Resonant oscillations of a weakly coupled, nonlinear layered system

Abstract

We analyze standing wave oscillations of a periodic system of infinite spatial extent composed of layers with cubic material nonlinearities that are coupled by weak linear stiffnesses. We show that nonlinear modal interactions in this system are unavoidable due to the existence of an infinite degenerate set of internal resonances. Considering only the dominant 1∶3 resonances, the standing wave problem is formulated in terms of a bi-infinite set of coupled nonlinear difference equations that govern the layer modal amplitudes. In the limit of weak coupling between layers this set of difference equations is analyzed by (i) direct perturbation analysis, and (ii) matched asymptotic expansions of the differential equations resulting from a continuum approximation. Strongly and weakly localized, as well as spatially extended (non-localized) solutions are computed. In addition, composite solutions derived by matching in-phase and out of phase segments of the aforementioned solutions can also be constructed. In the limit of weak coupling between layers and/or strong material nonlinearities, the solutions develop sensitive dependence on initial conditions, and the possibility of spatial chaos in the system exists.