| Sensitivity to initial conditions
is part of the definition of
chaos, described in our study of
one-dimensional dynamics. |
| This definition can be applied directly to cellular automata. |
| For a given rule, generate two
patterns, the second from changing some small number of initial states of the first. |
| If the difference between the two patterns grows, on average, with each generation, we say the CA
exhibits sensitivity to initial conditions. |
| For example, consider the rule of our first class III example. |
| On the left we see the (red) pattern evolving from a random initial distribution
of live cells. |
| On the right is the (blue) pattern of the differences between the red pattern and
the one that evolves when a single initial value is changed. |
| That is, a cell is painted blue if the two patterns differ at that location. |
| The growing blue pattern shows the effect of this small change
appears to grow without bound, and so we say this CA exhibits sensitivity to initial conditions. |
|
| For comparison, here is the same experiment, but done with a periodic CA. |
| Note the blue difference plot grows initially, but eventually reaches bounds beyond which it
does not propagate. |
| The number of generations needed to reach these bounds is about the same
as the number of generations before the red plot settles down into its periodic behavior. |
| That is, about the same as the duration of the initial chaotic transient. |
|
