Learning interpretable representations of neural population dynamics is a crucial step to understanding how brain activity relates to behavior. Models of neural dynamics often focus on either low-dimensional projections that overlook the temporal relationships within the data, oversimplify the dynamics to linear and stationary patterns, or provide un-interpretable representations. Here, we consider dynamical systems as representative of flows on a low-dimensional manifold, and propose a new decomposed Linear Dynamical Systems (dLDS) model that captures complex nonstationary dynamics. dLDS models the latent state's evolution as following a sparse combination of simple interpretable components identified through a dictionary learning procedure. Importantly, the decomposed nature of the dynamics enables identifying overlapping co-active processes—a feature unavailable to other methods. Through several examples, we demonstrate our model's ability to learn interpretable representations of multiple systems and demix population dynamics of multiple sub-networks. Finally, when applying our model to neural recordings of C. elegans, we identified unique patterns of dynamics emerging across behavioral states, which are obscured by other methods.