# Center problem for generalized lambda-omega differential systems

2018-11-14

## Authors

Llibre, Jaume
Ramirez, Rafael
Valentin, Ramirez

## Publisher

Texas State University, Department of Mathematics

## Abstract

Λ-Ω differential systems are the real planar polynomial differential equations of degree m of the form ẋ = -y(1 + Λ) + xΩ, ẏ = x(1 + Λ) + yΩ, 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 Λ = μ(α2x - α1y), and Ω = α1x + α2y + Σm-1j=2 Ωj, where μ, α1, α2 are constants and Ωj = Ωj(x, y) is a homogeneous polynomial of degree j, for j = 2,…, m - 1. We prove the following results. Assuming that m = 2, 3, 4, 5 and (μ + (m - 2)) (α21 + α2 2) ≠ 0 and ∑m-2j=2 Ωj ≠ 0 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 transformations (X, Y, t) → (-X, Y, -t). If (μ + (m - 2)) (α21 + α2 2) = 0 and ∑m-2j=1 Ωj = 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.

## Keywords

Linear type center, Darboux first integral, Weak center, Poincare-Liapunov theorem, Reeb integrating factor

## Citation

Llibre, J., Ramírez, R., & Ramírez, V. (2018). Center problem for generalized lambda-omega differential systems. Electronic Journal of Differential Equations, 2018(184), pp. 1-23.