Patterns produced by the rule 22 cellular automaton starting from random initial conditions and from an initial condition containing a single black cell.

Is it only cellular automata with very specific underlying rules that produce such behavior?

And indeed in Examples of nested patterns created by following the two-dimensional substitution rules shown.

So what about systems like cellular automata that have definite rules for evolution?

But presumably there is at least some feedback to previous layers, yielding in effect iteration of rules like the ones used in the main text.

Thus, for example, with the rule {{1, 0}  {1, 1, -1}, {1, 1}  {2, 1, 1}, {2, 0}  {1, 0, -1}, {2, 1}  {1,0,1}} the head moves to the right whenever the initial condition consists of odd-length blocks of 1's separated by single 0's; otherwise it stays in a fixed region.

1D cellular automata In a cellular automaton with k colors and r neighbors, configurations that are left invariant after t steps of evolution according to the cellular automaton rule are exactly the ones which contain only those length 2r + 1 blocks in which the center cell is the same before and after the evolution.

may boil down to issues analogous to whether it is possible to construct a configuration that has a certain property in, say, the rule 110 cellular automaton.

For many rules, this converges rapidly to a definite value; but for some rules it will wiggle forever as more and more initial conditions are included in the average.

Michael Beeler in 1973 used a computer at MIT to investigate all 1296 possible worms with rules of the simplest type on a hexagonal grid, and he found several with fairly complex behavior. … The specific 4-state rule {s_, c_}  With[{sp = s (2c - 1)  }, {sp, 1 - c, {Re[sp], Im[sp]}}] has been called Langton's ant, and various studies of it were done in the 1990s.