Identifying topological chaos using set-oriented methods
Mark Stremler
Virginia Tech
Abstract:
Identifying topological chaos using the Thurston-Nielsen classification theorem (TNCT) is a powerful approach to quantifying and predicting chaos in a variety of dynamical systems. When three or more periodic orbits have (2+1)-dimensional trajectories with topology of pseudo-Anosov (pA) type, the TNCT establishes a quantitative lower bound on the stretching rate in (a subset of) the domain. This approach is easily applied to fluid 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 orbits that 'stir' the domain, 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 in practice. We will discuss the use of set-ori! ented statistical methods for identifying almost-periodic regions in the domain having high local residence time. These 'almost-cyclic sets' can reveal the underlying topology of the system, 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.