A kind of bifurcation of limit cycle from nilpotent critical point

TL;DR

This paper investigates a novel bifurcation phenomenon where limit cycles emerge from nilpotent nodes or foci by altering their stability, establishing an upper bound for the number of bifurcated cycles in polynomial systems.

Contribution

It introduces a new bifurcation mechanism from nilpotent critical points and proves an upper bound for the number of limit cycles in polynomial systems of specific degrees.

Findings

Up to $n^2+n-1$ limit cycles can bifurcate from nilpotent points.

The upper bound for bifurcated limit cycles is sharp and achievable.

Examples demonstrate the maximum number of limit cycles for degrees 3 and 5.

Abstract

In this paper, an interesting and new bifurcation phenomenon that limit cycles could be bifurcated from nilpotent node (focus) by changing its stability was investigated. It is different from lowing its multiplicity in order to get limit cycles. We prove that $n^2+n-1$ limit cycles could be bifurcated by this way for $2n+1$ degree system. Moreover, this upper bound could be reached. At last, we give two examples to show that $N(3)=1$ and $N(5)=5$.