3D Printing of Invariant Manifolds in Dynamical Systems

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.