Séminaire
Poincaré first described the way in which a dynamical system’s properties depend upon its topology. In this work, we study the temporal evolution of the topological structure of the branched manifold associated with the Lorenz model’s random attractor (LORA), when driven by multiplicative noise. While the attractor associated with the classical, deterministic Lorenz (1963) model is “strange” but fixed in time, LORA is a pullback attractor that changes in time in a rather spectacular fashion.
LORA’s topological structure can be studied by using homology groups Hn. These groups are a key tool of algebraic topology that characterizes the number of n-dimensional holes of a branched manifold: the connected components (0- dimensional holes), the cycles (1-dimensional holes), the cavities (2- dimensional holes), and so on. Our work shows that LORA is a 2-dimensional branched manifold whose number of 1-dimensional holes changes over time, i.e., its homology group H1 is distinct from the fixed one of Lorenz’s “butterfly” and cycles are created or destroyed by the noise.
helene.rouby@ens.fr