Time Quasicrystals in Dissipative Dynamical Systems

1) Good introduction to quasi-crystals and symbolic dynamics
2) Creative and interesting idea to combine the two concepts

1) Connection to time-crystals not sufficiently clarified (see report)

Generally, dissipative driven systems can have periodic orbits as stable attractors. Such periodic orbits can be stable (with respect to small parameter changes in the dynamical system) and attractive. If these attractors are course-grained (e.g., the point after each period is labelled according to whether it is "left" or "right" of the origin of a one-dimensional coordinate) one obtains a sequence of labels (a "word") that characterises the dynamics. In the manuscript it is shown that there exist two sequences of words of increasing length ("Pell words" and "Clapeyron words"), each corresponding to stable and attractive orbits, which converge to one-dimensional quasicrystals when "L" and "R" are identified with the short and long cells in the quasicrystal. In the manuscript these sequences are termed time-quasicystals.

However, in the present form of the manuscript I do not see a strong connection to "time-crystals":

In the recent literature, a time-crystal refers to the breaking of (discrete) time-translational invariance in a periodically driven quantum system. In the present manuscript, the notion is extended to classical dissipative systems. Before going to time-quasicrystals, a clear definition should therefore be given what is a normal time-crystal in the present setting. Should any periodic orbit in a driven system be considered a time-crystal if it is an attractive trajectory which is stable against perturbations of the parameters? With this, "time-crystals" would to be a common phenomena in dynamical systems which does not really need a new terminology. Is there an additional rigidity that characterises this "phase of matter"?

The aperiodic orbits of quasi-crystalline order which are discovered in the manuscript seem to satisfy even less the requirements for being a "stable phase", as they are not even stable against parameter changes of the dynamical system. While it is shown in the manuscript that any periodic approximation of the discussed time-sequences is stable throughout an open set of parameter space, in the discussed examples this stability range decreases strongly with length (e.g., in the logistic map the period-12 approximation to the Pell quasicrystal is stable only in a parameter range of $10^{-5}$). Even though, as the author says, there is no generic proof that the stability range would decrease with the length of the approximation, no example for a "time-quasicrystal" which is stable in a finite parameter regime is presented. Instead, the infinite time-quasicrystal seems to be embedded in a chaotic region of the dynamics which would be destroyed by infinitesimal variations of the parameters of the dynamical system.

I believe that a notion of rigidity should be added to the definition of time-crystals in dynamical systems, otherwise the term should probably be avoided (also in the title). In the paper it would still be useful to add a proper discussion of the differences between time crystals and periodic orbits in dynamical systems, which would make the paper still appealing for researchers working on time quasicrystals.

Furthermore, I have one minor technical question: The manuscript states that *only two* out of ten possible quasicrystals an be realised as course-grained time-series in the logistic map. Following the definitions given in section 3.4, I understand that if inflation rules that lead to a quasicrystal respect the the generalized composition criteria, we have managed to find a sequence of stable and attractive orbits of increasing length which finally gives a quasi-crystal order. While this can be used for a constructive proof for the existence of the two classes (Pell and Clapeyron), I do not yet see how it excludes other classes (probably because of my limited knowledge on symbolic dynamics). The generalized composition rules are only stated as a *sufficient* criterion to produce a sequence of maximal words, which does not exclude that inflation rules which do not satisfy them can produce an infinite sequence of maximal words (as I understand, it is not needed for every word in the series to be maximal in order to produce a series which approaches the quasi-crystal?).

In any case, the paper establishes a connection between two different concepts (quasi-crystalline order and classical dissipative driven systems). I found this an interesting observation, and it was a nice to think about it. The discussion is also presented in a detailed and pedagogic manner, which makes the manuscript accessible to readers who are not from the field of dynamical systems. Hence I believe the paper should be published if the discussion on time crystals is adapted.

see report

1. The paper adresses a timely subject.
2. The presentation is very clear, I actually enjoyed reading it.
3. It gives a thorough discussion on time quasicrystals

1. In terms of experimental implications, the discussion is too concise.
2. It is not clear to me how these findings would appear for the quantum version of time quasicrystals. This, I feel, is an important issue as existing experiments for time crystals stem from quantum systems.

The author discusses time quasicrystals in dynamical dissipative systems. This is a very timely subject, and since the original proposal for quantum and classical time crystals, there was an explosion of interest in this direction, followed by the experimental realization of such systems last year. Similarly to how crystals and quasicrystals exist in solid state physics, the author generalizes the concept of time crystals to time quasicrystals as stable trajectories in dissipative dynamical systems. Using the generalized composition rules, two admissible quasicrystals coined as Pell and Clapeyron quasicrystals are identified. In terms of physical models, discrete-time maps as well as periodically-driven continuous-time dynamical systems are analyzed.

1. Although a short listing of experiments is given in terms of where time quasicrystals could be looked for, a bit more detailed discussion about more specific experiments and signatures would be helpful for a general reader.
2. The author identifies two classes of quasicrystals (Pell and Clapeyron), characterized by different ratios of the cell length as 1+sqrt(2) and 2+sqrt(3).
Just by looking at these numbers, is it possible to have a quasicrystal with ratio for word length 4+sqrt(5) or something similar?
3. A discussion on the quantum version of time quasicrystals and their respective signatures would also be interesting.