Me, Myself and Math, a six-part series by Steven Strogatz, looks at us through the lens of math.
I don’t know much about camels, having ridden one only once (and that was enough). But from what I’ve been told, you can usually add another piece of straw to a camel’s burden without ill effect.
Except, of course, when it’s the last straw.
The ancient proverb about the straw that broke the camel’s back is meant as a lesson about the nature of precipitous change. It reminds us that big changes don’t necessarily require big forces. If the conditions are just right (or wrong), a tap can push a system over the brink.
In the mid-20th century, mathematicians updated this proverb by turning it into a picture, a graph of the interplay between input and output, force and response. 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.
Here we’ll take a look at the simplest picture from catastrophe theory. It’ll help you make sense of the economy we’ve been stuck in, and — bringing it back to you — the sleep you’ve been losing as a result.
The idea is that the stage is set for catastrophe whenever a line intersects a folded curve tangentially.
This kind of intersection is said to be “tangential” because the line is tangent to the curve: they touch at one point without crossing and they have the same slope there. It’s the gentlest possible contact. They barely graze each other.
You saw similar pictures in high school geometry when you studied tangent lines to a circle:
But back then your teacher never said anything about an impending catastrophe, and even now these pictures may strike you as harmless.
To see the potential for calamity within them, imagine that the line pulls away from the curve or the circle.
Poof! Goodbye, intersection.
Why does this matter? Because intersections 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.
An animation of a tangential intersection reveals its inner violence. Look at what happens as we slide a line across a circle:
Little by little the two points of intersection approach each other. At the moment of tangency they smash head on. After impact, the line no longer crosses the circle at all. It’s as if the intersections annihilated each other, like a particle and anti-particle.
This is the “fold catastrophe,” 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.
A leap like this occurs in our own bodies — specifically, in how long we tend to sleep after being awake for many hours. Imagine that you’ve just stayed up very late. Maybe you’ve even pulled an all-nighter. When you finally get to bed, will you sleep more than usual, the same, or less? Assume that you can sleep undisturbed until you wake up naturally.
It turns out that the longer you stay up, the less sleep you’ll get afterward. If you haven’t experienced this, it sounds unbelievable, but it’s been confirmed in several experiments and field studies (see the endnotes for references). Every extra hour awake will cost you about 20 or 30 minutes of sleep — assuming you go to bed before noon the next day. The leap occurs if you stay up even longer. Suppose you don’t get to bed till around 3 the following afternoon, having been awake for something like 32 straight hours. How long do you think you’ll sleep now? A 1996 study found that half the subjects woke up after only a lousy little three-hour nap, while the other half corked off for 11 hours!
Let me repeat this since it’s so strange. If you’ve been awake long enough and you fall asleep when your body thinks it’s siesta time, the amount of sleep you’ll get can be either very short or very long. At that time of day, sleep duration jumps abruptly.
To help us understand these results, consider a simple pictorial model. The sloping line represents the rise in your restedness as you accumulate sleep. When restedness reaches a threshold level shown by the wavy curve, it becomes hard to sleep any more and you wake up spontaneously.
The threshold in the model bobs down and up periodically like a wave to mimic the action of your internal circadian clock. Anyone who’s ever had jet lag or worked the night shift has felt these waves. For example, have you ever noticed that when you pull an all-nighter, you feel more and more tired but then get a second wind? That’s your internal alarm clock kicking in. This same circadian clock also affects how long you can sleep. When the internal alarm is ringing loudest, the threshold drops almost to its lowest point and cuts off sleep prematurely. (Such truncation of sleep caused by a misaligned circadian clock is a chronic frustration for police officers, nurses, nuclear power plant operators, train drivers and many other shift workers.)
The experimental results fall into place when viewed through this model. The picture correctly predicts that the later you fall asleep, the less sleep you’ll get …
… but only up to a point. At a critical phase of bedtime, sleep duration lengthens suddenly. The leap occurs at a fold catastrophe where the sloping line intersects the wave tangentially.
Animating the model makes these effects seem obvious.
In economics, a similar mechanism underlies the jumps predicted by a model of the business cycle. The idea goes back to Nicholas Kaldor, a disciple of Keynes, in the 1940s, and was recast in the framework of catastrophe theory in 1979 by Hal Varian, currently the chief economist at Google. The assumptions behind the model — and their limitations — are explained in detail here and in Varian’s elegant paper, so let me focus on the model’s tangential intersections and what they suggest about the economy.
