Dynamics of a chain with four particles, alternating masses and nearest neighbor interaction

Abstract

We formulate the periodic FPU problem with four alternating masses which is the simplest nontrivial version. The analysis involves normal form calculations to second order producing integrable normal forms with three timescales. In the case of large alternating mass the system is an example of dynamics with widely separated frequencies and three timescales. The presence of approximate integrals and the stability characteristics of the periodic solutions lead to weak interaction of the modes of the system.