{'fraction_non_alphanumeric': 0.011589403973509934, 'fraction_numerical': 0.0, 'mean_word_length': 5.875, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Smooth Lyapunov functions for multistable hybrid systems on manifolds

For a broad class of complete hybrid systems evolving on Riemannian manifolds and satisfying mild regularity conditions on the data we introduce a notion of multistability based on the existence of a finite number of compact, globally attractive, and weakly invariant sets. Such notion not only generalizes the standard global uniform asymptotic stability requirement, but can also be characterized in terms of equivalent asymptotic properties, existence of smooth Lyapunov functions, and intrinsic robustness to small perturbations.