In these diagrams of the economy, the horizontal axis shows the level of economic activity, as reflected by the national income. Values on the far right mean a booming economy, while values on the left mean a sluggish economy like the one we’ve been in for the past few years. The model says that the economy will be in equilibrium when the supply of funds available for investment matches the demand for such funds. On the graph, equilibrium occurs where the “savings line” crosses the “investment curve.”
Depending on how the line and curve are situated, one, two or three intersections can occur. The upper intersection (the green dot) represents a strong economy with high levels of national income. The lower equilibrium (solid red dot) depicts an economy stuck in the doldrums. The middle equilibrium (open red circle) turns out to be unstable and acts like a watershed; when economic conditions are near it, they drift away toward one of the other two equilibriums.
Now comes the crucial idea. The amount of investment doesn’t depend only on national income but also on how much investment has already been accumulated. At some point enough is enough. During the housing boom, for example, increases in income fueled the demand for housing investment. But as the stock of housing rose, the demand for investment dropped. In the model this drags the S-shaped investment curve down. It’s like the straw being added to the camel’s back.
As the next animation shows, that little straw — that gradual lowering of the demand for investment — can suddenly tip a strong economy into recession:
The national income level crashes from the green dot at the upper right to the red dot at the lower left. The triggering event was a tangent intersection as the curve pulled away from the line. When that upper equilibrium disappeared, the economy had no place to go but down.
Remarkably, scientists in disparate fields have uncovered this same general picture, again and again. It’s there in the thermodynamics of water heated past the boiling point; in the optical focusing that creates intense webs of light at the bottom of a swimming pool; in sociological models of mobs and mass movements like the sudden revolts that became the Arab Spring; and in ecological models for the collapse of a forest from an outbreak of insects.
In some of these cases (boiling water, optical patterns), the picture from catastrophe theory agrees rigorously with observations. But when applied to economics, sleep, ecology or sociology, it’s more like the camel story — a stylized scenario that shouldn’t be taken for more than it is: a speculation, a hint of something deeper, a glimpse into the darkness.
1. The expression “the straw that broke the camel’s back” may have originated in an Arabic proverb. Charles Dickens refers specifically to the “last straw” in his novel “Dombey and Son” (1848): “As the last straw breaks the laden camel’s back, this piece of underground information crushed the sinking spirits of Mr. Dombey.”
2. For a readable and wide-ranging introduction to catastrophe theory, I’d recommend Tim Poston and Ian Stewart, “Catastrophe Theory and Its Applications” (Dover Publications, 2012). The pioneers of the theory on the pure mathematical side were René Thom and Vladimir Arnold, while the more applied aspects of the subject were enthusiastically developed and popularized by E.C. Zeeman.
3. Contrary to its ordinary meaning, a “catastrophe” in the mathematical sense is value-neutral. A discontinuous jump can be good or bad, depending on context.
4. Catastrophe theory has been controversial ever since its emergence as a pop sensation in the 1970s. Zeeman’s article “Catastrophe Theory,” Scientific American, Vol. 234 (April 1976), pp. 65–70, 75–83, showcased the theory’s exciting promise but was seen by many as overreaching. Predictably, the backlash it provoked also went too far, as exemplified by the screed of R.S. Zahler and H.J. Sussman, “Claims and Accomplishments of Applied Catastrophe Theory,” Nature, Vol. 269 (1977), pp. 759-763. For a more measured assessment, see J. Guckenheimer, “The Catastrophe Controversy,” The Mathematical Intelligencer, Vol. 1, No. 1 (1978), pp. 15–20.
5. For experimental evidence that sleep duration shortens as bedtime is delayed, but then jumps up at a critical circadian phase in the following afternoon, see T. Akerstedt and S. Folkard, “Predicting Duration of Sleep From the Three Process Model of Regulation of Alertness,” Occupational and Environmental Medicine, Vol. 53 (1996), pp. 136–141.
