Limit cycles and Integrability of a continuous system with a line of equilibrium points

TL;DR

This paper investigates a 3D chaotic differential system with a line of equilibrium points, establishing conditions for limit cycle bifurcation and analyzing the system's integrability, including non-existence of certain first integrals and classification of Darboux polynomials.

Contribution

It provides new conditions for limit cycle bifurcation from a zero-Hopf equilibrium and analyzes the integrability, showing the absence of polynomial, rational, or Darboux first integrals.

Findings

Existence conditions for limit cycles bifurcating from the origin.

Proof that the systems lack polynomial, rational, or Darboux first integrals.

Construction of formal series and analytic first integrals.

Abstract

We focus on a chaotic differential system in 3-dimension, including an absolute term and a line of equilibrium points. Which describes in the following This system has an implementation in electronic components. The first purpose of this paper is to provide sufficient conditions for the existence of a limit cycle bifurcating from the zeroHopf equilibrium point located at the origin of the coordinates. The second aim is to study the integrability of each differential system, one defined in half-space y and the other in half-space y. We prove that these two systems have no polynomial, rational, or Darboux first integrals for any value of a, b, and c. Furthermore, we provide a formal series and an analytic first integral of these systems. We also classify Darboux polynomials and exponential factors.