  It has recently been speculated that statistical properties of chaos may be
captured by weighted sums over unstable invariant tori embedded in the chaotic
attractor of hyperchaotic dissipative systems; analogous to sums over periodic
orbits formalized within periodic orbit theory. Using a novel numerical method
for converging unstable invariant 2-tori in a chaotic PDE, we identify many
quasiperiodic, unstable, invariant 2-torus solutions of a modified
Kuramoto-Sivashinsky equation exhibiting hyperchaotic dynamics with two
positive Lyapunov exponents. 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
statistics for the chaotic dynamics. These results are a step towards including
higher-dimensional invariant sets in a generalized periodic orbit theory for
hyperchaotic systems including spatio-temporally chaotic PDEs.
