Noise-driven topological changes in chaotic dynamics

Abstract

Noise modifies the behavior of chaotic systems in both quantitative and qualitative ways. To study these modifications, the present work compares the topological structure of the deterministic Lorenz (1963) attractor with its stochastically perturbed version. The deterministic attractor is well known to be “strange” but it is frozen in time. When driven by multiplicative noise, the Lorenz model’s random attractor (LORA) evolves in time. Algebraic topology sheds light on the most striking effects involved in such an evolution. In order to examine the topological structure of the snapshots that approximate LORA, we use branched manifold analysis through homologies—a technique originally introduced to characterize the topological structure of deterministically chaotic flows—which is being extended herein to nonlinear noise-driven systems. The analysis is performed for a fixed realization of the driving noise at different time instants in time. The results suggest that LORA’s evolution includes sharp transitions that appear as topological tipping points.

Review Article: Dynamical Systems, Algebraic Topology, and the Climate Sciences

“…The deterministic concept of branched manifold (Williams, 1974) was extended to the stochastic framework by redefining it locally as an integer-dimensional set in phase space that robustly supports the point cloud associated with the system's invariant measure at each time instant. The numerical results show that BraMAH captures LORA's timeevolving homologies (Charó et al, 2021b), as shown here in Fig. 18.…”

“…In Sec. 3.1, we present the rather novel approach of Branched Manifold Analysis through Homologies (BraMAH) (Sciamarella and Mindlin, 2001;Charó et al, 2021b) for approximating the branched manifolds (Birman and Williams, 1983a, b) of dynamical systems by a cell complex that allows one to characterize the manifold by its homology group in phase space (Poincaré, 1895;Sciamarella and Mindlin, 1999). The detection and description of localized coherent sets (LCSs) in twodimensional flows in physical space by BraMAH-based methods is reviewed in Sec.…”

“…(2011) motivated much of the work reviewed in Sec. 3 below, starting with LORA's topological study by Charó et al (2021b).…”

“…In the stochastically perturbed Lorenz (1963a) model's random attractor, termed LORA (Chekroun et al, 2011), the stretching and folding mechanisms shape the flow in phase space yielding a time-evolving branched manifold, which must be analyzed accordingly. Nothing prevents one from applying BraMAH to successive point clouds, each of which corresponds to a single snapshot, and comparing the topological properties of these instantaneous cell complexes, as done for the first time by Charó et al (2021b).…”

“…Some of them can be found to split or merge. Such changes are associated with what we call hereafter a Topological Tipping Point (TTP) (Charó et al, 2021b(Charó et al, , 2022b. Since the Betti numbers are integers, any changes in them must be sudden.…”