Towards the Full Lyapunov Spectrum of Elementary Cellular Automata

Abstract

Throughout the decades following the postulation of cellular automata (CA) halfway the previous century, numerous studies have been conducted to gain insight into the dynamical properties of these uttermost discrete dynamical systems. Mostly, these studies were motivated by the fact that CA turned out capable of evolving intriguing spatio‐temporal dynamics notwithstanding their intrinsically simple nature. Though many measures have been proposed for gaining insight into CA dynamics, the use of Lyapunov exponents that measure how their phase space trajectories evolve with respect to each other has proved particularly fruitful. Yet, to this day, in all studies relying on this measure the conclusions are drawn upon the so‐called maximum Lyapunov exponent (MLE), whereas the determination of the full Lyapunov spectrum has been neglected despite its demonstrated usefulness in elucidating the dynamics of continuous dynamical systems and maps. In this preliminary study, we outline how the Lyapunov spectrum of so‐called elementary CA can be obtained, we show the validity of the proposed computational methodology and, finally, we present the full spectra of some renowned elementary CA.