In this paper we determine multiple resonant operating points (ROPs) of inductive power transfer (IPT) systems and perform the corresponding stability analysis of a series-tuned IPT system, which is taken for example, through using nonlinear dynamics theories. The stroboscopic mapping model of the system is built and a piecewise analytical function of the steady-state response is derived with the fixed-point theory. Then a criterion for assessing the system ROPs is given mathematically. The stability analysis of ROPs is achieved according to the locations of the eigenvalues of the Jacobi matrix of the Poincare mapping model of the system. A case study of the phenomenon of multiple ROPs is conducted, and both simulation and experimental results verify the theoretical results of the proposed method. Furthermore, the proposed method can provide useful theoretical reference for modeling and steady-state analysing other similar resonant converters.