Period-doubling and period-tripling in growing bilayered systems

Abstract

Growing layers on elastic substrates are capable of creating a wide variety of surface morphologies. Moderate growth generates a regular pattern of sinusoidal wrinkles with a homogeneous energy distribution. While the critical conditions for periodic wrinkling have been extensively studied, the rich pattern formation beyond this first instability point remains poorly understood. Here, we show that upon continuing growth, the energy progressively localizes and new complex morphologies emerge. Previous studies have often overlooked these secondary bifurcations; they have focused on large stiffness ratios between layer and substrate, where primary instabilities occur early, long before secondary instabilities emerge. We demonstrate that secondary bifurcations are particularly critical in the low stiffness ratio regime, where the critical conditions for primary and secondary instabilities move closer together. Amongst all possible secondary bifurcations, the mode of period-doubling plays a central role – it is energetically favourable over all other modes. Yet, we can numerically suppress period-doubling, by choosing boundary conditions, which favour alternative higher order modes. Our results suggest that in the low stiffness regime, pattern formation is highly sensitive to small imperfections: surface morphologies emerge rapidly, change spontaneously and quickly become immensely complex. This is a common paradigm in developmental biology. Our results have significant applications in the morphogenesis of living systems where growth is progressive and stiffness ratios are low.