Energy estimates for wave amplification in quasiperiodic Fibonacci time-modulated media

We show that Fibonacci time quasicrystals can be approximated by temporal supercells, which capture the nonperiodic but self-similar modulation in an analytically tractable form. This reveals a fractal hierarchy of instability gaps, indexed by the wave number <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>k</a:mi> </a:math> , in which wave energy is amplified exponentially. These estimates are validated by the observation of super <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>k</b:mi> </b:math> gaps: instability regions that are independent of the duration of the temporal supercell. This approach predicts the regions of parametric amplification and provides arbitrarily accurate estimates of the energy growth rate.