Stability for a family of planar systems with nilpotent critical points

2024/06/04 by Zhuang, Ziwei, Liu, Changjian
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2406.02226

Abstract

Consider a family of planar polynomial systems x = y^2l-1 - x^2k+1, y =-x +m y^2s+1, where l,k,s∈\mathbbN^*, 2≤ l ≤ 2s and m∈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.

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