  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.
