Integrability of complex planar systems with homogeneous nonlinearities

The Poincaré center‐focus problem for a class of higher order polynomial differential systems

“…The conditions for a center in the quadratic system have been obtained in Kapteyn, 1,2 and in Dulac, 3 Lunkevich and Sibirskii, 4 and Malkin, 5 the problem has been solved for systems in which p and q are cubic polynomials without quadratic terms. The problem is also solved for some families of cubic systems and systems in the form of the linear center perturbed by homogeneous quartic and quintic nonlinearities, see eg, previous studies [6][7][8][9][10][11][12][13] and references given there. The problem of distinguishing between a center and a focus for polynomial systems (1.1) has an analog for the corresponding periodic differential equations.…”

Integrability conditions of a resonant saddle perturbed with homogeneous quintic nonlinearities

“…In [6] the integrability conditions for the subfamily with a −15 = b 5,−1 = 0 were given. If a −15 = 0 almost a complete study of the cases when we have an integrable saddle at the origin of system (3) is given in [4]. For the case a −15 b 5,−1 ̸ = 0, by a linear transformation system (3) can be written aṡ…”

“…The case (C 2 )(γ) was solved in [4] using a new method to prove the sufficiency for homogeneous differential systems. As pointed out in [3], system (4) with conditions (C 2 )(δ) can be transformed into system (4) with conditions (C 2 )(γ) by the substitution x → αx, y → βy where α = β 5 and β = (b 31 − 1) −1/8 .…”

Center conditions for generalized polynomial Kukles systems

“…Although several methods exist to find the necessary conditions to have a center, the sufficient conditions are proved using different methods and in some cases ad hoc methods for each center case. However sometimes some cases remain open and all the known methods fail, see [9] and references therein for the equivalent case of a resonant saddle.…”

“…The sufficiency looking for a first integral is also approached by several methods, see [9,11,21] but some particulary examples remain open. Anyway there are some concrete families for which the center problem is fully understood, see [5,17].…”