pith. sign in

arxiv: 2606.25766 · v1 · pith:HAJQ37BXnew · submitted 2026-06-24 · 🌊 nlin.CD · math-ph· math.DS· math.MP

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

Govind S. Krishnaswami , Anirudh Rameshan This is my paper

Pith reviewed 2026-06-25 19:06 UTC · model grok-4.3

classification 🌊 nlin.CD math-phmath.DSmath.MP
keywords three-rotor problemsymbol sequencesword complexitytopological entropysubshift of finite typechaotic orbitsquasiperiodic sequencesJosephson junctions
0
0 comments X

The pith

Coincidence symbol sequences from three-rotor motions have word complexity growing as 3 times 2 to the n-1 for chaotic orbits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper digitizes three-rotor dynamics on a circle by recording isolated pair and triple coincidences with four symbols. This produces sequences whose word-complexity, the count of distinct substrings of length n, separates different dynamical regimes. For quasiperiodic ultra-high-energy orbits with irrational slopes, complexity grows linearly as C_n equals n plus three with zero topological entropy, and the sequences behave like Sturmian words. Numerical checks on chaotic orbits in the global chaos band lead to the conjecture that C_n equals three times two to the n minus one, giving entropy log two, with the sequences obeying grammar rules of a subshift of finite type. A reader would care because the coarse encoding still distinguishes ordered from chaotic motion in a system that models coupled Josephson junctions.

Core claim

In the three-rotor problem, replacing trajectories by four-symbol sequences based on isolated rotor coincidences captures key qualitative features. Quasiperiodic sequences from irrational slopes have C_n = n+3 and zero entropy by mapping to Sturmian sequences, while chaotic sequences are conjectured to satisfy C_n = 3 imes 2^{n-1} with entropy log 2 and to be generated by a subshift of finite type, complete with explicit grammar rules that the quasiperiodic case lacks.

What carries the argument

The four-symbol digitization that records successive isolated pair and triple coincidences, partitioning the configuration torus into ordered-rotor cells whose boundaries mark the symbols.

If this is right

  • Periodic orbits produce sequences whose word-complexity stops growing once n reaches the period.
  • Quasiperiodic sequences have zero topological entropy and cannot be realized as a topological Markov shift.
  • Chaotic sequences obey a finite set of grammar rules that allow complete description by a subshift of finite type.
  • The encoding distinguishes quasiperiodic from chaotic dynamics even though it is coarse.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same coincidence digitization could be used to forecast when chains of coupled Josephson junctions enter chaotic regimes.
  • Proving the exact growth rate would require constructing the transition matrix of the conjectured subshift.
  • The linear complexity of the quasiperiodic case shows that the encoding preserves the rigid ordering constraints absent in chaos.

Load-bearing premise

The four-symbol recording of isolated coincidences is detailed enough to capture the full grammar rules and complexity growth that separate quasiperiodic from chaotic regimes.

What would settle it

A long chaotic orbit whose symbol sequence yields a number of distinct n-letter words other than 3 imes 2^{n-1} for large n, or contains a word forbidden by the conjectured grammar.

Figures

Figures reproduced from arXiv: 2606.25766 by Anirudh Rameshan, Govind S. Krishnaswami.

