Limit cycle bifurcations from a nilpotent focus or center of planar systems

TL;DR

This paper investigates the bifurcation of limit cycles from nilpotent foci or centers in planar systems by analyzing Poincare return maps and focal values, providing new bounds on their maximum number.

Contribution

It introduces new analytical methods to estimate the number of limit cycles near nilpotent foci or centers, advancing understanding of bifurcation phenomena in planar systems.

Findings

Derived bounds for the maximum number of limit cycles

Analyzed the role of focal values and return maps in bifurcation

Provided new theoretical results on limit cycle bifurcations

Abstract

In this paper, we study the analytical property of the Poincare return map and the generalized focal values of an analytical planar system with a nilpotent focus or center. Then we use the focal values and the map to study the number of limit cycles of this kind of systems with parameters, and obtain some new results on the lower and upper bounds of the maximal number of limit cycles near the nilpotent focus or center.