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Identifying topological chaos using set-oriented methods

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Topological Dynamics in the Physical and Biological Sciences

Identifying topological chaos using the Thurston-Nielsen classification theorem (TNCT) is a powerful approach to quantifying and predicting chaos in a variety of fluid systems. This approach is most easily applied to systems stirred by physical rods, since the rods can be prescribed to move on sufficiently complex space-time trajectories. In many cases, however, an analysis based solely on the motion of physical rods cannot capture the full complexity of the flow. Consideration of ‘ghost rods’, or material particles that ‘stir’ the fluid, can provide the missing information needed for an accurate topological representation. Unfortunately, even when such low-order periodic orbits exist, they can be difficult to identify. We will discuss the use of set-oriented, or mapping-based, statistical methods for identifying periodic regions in the domain having high local residence time. These ‘almost-cyclic sets’ can reveal the underlying topology of the s ystem, enabling application of the TNCT even in the absence of low-order periodic orbits. Viscous flow examples show that this approach can provide a good representation of system behavior over a range of parameters.

This talk is part of the Isaac Newton Institute Seminar Series series.

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