Pell and Clapeyron Words as Stable Trajectories in Dynamical Systems

TL;DR

This paper demonstrates the existence of stable, aperiodic time tilings called Pell and Clapeyron words in dynamical systems, revealing new stable trajectories with specific growth patterns in nonlinear systems.

Contribution

It introduces the concepts of Pell and Clapeyron words as stable trajectories in dynamical systems, extending the understanding of time quasicrystals and their systematic approximations.

Findings

Two classes of time quasicrystals can be stable in dynamical systems.

Stable trajectories grow according to Pell and Clapeyron numbers.

Results apply broadly to dissipative nonlinear systems, including maps and continuous systems.

Abstract

We establish the existence of `time quasicrystals', tilings of the time axis with two unit cells of different duration. These aperiodic tilings can be constructed as slices through regular tilings of a space spanned by two orthogonal time directions. We establish the result rigorously using the tools of symbolic dynamics. We show that, of the ten physically-relevant classes of one-dimensional quasicrystal, precisely two can appear as stable, attracting trajectories in dynamical systems, which we term the infinite Pell and Clapeyron words. These grow, via a generalization of the period-doubling cascade, as a sequence of stable orbits with periods increasing as the Pell and Clapeyron numbers, providing systematic approximations which can be experimentally implemented. The results apply to a wide universality class of dissipative nonlinear systems: we consider discrete-time maps, and continuous-time dynamical systems, both autonomous and periodically driven. This Paper proves and extends the results of a companion Letter, as well as providing a pedagogical background.