Identifying topological tipping points in noise-driven chaotic dynamics using random templexes

Abstract

Random attractors are the time-evolving pullback attractors of stochastically perturbed, deterministically chaotic dynamical systems. These attractors have a structure that changes in time, and that has been characterized recently using BraMAH cell complexes and their homology groups (Chaos, 2021, doi:10.1063/5.0059461). A more complete description is obtained for their deterministic counterparts if the cell is endowed with a directed graph (digraph) that prescribes cell connections in terms of the flow direction. Such a topological description is given by a templex, which carries the information of the structure of the branched manifold, as well as information on the flow (Chaos, 2022, doi:10.1063/5.0092933). The present work (Chaos, 2023, arXiv:2212.14450 [nlin.CD]) introduces the stochastic version of a templex. Stochastic attractors in the pullback approach, like the LOrenz Random Attractor (LORA), include sharp transitions in their branched manifold. These sharp transitions can be suitably described using what we call here a random templex. In a random templex, there is one cell complex per snapshot of the random attractor and the cell complexes are such that changes can be followed in terms of how the generators of the homology groups, i.e., the “holes” of these complexes, evolve. The nodes of the digraph are the generators of the homology groups, and its directed edges indicate the correspondence between holes from one snapshot to the next. Topological tipping points can be identified with the creation, destruction, splitting or merging of holes, through a definition in terms of the nodes in the digraph.