In certain two-dimensional time-dependent flows, the braiding of periodic
orbits provides a way to analyze chaos in the system through application of the
Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work
that introduced the application of the TNCT to braiding of almost-cyclic sets,
which are individual components of almost-invariant sets [Stremler, Ross,
Grover, Kumar, Topological chaos and periodic braiding of almost-cyclic sets.
Physical Review Letters 106 (2011), 114101]. In this context, almost-cyclic
sets are periodic regions in the flow with high local residence time that act
as stirrers or `ghost rods' around which the surrounding fluid appears to be
stretched and folded. In the present work, we discuss the bifurcation of the
almost-cyclic sets as a system parameter is varied, which results in a sequence
of topologically distinct braids. We show that, for Stokes' flow in a
lid-driven cavity, these various braids give good lower bounds on the
topological entropy over the respective parameter regimes in which they exist.
We make the case that a topological analysis based on spatiotemporal braiding
of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence we
further develop a connection between set-oriented statistical methods and
topological methods, which promises to be an important analysis tool in the
study of complex systems.Comment: Submitted to Chao
