Relative Equilibria of a 4-Particle Ring Close to the 1: 2: 4 Resonance

Abstract

A 4–particle ring with different masses in nearest-neighbour interaction generalizes the spatially periodic Fermi–Pasta–Ulam chain where all masses are equal. For appropriate mass ratios the system is in 1 : 2 : 4 resonance and the 4–particle ring provides for a versal detuning of the 1 : 2 : 4 resonance. The normal form of the system is not integrable, but can be reduced to two degrees of freedom. We determine the relative equilibria and how these behave under detuning. The reduced phase space consists of a singular part in one degree of freedom and a regular part in two degrees of freedom. On the latter the normal form of the 4–particle ring has at most 4 relative equilibria as these are given by the roots of a single quartic polynomial F in one variable. We find a rich bifurcation scenario, with relative equilibria undergoing Hamiltonian flip bifurcations, centre-saddle bifurcations and Hamiltonian Hopf bifurcations. These bifurcations are both approached from a theoretical point of view for general detuned 1 : 2 : 4 resonances and practically compiled to the set of local bifurcations for the normal form of a 4–particle ring passing through the 1 : 2 : 4 resonance.