Predicting chaotic statistics with unstable invariant tori

It has recently been speculated that long-time average quantities of hyperchaotic dissipative systems may be approximated by weighted sums over unstable invariant tori embedded in the attractor, analogous to equivalent sums over periodic orbits, which are inspired by the rigorous periodic orbit theory and which have shown much promise in fluid dynamics. Using a new numerical method for converging unstable invariant two-tori in a chaotic partial differential equation (PDE), and exploiting symmetry breaking of relative periodic orbits to detect those tori, we identify many quasiperiodic, unstable, invariant two-torus solutions of a modified Kuramoto–Sivashinsky equation. The set of tori covers significant parts of the chaotic attractor and weighted averages of the properties of the tori—with weights computed based on their respective stability eigenvalues—approximate average quantities for the chaotic dynamics. These results are a step toward exploiting higher-dimensional invariant sets to describe general hyperchaotic systems, including dissipative spatiotemporally chaotic PDEs.