Scaling and interleaving of subsystem Lyapunov exponents for spatio-temporal systems

The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the spectrum of a subsystem by a suitable rescaling in a very cost effective way. We compute the Lyapunov spectrum for the subsystem by truncating the original Jacobian without modifying the original dynamics and thus taking into account only a portion of the information of the entire system. In doing so we notice that the Lyapunov spectra for consecutive subsystem sizes are interleaved and we discuss the possible ways in which this may arise. We also present a new rescaling method, which gives a significantly better fit to the original Lyapunov spectrum. We evaluate the performance of our rescaling method by comparing it to the conventional rescaling (dividing by the relative subsystem volume) for one- and two-dimensional lattices in spatio-temporal chaotic regimes. Finally, we use the new rescaling to approximate quantities derived from the Lyapunov spectrum (largest Lyapunov exponent, Lyapunov dimension, and Kolmogorov–Sinai entropy), finding better convergence as the subsystem size is increased than with conventional rescaling.