Using heteroclinic orbits to quantify topological entropy in fluid flows
Topological approaches to mixing are important tools to understand chaotic fluid flows, ranging from oceanic transport to the design of micro-mixers. Typically, topological entropy, the exponential growth rate of material lines, is used to quantify topological mixing. Computing topological entropy from the direct stretching rate is computationally expensive and sheds little light on the source of the mixing. Earlier approaches emphasized that topological entropy could be viewed as generated by the braiding of virtual, or “ghost,” rods stirring the fluid in a periodic manner. Here, we demonstrate that topological entropy can also be viewed as generated by the braiding of ghost rods following heteroclinic orbits instead. We use the machinery of homotopic lobe dynamics, which extracts symbolic dynamics from finite-length pieces of stable and unstable manifolds attached to fixed points of the fluid flow. As an example, we focus on the topological entropy of a bounded, chaotic, two-dimensional, double-vortex cavity flow. Over a certain parameter range, the topological entropy is primarily due to the braiding of a period-three orbit. However, this orbit does not explain the topological entropy for parameter values where it does not exist, nor does it explain the excess of topological entropy for the entire range of its existence. We show that braiding by heteroclinic orbits provides an accurate computation of topological entropy when the period-three orbit does not exist, and that it provides an explanation for some of the excess topological entropy when the period-three orbit does exist. Furthermore, the computation of symbolic dynamics using heteroclinic orbits has been automated and can be used to compute topological entropy for a general 2D fluid flow.