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Subject:  Catastrophe theory, instability, risk Date:  10/9/2012  2:13 PM
Author:  WendyBG Number:  405636 of 484819

Dangerous Intersection

A field known as catastrophe theory explores how slow continuous changes in the force applied to a system (like the gradually increasing load on a camel’s back) can trigger rapid discontinuous jumps in its response....

...intersections [between a line and a curve] often represent answers. Solutions. States of equilibrium. In mathematical models of economies, or ecosystems, or other kinds of dynamical systems, intersections are where variables come to rest and settle down. In economic models, for example, the equilibrium price of an item is set by the intersection of supply and demand curves. If that intersection suddenly vanishes, the price has to jump.

What’s especially worrisome is that the jump occurs without warning. An intersection, by its very nature, doesn’t fade away. It exists until it doesn’t...

....the “fold catastrophe” is the most basic scenario in catastrophe theory. It’s important because in its aftermath there are no other intersections in sight. Whatever the system is going to do next, it’s going to be something radically different. It has to leap to a different state....
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This article reminds me of the striking, memorable 2006 article by John Mauldin:

Fingers of Instability

By John Mauldin

August 25, 2006

This week we revisit some ideas on how change occurs. We are in a transition in the world economy, and it sometimes helps to think about how these transitions take place. What is the mechanism for change? Can we see it coming soon enough to avoid the problems and take advantage of them? ...

How Change Happens

By John Mauldin

August 17, 2012

Imagine, Buchanan says, dropping one grain of sand after another onto a table. A pile soon develops. Eventually, just one grain starts an avalanche. Most of the time it is a small one, but sometimes it builds on itself and it seems like one whole side of the pile slides down to the bottom.

They learned some interesting things [about nonequilibrium systems]. What is the typical size of an avalanche? After a huge number of tests with millions of grains of sand, they found that there is no typical number. "Some involved a single grain; others, ten, a hundred or a thousand. Still others were pile-wide cataclysms involving millions that brought nearly the whole mountain down. At any time, literally anything, it seemed, might be just about to occur."

The piles were indeed completely chaotic in their unpredictability.

Imagine peering down on the pile from above, and coloring it in according to its steepness. Where it is relatively flat and stable, color it green; where steep and, in avalanche terms, ‘ready to go,’ color it red.

What do you see? They found that at the outset the pile looked mostly green, but that, as the pile grew, the green became infiltrated with ever more red. With more grains, the scattering of red danger spots grew until a dense skeleton of instability ran through the pile. Here then was a clue to its peculiar behavior: a grain falling on a red spot can, by domino-like action, cause sliding at other nearby red spots.

If the red network was sparse, and all trouble spots were well isolated one from the other, then a single grain could have only limited repercussions. But when the red spots come to riddle the pile, the consequences of the next grain become fiendishly unpredictable. It might trigger only a few tumblings, or it might instead set off a cataclysmic chain reaction involving millions. The sandpile seemed to have configured itself into a hypersensitive and peculiarly unstable condition in which the next falling grain could trigger a response of any size whatsoever."

Scientists refer to this as a critical state....