Symbol sequences from three-rotor coincidences and their word-complexity

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. Ultrahigh-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 ultrahigh energies, which are not topological Markov chains.