1. Topological chaos, braiding and bifurcation of almost-cyclic sets
- Author
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Mark A. Stremler, Shane D. Ross, Piyush Grover, Pankaj Kumar, Biomedical Engineering and Mechanics, and Virginia Tech
- Subjects
2-dimensional maps ,Pure mathematics ,Entropy ,Complex system ,FOS: Physical sciences ,General Physics and Astronomy ,Transport ,Context (language use) ,Topological entropy ,Invarient sets ,01 natural sciences ,34C28, 37N10 ,010305 fluids & plasmas ,0103 physical sciences ,Braid ,Classification theorem ,Fluid mechanics ,Dynamical-systems ,0101 mathematics ,Manifolds ,Approximation ,Mathematical Physics ,Bifurcation ,Mathematics ,Sequence ,Applied Mathematics ,010102 general mathematics ,Statistical and Nonlinear Physics ,Nonlinear Sciences - Chaotic Dynamics ,Coherent structures ,Flow (mathematics) ,Advection ,Chaotic Dynamics (nlin.CD) - 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., Comment: Submitted to Chaos
- Published
- 2012