Global bifurcations of limit cycles in the Kukles cubic system

TL;DR

This paper investigates the maximum number and arrangement of limit cycles in the Kukles cubic system, employing bifurcational geometric methods and the Wintner-Perko termination principle to advance understanding of polynomial dynamical systems.

Contribution

It introduces a bifurcational geometric approach and applies the Wintner-Perko principle to determine limit cycle configurations in the Kukles system.

Findings

Maximum number of limit cycles identified

Distribution patterns of limit cycles established

Methodological framework for similar systems developed

Abstract

In this paper, using our bifurcational geometric approach, we solve the problem on the maximum number and distribution of limit cycles in the Kukles system representing a planar polynomial dynamical system with arbitrary linear and cubic right-hand sides and having an anti-saddle at the origin. We also apply alternatively the Wintner-Perko termination principle to solve this problem.