Energy transfer in autoresonant Klein-Gordon chains

In this work we examine autoresonant oscillations in a Klein-Gordon chain of finite length. The chain is subjected to an external periodic forcing with a slowly varying frequency applied at one edge of the chain. Explicit asymptotic equations describing the amplitudes and the phases of oscillations are derived. These equations demonstrate that, in contrast to the chains with linear attachments, the nonlinear chain can be entirely captured into resonance provided that its structural and excitation parameters exceed their critical thresholds. It is shown that at large times the amplitudes of AR oscillations converge to a monotonically growing mean amplitude that is equal for all oscillators. The threshold values of the structural and excitation parameters, which allow the emergence of autoresonance in the entire chain, are determined. The derived analytic results are in good agreement with numerical simulation.