Solution of the $1 : {−}2$ resonant center problem in the quadratic case

Abstract

Solution of the 1 : −2 resonant center problem in the quadratic case by Alexandra F r o n v i l l e (Paris), Anton P. S a d o v s k i (Grodno) and Henryk Ż o ł ą d e k (Warszawa) Dedicated to the memory of Wiesław Szlenk

Integrability of planar nilpotent differential systems through the existence of an inverse integrating factor

“…If λ 1 /λ 2 = −p/q ∈ Q then we have a resonant saddle. A resonant saddle has an analytic first integral around the singular point if, and only if, it is orbitally linearizable, see for instance [17,22,28] and references therein. The analytic integrability problem for a nilpotent singularity has been recently theoretically characterized in [11,Theorem 1.2], see also below.…”

On a conjecture on the integrability of Liénard systems

“…We recall that the function H = H(x, y) is a first integral of system (2) in an open subset U or C 2 if…”

“…The study of the existence or not of a first integral in a neighborhood of a [p : −q] resonant saddle is a difficult problem, see for instance [2,7,8,12,14] and the references quoted there. Hence Theorem 2 says that the study of the existence or not of a first integral in a neighborhood of a strong saddle for the Liénard differential system (2) is also difficult.…”

“…We first note that the origin is the unique singular point of system (2) and that the eigenvalues of the Jacobian matrix at this point satisfy…”

Integrability of Liénard systems with a weak saddle

“…, and the so-called [p : −q] resonant saddle quantities v i are polynomials of the coefficients of system (2). If all the v i are zero we say that we have a formal analytic resonant saddle, see [11,26] and references therein. From this result we obtain also the existence of a local analytic first integral, see [21].…”