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We study a weak stability property called recurrence for a class of hybrid systems. An open set is recurrent if there are no finite escape times and every complete trajectory eventually reaches the set. Under sufficient regularity properties for the hybrid system we establish that the existence of a smooth, radially unbounded Lyapunov function that decreases along solutions outside an open, bounded set is a necessary and sufficient condition for recurrence of that set. Recurrence of open, bounded sets is robust to sufficiently small state dependent perturbations and this robustness property is crucial for establishing the existence of a Lyapunov function that is smooth. We also highlight some connections between recurrence and other well studied properties like asymptotic stability and ultimate boundedness. I. INTRODUCTION Hybrid systems are a class of dynamical systems that combine continuous-time evolution and discrete-time events. Several frameworks have been proposed in the literature for analysis of hybrid systems. We refer the reader to [1], [2] and [3] for details. Converse Lyapunov theorems are used to establish the equivalence between asymptotic stability properties and the existence of Lyapunov-like functions that satisfy certain decrease conditions along solutions. Applications of converse theorems in stabilization and robust stability analysis can be found in [4], [5] and [6]. For continuous-time systems converse theorems related to asymptotic stability are established in [6], [7] and [8]. See [9], [10] and [11] for similar results in the discrete-time case. Converse theorems for asymptotic stability of compact sets for a class of hybrid systems has been established in [12] and [13]. In this paper, we establish a converse theorem for a similar class of hybrid systems considered in [12] but for a weaker property called recurrence. Recurrence of an open, bounded set is a weak stability property that is frequently studied in the literature for stochastic systems. We refer the reader to [14], [15] and [16] for details. Loosely speaking, recurrence of an open, bounded set implies that solutions visit the set infinitely often with probability one. It is a weaker notion of stability compared to probabilistic notions of asymptotic stability but nevertheless useful in many applications. In particular, for the problem of RF-source seeking using Micro Aerial Vehicles (MAV) considered in [17] and [18], it is shown The authors are with the Electrical and

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