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Topological chaos, braiding and bifurcation of almost-cyclic sets
- Source :
- Chaos 22, 043135 (2012)
- Publication Year :
- 2012
-
Abstract
- 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.<br />Comment: Submitted to Chaos
- Subjects :
- Nonlinear Sciences - Chaotic Dynamics
34C28, 37N10
Subjects
Details
- Database :
- arXiv
- Journal :
- Chaos 22, 043135 (2012)
- Publication Type :
- Report
- Accession number :
- edsarx.1206.2321
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1063/1.4768666