Limit Cycles of a Quadratic System with Two Parallel Straight Line-Isoclines

TL;DR

This paper proves that a specific quadratic system with two parallel line-isoclines can have at most two limit cycles, using geometric and bifurcation methods, contributing to the understanding of limit cycle configurations.

Contribution

The paper establishes the maximum number of limit cycles in a quadratic system with two parallel line-isoclines as two, employing novel geometric and bifurcation techniques.

Findings

Maximum of two limit cycles in the system

Application of geometric properties of spirals

Use of Wintner-Perko termination principle

Abstract

In this paper, a quadratic system with two parallel straight line-isoclines is considered. This system corresponds to the system of class II in the classification of Ye Yanqian. Using the field rotation parameters of the constructed canonical system and geometric properties of the spirals filling the interior and exterior domains of its limit cycles, we prove that the maximum number of limit cycles in a quadratic system with two parallel straight line-isoclines and two finite singular points is equal to two. Besides, we obtain the same result in a different way: applying the Wintner-Perko termination principle for multiple limit cycles and using the methods of global bifurcation theory developed earlier by the author.