{'fraction_non_alphanumeric': 0.013986013986013986, 'fraction_numerical': 0.0, 'mean_word_length': 5.432584269662922, '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}}
A dynamical approach to the complex coefficients of hysteretic responses

To describe the field-dependent hysteretic responses of dielectric materials when the applied electrical loadings beyond small field ranges, the Rayleigh's law and the complex dielectric constant notation have been introduced with some success. In the current paper, we investigate the field-dependent hysteretic responses on a dynamical point of view. It is shown that the hysteretic responses can be modeled by the Duffing equation, and the field-dependency feature can be naturally captured by the model. The complex dielectric coefficients and the Rayleigh's law can be regarded as special cases of the dynamical system approach. Numerical comparisons among the three models illustrate that the proposed nonlinear differential model is able to capture more accurately the hysteretic behavior of dielectric materials. It is also much easier to model the dielectric losses that presented in all dielectric materials. In the proposed dynamical approach, the loss tangent and the hysteretic behavior are both influenced by the frequency of the applied external loadings.