The rotor of PMSM-based electromechanically driven system is a typical electromechanically coupled system. In this paper, we analyzed the nonlinear magnetic interaction torque of permanent magnet synchronous motor (PMSM) and deduced the nonlinear electromechanically coupled equation of PMSM-based electromechanical driven system using Lagrange–Maxwell theory. We determined the equation of the movement of the dynamic system from his asymmetric double well potential. The unperturbed system was classified to several categories based on the shapes of potential functions and phase portraits. An analytical criterion for homoclinic chaos is written in terms of the system parameters by means of Melnikov’s method. Detailed numerical studies including phase portrait, Poincare map, and bifurcation diagram confirm the analytical prediction and reveal the effect of excitation amplitude and damp on the system transition to chaos. The conclusion can provide reference for deeply research the dynamic behaviors of mechanical drive system.