A new sort of Chaos?

I've been thinking. I've also been reading this book, which is about some of the great equations of modern science. It covers a range of fields; so far it has covered quantum theory, exobiology, nuclear energy, and chaos. Or, for the scientists, E=hf, the Drake equation, E=mc2, and x(n+1)=ax(1-x). The latter two desperately need some subscripts and superscripts, but (sadly) Blogger is frustrating me on that score.

The latter equation is fascinating (Stay with me on this - I'll return to a familiar theme soon). It was developed to model animal populations. The constant 'a' represents a growth rate - roughly representing a number of offspring. So if 'x' is a number between 0 and 1 representing the fraction which the current population represents of the total possible population that can be sustained in the locality concerned, 'ax' is the number after a generation. Of course, some will die; generally, the higher the population, the more disease will spread, the more attractive the colony will be to predators, and so on. So the (1-x) factor limits the population in this way; as the population grows, (1-x) shrinks and this the next generation is reduced by this factor.

The result is weird. If a is small (less than 3), then the population grows steadily to a stable point and then stays there. For values of a between 3 and 3.57, the population oscillates between two (or more) stable values. Above 3.57, the population varies widly between generations and becomes wholly unpredictable - i.e. chaos.

This is usually shown in a graph known as the "Logistic Map", in which the eventual stable point is shown (on the y axis) varying with the chosen value of a (on the x axis):


So, in other words, at low growth rates, things are stable. Push the growth rate, and things start to oscillate, alternately growing and crashing. Push harder still, and the whole thing goes haywire.

Does this sound familiar?

When I read that, I wondered if it could be applied to economics. Go for gentle growth,and things can be stable albeit disappointing. Push harder, and the good times will be interspersed with bad times. Push harder still, and all bets are off; anything might happen.

Maybe, when our Glorious Leader And Saviour Of The World's Economies said he had put an end to boom and bust, he really had. If so, his mistake was to push the economy even harder; by taking more out of the productive parts of the economy to fund the unproductive public sector, he forces the private sector to work correspondingly harder in order to maintain its position. And the result has indeed begun to look like the right hand side of the graph, I have to say.

Guess I'm just a frustrated economist, really....