The fundamental concept underlying topological phenomena posits the geometric phase associated with eigenstates. In contrast to this prevailing notion, theoretical studies on time-varying Hamiltonians allow for a new type of topological phenomenon, known as topological dynamics, where the evolution process allows a hidden topological invariant associated with continuous flows. To validate this conjecture, we study topological chiral and nonreciprocal dynamics by encircling the exceptional points (EPs) of non-Hermitian Hamiltonians in a trapped ion system. These dynamics are topologically robust against external perturbations even in the presence dissipation-induced nonadiabatic processes. Our findings indicate that they are protected by dynamical vorticity -- an emerging topological invariant associated with the energy dispersion of non-Hermitian band structures in a parallel transported eigenbasis. The symmetry breaking and other key features of topological dynamics are directly observed through quantum state tomography. Our results mark a significant step towards exploring topological properties of open quantum systems.
