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

Chaos 1997 Mar;7(1):125-138

Institute for Physical Science and Technology, University of Maryland at College Park, College Park, Maryland 20742and Departamento de Fisica e Instalaciones Aplicadas, E.T.S. de Arquitectura, Universidad Politecnica de Madrid, 28040 Madrid, Spain.

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 Poincare map. Then we add a small amount of noise, so that on each iteration the Poincare map is perturbed smoothly, but differently for each time cycle. Fix an x coordinate x(0) and an initial time t(0). We discuss when the set of initial points at a time t(0) whose trajectory (x(t),y(t)) is semibounded (i.e., x(t)>x(0) for 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. (c) 1997 American Institute of Physics.

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http://dx.doi.org/10.1063/1.166244DOI Listing
March 1997
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References

(Supplied by CrossRef)

Barge M. et al.
Proc. Am. Math. Soc. 1987

Bing R. H. et al.
Duke Math. J. 1951

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