The 1:2:4 resonance in a particle chain

We consider four masses in a circular configuration with nearest-neighbour interaction, generalizing the spatially periodic Fermi--Pasta--Ulam-chain where all masses are equal. We identify the mass ratios that produce the $1{:}2{:}4$~resonance --- the normal form in general is non-integrable already at cubic order. Taking two of the four masses equal allows to retain a discrete symmetry of the fully symmetric Fermi--Pasta--Ulam-chain and yields an integrable normal form approximation. The latter is also true if the cubic terms of the potential vanish. We put these cases in context and analyse the resulting dynamics, including a detuning of the $1{:}2{:}4$~resonance within the particle chain.