Figure 1
Figure 1. Figure 1: Fundamental domain of φ1 -φ2 configuration torus. The rotors are ordered anticlockwise (A) and clockwise (C) in the shaded and unshaded regions. The thick black boundaries correspond to pairwise coincidences (1, 2 and 3), and triple coincidence (0) at the origin. The static solutions G, D and T are indicated. Librational and rotational periodic pendula lie along the coincidence lines 1, 2 and 3, while the … view at source ↗
Figure 2
Figure 2. Figure 2: Rotor positions and inter-rotor vectors to find the rotor order (anticlockwise in this example). The order of rotors is determined by the sign of a certain scalar triple product. Suppose the rotors lie on the x-y plane with ˆz pointing out of the plane as in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Two periodic UHE trajectories on the [0, 2π) 2 fundamental domain of the φ1 -φ2 configuration space: (a) σ = 3/2 and (b) σ = −5/2. Arrows indicate the direction of time assuming ω2 > 0 with trajectories beginning at the origin. The solid red lines are unbroken lines that cross the dotted black 3-1 coincidence curve. The solid green lines are unbroken lines that do not involve 3-1 pair coincidences. Dashed … view at source ↗
Figure 4
Figure 4. Figure 4: φ1 -φ2 configuration space showing some high-energy trajectories: (a) σ = 1/2, (b) σ = −1/2, (c) σ = −3/5, (d) σ = 2/3 and (e) σ = 8/5 = 1.6, plotted over one period, and (f) σ = (1 + √ 5)/2 = 1.618 . . . (golden ratio), plotted up to 10 cycles of φ2 . The symbol sequences for (e) and (f) are the same up to 8 φ2 cycles but differ thereafter. In (e), for σ = 8/5, the two consecutive unbroken lines (solid re… view at source ↗
Figure 5
Figure 5. Figure 5: (a) Vertex-labeled Markov graph for grammar rules (1) and (2) has four vertices v0, v1, v2, v3 corresponding to the symbols 0, 1, 2, 3 with bidirectional edges connecting v2 to every other vertex. (b, c) Edge-labeled Markov graphs for rules (1,2) and (1,2,3). To incorporate rule (4) for irrational σ, we remove the top edge in (c) with label 2. (3) For σ > 1, a trajectory starting on the wall must hit the c… view at source ↗
Figure 6
Figure 6. Figure 6: Word-complexity Cn(l) plotted against the logarithm of the sequence length l for words of length n = 2, 3, · · · , 8 in the symbol sequence of a chaotic trajectory with E = 5.5 and ICs {φ1(0) = 0, φ2(0) = p2(0) = 1} evolved up to t = 6 × 106 in units where m = r = g = 1. The initial 0 in the symbol sequence is omitted. The horizontal lines are at the conjectured asymptotic word-complexities Cn = 3 × 2 n−1 … view at source ↗
Figure 7
Figure 7. Figure 7: Proposed vertex- and edge-labeled Markov graphs for chaotic sequences in globally chaotic band. Grammar rules and adjacency matrix. Examination of the resulting symbol sequences of such chaotic trajectories leads us to the following observations. (i) Triple coincidences do not occur other than possibly at t = 0 (which happens only if the IC involves a triple coincidence). In what follows, we only record sy… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes a four-symbol digitization of the three-rotor dynamics based on isolated pair and triple coincidences. For ultra-high-energy quasiperiodic orbits it rigorously establishes word complexity C_n = n+3 and zero topological entropy via a mapping to Sturmian sequences. For chaotic orbits in the band of global chaos, numerical examination of grammar rules leads to the conjecture that C_n = 3 × 2^{n-1} (topological entropy log 2) and that the sequences are generated by a subshift of finite type.

Significance. If the conjecture holds, the work supplies a concrete symbolic dynamics for the three-rotor problem that distinguishes quasiperiodic from chaotic regimes on the configuration torus and yields a falsifiable prediction for word counts in the chaotic case. The rigorous Sturmian mapping for the quasiperiodic regime is a clear strength, providing a parameter-free derivation and explicit grammar rules.

major comments (2)
  1. [Abstract and chaotic-orbits section] Abstract and chaotic-orbits section: the conjecture that the observed grammar rules are exhaustive and that the four-symbol partition induces a subshift of finite type with exactly C_n = 3 × 2^{n-1} is load-bearing for the central claim; the manuscript supplies no analytic argument or exhaustive enumeration (e.g., of all admissible words up to length 12) showing that no further forbidden words arise from the underlying flow on the torus.
  2. [Digitization and chaotic-orbits section] Digitization and chaotic-orbits section: the claim that the coarse coincidence partition suffices to capture the qualitative distinction between quasiperiodic and chaotic dynamics (including the full set of grammar rules) is not independently verified; an explicit check that the extracted transition matrix reproduces the numerically observed periodic-orbit counts or entropy estimate would be required to support the modeling statement.
minor comments (3)
  1. [Digitization paragraph] The four symbols are introduced in the digitization paragraph but lack an accompanying table that lists the geometric meaning of each symbol together with the corresponding coincidence type.
  2. [Quasiperiodic section] In the quasiperiodic section the right-special words are discussed but the explicit recurrence or substitution rule that generates them is not written as an equation.
  3. Figure captions for the complexity plots should state the length of the symbol sequences used and the number of sampled orbits.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The two major comments correctly identify that the chaotic-orbit results rest on a numerically supported conjecture rather than a complete analytic proof. We respond to each point below and will revise the manuscript to incorporate additional numerical checks.

read point-by-point responses

  1. Referee: [Abstract and chaotic-orbits section] Abstract and chaotic-orbits section: the conjecture that the observed grammar rules are exhaustive and that the four-symbol partition induces a subshift of finite type with exactly C_n = 3 × 2^{n-1} is load-bearing for the central claim; the manuscript supplies no analytic argument or exhaustive enumeration (e.g., of all admissible words up to length 12) showing that no further forbidden words arise from the underlying flow on the torus.

