Energy Transport and Localization in Weakly Dissipative Resonant Chains

Abstract

In this paper we examine the effect of dissipation on the emergence of resonance in a weakly dissipative Klein-Gordon chain subjected to harmonic forcing applied to the first oscillator. Both asymptotic approximations and numerical simulations prove that weak linear dissipation counteracts resonant oscillations in the entire chain even if a similar undamped array exhibits resonance. Stable resonance may occur either in a short-length chain or in an initial segment of a long-length weakly damped chain but motion of distant oscillators becomes non-resonant. Furthermore, an increase of dissipation diminishes the localization length for resonant oscillations. The conditions of the emergence of resonance as well as an expected length of localization are obtained from the equations for the steady solutions of the system under consideration. The closeness of the approximate solutions to exact (numerical) results is demonstrated.