Stability for a family of planar systems with nilpotent critical points

Abstract

Consider a family of planar polynomial systems $\dot{x}=y^{2l-1}-x^{2k+1},\dot{y}=-x+m{y}^{2s+1},$ where $l,k,s\in {\mathbb{N}}^{\ast },$ $2\le l\le 2s$ and $m\in \mathbb{R}.$ We study the center-focus problem on its origin which is a monodromic nilpotent critical point. By directly calculating the generalized Lyapunov constants, we find that the origin is always a focus and we complete the classification of its stability. This includes the most difficult case: $s=kl$ and $m=(2k+1)!!/(2kl+1){!}_{(2l)}.$ In this case, we prove that the origin is always unstable. Our result extends and completes a previous one.