    Authors: We agree that the conjecture is load-bearing and that the manuscript presents it on the basis of numerical observation of grammar rules rather than an analytic demonstration of exhaustiveness. No analytic argument is supplied because a rigorous proof that the observed rules capture all constraints imposed by the flow would require a complete topological analysis of the three-rotor system, which lies outside the scope of the present work. In the revision we will add an explicit enumeration of admissible words up to length 12 obtained from long chaotic trajectories; this will confirm that no additional forbidden words appear within the resolution of the simulations. The conjecture status of the claim will be stated more explicitly. revision: partial

  2. Referee: [Digitization and chaotic-orbits section] Digitization and chaotic-orbits section: the claim that the coarse coincidence partition suffices to capture the qualitative distinction between quasiperiodic and chaotic dynamics (including the full set of grammar rules) is not independently verified; an explicit check that the extracted transition matrix reproduces the numerically observed periodic-orbit counts or entropy estimate would be required to support the modeling statement.

    Authors: We accept that an independent consistency check between the proposed subshift and the underlying flow is needed. In the revised manuscript we will extract the transition matrix implied by the observed grammar rules, compute the number of periodic sequences of periods 2 through 10, and compare these counts (as well as the topological entropy given by the logarithm of the largest eigenvalue) with the corresponding quantities obtained by direct numerical integration of the three-rotor equations. This verification will be reported in a new subsection of the chaotic-orbits section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results are numerical conjecture plus independent Sturmian mapping

full rationale

The paper's analytic component for ultra-high-energy quasiperiodic orbits derives C_n = n+3 and zero entropy via an explicit mapping to Sturmian sequences on limiting slopes, which is independent of the target result and relies on standard combinatorial word theory. The chaotic case is presented explicitly as a numerical conjecture C_n = 3×2^{n-1} obtained by direct counting of words in generated coincidence sequences, with grammar rules identified from the same sequences but not claimed as a closed derivation. No parameter is fitted to a subset and then relabeled a prediction, no self-definition equates X to Y, and no load-bearing step reduces to a self-citation. The work is therefore self-contained against external benchmarks (Sturmian theory and direct enumeration) with no circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard Newtonian model of three point masses on a circle with pairwise cosine potentials and on the mathematical framework of symbolic dynamics; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The three-rotor system is governed by Newtonian mechanics with attractive pairwise cosine potentials.
    This is the defining physical model stated in the abstract.
  • domain assumption Isolated rotor coincidences correspond to crossings in the configuration space partition.
    Invoked to justify the four-symbol encoding.

pith-pipeline@v0.9.1-grok · 5841 in / 1295 out tokens · 25607 ms · 2026-06-25T19:06:48.760447+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

30 extracted references

  1. [1]

    N.,Deterministic nonperiodic flow, J

    Lorenz, E. N.,Deterministic nonperiodic flow, J. Atmos. Sci.20, 130 (1963). 2

    1963

  2. [2]

    Hadamard, J.,Les surfaces ` a courbures oppos´ ees et leurs lignes g´ eod´ esiques, Journal de Math´ ematiques Pures et Appliqu´ ees4, 27-73 (1898). 2

  3. [3]

    D.,Dynamical Systems, American Mathematical Society (1927)

    Birkhoff, G. D.,Dynamical Systems, American Mathematical Society (1927). 2

    1927

  4. [4]

    and Hedlund, G

    Morse, M. and Hedlund, G. A.,Symbolic dynamics, Amer. J. Math.60, 815–866 (1938). 2, 21

    1938

  5. [5]

    and Hedlund, G

    Morse, M. and Hedlund, G. A.,Symbolic dynamics II. Sturmian trajectories, Amer. J. Math.62, 1-42 (1940). 2, 4, 10, 21

    1940

  6. [6]

    Hao, B.,Symbolic dynamics and characterization of complexity, Physica D51, 161-176 (1991) 2

    1991

  7. [7]

    Arioli, G.,Periodic orbits, symbolic dynamics and topological entropy for the restricted 3-body problem, Commun. Math. Phys.231, 1-24 (2002). 2

    2002

  8. [8]

    and Mees, A

    Hirata, Y. and Mees, A. I.,Estimating topological entropy via a symbolic data compression technique, Phys. Rev. E67, 026205 (2003). 2

    2003

  9. [9]

    and Hiraide, K.,Computation of entropy and Lyapunov exponent by a shift transform, Chaos 25(10), 103110 (2015)

    Matsuoka, C. and Hiraide, K.,Computation of entropy and Lyapunov exponent by a shift transform, Chaos 25(10), 103110 (2015). 2

    2015

  10. [10]

    and Trevino, R.,Algorithms for rigorous entropy bounds and symbolic dynamics, SIAM J

    Day, S., Frongillo, R. and Trevino, R.,Algorithms for rigorous entropy bounds and symbolic dynamics, SIAM J. Applied Dynamical Systems7(4), 477-1506(2008). 2

    2008

  11. [11]

    and Soloviev, I.,On a method of applied symbolic dynamics for investigation of dynamical systems, Vibroengineering Procedia25(4), 128-134 (2019)

