Optimal Projection in a Koopman-Based Sorting-Free Hill Method

In this chapter, the performance of a novel method to determine the stability of periodic solutions is analyzed. Using the Koopman framework, the linear time-periodic nonautonomous perturbed dynamics around a periodic solution can be approximated by a linear autonomous system of higher order, whose system matrix is the well-known Hill matrix. The evaluation of the closed-form solution reveals that the monodromy matrix can be approximated by the Hill matrix, using only a matrix exponential and a projection instead of solving a large eigenvalue problem or integrating numerically over one period. There does not exist a unique choice for the projection of the Hill matrix to the monodromy matrix. In this chapter, various ways to obtain a suitable projection are discussed. The computational efficiency of the novel method is analyzed for the vertically excited multiple pendulum, comparing the effect of the considered projection options.