Ensemble-based Topological Entropy Calculation in Three Dimensions

Topological entropy measures the number of distinguishable orbits in a dynamical system, thereby quantifying the complexity of chaotic dynamics. Such knowledge aids greatly in a wide variety of natural and industrial fluid systems, including the rapidly developing field of microfluidics and the large-scale dispersion of pollutants in the Earth's atmosphere and oceans. We introduce a computational geometry framework for estimating a three dimensional flow's topological entropy from the collective motion of an ensemble of system trajectories. This work is analogous to the entropy calculation extracted from the "braiding" of system trajectories in two dimensions and is a first step towards building a triangulation-based method for computing topological entropy from an ensemble of trajectory data in three dimensions and higher. In it, we consider a two-dimensional rubber sheet stretched around a collection of points in a three-dimensional flow. A 3D triangulation may be used to track point-face or edge-edge collisions and the rubber sheet may be chosen as one of the faces in the initial triangulation. As the points evolve in time, they carry the sheet along with them, stretching and folding it so that its growth reflects the flow complexity.