  We focus on the recent development of the investigation on the equilibria of
the Frenkel-Kontorova models subjected to potentials generated by
quasi-crystals. We present a specific one-dimensional model with an explicit
equivariant potential driven by the Fibonacci quasi-crystal. For a given
positive number {\theta} , we show that there are multiple equilibria with
rotation number {\theta} , e.g., a minimal configuration and a non-minimal
equilibrium configuration. Some numerical explorations of of finding these
equilibira are provided.
