Four Limit Cycles in Three-Dimensional Competitive Lotka-Volterra Systems of Class 28 in Zeeman's Classification

TL;DR

This paper constructs a three-dimensional competitive Lotka-Volterra system with four limit cycles for class 28, demonstrating the existence of multiple limit cycles across several classes in Zeeman's classification, expanding understanding of system dynamics.

Contribution

It introduces a new system with four limit cycles for class 28 and shows such systems exist across classes 26 to 29, broadening the known dynamics in these classifications.

Findings

Existence of four limit cycles in class 28 system.

Systems with multiple limit cycles exist in classes 26-29.

Expands the understanding of limit cycle behavior in competitive systems.

Abstract

In this paper, a three-dimensional competitive Lotka-Volterra system with four limit cycles is constructed for class 28 in Zeeman's classification. Combined with existing results -- from Gyllenberg and Yan (2009) for class 27, from Wang, Huang and Wu (2011) for class 29, and from Yu, Han and Xiao (2016) for class 26 -- our finding indicates that there exist systems with at least four limit cycles for each class among classes 26 $-$ 29.