Family Structure from Periodic Solutions of an Improved Gap Equation
  Fermion mass models usually contain a horizontal symmetry and therefore fail to predict the exponential mass spectrum of the Standard Model in a natural way. In dynamical symmetry breaking there are different concepts to introduce a fermion mass spectrum, which automatically has the desired hierarchy. In constructing a specific model we show that in some modified gap equations periodic solutions with several fermion poles appear. The stability of these excitations and the application of this toy model are discussed. The mass ratios turn out to be approximately $e^\pi$ and $e^{2\pi}$. Thus the model explains the large ratios of fermion masses between successive generations in the Standard Model without introducing large or small numbers by hand. 
Andreas Blumhofer and Marcus Hutter