Bright light

The Edge of Chaos

Often there is a boundary between order and chaotic regions. In some cases the boundary is relatively simple and clearly defined.

The well-known bifurcation diagram representing the logistic map - illustrates the nature of the onset or chaos.

Bifurcation diagram

Here is the graph resulting from iterating xn+1 = a xn(1 - xn) - and plotting x200 against a.

Ordered and stable states occur on the left. Chaotic and disordered ones appear on the right - and there is a transition between them.

Norman Packard coined the idea of "adaptation to the edge of chaos" [3] in 1988 and Chris Langton authored papers on the subject in the early 1990s [1,2].

The idea has come in for some criticism [4]. This is references a paper by Melanie Mitchell [5].

The conclusion I tend to draw from Melanie's work is that Langton's "Lambda" parameter is not terribly useful.

However I think the notion that living systems approach "the edge of chaos" has at least some remaining utility.

The order-chaos spectrum can be mapped onto the quantity of order present in the environment - i.e. onto total entropy.

By acting in such a way that entropy increases, living systems would be pushed away from highly ordered states - and towards the chaotic region.

They attemt to avoid going into the chaotic regions - since that represents a zone where all structure is destroyed.

Thus they gradually take their ecosystem towards "the edge".

References

1. Computation at the edge of chaos : Phase transitions and emergent computation - Chris Langton, 1990 - Physica D, 42:12-37;

2. Life at the Edge of Chaos - Chris Langton, 1991 - Proceedings of the 2nd Conf. on Artificial Life;

3. Adaptation towards the edge of chaos - Norman Packard, 1988, In: Kelso, J. A. S.; Mandell, A. J. & Shlesinger, M. F. (eds.), Dynamic Patterns in Complex Systems, World Scientific. pp. 293-;

4. ``The Edge of Chaos''

5. Revisiting the Edge of Chaos: Evolving Cellular Automata to Perform Computations - Melanie Mitchell, Peter T. Hraber, and James P. Crutchfield, 1993.