  Topological entropy measures the number of distinguishable orbits in a
dynamical system, thereby quantifying the complexity of chaotic dynamics. One
approach to computing topological entropy in a two-dimensional space is to
analyze the collective motion of an ensemble of system trajectories taking into
account how trajectories "braid" around one another. In this spirit, we
introduce the Ensemble-based Topological Entropy Calculation, or E-tec, a
method to derive a lower-bound on topological entropy of two-dimensional
systems by considering the evolution of a "rubber band" (piece-wise linear
curve) wrapped around the data points and evolving with their trajectories. The
topological entropy is bounded below by the exponential growth rate of this
band. We use tools from computational geometry to track the evolution of the
rubber band as data points strike and deform it. Because we maintain
information about the configuration of trajectories with respect to one
another, updating the band configuration is performed locally, which allows
E-tec to be more computationally efficient than some competing methods. In this
work, we validate and illustrate many features of E-tec on a chaotic lid-driven
cavity flow. In particular, we demonstrate convergence of E-tec's approximation
with respect to both the number of trajectories (ensemble size) and the
duration of trajectories in time.