    Ampilova, N. and Soloviev, I.,On a method of applied symbolic dynamics for investigation of dynamical systems, Vibroengineering Procedia25(4), 128-134 (2019). 2

    2019

  12. [12]

    and Marcus, B.An Introduction to Symbolic Dynamics and Coding, 2nd Ed., Cambridge University Press (1995)

    Lind, D. and Marcus, B.An Introduction to Symbolic Dynamics and Coding, 2nd Ed., Cambridge University Press (1995). 2, 4, 20, 21

    1995

  13. [13]

    P.,Substitutions in Dynamics, Arithmetics and Combinatorics, Vol

    Fogg, N. P.,Substitutions in Dynamics, Arithmetics and Combinatorics, Vol. 1794 of Lecture Notes in Mathe- matics, Eds. Berth´ e, V., Ferenczi S., Mauduit, C. and Siegel A., Springer-Verlag (2002). 2, 17, 20, 21

    2002

  14. [14]

    and Williams, S.,Symbolic dynamics, Scholarpedia3(11):2923 (2008)

    Marcus, B. and Williams, S.,Symbolic dynamics, Scholarpedia3(11):2923 (2008). 2

    2008

  15. [15]

    and Zheng, W.,Applied Symbolic Dynamics and Chaos, 2nd Ed., World Scientific (2018)

    Hao, B. and Zheng, W.,Applied Symbolic Dynamics and Chaos, 2nd Ed., World Scientific (2018). 2

    2018

  16. [16]

    2, 4, 7, 8, 10, 20, 21, 22

    Bruin, H.,Topological and ergodic theory of symbolic dynamics, American Mathematical Society (2022). 2, 4, 7, 8, 10, 20, 21, 22

    2022

  17. [17]

    and Amigo, J

    Hirata, Y. and Amigo, J. M.,A review of symbolic dynamics and symbolic reconstruction of dynamical systems, Chaos33(5), 052101 (2023). 2

    2023

  18. [18]

    Montgomery, R.,Infinitely many syzygiesArch. Ration. Mech. Anal.164311-40 (2002). 2

    2002

  19. [19]

    Dullin, H. R. and Montgomery, R.,Syzygies in the two center problem, Nonlinearity29, 1212-1237 (2016). 2, 4, 21

    2016

  20. [20]

    and Misiurewicz, M.,Topological entropy, Scholarpedia,3(2):2200 (2008)

    Adler, R., Downarowicz, T. and Misiurewicz, M.,Topological entropy, Scholarpedia,3(2):2200 (2008). 2

    2008

  21. [21]

    Catalan, T.,A link between topological entropy and Lyapunov exponents, Ergodic Theory and Dynamical Sys- tems39(3), 620-637 (2019). 2

    2019

  22. [22]

    Krishnaswami, G. S. and Senapati, H.,Classical three rotor problem: periodic solutions, stability and chaos, Chaos29(12), 123121 (2019). 2, 3, 8, 9 25

    2019

  23. [23]

    P., Mooij, J

    Orlando, T. P., Mooij, J. E., Tian, L., van der Wal, C.H., Levitov, L.S., Lloyd, S. and Mazo, J. J.,Supercon- ducting persistent-current qubit, Phys. Rev. B60, 15398 (1999). 2

    1999

  24. [24]

    Krishnaswami, G. S. and Yadav, A.,Bifurcation cascade, self-similarity, and duality in the three-rotor problem, Chaos33, 083101 (2023). 2, 3, 10, 22

    2023

  25. [25]

    Senapati, H.,Instabilities and chaos in the classical three-body and three-rotor problems, PhD thesis, Chennai Mathematical Institute (2020). 3

    2020

  26. [26]

    Krishnaswami, G. S. and Senapati, H.,Ergodicity, mixing and recurrence in the three rotor problem, Chaos30 (4), 043112 (2020). 3, 22

    2020

  27. [27]

    Krishnaswami, G. S. and Senapati, H.,Quantum three-rotor problem in the identity representation, Phys. Rev. E111, 014221 (2025). 3, 5, 13

    2025

  28. [28]

    4, 20, 21, 22

    Lothaire, M.,Algebraic Combinatorics on Words, Cambridge University Press (2002). 4, 20, 21, 22

    2002

  29. [29]

    thesis, UM-DAE Centre for Excellence in Basic Sciences, University of Mumbai (2024)

    Rameshan, A.,Symbol sequences from three rotor coincidences, M.Sc. thesis, UM-DAE Centre for Excellence in Basic Sciences, University of Mumbai (2024). 4

    2024

  30. [30]

    and Hasselblatt, B.,Introduction to modern theory of dynamical systems, Cambridge University Press (1995)

    Katok, A. and Hasselblatt, B.,Introduction to modern theory of dynamical systems, Cambridge University Press (1995). 18, 20 26

    1995