Virginia TechGrover, PiyushRoss, Shane D.Stremler, Mark A.Kumar, Pankaj2013-12-042013-12-042012-12-01Grover, Piyush and Ross, Shane D. and Stremler, Mark A. and Kumar, Pankaj, “Topological chaos, braiding and bifurcation of almost-cyclic sets,” Chaos 22, 043135 (2012), DOI:http://dx.doi.org/10.1063/1.47686661054-1500http://hdl.handle.net/10919/24395In 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 et al., "Topological chaos and periodic braiding of almost-cyclic sets," Phys. Rev. Lett. 106, 114101 (2011)]. 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.application/pdfenIn CopyrightDynamical-systemsInvarient setsCoherent structures2-dimensional mapsFluid mechanicsEntropyTransportAdvectionApproximationManifoldsArticle - Refereedhttp://scitation.aip.org/content/aip/journal/chaos/22/4/10.1063/1.4768666https://doi.org/10.1063/1.4768666