We propose a hyperbolicity analysis method for multiple equilibria of a nonlinear system with dynamic uncertainties. Since dynamic uncertainties alter the location and even the existence of equilibria, it is difficult to evaluate the hyperbolicity of equilibria for such a system. In this paper, we provide an analysis method to determine whether all equilibria robustly maintain their hyperbolicity under the existence of a norm-bounded class of dynamic uncertainties. The displacement of the equilibria is considered explicitly to evaluate the hyperbolicity. Finally, the proposed method is applied to a genetic network model of induced pluripotent stem cells (iPSCs), to explain how one reprogramming mechanism of iPSCs is robust against various uncertainties. Hyperbolicity of multiple equilibria plays a crucial role in the reprogramming mechanism.