NP Division Campfire: Perspectives on Climate featuring Denisse Sciamarella

This week Denisse Sciamarella (CIMA-FCEN) with discuss the “Topology of Chaos”.

Packard et al. (1980) first attempted to identify the key features of a dynamical system from observational or experimental data. They used time series to reconstruct a finite-dimensional phase-space picture of the sampled system’s time evolution within an embedding space, and to characterize it geometrically. Topological properties can replace geometrical ones and have the advantage of providing information about the mechanisms that act in phase space to generate the flow. These mechanisms — stretching, squeezing, tearing, and folding — are topological in nature, and they are intimately related to the governing equations. The duality of dynamics and topology opens several doors. In the deterministic realm, it can provide clear-cut definitions of categories such as “coherence” and “regime.” Recent research shows, for instance, how to rely on topological tools to unravel coherent sets from sparse data in fluid flows. These tools can also help validate, emulate or refute models from data, as well as in comparing data sets. Enlightening surprises arise if one takes one step beyond, and extends the topological perspective to random dynamical systems, which provide the appropriate mathematical framework to tackle ocean–atmosphere coupling and climate change. Noise modifies the behavior of a random attractor: at each instant in time, though, the random attractor’s structure is still well represented by a branched manifold, defined as an integer-dimensional subset of phase space that supports the invariant sample measure. Different “stages” in the “life” of a random attractor can be identified by monitoring the abrupt changes of the branched manifold’s topology. These findings hold promise for the understanding of the climate system’s “tipping points” in unprecedented ways.

This talk will present joint work with Gisela Charó, Mickaël Chekroun and Michael Ghil.

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