{'fraction_non_alphanumeric': 0.019916142557651992, 'fraction_numerical': 0.0020964360587002098, 'mean_word_length': 5.452702702702703, '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}}
Fold-pitchfork bifurcation for maps with Z(2) symmetry in pipe flow.

This study aims to provide a better understanding of recently identified transition scenarios exhibited by traveling wave solutions in pipe flow. This particular family of solutions are invariant under certain reflectional symmetry transformations and they emerge from saddle-node bifurcations within a two-dimensional parameter space characterized by the length of the pipe and the Reynolds number. The present work precisely provides a detailed analysis of a codimension-two saddle-node bifurcation arising in discrete dynamical systems (maps) with Z(2) symmetry. Normal form standard techniques are applied in order to obtain the reduced map up to cubic order. All possible bifurcation scenarios exhibited by this normal form are analyzed in detail. Finally, a qualitative comparison of these scenarios with the ones observed in the aforementioned hydrodynamic problem is provided.