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.

中文翻译：

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生长双层系统中的倍周期和三倍周期

弹性基材上的生长层能够产生多种表面形态。适度的生长会产生规则的正弦皱纹图案，并具有均匀的能量分布。虽然周期性起皱的关键条件已被广泛研究，但超出第一个不稳定点的丰富图案的形成仍然知之甚少。在这里，我们表明，在持续生长时，能量逐渐局部化，并出现新的复杂形态。以前的研究经常忽视这些次要分歧。他们专注于层和基底之间的大刚度比，其中初级不稳定性很早就出现，远早于次级不稳定性出现。我们证明，二次分叉在低刚度比状态下尤其重要，其中一次和二次不稳定性的临界条件更加接近。在所有可能的二次分岔中，倍周期模式起着核心作用——它比所有其他模式在能量上更有利。然而，我们可以通过选择有利于替代高阶模式的边界条件来在数值上抑制倍周期。我们的结果表明，在低刚度状态下，图案形成对小缺陷高度敏感：表面形态迅速出现，自发变化并迅速变得极其复杂。这是发育生物学中的常见范例。我们的结果在生长渐进且刚度比较低的生命系统的形态发生中具有重要应用。