  We use persistent homology to build a quantitative understanding of large
complex systems that are driven far-from-equilibrium; in particular, we analyze
image time series of flow field patterns from numerical simulations of two
important problems in fluid dynamics: Kolmogorov flow and Rayleigh-B\'enard
convection. For each image we compute a persistence diagram to yield a reduced
description of the flow field; by applying different metrics to the space of
persistence diagrams, we relate characteristic features in persistence diagrams
to the geometry of the corresponding flow patterns. We also examine the
dynamics of the flow patterns by a second application of persistent homology to
the time series of persistence diagrams. We demonstrate that persistent
homology provides an effective method both for quotienting out symmetries in
families of solutions and for identifying multiscale recurrent dynamics. Our
approach is quite general and it is anticipated to be applicable to a broad
range of open problems exhibiting complex spatio-temporal behavior.
