There is much lamentation these days about the extremes in politics, in our culture and in our individual choices and preferences. We also talk about extremes in our weather, possible climate changes and the general mathematics of fluid flow (atmosphere or ocean e.g.).
I have studied fluid dynamic turbulence, defined as random, unpredictable flow patterns, by considering an equilibrium flow that is subject to a very small perturbation. Last week I mentioned the change in equilibrium, from glacial to interglacial, that arose from the very small perturbation in the earth’s orbit.
A problem becomes difficult, even intractable, as the perturbation grows, and the mathematical assumptions fail. So we can tell when a system is unstable, but we can’t tell where it is going.
It is reasonable to assume that the perturbation doesn’t grow to infinity, and something happens along the way to stop it and establish another equilibrium (or head back to the former one?). In the rare cases that we can determine another equilibrium (solve the nonlinear equations) it is called Chaos Theory (although ‘Complexity’ would be more apt). The theory emerged from the study of weather predictability and has evident applications in climate theory. The single second order nonlinear solution for “Coherent Structures” embedded in a linear mean flow exists in the Planetary Boundary Layer flow solution. So we understand a lot about this theory in the atmospheric sciences.
Getting back to the cultural applications of Chaos Theory, there have been many attempts in philosophy and economics and elsewhere to obtain Chaos solutions, without much success. Still, the basic ideas have tempting analogs to talk around (Feigenbaum, Mitchell (July 1978). “Quantitative universality for a class of nonlinear transformations”, Jn of Statistical Physics 19 (1); and Chaos: Making a New Science, Viking Penguin, J. Gleick).
The basic conclusions from these studies is that we live in a “Chaotic” world, and we must become accustomed to:
1) Flexibility in understanding and expectations of the world we live in — the weather, climate, fashion, stock market, and political stabilities to name a few — is necessary for survivability. The better the understanding of the environment we have, the more likely we are to accommodate to, or perhaps just survive the changes.
2) Some predictions are good, particularly of the stability or instability of a given equilibrium.
3) Predictability of transition to a new complex equilibrium is extremely difficult. Success varies from adequate to poor. It is very difficult to anticipate every important effect, particularly the feedbacks.
4) We should do the best we can to anticipate the new state, since failure to anticipate can result in economic disasters, wars, biological evolution, and global warming to name a few.
5) Nothing lasts forever.