<p> Coupled mechanical oscillators have long been an archetypical system for understanding eigcnmodes and coupled dynamics. But in the last few decades, the study of open systems (i.e. those open to loss or gain) has brought a fresh interest and perspective to such simple systems, revealing a surprisingly rich set of physical phenomena. Specifically, it was realized that degeneracies in open systems ('exceptional points', or EPs) possess a non-trivial topology, with interesting implications for closed adiabatic cycles. The theoretical properties of EPs have been made increasingly clear over the last 20 years, but experimental progress has generally been limited to spectroscopy, with no demonstrations of the predicted dynamical behavior. Here, I'll present work in which we use a cavity optomechanical system as a convenient, highly tunable platform for studying this multimode physics.</p><p> I'll begin with a pedagogical introduction to cavity optomechanics, including our particular experimental realization: a Si₃N₄ membrane coupled to a high-finesse optical cavity. Then, the physics of exceptional points will be reviewed using a toy model, before seeing how these features are accessible in our optomechanical system. I'll then present our study of time-dependent perturbations of this system, which provided the first experimental demonstration of adiabatic (and non-adiabatic) behavior near, an EP. These perturbations can be used to affect energy transfer which is both topology-dependent and non-reciprocal. This demonstration relies on a somewhat fortunate symmetry in our system, but in the final chapter, we'll see that this restriction can be lifted, to enable this energy transfer in a broad class of systems.</p><p>