With the continuous integration of new energy sources, the power system gradually begins to present the characteristics of a weak power grid. The system’s inertia decreases, leading to problems in the stability of the power grid. In this paper, a virtual synchronous generator model with a supercapacitor is analyzed. The supercapacitor provides additional virtual inertia to the system and suppresses system frequency disturbances more quickly. Bifurcation theory is used to analyze the nonlinear dynamics of the model. The bifurcation diagram of input active power is given in this paper, and the phase portraits and sequence diagrams of the frequency and power angle are presented to verify that, if the initial value of the system falls inside the stability region, the system can remain stable. If the initial value of the system falls outside the stability region, conversely, the system will lose stability. Finally, the simulation verifies the influence of the supercapacitor on the system inertia. The results show that the recovery speed of a small capacitance system is faster than that of a large capacitance system when disturbance occurs. It is concluded that, the smaller the supercapacitor is, the greater the virtual inertia it can provide.