Integrability conditions for complex kukles systems

ABSTRACT In this paper, we provide necessary and sufficient conditions for the existence of local analytic first integrals for a seventh-parameter family of complex cubic systems called the complex Kukles systems.


Introduction and statement of the main results
The integrability problem is one of the main problems in the qualitative theory of planar polynomial differential systems. Although integrability is a restrictive condition and generically a differential system is not integrable, the existence of a first integral allows to know the phase portrait of the planar differential system. These non-generic integrable differential systems are used to analyse a more big family of differential systems that is described as perturbations of these non-generic integrable systems. Closely related to the existence of a local analytic first integral is the so-called centre-focus problem, i.e. the study of when a planar polynomial real systems of the forṁ x = −y + P(x, y),ẏ = x + Q(x, y) (1) being P and Q polynomials admit a local analytic first integral of the form where each X k is a homogenous polynomial of degree k andz = x − iy. It turns easier and more convenient from the computational point of view to study the integrability problem not only for Equation (3) but also for the more general complex systemṡ where each X jk and Y jk are complex parameters. If we apply the change of time t →it, system (4) becomes a system of the forṁ a jk z jzk = z +P(z,z), For system (5), one can always find a function of the form such that˙ where g ss are polynomials of a jk and b jk with rational coefficients. In the case that˙ = 0, we say that the origin is a complex centre. Note that in this case, is a first integral of system (5). If the differential system (5) is a complexification of the differential system (1), then going back to the original coordinates we obtain from a first integral of (1) of the form (2). Conditions for the existence of a complex centre (that is for the existence of a first integral such as in (6)) have been found for the case of quadratic systems (i.e. for n = 2 in (5)), see for instance [1,2]. Also, whenP andQ are homogeneous polynomials of degree 3, see for instance [3]. However, the case whenP andQ are polynomials of degree three but are not homogenous is an open and difficult problem, see [4]. However, several authors have studied some particular cases, see for instance [4][5][6][7] and references therein.
In this direction, we study the following family that exhibits properties and issues which are important in the problem of full classification of complex cubic differential systems with a complex centre. More precisely, we consider the cubic planar differential system of the formẋ = x − a 10 x 2 − a 01 xy − a 12 y 2 − a 20 x 3 − a 11 x 2 y − a 02 xy 2 − a 13 y 3 , where the dot denotes the derivative with respect to the time t and a 10 , a 01 , a 12 , a 20 , a 11 , a 02 , a 13 are parameters. These systems are called complex Kukles systems since systems of this form but in R 2 and with the linear part with eigenvalues ±i were studied first by Kukles in [8], who was the first in study a linear centre perturbed by cubic nonhomogeneous nonlinearities. More specifically, Kukles in [8] initiated the study of the necessary and sufficient conditions for the existence of a real centre in differential systems of the forṁ x = y, y = −x + a 1 x 2 + a 2 xy + a 3 y 2 + a 4 x 3 + a 5 x 2 y + a 6 xy 2 + a 7 y 3 .
The centre problem for system (8) with a 2 = 0 was resolved in [9] and with a 7 = 0 in [10]. The first complete solution of the centre-focus problem for Kukles system (8) was obtained in [11]. In [12] was also given the complete solution using the Cherkas' method of passing to a Liénard equation, see also the works [10,[13][14][15][16]. In [17] were studied the Kukles homogeneous systems. In this paper, we provide necessary and sufficient conditions for the existence of a complex centre at the origin of system (7). The main result of this paper is the following. The proof of Theorem 1.1 is given in Section 3.
In order to determine the necessary conditions to have a centre, we propose the Poincaré series where H 2 (θ) = 1/2 and H n (θ) are homogeneous trigonometric polynomials respect to θ of degree n. Imposing that this power series is a formal first integral of system (10), we obtaiṅ where V 2k are the saddle constants that depend on the parameters of system (9). Indeed, it is easy to see by the recursive equations that generate the V 2k that these saddle constants are polynomials in the parameters of system (9), see [18]. Taking into account that system (9) depends on a finite number of parameters, due to the Hilbert Basis theorem, the ideal J = <V 4 , V 6 , ... > generated by the saddle constants is finitely generated. The set of coefficients for which all the saddle constants vanish is called the complex centre variety of the family and it is an algebraic set. We assume that these generators are the first k saddle constants that define the ideal J k = <V 4 , V 6 , ..., V 2k >. It is clear that V(J)࣪V(J k ), being V the variety of the ideals J and J k , respectively. Hence, we determine a number of saddle constants thinking that inside these number there is a set of generators of J. The first saddle constants is V 4 = a 01 a 10 + a 11 . We vanish this constant taking a 11 = −a 01 a 10  From here, we compute V 10 , V 12 , V 14 , V 16 that we do not write here because of their length. We define The much harder problem is decompose this ideal into its irreducible components. We must to use a computer algebra system. The computational tool which we use is the routine minAssGTZ [19] of the computer algebra system Singular [20] which is based on the Gianni-Trager-Zacharias algorithm [21]. The computations have been completed in the field of rational numbers; therefore, we know that the decomposition that we encountered is complete. Finally, we have obtained the decomposition given in Theorem 1.1. In the following, we provide the sufficiency to have a complex centre for each component.
To prove the sufficiency condition, we will use the Darboux theory of integrability. This theory is one of the best tools to prove the existence of first integrals for polynomial differential systems. The Darboux theory relies on the existence of a sufficient number of invariant algebraic curves and of the so-called exponential factors in order to prove the existence of first integrals. More precisely, if we have a planar polynomial differential system of the formẋ we say that f (x, y) ∈ C[x, y] \ C is a Darboux polynomial of the differential system (12), if it satisfies being K the cofactor of f. The function F = e g/h with g, h ∈ C[x, y] being coprime is an exponential factor of the differential system (12) being L the cofactor of F. An inverse integrating factor of the differential system (12) on an open set is a differentiable function V(x, y) on such that where div (P 1 , Q 1 ) stands for the divergence of system (12). It follows from the Darboux theory of integrability the following: assume that system (12) admits p Darboux polynomials f i and q exponential factors F j wit cofactors K i and L j , respectively. Then, there exists λ i , μ j ∈ C not all zero such that is an inverse integrating factor of system (12).
Moreover, it follows from the Darboux theory of integrability that if there exists an integrating factor for system (12) that is non-zero well defined in a singular point of system (12), then system (12) has an analytic first integral at such singular point, see for instance [1,4].

