Limit cycles and Integrability of a class of 3-dimension chaotic systems

Abdulkareem, Aram A; Amen, Azad I.; Hussein, Niazy A.

doi:10.24200/sci.2025.64386.8917

Limit cycles and Integrability of a class of 3-dimension chaotic systems

Articles in Press

Document Type : Research Article

Authors

Aram A Abdulkareem ¹

Azad I. Amen ²

Niazy A. Hussein ³

¹ Department of Mathematics, Faculty of Education, Soran University, Erbil-Soran, Iraq

² - Department of Mathematics, College of Basic Education, Salahaddin University-Erbil, Erbil, Iraq - Department of Mathematics, Basic Education College, Raparin University - Ranya, Iraq - Department of Mathematics, College of Science, Duhok University, Iraq

³ Department of Mathematics, College of Education, Salahaddin University-Erbil, Erbil, Iraq

10.24200/sci.2025.64386.8917

Abstract

We focus on a chaotic differential system in 3-dimension, including an absolute term and a line of equilibrium points. Which describes as 𝑥⁄ = 𝑦 , 𝑦⁄ = −𝑎𝑥 + 𝑦𝑧 , 𝑧⁄ = 𝑏|𝑦| −𝑐𝑥𝑦 −𝑥2 . This system has an implementation by using electronic components. The first purpose of this paper is to provide sufficient conditions for the existence of a limit cycle bifurcating from the zero-Hopf 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 ≥ 0 and the other in half-space 𝑦 < 0. 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
find Darboux polynomials and exponential factors.

Keywords

Chaos system

Limit cycle

Zero-Hopf bifurcation

Darboux integrability

First integral

Subjects