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

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.