The concepts and tools of algebraic topology can be applied to the evolution of systems in both phase space and physical space, as well as to the interesting back-and-forth excursions between these two spaces. The way that dynamics and topology interact is at the core of the present course.
Starting with the early contributions of knot theory to nonlinear dynamics, we introduce the templex, a novel concept in algebraic topology that considers a flow in physical or phase space with no restrictions to its dimensions, drawing on both homology groups and graph theory. The templex approach is illustrated through its application to paradigmatic chaotic attractors – like the Lorenz or Rössler attractors – as well as to non-chaotic flows. Applications to kinematic and dynamic models of the ocean gyres and to idealized models of the Atlantic Meridional Overturning Circulation (AMOC) are presented, along with the topological analysis of oceanographic time series derived from altimetric velocity fields. Lagrangian ocean analysis is a key element of the course.
The extension of the templex concept to the noise-perturbed chaotic attractors of random dynamical systems theory is presented, leading to the definition of topological tipping points (TTPs). TTPs enable the study of successive bifurcations of climate models beyond those known from the classical theory of autonomous dynamical systems, as well as of those more recently added by consideration of tipping points in nonautonomous systems.
We thus propose to start a journey through the mathematical concepts and tools that characterize the topological approach to nonlinear dynamics. This approach goes beyond purely metric, i.e., non-topological, descriptions of the mechanisms that are responsible for higher and higher versions of irregular behavior, from deterministic chaos to various forms of turbulence. These novel tools provide challenging and promising inroads for understanding the effects of anthropogenic forcing on the climate system’s intrinsic variability.