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 distinct 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 chaotic orbits in the band of global chaos leads us to conjecture an exponentially growing word-complexity $C_n=3\times 2^{n-1}$, corresponding to a topological entropy $\log 2$. We identify their grammar rules and model them by a subshift of finite type, unlike the quasiperiodic ultra-high-energy sequences which cannot be modeled as a topological Markov shift.

26 pages and 16 figure files