Spatiotemporal Tiling of the Kuramoto-Sivashinsky equation

Abstract

Motivated by space-time translational invariance and exponentially unstable dynamics, `spatiotemporally chaotic' or `turbulent' flows are recast as a (D+1)-dimensional spatiotemporal theory which treats space and time equally. Time evolution is replaced by a repertoire of spatiotemporal patterns taking the form of (D+1) dimensional invariant tori (periodic orbits). Our claim is that the entirety of space-time can be described as the shadowing of a finite collection of `fundamental orbits'. We demonstrate that not only can fundamental orbits be extracted from larger orbits, they can also be used as the `building blocks' of turbulence. In the future we aim to explain all of these results by constructing a (D+1)-dimensional symbolic dynamics whose alphabet is the set of fundamental orbits, however, in order to do so we must first find all fundamental orbits. These ideas are investigated in the context of the 1+1 dimensional space-time of the Kuramoto-Sivashinsky equation using the independently developed Python package 'orbithunter'.