Higher Order Transverse Discontinuity Mapping for Hybrid Dynamical Systems

This chapter presents a higher order correction to the transverse discontinuity mapping (TDM) of trajectories undergoing border collisions in hybrid dynamical systems. To demonstrate its usefulness, a 4-dimensional hybrid system representing the simplest case of fluid–structure interaction (FSI) undergoing impacts is studied. The model simulates a structure undergoing vortex-induced vibrations (VIVs) in a uniform flow. Non-smoothness manifests when a barrier obstructs the structure’s motion, resulting in discontinuity-induced bifurcations (DIBs) like period-adding cascades and chaos. The derived TDM, compared to the first-order map, reveals significant improvements in prediction of the dynamics and offers a better estimation of the states post impact. The necessity of a higher order mapping to accurately predict the outcome of trajectories in vibro-impact systems is highlighted.