Limit cycles for planar semi-quasi-homogeneous polynomial vector fields

TL;DR

This paper investigates the existence and maximum number of limit cycles in planar semi-quasi-homogeneous polynomial vector fields, providing explicit criteria and bounds, and extending previous results for polynomial semi-homogeneous systems.

Contribution

It introduces explicit criteria for limit cycle existence and nonexistence, and establishes bounds on their maximum number, generalizing prior work on semi-homogeneous systems.

Findings

Explicit criteria for limit cycle existence and nonexistence

Lower bounds for the maximum number of limit cycles

Generalization of results to broader semi-quasi-homogeneous systems

Abstract

This paper is concerned with the limit cycles for planar semi-quasi-homogeneous polynomial systems. We give some explicit criteria for the nonexistence and existence of periodic orbits. Let $N=N(p,q,m,n)$ be the maximum number of limit cycles of such system. A lower bound is given for $N=N(p,q,m,n)$. The cyclicity and center problem are studied for some subfamilies of semi-quasi-homogeneous polynomial systems. Our results generalize those obtained for polynomial semi-homogeneous systems.