Global dynamics of a polynomial Liénard differential system with arbitrary degree

In this paper, we study the polynomial Liénard differential system of arbitrary degree: \( \dot{x}=y, \ \dot{y}=a_1x+a_2x^{2\,m}+(a_3+a_4x^{2n})y \) $\dot{x}=y,\dot{y}=a_{1}x+a_2{x}^{2m}+(a_3+a_4{x}^{2n})y$ applying to a Duffing–Van der Pol oscillator. We prove that it has abundant dynamics, such as the generalized transcritical bifurcation, Hopf bifurcation, heteroclinic bifurcation, homoclinic bifurcation and double limit cycle bifurcation. The associated global bifurcation diagram in the parameter space and the corresponding global phase portraits in the Poincaré disc are presented. These analytic results are also demonstrated by concrete examples via numerical simulations.