Topological study of the Lorenz convection model's random attractor (LORA)

MICHAEL GHIL; CHARÓ, GISELA D.; DENISSE SCIAMARELLA

San Francisco

Congreso; AGU Fall Meeting 2019; 2019

American Geophysical Union

This paper examines the topological structure of the snapshots that approximate the evolution in time of the global random attractor associated with the Lorenz (1963) model driven by multiplicative noise; see Chekroun et al. (Physica D, 240, 2011). and the figure below. We focus on changes in time of LORA's topology and compare it with that of the classical deterministic attractor. The methodology of Branched Manifold analysis through Homologies (BraMaH: Sciamarella & Mindlin, Phys. Rev. Lett., 82, 1999; Phys. Rev. E, 64, 2001) is extended here for the first time, to the best of our knowledge, from deterministically chaotic flows to nonlinear noise-driven systems. The algorithm starts from a cloud of points given by a large number of orbits and it builds a rough skeleton of the underlying branched manifold on which the points lie. This construction is achieved by local approximations of the manifold that use Euclidean closed sets, which depend on the topological dimension, e.g., segments or disks. The skeleton is mathematically expressed as a complex of cells, whose topology is analyzed by computing its homology groups. The analysis is performed for a fixed realization of the driving noise at different time instants. We show that the topology of LORA changes in time and that it is quite distinct from the time-independent one of the classical Lorenz (1963) strange attractor, for the same values of the parameters.