Forced localization in a periodic chain of non-linear oscillators

Forced localization in a periodic system consisting of an infinite number of coupled non-linear oscillators is examined. A "continuum approximation" is used to reduce the infinite set of ordinary differential equations of motion to a single approximate, non-linear partial differential equation. The structure of the propagation and attenuation zones of the linearized system is found to affect the non-linear localization. Harmonic excitations with general spatial distributions are considered and the localized responses of the chain are studied using exact and asymptotic techniques. Only certain classes of forcing distributions lead to spatial confinement of the forced responses, whereas other types of excitation give rise to spatially periodic or even chaotic harmonic motions of the chain. Systems with weak coupling between particles and/or strong non-linear effects have more profound localization characteristics. The theoretical predictions of the analysis are verified by direct numerical simulations of the equations of motion.