On the Periodic Orbits of Four-Particle Time-Dependent FPU Chains

Abstract

We study the existence of the periodic orbits for four-particle time-dependent FPU chains. To begin with, taking an orthogonal transformation and dropping two variables, we put the system into a lower dimensional system. In this case, the unperturbed system has the linear spectra 2 and √ 2, which are rational independent. Therefore, the classical averaging theory cannot be applied directly, and the existence and multiplicity of periodic orbits for four-particle FPU chains is proved by using a new result from averaging theory by Buicȃ, Francoise, and Llibre.