A Fast Semi-Implicit Level Set Method for Curvature Dependent Flows with an Application to Limit Cycles Extraction in Dynamical Systems (Communications in Computational Physics, 2018)
We propose a new semi-implicit level set approach to a class of curvaturedependent flows. The method generalizes a recent algorithm proposed for the motionby 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 caneasily extend and apply the method to other curvature dependent motions. Since thederivation is based on a semi-implicit time discretization, this suggests that the numericalscheme is stable even using a time-step significantly larger than that of the correspondingexplicit method. As an interesting application of the numerical approach, wepropose a new variational approach for extracting limit cycles in dynamical systems.The resulting algorithm can automatically detect multiple limit cycles staying insidethe initial guess with no condition imposed on the number nor the location of the limitcycles. Further, we also propose in this work an Eulerian approach based on the levelset method to test if the limit cycles are stable or unstable.