A similar shortening of sleep with later bedtime was reported in a field study of train drivers working irregular schedules; see J. Foret and G. Lantin, “The Sleep of Train Drivers: An Example of the Effects of Irregular Work Schedules on Sleep,” in “Aspects of Human Efficiency,” edited by W.P. Colquhoun (English University Press, 1972), pp. 272–282.
This counterintuitive trend in sleep duration was also observed in “time-isolation” studies, in which brave subjects volunteered to live alone for months in underground caves or soundproofed, windowless apartments. For the story of these remarkable experiments — and the surprises they revealed about human sleep and circadian rhythms — see Chapter 3 of S. Strogatz, “Sync” (Hyperion, 2003). The experimental data are analyzed in S.H. Strogatz, R.E. Kronauer and C.A. Czeisler, “Circadian Regulation Dominates Homeostatic Control of Sleep Length and Prior Wake Length in Humans,” Sleep, Vol. 9, No. 2 (June 1986), 353–364.
6. Art Winfree proposed the deliberately simplified model of sleep duration discussed above, in “Impact of a Circadian Clock on the Timing of Human Sleep,” American Journal of Physiology, Vol. 245, No. 4 (Oct. 1983), pp. R497–R504. He linked the model to catastrophe theory in A.T. Winfree, “Exploratory Data Analysis: Published Records of Uncued Human Sleep-Wake Cycles,” in “Mathematical Models of the Circadian Sleep-Wake Cycle,” edited by M.C. Moore-Ede and C.A. Czeisler (Raven Press, 1984), pp. 187–199. The Akerstedt and Folkard paper cited above in Note 5 presents a more refined model that matches experimental data quantitatively.
7. Hal Varian explored the application of catastrophe theory to Kaldor’s model of recessions and depressions in H.R. Varian, “Catastrophe Theory and the Business Cycle,” Economic Inquiry, Vol. 17, Issue 1 (Jan. 1979), pp. 14–28. The paper is careful and intellectually honest, and clarifies which parts of the model are speculative. Kaldor’s original paper is N. Kaldor, “A Model of the Trade Cycle,” The Economic Journal, Vol. 50, No. 197 (Mar., 1940), pp. 78–92.
I have glossed over the economic reasoning behind the model. For readers who are wondering, for example, why the investment demand curve should flatten out at both low and high levels of economic activity, Varian quotes Kaldor: the marginal propensity to invest
“will be small for low levels of activity because when there is a great deal of surplus capacity, an increase in activity will not induce entrepreneurs to undertake additional construction: the rise in profits will not stimulate investment.… But it will also be small for unusually high levels of activity because rising costs of construction, increasing costs and increasing difficulty of borrowing will dissuade entrepreneurs from expanding still faster — at a time when they already have large commitments.” p. 81, Kaldor
For further information, see “Kaldor’s Non-Linear Cycle.” And for a comprehensive survey of how catastrophe theory has been applied in other parts of economics and social science, see J.B. Rosser, “From Catastrophe to Chaos”, 2nd edition (Kluwer Academic Publishers, 2000).
8. The connections between catastrophe theory and phase transitions are discussed in Chapter 14 of Poston and Stewart’s book, mentioned above in Note 2.
9. The webs of light at the bottom of a swimming pool have been studied by the physicist Michael Berry and his colleagues in their work on “catastrophe optics.” See, for example, M.V. Berry and J.F. Nye, “Fine Structure in Caustic Junctions,” Nature, Vol. 267, No. 5606 (1977), pp. 34-36, and the papers and posters available at Berry’s Web page.
10. The sociologist Mark Granovetter didn’t explicitly refer to catastrophe theory in his classic paper on riots and mobs, “Threshold Models of Collective Behavior,” American Journal of Sociology, Vol. 83, No. 6 (1978), pp. 1420–1443. But tangent intersections underlie the discontinuities in crowd behavior that the model predicts.
11. For a lucid application of catastrophe theory to an important problem in ecology and forest management, see D. Ludwig, D.D. Jones and C.S. Holling, “Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and Forest,” Journal of Animal Ecology, Vol. 47, No. 1 (Feb. 1978), pp. 315–332.
Thanks to Diarmuid Cahalane for creating the animations; Margaret Nelson for preparing the illustrations; and Bob Frank, Paul Ginsparg, Bobby Kleinberg, Tim Novikoff, Barkley Rosser, Andy Ruina, