Karhunen-Loève Decomposition of Extensive Chaos

Abstract

We show that the number of KLD (Karhunen-Lo\`eve decomposition) modes ${D}_{\mathrm{KLD}}$ needed to capture a fraction $f$ of the total variance of an extensively chaotic state scales extensively with subsystem volume $V$. This allows a correlation length ${\ensuremath{\xi}}_{\mathrm{KLD}}$ to be defined that is easily calculated from spatially localized data. We show that ${\ensuremath{\xi}}_{\mathrm{KLD}}$ has a parametric dependence similar to that of the dimension correlation length and demonstrate that this length can be used to characterize high-dimensional inhomogeneous spatiotemporal chaos.