{'fraction_non_alphanumeric': 0.04413472706155633, 'fraction_numerical': 0.010452961672473868, 'mean_word_length': 4.708609271523179, 'pattern_counts': {'":': 0, '<': 1, '<?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}}
Scaling properties of a simplified bouncer model and of Chirikov's standard map

Scaling properties of Chirikov's standard map are investigated by studying the average value of I2, where I is the action variable, for initial conditions in (a) the stability island and (b) the chaotic component. Scaling behavior appears in three regimes, defined by the value of the control parameter K: (i) the integrable to non-integrable transition (K ≈ 0) and K < Kc (Kc ≈ 0.9716); (ii) the transition from limited to unlimited growth of I2, K ≳ Kc; (iii) the regime of strong nonlinearity, K ≫ Kc. Our scaling results are also applicable to Pustylnikov's bouncer model, since it is globally equivalent to the standard map. We also describe the scaling properties of a stochastic version of the standard map, which exhibits unlimited growth of I2 even for small values of K.