We propose a new semi-implicit level set approach to a class of curvature dependent flows. The method generalizes a recent algorithm proposed for the motion by mean curvature where the interface is updated by solving the Rudin-Osher-Fatemi (ROF) model for image regularization. Our proposal is general enough so that one can easily extend and apply the method to other curvature dependent motions. Since the derivation is based on a semi-implicit time discretization, this suggests that the numerical scheme is stable even using a time-step significantly larger than that of the corresponding explicit method. As an interesting application of the numerical approach, we propose a new variational approach for extracting limit cycles in dynamical systems. The resulting algorithm can automatically detect multiple limit cycles staying inside the initial guess with no condition imposed on the number nor the location of the limit cycles. Further, we also propose in this work an Eulerian approach based on the level set method to test if the limit cycles are stable or unstable.