  Through numerical simulations of the Kuramoto equation, which displays
high-dimensional dissipative chaos, we find a quantity representing the cost
for maintenance of a spatially non-uniform structure that appears in the phase
turbulence of chemical oscillatory waves. We call this quantity the generalized
entropy production and demonstrate that its distribution function possesses a
symmetry expressed by a fluctuation theorem. We also report a numerical result
which suggests a relation between this generalized entropy production rate and
the Kolmogorov-Sinai entropy.
