On first integrals of a family of generalized Lorenz-like systems

Abstract

In this paper, we study a seven-parameter family of generalized Lorenz-like systemsx˙=a(y−x),y˙=bx+cy−dxz,z˙=ez+fxy+gx2,from the view of integrability, which includes many well-known chaotic nonlinear systems. Due to multi-parameters, this nonlinear system has rich integrability properties. We investigate the existence or non-existence of its global analytic, local C1 and local rational first integrals. In addition, we also find eight cases when the systems have invariant algebraic surfaces(Darboux polynomials). Our results can help to understand the complex behaviour of these systems and give new evidences about the connection between the non-integrability and the chaotic phenomenon.