Analytic integrability of two lopsided systems

TL;DR

This paper investigates the analytic integrability of two classes of lopsided systems, establishing conditions for integrability and identifying the number of small-amplitude limit cycles around the origin.

Contribution

It introduces new classes of lopsided systems and determines their integrability conditions and maximum number of small-amplitude limit cycles.

Findings

For the first class, n+4 small-amplitude limit cycles for n ≥ 2.

For the first class, 10 limit cycles when n=1.

For the second class, n+4 small-amplitude limit cycles for n ≥ 2.

Abstract

In this paper, we present two classes of lopsided systems and discuss their analytic integrability. The analytic integrable conditions are obtained by using the method of inverse integrating factor and theory of rotated vector field. For the first class of systems, we show that there are n + 4 small-amplitude limit cycles enclosing the origin of the systems for n ? 2, and 10 limit cyclesfor n = 1. For the second class of systems, we prove that there exist n+4 small-amplitude limit cycles around the origin of the systems for n ? 2, and 9 limit cycles for n = 1.