管状折纸动力系统中的不变曲线与变分结构
Invariant Curves and the Variational Structure in Tubular Origami Dynamical Systems
AI总结：
本文通过将逆模数视为KAM理论中的摄动参数，分析了管状折纸拼图的动力学，证明了在大模数极限下保守动力学收敛到具有变分结构的可积映射，并利用不变曲线的持久性解释了折叠模式的控制。
摘要：
我们通过将逆模数$N^{-1}$视为Kolmogorov--Arnold--Moser (KAM)理论框架中的摄动参数，对管状折纸拼图进行了理论和数值动力学系统分析。在大模数极限（$N \to \infty$）下，我们证明保守动力学形式收敛到具有变分结构的可积映射，其生成函数对应于总离散平均曲率。从KAM理论的角度看，可积极限的非共振不变曲线预期在$N$足够大时持续存在。与此预期一致，随着$N$增大的数值计算表明，相空间的大片区域充满了看似不变曲线的结构。通过调整山-谷折痕分配和折痕长度，系统可转化为具有多个零频率的非扭转映射。可积极限下的频率分布和不变曲线的持久性使我们能够控制作为共存椭圆岛出现的稳定折叠区域的数量和排列。这些岛提供了不同折叠模式的相空间解释，它们由作为连续变形几何障碍的不变曲线分隔，并阻碍无自交的转变。最后，我们在共形辛系统框架下分析了折纸结构的膨胀和收缩动力学。通过引入虚拟辅助折痕作为漂移控制机制，我们数值证实了稳定准周期吸引子的存在。
英文摘要：
We present a theoretical and numerical dynamical-systems analysis of tubular origami tessellations by identifying the inverse module number, $N^{-1}$, as a perturbation parameter within the framework of Kolmogorov--Arnold--Moser (KAM) theory. In the large-module limit ($N \to \infty$), we show that the conservative dynamics formally converges to an integrable map with a variational structure, whose generating function corresponds to the total discrete mean curvature. From the viewpoint of KAM theory, nonresonant invariant curves of the integrable limit are expected to persist for sufficiently large $N$. Consistent with this expectation, numerical computations with increasing $N$ show that large regions of the phase space are filled with structures that appear to be invariant curves. By adjusting mountain-valley fold assignments and fold lengths, the system can be transformed into a nontwist map that exhibits multiple zero frequencies. The frequency profile in the integrable limit and the persistence of invariant curves allow us to control the number and arrangement of stable folding regions appearing as coexisting elliptic islands. These islands provide a phase-space interpretation of distinct folding modes, separated by invariant curves that act as geometric barriers to continuous deformation and obstruct transitions without self-intersection. Finally, we analyze the expanding and contracting dynamics of the origami structure within the framework of conformally symplectic systems. By introducing a virtual auxiliary fold as a drift control mechanism, we numerically confirm the existence of stable quasi-periodic attractors.
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