{'fraction_non_alphanumeric': 0.02321083172147002, 'fraction_numerical': 0.0019342359767891683, 'mean_word_length': 6.1875, '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}}
Bifurcation analysis of nonlinear time‐periodic time‐delay systems via semidiscretization

Bifurcations of the periodic stationary solutions of nonlinear time‐periodic time‐delay dynamical systems are analyzed. The solution operator of the governing nonlinear delay‐differential equation is approximated by a sequence of nonlinear maps via semidiscretization. The subsequent nonlinear maps are combined to a single resultant nonlinear map that describes the evolution over the time period. Fold, flip, and Neimark‐Sacker bifurcations related to the fixed point of this map are analyzed via center manifold reduction and normal form theorems. The analysis unfolds the approximate stability properties and bifurcations of the stationary solution of the delay‐differential equation and, at the same time, allows the approximate computation of the arising period‐1, period‐2, and quasi‐periodic solution branches. The method is demonstrated for the delayed Mathieu‐Duffing equation, and the results are verified by numerical continuation.