**Abstract**: In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. First, we consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Second, we provide a generalization of Lyapunov’s Direct Method that allows for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent (instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize discussing future directions and possible extensions for control.

**Bio**: Enrique Mallada is an Associate Professor of Electrical and Computer Engineering at Johns Hopkins University since 2022. Prior to joining Hopkins in 2016, he was a Post-Doctoral Fellow in the Center for the Mathematics of Information at Caltech from 2014 to 2016. He received his Ingeniero en Telecomunicaciones degree from Universidad ORT, Uruguay, in 2005 and his Ph.D. degree in Electrical and Computer Engineering with a minor in Applied Mathematics from Cornell University in 2014. Dr. Mallada has received several awards, including the Johns Hopkins Alumni Association Teaching Award in 2021, the NSF CAREER award in 2018, the Center for the Mathematics of Information (CMI) Fellowship from Caltech in 2014, and the ECE Director’s Ph.D. Thesis Research Award for his dissertation in 2014. His research interests lie in the areas of control, dynamical systems, and optimization, with applications to engineering networks.