Topological data analysis of three dimensional orbits in a convective flow

We study the topological properties of the Lagrangian orbits of natural convection in a cube with Rayleigh numbers corresponding to steady-state motion. The Lagrangian orbits are considered clusters of points distributed in a three-dimensional space. This leads to the natural application of topological data analysis (TDA) tools that include alpha complexes, 0-, 1- and 2-persistent homologies, persistence diagrams, and the bottleneck metric. For low Rayleigh numbers, the analysis is applied to orbits that correspond to a region in a Poincaré map around an elliptic point. This leads to the conclusion that the points comprising individual Lagrangian orbits are embedded on the surfaces of nested tori. For larger Rayleigh numbers, the Poincaré map includes two elliptic points separated by a chaotic region. The points composing the orbits in the three dimensional space, form two nested tori structures separated by complex and difficult to classify orbits. The bottleneck metric applied to the 0-, 1- and 2-persistence diagrams, captures a smooth evolution of the structures’ geometrical properties, suggesting an order of the orbits present in the chaotic region. Interestingly, in the region between the two nested tori structures, an orbit with topological properties similar to a trivalent 2-stratifold was found.