Kauffman Cellular Automata on Quasicrystal Topology

Abstract

In this paper we perform numerical simulations to study Kauffman cellular automata (KCA) on quasiperiod lattices. In particular, we investigate phase transition, magnetic entropy and propagation speed of the damage on these lattices. Both the critical threshold parameter $p_{c}$ and the critical exponents are estimated with good precision. In order to investigate the increase of statistical fluctuations and the onset of chaos in the critical region of the model, we have also defined a magnetic entropy to these systems. It is seen that the magnetic entropy behaves in a different way when one passes from the frozen regime ($p<p_{c}$) to the chaotic regime ($p>p_{c}$). For a further analysis, the robustness of the propagation of failures is checked by introducing a quenched site dilution probability $q$ on the lattices. It is seen that the damage spreading is quite sensitive when a small fraction of the lattice sites are disconnected. A finite-size scaling analysis is employed to estimate the critical exponents. From these numerical estimates, we claim that on both pure ($q=0$) and diluted ($q=0.05$) quasiperiodic lattices, the KCA model belongs to the same universality class than on square lattices. Furthermore, with the aim of comparing the dynamical behavior between periodic and quasiperiodic systems, the propagation speed of the damage is also calculated for the square lattice assuming the same conditions. It is found that on square lattices the propagation speed of the damage obeys a power law as $v\sim (p-p_{c})^{\alpha}$, whereas on quasiperiod lattices it follows a logarithmic law as $v \sim \ln(p-p_{c})^\alpha$.