Noise-driven Topological Changes in Chaotic Dynamics

TL;DR

This paper investigates how multiplicative noise influences the topological structure of chaotic Lorenz systems over time, revealing sharp transitions and topological tipping points using algebraic topology and Branched Manifold Analysis.

Contribution

It extends Branched Manifold Analysis through Homologies (BraMAH) to analyze the evolving topological structure of noise-driven chaotic systems.

Findings

LORA exhibits sharp topological transitions over time.

Noise induces dynamic topological changes in chaotic attractors.

Topological tipping points correspond to significant system transitions.

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 (BraMAH) -- 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.