Phonon spectra of a Fibonacci chain

Based on more realistic physics we study the phonon spectra of the Fibonacci chain by taking into account a nonlinear resistance. It is found that the nonlinear force should be very weak and consequently, the continuity, range and gaps of the phonon spectra would be still controlled dominantly by the relative strength of spring constants and chain length. It means that even if no additional nonlinear resistance was taken into account, the conventional results of phonon spectra are exactly correct. On the other hand, in the framework of a conventional model we investigated the relationship between the biggest gaps of phonon spectra and defects of Fibonacci-like aperiodic chains. By means of numerical calculations one can obtain quantitatively the maximum of the length of a one-dimensional aperiodic chain sensitive to boundaries. This method would be useful for the calculation of quasiperiodic and aperiodic lattices.