Center problem for generalized Λ-Ω differential systems

© 2018 Texas State University. Λ-Ω differential systems are the real planar polynomial differential equations of degree m of the form (Formula presented), where Λ = Λ(x, y) and Ω = Ω(x, y) are polynomials of degree at most m − 1 such that Λ(0, 0) = Ω(0, 0) = 0. A planar vector field with linear type center can be written as a Λ-Ω system if and only if the Poincaré-Liapunov first integral is of the form F =1/2 (x2 + y2)(1 + O(x, y)). The main objective of this article is to study the center problem for Λ-Ω systems of degree m with (Formula presented), where µ, a1, a2 are constants and Ωj = Ωj (x, y) is a homogenous polynomial of degree j, for j = 2, …, m−1. We prove the following results. Assuming that m = 2, 3, 4, 5 and (Formula presented) the Λ-Ω system has a weak center at the origin if and only if these systems after a linear change of variables (x, y) → (X, Y) are invariant under the transformationsP (X, Y, t) → (−X, Y, −t). If (µ + (m − 2))(a21 + a22) = 0 and (Formula presented) = 0 then the origin is a weak center. We observe that the main difficulty in proving this result for m > 6 is related to the huge computations.