Spatiotemporal tiling of the Kuramoto-Sivashinsky equation

Numerical simulations play a very important role in the study of chaotic partial differential equations due to the lack of analytic solutions. In the limit of strong chaos and or turbulence, these computations become very challenging if not completely intractible. In an attempt to circumvent these difficulties, we recast time dynamical systems as purely spatiotemporal problems in (d+1) dimensional spacetime. Specifically, the focus of this study will be on the spatiotemporal Kuramoto-Sivashinsky equation, a (1+1) dimensional system. Our main hypothesis is that spatiotemporal recurrences resultant from shadowing of invariant 2-tori are of critical import. This intuition is a spatiotemporal parody derived from the theory of cycle expansions [1]. By developing a (1+1) dimensional symbolic dynamics with invariant 2-tori as the fundamental building blocks, we hope to quantitatively characterize infinite spacetime solutions.
[1] Cvitanović, Predrag. Invariant measurement of strange sets in terms of cycles. Phys. Rev. Lett. 61. p. 2729. 1998