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 ẋ = y , ẏ = −ax + yz , ż = by − cxy − x2. 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 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 classify Darboux polynomials and exponential factors.