Stability for a family of planar systems with nilpotent critical points

TL;DR

This paper analyzes the stability of a family of planar polynomial systems with nilpotent critical points, determining conditions under which the origin is a focus or unstable, thus completing the classification of their stability.

Contribution

It provides a complete classification of the stability of the origin for a family of planar systems with nilpotent critical points, including the challenging case where parameters satisfy specific relations.

Findings

The origin is always a focus for the studied systems.

The origin is always unstable in the case where s=kl and m equals a specific factorial ratio.

The results extend and complete previous stability classifications.

Abstract

Consider a family of planar polynomial systems $\dot x = y^{2l-1} - x^{2k+1}, \dot y =-x +m y^{2s+1},$ where $l,k,s\in\mathbb{N^*},$ $2\le l \le 2s$ 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.