Indecomposable continua in dynamical systems with noise: Fluid flow past an array of cylinders

Abstract

Standard dynamical systems theory is based on the study of invariant sets. However, when noise is added, there are no bounded invariant sets. Our goal is then to study the fractal structure that exists even with noise. The problem we investigate is fluid flow past an array of cylinders. We study a parameter range for which there is a periodic oscillation of the fluid, represented by vortices being shed past each cylinder. Since the motion is periodic in time, we can study a time-1 Poincaré map. Then we add a small amount of noise, so that on each iteration the Poincaré map is perturbed smoothly, but differently for each time cycle. Fix an x coordinate x0 and an initial time t0. We discuss when the set of initial points at a time t0 whose trajectory (x(t),y(t)) is semibounded (i.e., x(t)>x0 f°r all time) has a fractal structure called an indecomposable continuum. We believe that the indecomposable continuum will become a fundamental object in the study of dynamical systems with noise. © 1997 American Institute of Physics.