Invariant Curves and the Variational Structure in Tubular Origami Dynamical Systems

Abstract

We present a rigorous 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 prove that the conservative dynamics converges to an integrable map with a variational structure, whose generating function corresponds to the total discrete mean curvature. Although the geometric interpretation of the generating function becomes more complex under perturbations, it is straightforward in the integrable limit, where its structure can be clearly understood. This limit also provides a fundamental framework for characterizing the global behavior of the system. The KAM-predicted persistence of invariant curves is supported by numerical results showing a phase space densely populated with such 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 resonance associated with these zero frequencies leads to the emergence of new stable foldable regions in phase space, appearing as elliptic islands. These regions enable the design of foldable configurations that are inaccessible within standard twist regimes. 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.

Figures (15)

• Figure 1: Schematic illustration of the waterbomb tube origami tessellation. Left: The crease pattern where red, blue, and black lines represent mountain folds, valley folds, and boundary lines, respectively. The top and bottom edges are identified. Right: The 3D cylindrical structure obtained by nonuniform-folding. The light green polygon indicates a single module, and the yellow region highlights a ring consisting of $N$ modules. These colored regions correspond to each other in both figures.
• Figure 2: Ring and zigzags as constitutive elements. (Left) The crease pattern of the ring shown in Fig. \ref{['fig:origami_model']} and its corresponding folded configuration. (Right) Structural decomposition into zigzags. The solid lines in pink, light blue, and yellow-green represent adjacent zigzags (the 0-th, 1-st, and 2-nd zigzags, respectively), illustrating how the global structure is formed by their concatenation along the cylindrical axis.
• Figure 3: Parametrization of the zigzag within a module. The zigzag segment defined by vertices $U_1, U_2$, and $U_3$ (the $i$-th zigzag) is uniquely determined by the state variables $(\theta, I)$. Here, $I$ is defined as half the Euclidean distance between $U_1$ and $U_3$, and $\theta$ is defined as the angle formed by the vector $\overrightarrow{U_M U_2}$ relative to the central cylinder axis ($X$-axis). The position of the vertex $V_\sigma$ in the subsequent ($i+1$)-th zigzag is determined by the intersection of three spheres centered at $U_1, U_2$, and $U_3$.
• Figure 4: Two distinct patterns of zigzag connectivity. (A) The configuration involving a circumferential shift, which corresponds to the waterbomb tube structure. (B) The configuration where $V_\sigma$ is constructed directly from the original zigzag without a shift. In this pattern, a re-parametrization of the state variables $(\theta, I)$ is required.
• Figure 5: Crease patterns for two representative module configurations. (A) A configuration featuring six-valent vertices, which corresponds to a generalization of the waterbomb tessellation. (B) A configuration characterized by four- or eight-valent vertices. In both panels, different colors indicate distinct zigzags, and the gray shaded region represents the fundamental domain of a single module.