Symbol sequences from three-rotor coincidences and their word-complexity
Abstract
In the three-rotor problem, three equally massive point particles move on a circle interacting via attractive pairwise cosine potentials. Rotors can represent superconducting phases of metallic segments in a chain of coupled Josephson junctions. We propose a digitization of the classical dynamics that records successive pair and triple coincidences of rotors using four symbols. Rotor coincidences correspond to boundaries in a disjoint partition of the configuration torus into cells where the rotors are ordered clockwise and anticlockwise. It is shown that isolated rotor coincidences must be crossings. Despite being a rather coarse digitization, we find that replacing trajectories by coincidence symbol sequences captures significant qualitative features of the dynamics through word statistics. Word-complexity Cn measures the diversity of n-letter words in the symbol sequence while topological entropy governs asymptotic exponential growth of Cn. Sequences from periodic orbits have a word-complexity that saturates at the period. Ultra-high-energy trajectories with irrational `slope' are quasiperiodic. We show that they have zero entropy and Cn = n+3 by examining limiting slopes and by a mapping to Sturmian sequences. We examine their grammar rules and propose how their right-special words may be identified. On the other hand, numerical investigation of sequences from globally chaotic orbits leads us to conjecture an exponentially growing word-complexity Cn = 3 × 2n−1 with a topological entropy log 2. We identify grammar rules that imply this conjectured word-complexity and model the resulting sequences as a subshift of finite type, unlike those at ultra-high energies which are not topological Markov chains.
Versions
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BibTeX
@misc{krishnaswami2026symbol,
title = {{Symbol sequences from three-rotor coincidences and their word-complexity}},
author = {Govind S. Krishnaswami and Anirudh Rameshan},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.25766},
eprint = {2606.25766},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.25766},
}
RIS
TY - UNPB
TI - Symbol sequences from three-rotor coincidences and their word-complexity
AU - Govind S. Krishnaswami
AU - Anirudh Rameshan
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2606.25766
UR - https://arxiv.org/abs/2606.25766
AB - In the three-rotor problem, three equally massive point particles move on a circle interacting via attractive pairwise cosine potentials. Rotors can represent superconducting phases of metallic segments in a chain of coupled Josephson junctions. We propose a digitization of the classical dynamics that records successive pair and triple coincidences of rotors using four symbols. Rotor coincidences correspond to boundaries in a disjoint partition of the configuration torus into cells where the rotors are ordered clockwise and anticlockwise. It is shown that isolated rotor coincidences must be crossings. Despite being a rather coarse digitization, we find that replacing trajectories by coincidence symbol sequences captures significant qualitative features of the dynamics through word statistics. Word-complexity C_n measures the diversity of n-letter words in the symbol sequence while topological entropy governs asymptotic exponential growth of C_n. Sequences from periodic orbits have a word-complexity that saturates at the period. Ultra-high-energy trajectories with irrational `slope' are quasiperiodic. We show that they have zero entropy and C_n = n+3 by examining limiting slopes and by a mapping to Sturmian sequences. We examine their grammar rules and propose how their right-special words may be identified. On the other hand, numerical investigation of sequences from globally chaotic orbits leads us to conjecture an exponentially growing word-complexity C_n = 3 × 2^n−1 with a topological entropy log 2. We identify grammar rules that imply this conjectured word-complexity and model the resulting sequences as a subshift of finite type, unlike those at ultra-high energies which are not topological Markov chains.
ER -