Proof of Theorem 1.1
In this section, we prove the sufficiency of all conditions of Theorem 1.1 by doing a caseby-case analysis of all four cases.

Proof of statement (a):
In this case, system (7) becomeṡ It has three invariant lines l 1 = x, l 2 = y, and l 3 = 1 − a 10 x − a 20 x 2 and an inverse integrating factor is of the form V = l 1 l 2 l 3 . This inverse integrating factor is well-defined at the origin but V(0, 0) = 0. However, from this inverse integrating factor, we can compute a first integral for system (7), which is analytic around the origin, given by Here, the corresponding system (7)  Since the inverse integrating factor is non-zero well defined at the origin, system (7) has an analytic first integral at the origin.

Proof of statement (d):
In this case, system (7) becomeṡ x = x − a 10 x 2 − a 01 xy + a 01 a 10 x 2 y − a 12 y 2 − a 02 xy 2 − a 13 y 3 , To prove the existence of a first integral, we will use the so-called sum method, see [4,22]. We look for a formal first integral expressed in the form Note that by (13), imposing that˙ = 0 we get f 0 = 0, and for k ࣙ 3, the functions f k are determined recursively by the first order differential equation Solving the first linear differential equation in (15) and using that the integration constant is equal to one, we get Moreover, using the second and third linear differential equations in (15) and using that the integration constants are zero, we obtain Let p i (x) denote a polynomial of degree i. Using the induction over k ࣙ 2, we will show that Assume by induction that f k is as in (18) for k = 2, …, n − 1 and we will show it for n. We can assume that n ࣙ 5 since for k = 2, 3 we have showed it in (17).

Conclusions
We have determined the necessary and sufficient conditions to have a complex centre for the complex Kukles system. The necessary conditions are found imposing the existence of a formal first integral. The sufficiency is derived from the existence of invariant algebraic curves that allows to compute inverse integrating factors and first integral for each case except the last one where the sum method developed in [4,22] is used. Remains open the same problem for a general complex cubic system which is outside of our current computing facilities.