2. For the particular case A+b= 0 the decomposition is different. We have (30) P=1+c2 c2 ∂(V1V2) ∂y +P1V2+P2V1, Q=−1+c2 c2 ∂(V1V2) ∂x +Q1V2+Q2V1, where c= 0 and (P1,Q 1)=(b, c),with V1=1−by +cx and (31) (P2,Q 2)=3b+2bc2−bcx −3b2y−c2y−2b2c2y c2,1−by c, with V2=(1−by)2. The case A+b=c= 0 is a particular case of (ii). Finally, for condition (iv) we find the decomposition (32) P=∂(V1V2) ∂y +P1V2+P2V1, Q=−∂(V1V2) ∂x +Q1V2+Q2V1, 56 J. Chavarriga, H. Giacomini, J. Gin´ e 1991 Mathematics subject classifications: Primary 34C05; Secondary 34A05. J. Chavarriga: Departament de Matem`atica Universitat de Lleida Pla¸ca Victor Siurana 1 25003 Lleida SPAIN e-mail: chav[email protected] H. Giacomini: Laboratoire de Math´ematiques et Physique Th´eorique CNRS UPRES A6083 Facult´e des Sciences et Techniques Parc de Grandmont 37200 Tours FRANCE e-mail: [email protected] J. Gin´e: Departament de Matem`atica Universitat de Lleida Pla¸ca Victor Siurana 1 25003 Lleida SPAIN e-mail: [email protected] Primera versi´o rebuda el 30 de Novembre de 1996, darrera versi´o rebuda el 8 d’Abril de 1997