{'fraction_non_alphanumeric': 0.014545454545454545, 'fraction_numerical': 0.0, 'mean_word_length': 5.333333333333333, '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}}
Nonlinear boundary feedback control of the one-dimensional wave equation

In this paper, we analyze the dynamical behavior of the linear wave equation on an interval, where the right endpoint has a van der Pol type nonlinearity or boundary controller, while the left endpoint has a boundary condition involving displacement. The asymptotic behavior of the system can be classified into two basic types: classical unbounded instability, or spatial pointwise convergence to periodic points of a nonlinear map corresponding to the van der Pol condition.