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Van Hove singularities in the density of states of a chaotic dynamical system

Bryn Davies

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Phys. Rev. E 113, 064211 – Published 26 June, 2026

DOI: https://doi.org/10.1103/5pjc-hwjw

Abstract

We show that the statistics of chaotic systems can be predicted by constructing an associated sequence of periodic differential operators and computing their densities of states. For such operators, the density of states is well understood and can be computed straightforwardly, often yielding explicit formulas. As a case study, we investigate a nonlinear recursion relation that maps naturally onto a family of periodic operators generated by a Fibonacci tiling rule. This correspondence enables us to derive an explicit formula for the limiting statistics of the chaotic system and to demonstrate that the clustering near critical values is equivalent to the van Hove singularities in the operators' densities of states.

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Phys. Rev. E 113, 064211– Published 26 June, 2026
  • Received 23 September 2025
  • Accepted 10 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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