Darboux integrability of a Mathieu-van der Pol-Duffing oscillator (Q2112602)

In this paper, the author studied the Darboux integrability of a three-dimensional system of ordinary differential systems \[ \dot{x}=y,~~\dot{y}=mx+\alpha y+\chi z-\gamma x^3-\beta x^2y,~~\dot{z}=-ky+\bar{n}z \] which describes the Mathieu-van der Pol-Duffing oscillator coupled to an electrical circuit by a piezoelectric device. They provided a complete classification of Darboux polynomials of this model, which yields all possible rational or Darboux first integrals. The method is based on the careful analysis of the polynomial solution of a linear partial differential equation, which may be used to deal with Darboux integrability of other simialr models.