Pell and Clapeyron Words as Stable Trajectories in Dynamical 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.