Figure 1

Spatiotemporal evolution of one-dimensional cellular automata rules representative of each of the Wolfram classes of increasing complexity. Wolfram's rules 254 (class 1), 232 (class 2), 30 (class 3), and 110 (class 4). Time flows from top to bottom.Reuse & Permissions

Figure 2

Spatiotemporal evolutions of (a) the rule Eq. (3) ($p=2$); (b) Wolfram's rule 110, Eq. (5), derived from the rule in (a) through a process of symmetry breaking illustrated in the inset (see main text). Time flows from top to bottom and the spatial dimension $i$ increases to the left in each panel.Reuse & Permissions

Figure 3

Spatiotemporal evolutions of the 16 rules with $p=2$, $r=1$, $l=0$ (range $\rho =2$). These rules constitute the 16 operators of Boolean logic [25]. Rule labeled 6 (and its class equivalent under global complementation, 9) are the only class 3 rules among the 16 rules, being as well the only Pascal rules in the set. All other rules are class 1 or 2. Time flows from top to bottom. A ring of 20 sites with periodic boundary conditions is considered.Reuse & Permissions

Figure 4

Spatiotemporal evolutions of (a) the Pascal rule given by Eq. (6), symmetric under addition modulo 2 and (b) the symmetry-breaking rule in Eq. (7). The detail shows the only configuration 01010 that is tuned compared to the symmetric Pascal rule (when the additional degree of freedom is added) in order to break the symmetry. Time flows from top to bottom. $i$ increases to the left. Shown is a window where $t\in [0,150]$ and $i\in [1,150]$.Reuse & Permissions

Figure 5

(a) Spatiotemporal evolution of Eq. (18) with $p=7$, $l=r=2$ ($\rho =5$), and $C=16$ for an initial condition consisting of 38 sites with value 1 surrounded by 0's on a ring of $N_s=800$ sites and for a time window $t\in [0,4000]$ time steps, with time flowing from top to bottom [note that the homogeneous background black and white regions contain actually 2000 oscillations between black and white that cannot be displayed in (a)]. (b) Detail of the stable incoherent structure. Snapshot (detail) of the spatial phase distribution for (c) $t=2000$ and (d) $t=2001$ time steps. (e) Local time series for an oscillator in the incoherent region corresponding to the cut in (b) where a duration spanning the recurrence time $T=1408$ of the full incoherent structure is indicated. (f), (g) Spatiotemporal evolutions (detail) for $C=30$ and $C=6$, respectively [other parameter values as in (a)].Reuse & Permissions