{'fraction_non_alphanumeric': 0.04932735426008968, 'fraction_numerical': 0.002242152466367713, 'mean_word_length': 4.258823529411765, '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}}
Dynamics of periodically forced parabolic equations on the circle

Abstract We consider the dynamics of scalar reaction-diffusion equations: We show that every prescribed vectorfield in the plane embeds into equation (1) for a suitable f = f (t, x, u, ux). In contrast, suppose f = f (t, u, ux) is independent of x and T-periodic in t. Then the ω-limit set of any bounded solution is a subset of some two-dimensional torus carrying a linear flow.