  The inhomogeneous Fermi-Pasta-Ulam chain is studied by identifying the mass
ratios that produce prominent resonances. This is a technically complicated
problem as we have to solve an inverse problem for the spectrum of the
corresponding linearized equations of motion. In the case of the inhomogeneous
periodic Fermi-Pasta-Ulam chain with four particles each mass ratio determines
a frequency ratio for the quadratic part of the Hamiltonian. Most prominent
frequency ratios occur but not all. In general we find a one-dimensional
variety of mass ratios for a given frequency ratio.
  For the resonance 1:2:3 a small cubic term added to the Hamiltonian leads to
a dynamical behaviour that shows a difference between the case that two masses
are equal and the more general case of four different masses. For two equal
masses the normalized system is integrable and chaotic behaviour is
small-scale. In the transition to four different masses we find a
Hamiltonian-Hopf bifurcation of one of the normal modes leading to complex
instability and Shilnikov-Devaney bifurcation. The other families of
short-periodic solutions can be localized from the normal forms together with
their stability characteristics. For illustration we use action simplices and
the behaviour with time of the H_2 integral of the normal forms.
