Will a time-varying complex system be stable?
Randomly-assembled dynamical systems are theoretically predicted to be unstable upon crossing a critical threshold of complexity, as first shown by May. Yet, empirical complex systems exhibit remarkable stability, indicating the presence of additional mechanisms playing a stabilizing role. The relation between complexity and stability is typically assessed by assuming fixed interactions, whereas real systems often evolve in intrinsically time-dependent states. To understand how this affects stability, we linearize a general non-autonomous dynamics around a reference operating state and model the resulting parameters as stochastic processes, which represent the minimal extension of static random interactions to time-varying ones. We derive exact stability bounds that generalize complexity-stability theory to dynamically varying systems. Notably, we find that temporal variability allows systems to remain stable even when their instantaneous Jacobian would predict instability. We compare our results against a non-linear neural network model, where our theory applies exactly, and the generalized Lotka-Volterra equations, where we numerically find that time-varying interactions systematically postpone the onset of replica-symmetry breaking. Overall, our results indicate that temporal variability systematically improves stability, demonstrating a general mechanism by which complex systems can violate classical complexity-stability bounds.