The nondegenerate center problem and the inverse integrating factor (Q2490043)

This paper deals with the center-focus problem for analytic planar differential systems and, in particular, polynomial differential systems. That is, given a planar analytic differential system defined in a neighborhood of the origin of the form \[ \dot{x} = -y + p(x,y), \quad \dot{y} = x + q(x,y), \tag{1} \] where \(p(x,y)\) and \(q(x,y)\) are of order greater or equal than two, we see that the linear part is a center. The Poincaré center-focus problem asks for criteria to determine whether the origin of system (1) is really a center. A characterization of centers is due to \textit{H. Poincaré} [Résal J. (3) VII. 375--422 (1881; JFM 13.0591.01); Résal J. (3) VIII. 51--296 (1882; JFM 14.0666.01)] and to \textit{Lyapunov} [Ann. of Math. Stud. 17 (1947)] and states that system (1) has a center at the origin if, and only if, there exists a local analytic first integral defined in a neighborhood of the origin and of the form \(H(x,y) = (x^2+y^2) + F(x,y),\) where \(F\) is of order greater or equal two. This characterization leads to an algorithm to solve the Poincaré center-focus problem which works in the following way: consider a formal power series \(H\) of the described form with unfixed coefficients and write it ordered by homogeneous polynomials of increasing order. Then, impose it to be a first integral of system (1) writing this imposition over the homogeneous polynomial of order \(j+1\) once the homogeneous polynomial of order \(j\) has been determined. In case a certain order cannot be computed due to an incompatibility, then the origin of system (1) is a focus. The incompatibility found is always of the form \[ \left( -y + p(x,y) \right) \frac{\partial H}{\partial x} + \left( x + q(x,y) \right) \frac{\partial H}{\partial y} = L_{2k} \left(x^2+y^2\right)^k + \cdots, \] where the dots denote terms of order greater that \(k\) and \(L_{2k}\) is a nonzero constant which cannot be avoided by any choice of \(H\). Strictly speaking, this procedure is an algorithm, that is it ends after a finite number of steps, only when the origin of system (1) is a focus. This is one of the main difficulties of the center-focus problem as well as the cumbersome computational effort to be overcome when applying this algorithm. However, this computation worths its effort since gives important information about the dynamics of the system and of its perturbations. Another criterion to characterize centers is due to \textit{W.-T. Wu} and \textit{G. Reeb} [Sur les espaces fibrés et les variétés feuilletées. I: Sur les classes caractéristiques des structures fibrées sphériques (Wu). II: Sur certaines propriétés topologiques des variétés feuilletées (Reeb). (French) Actualités scientifiques et industrielles. 1183. Publ. Inst. Math. Univ. Strasbourg. XI. Paris: Hermann \& Cie. (1952; Zbl 0049.12602)] and states that system (1) has a center at the origin if and only if there is a nonzero analytic integrating factor in a neighborhood of the origin. In the same way, we can ensure a center by the existence of a local analytic inverse integrating factor defined in a neighborhood of the origin and of the form \(V(x,y) = 1 + G(x,y)\), where \(G\) is of order greater or equal than one. This characterization gives an algorithm of the same flavor as the former one. In case the origin is not a center, it gives rise to an incompatibility of the same type as before, that is, an unavoidable nonzero constant \(\ell_{2k}\) is found which leads to the impossibility of determining any higher order of \(V\). The main result of the paper is to relate these two algorithms by giving the relation between the constants \(L_{2k}\) and \(\ell_{2k}\), for the particular case when \(p(x,y)\) and \(q(x,y)\) are homogeneous polynomials of degree \(s\), with \(s \geq 2\). In such a case, the main theorem of the paper states that \(L_{2k} = -(2k(s-1)+2) \ell_{2k}\). That is, we get the same restrictions when using any one of the two methods.