An Attractor Partition Described as Joined Manifolds for a Chaotic Beam System

Motivated by considering the application of a ship carrying either a crane(s), or perhaps fluids in internal tanks, here we model an elastic beam stressed under two separate loads, which we will see brings important dimensionality differences to the single load scenario we studied in [37], and modeled spatiotemporal evolution by principle curves methods. These beam models generally yields a Partial Differential Equation, (PDE), but under the complex load dynamics the beam yields chaotic solutions. Despite the temporal complexity of the solutions of the beam dynamics at any given site on the beam, when considering solutions of the PDE, the solutions show regularity in space, where the solution at one spatial site is compared to solutions at other spatial sites. Specifically, we find that in this spatially displaced coordinate representation, that there is an attractor like structure that displays a strong degree of simplicity in the form of an apparent two-dimensional structure which well describes the data. Interestingly this two dimensional structure appears as a union of two-dimensional manifolds, which we call a “joined” manifold, and we exploit this structure to partition the set into manifold segments, and a locus of joins. This partitioning allows us to uniquely identify states of the system as coordinates on an approximating manifold, for prediction of states at other sites in terms of observations on this unusual but useful low-dimensional representation.