High-low frequency interaction in alternating FPU alfa-chains

Abstract

One of the problems of periodic FPU-chains with alternating large masses is whether signifi- cant interactions exist between the so-called (high frequency) optical and (low frequency) acoustic groups. We show that for ↵-chains with 2n particles we have significant interactions caused by external forcing of the acoustic modes by a stable or unstable optical normal mode. In the proofs an embedding theorem plays a part; the analysis is straightforward in the case that n is even, a di↵erent approach using invariant manifolds with symmetry is needed if n is odd. For -chains the interactions are characterised by parametric excitation, there are indications that interaction is negligible.