Darboux polynomials and rational first integrals of the generalized Lorenz systems (Q414796)

By the generalized Lorenz system is meant the system of differential equations NEWLINE\[NEWLINE(GL) \;\dot x = a (y - x ),\dot y = b x + c y - x z,\dot z = d z + x y \;\mathrm {on} \;{\mathbb R}^3,NEWLINE\]NEWLINE where \(a\), \(b\), \(c\), and \(d\) are real parameters. It unifies the original Lorenz system and the Chen and Lü systems. If \({\mathcal X}\) denotes the vector field on \({\mathbb R}^3\) associated to \((GL)\) then a Darboux polynomial for the system is a polynomial \(f\) for which there exists a polynomial function \(k\) such that \({\mathcal X} \cdot f = kf\) (so that the zero set of \(f\) is an invariant surface). In this paper, the authors characterize members of the family \((GL)\) that admit a Darboux polynomial, a polynomial first integral, or a rational first integral. They fully describe these functions when they exist. They further show that the generalized Lorenz system never possesses two functionally independent algebraic first integrals.