Ziwei Zhuang | Searcharxiv

arXiv2024

Stability for a family of planar systems with nilpotent critical points

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.

math.DS↗

arXiv2022

A proof of the uniqueness of the limit cycle of a quasi-homogeneous system

A. Gasull shared a list of 33 open problems in low dimensional dynamical systems in his work in 2021. The second part of Problem 3 is about whether the limit cycle of a quasi-homogeneous system $ \dot{x}=y,\; \dot{y}=-x^3+αx^2y+y^3 $ is unique. In this paper, we give a positive answer to this question by analysing the uniqueness of the heteroclinic separatrix at infinity.

math.DS↗