{'fraction_non_alphanumeric': 0.04903677758318739, 'fraction_numerical': 0.005253940455341506, 'mean_word_length': 4.553398058252427, '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}}
Stability analysis for ODEs

1 Linear stability analysis Equilibria are not always stable. Since stable and unstable equilibria play quite different roles in the dynamics of a system, it is useful to be able to classify equilibrium points based on their stability. Suppose that we have a set of autonomous ordinary differential equations, written in vector form: ẋ = f(x). (1) Suppose that x∗ is an equilibrium point. By definition, f(x∗) = 0. Now suppose that we take a multivariate Taylor expansion of the right-hand side of our differential equation: ẋ = f(x)+ ∂f ∂x ∣