On the Formal Integrability Problem for Planar Differential Systems

Abstract

We study the analytic integrability problem through the formal integrability problem and we show its connection, in some cases, with the existence of invariant analytic (sometimes algebraic) curves. From the results obtained, we consider some families of analytic differential systems in C 2 , and imposing the formal integrability we find resonant centers obviating the computation of some necessary conditions.

Integrability Conditions for Complex Homogeneous Kukles Systems

“…Although integrability is a restrictive condition and a generic system is not integrable, the existence of a first integral allows to know the phase portrait of a planar differential system. These non-generic integrable differential systems are used to analyze a bigger family of differential systems that are described as perturbations of this non-generic system, see [1] and references therein. Closely related to the existence of a local analytic first integral is the so-called center-focus problem.…”

Integrability of Lotka–Volterra Planar Complex Cubic Systems

“…Sometimes it is possible to prove the existence of a formal first integral developing with respect to one of the algebraic invariant curves of the system, see for instance [Algaba et al, 2013]. In our case we can propose developments of the form…”

Integrability conditions of a resonant saddle in Liénard-like complex systems

“…In order to prove the existence of an analytic first integral for such system we follow the ideas developed in [1]. We look for a formal first integral expressed of the form H = ∑ ∞ k=1 h k (x)y k .…”

“…Taking into account that the first h i are polynomial in x of degree i, we assume that the first k − 1 are polynomial of degree k − 1, and from (8) we get that h k is also a polynomial of degree at most k. Hence, system (4) with a 1 = a 2 = 0, following [1], has a formal first integral which implies the existence of an analytic one whose power series expansion is H = xy + · · · .…”