Minicourse “Dynamical Systems, Algebraic Topology, and Climate”

By: Michael Ghil, ENS Paris, UCLA, & Imperial College London
Denisse Sciamarella, IFAECI (CNRS) & University of Buenos Aires

• Dates: Tues. 13 Feb.  14:00-16:00, Thurs. 15 Feb. 14:00-16:00, and Tues. 20 Feb  14:00-16:00.
• Platform: Online via Zoom
• Registration: The Zoom link will be shared with interested participants.
• Please let Stefano Galatolo know your interest in participating.

Abstract:

The definition of climate itself cannot be given without a proper understanding of the key ideas
of long-term behavior of a system, as provided by dynamical systems theory. Concepts and
methods of this theory have been applied to the climate sciences as early as the 1960s. The major
increase in public awareness of the socio-economic threats and opportunities of climate change
has led more recently to an increased understanding of the interplay between natural climate
variability and anthropogenically driven climate change.
In this minicourse, we shall introduce first the simplest concepts and methods of differentiable
dynamical systems (DDS) theory and of multi-stability and apply them to energy balance models
of Earth’s radiation budget and to their saddle-node bifurcations. Next, we shall introduce time dependent forcing, both deterministic and stochastic, and show how it can modify substantially
climate’s intrinsic variability. This presentation will include basic concepts of the theory of
nonautonomous and random dynamical systems (NDS and RDS), as well as an application to a
randomly perturbed version of the classical Lorenz convection model.
Finally, we shall introduce concepts from the field of chaos topology. Topological properties
provide detailed information about the fundamental mechanisms that act to shape a dynamical
system’s flow in state space. Topological invariants hence allow us to determine whether two
dynamics are equivalent, or whether a particular model is an adequate representation of the
dynamics underlying an observational or numerically simulated time series. These properties can
be encoded by Branched Manifold Analysis through Homologies (BraMAH) complex. Homology
and torsion groups can then be computed to describe the topology of DDS, NDS and RDS, and of
their relevance to climate dynamics.
References

1. Ghil, M., and Sciamarella, D., 2023: Review article: Dynamical systems, algebraic topology, and the
   climate sciences, Nonlin. Processes Geophys., 30(4), 399–434, https://doi.org/10.5194/npg-30-399-
   2023, doi:10.5194/npg-30-399-2023.
2. Ghil, M., 2021a: Mathematical Methods in the Climate Sciences, III: Energy balance models,
   paleoclimate & “tipping points,” Lecture Notes, v1.3.1, eprint doi:10.5281/zenodo.4765733.
3. Ghil, M., 2021c: Mathematical Methods in the Climate Sciences, IV: Nonlinear & stochastic models—
   Random dynamical systems, Lecture Notes, v1.3.2, eprint doi:10.5281/zenodo.4765766.

Lecure 1, video:

Lecture 1, slides:

Lecture 3, video:

Lecture 3, slides: