3D Printing of Invariant Manifolds in Dynamical Systems

TL;DR

The paper addresses visualizing and studying invariant manifolds in three-dimensional dynamical systems by producing tangible 3D-printed models. It combines the Parameterization Method for local manifolds with arclength-based global integration to generate complete manifolds, followed by thickening and meshing to create printable objects. Demonstrations on the Lorenz, Arneodo-Coullet-Tresser, and Langford systems showcase stable, unstable, and intersecting manifolds and illustrate parameter-dependent geometric changes. A practical workflow and an open-source code repository enable researchers and educators to reproduce and extend the pipeline, advancing tactile understanding and exploration of nonlinear dynamics.

Abstract

Invariant manifolds are one of the key features that organize the dynamics of a differential equation. We introduce a novel approach to visualizing and studying invariant manifolds by using 3D printing technology, combining advanced computational techniques with modern 3D printing processes to transform mathematical abstractions into tangible models. Our work addresses the challenges of translating complex manifolds into printable meshes, showcasing results for the following systems of differential equations: the Lorenz system, the Arneodo-Coullet-Tresser system, and the Langford system. By bridging abstract mathematics and physical reality, this approach promises new tools for research and education in nonlinear dynamics. We conclude with practical guidelines for reproducing and extending our results, emphasizing the potential of 3D-printed manifolds to enhance understanding and exploration in dynamical systems theory.

Figures (7)

• Figure 1: A 3D printed stable manifold of the origin for the Lorenz system \ref{['eq:lorenz']}.
• Figure 2: This picture shows 3D printed unstable manifolds of equilibria in the Langford system \ref{['eq:langford']}, displaying structural changes as the bifurcation parameter $\alpha$ varies. Left: The unstable manifold of $p_1 \approx 1.94$ for $\alpha=0.95$ exhibits a complex structure with multiple lobes. Creating a cut-out allows the viewer to see the internal structure of the manifold. Top-right and bottom: Two views of the unstable manifold of $p_1 \approx 1.84$ for $\alpha=0.806$, revealing its spiral structure from above (top-right) and its three-dimensional profile (bottom). Printing using a mesh structure is an alternative to the cut-out for revealing the manifold's intricate details.
• Figure 3: Different perspectives of the 3D-printed stable manifold of the Lorenz system \ref{['eq:lorenz']} at the origin. Top left and right: Two views of the larger model showing the characteristic spiral structure and complex geometry of the stable manifold. Bottom: Comparison with a smaller version, demonstrating that the same data can be used to print in a variety of sizes. The small version is held in hand for size reference.
• Figure 4: Physical realization of the unstable manifold in the Arneodo-Coullet-Tresser system \ref{['eq:arneodo']} with $\beta = 0.4$ and $\mu = 0.863$.
• Figure 5: Physical realization of the intersecting global manifolds in the Langford system. Top left: The 2D unstable manifold of the equilibrium point $p_1$. Top right: 2D stable manifold of the equilibrium point $p_2$. Bottom: A view highlighting the intersection between these two invariant manifolds, with a portion of one manifold protruding through the other. The parameter values for the Langford system used in these models are given in Section \ref{['sec:results']}.

Theorems & Definitions (5)

• Definition 2.1: Stable and unstable manifolds
• Theorem 2.2: Stable Manifold Theorem Guck
• Theorem 2.3: Hartman-Grobman
• Remark 3.1
• Remark 3.2