A Low-Dimensional Principal Manifold as the "Attractor Backbone" of a Chaotic Beam System

We model an elastic beam subject to a contact load which displaces under a chaotic external forcing, motivated by application of a ship carrying either a crane, or fluids in internal tanks. This model not only has rich dynamics and relevance in its own right, it gives rise to a Partial Differential Equation (PDE) whose solutions are chaotic, with an attractor whose points lie “near” a low-dimensional curve. This form identifies a data-driven dimensionality reduction which encapsulates a Cartesian product, approximately, of a principal manifold, corresponding to spatial regularity, against a temporal complex dynamics of the intrinsic variable of the manifold. The principal manifold element serves to translate the complex information at one site to all other sites on the beam. Although points of the attractor do not lie on the principal manifold, they lie sufficiently close that we describe that manifold as a “backbone” running through the attractor, allowing the manifold to serve as a suitable space to approximate behaviors.