Characterization of stability of dynamic particle ensemble systems using topological data analysis

Holes are ubiquitous structures in phase space, and their time evolution could indicate an instability in the dynamics of the system. However, the properties of these holes are difficult to study directly due to their theoretical complexity and lack of computational tools. This study proposes the use of persistent homology (PH), a technique from topological data analysis, as a computational tool for analyzing the properties of these phase-space holes, or more formally the H1 homology class according to PH. Initially, by using a toy data set, it is shown that the time evolution and the growth rate of a H1 class in phase space could be obtained by PH. For further validation, PH is applied to particle ensemble systems, such as the Hamiltonian flow and the two-stream instability (TSI). Both the stable case, where no H1 forms, and the unstable case, where H1 forms, were analyzed. It was shown that PH can distinguish between the stable and unstable cases purely from the phase-space time evolution plots. In unstable TSI, the PH also distinguished the transition of the H1 class from linear to non-linear growth. The growth rate, thus, obtained is in excellent agreement with the growth rate of the particle energy in the TSI system.