Normal modes on average for purely stochastic systems

We study a class of non-integrable systems, linear chains with homogeneous attractive potentials and periodic boundary conditions, which are not pertu rbations of the harmonic chain. In particular, we deal with the system H-4 with a purely quartic potential, which may be shown to be stochastic withou t any transition. For this model we prove the following pseudo-harmonic pro perties: (1) the existence of a spectrum of frequencies which are proportio nal to the harmonic ones, according to a well defined law; (2) the separabi lity on average of the Hamiltonian function among normal modes with these f requencies. Moreover, as far as stochasticity and pseudo-harmonicity are co ncerned, H-4 is the limit of the Fermi-Past-Ulam (FPU) chain when the energ y density tends to infinity. In this frame, the same results as previously obtained for the FPU chain at high energy density are proven to be independ ent of the presence of the harmonic potential, and to hold at arbitrarily h igh energies. As a byproduct, we have a stochasticity indicator based on co rrelations which proves to be very efficient and reliable.