between class 2 and class 3 in terms of what one might think of as overall activity.
The point is that class 1 and 2 systems rapidly settle down to states in which there is essentially no further activity. But class 3 systems continue to have many cells that change at every step, so that they in a sense maintain a high level of activity forever. Class 4 systems are then in the middle: for the activity that they show neither dies out completely, as in class 2, nor remains at the high level seen in class 3.
And indeed when one looks at a particular class 4 system, it often seems to waver between class 2 and class 3 behavior, never firmly settling on either of them.
In some respects it is not surprising that among all possible cellular automata one can identify some that are effectively on the boundary between class 2 and class 3. But what is remarkable about actual class 4 systems that one finds in practice is that they have definite characteristics of their own—most notably the presence of localized structures—that seem to have no direct relation to being somehow on the boundary between class 2 and class 3.
And it turns out that class 4 systems with the same general characteristics are seen for example not only in ordinary cellular automata but also in such systems as continuous cellular automata.
For different values of this parameter, the behavior one sees is different. But it seems that this behavior falls into essentially the same four classes that we have already seen in ordinary cellular automata. And indeed there are even quite direct analogs of for example the triangle structures that we saw in ordinary class 3 cellular automata.
But since continuous cellular automata have underlying rules based on a continuous parameter, one can ask what happens if one smoothly varies this parameter—and in particular one can ask what sequence of classes of behavior one ends up seeing.
The answer is that there are normally some stretches of class 1 or 2 behavior, and some stretches of class 3 behavior. But at the transitions