Mu Xu

Analysis of Kolmogorov Flow and Rayleigh-B enard Convection using Persistent Homology

We use persistent homology to build a quantitative understanding of large complex systems that are driven far-fromequilibrium; in particular, we analyze image time series of ow eld patterns from numerical simulations of two important problems in uid dynamics: Kolmogorov ow and Rayleigh-B enard convection. For each image we compute a persistence diagram to yield a reduced description of the ow eld; by applying dierent metrics to the space of persistence diagrams, we relate characteristic features in persistence diagrams to the geometry of the corresponding ow patterns. We also examine the dynamics of the ow patterns by a second application of persistent homology to the time series of persistence diagrams. We demonstrate that persistent homology provides an eective 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.