Periodically driven anharmonic chain: Convergent power series and numerics

Abstract. We investigate the long time behavior of a pinned chain of 2N +1 oscillators, indexed by x ∈ {−N,...,N}. The system is subjected to an external driving force on the particle at x = 0, of period θ = 2π/ω, and to frictional damping γ > 0 at both endpoints x = −N and N. The oscillators interact with a pinned and nearest neighbor harmonic plus anharmonic potentials of the form, with V′′ and U′′ bounded andν ∈ R. We recall the recently proven convergence and the global stability of a perturbation series in powers of ν for ∣ν∣ < ν0, yielding the long time periodic state of the system. Here ν0 depends only on the supremum norms of V′′ and U′′ and the distance of the set of non-negative integer multiplicities of ω from the interval - the spectrum of the infinite harmonic chain for ν = 0. We describe also some numerical studies of this system going beyond our rigorous results.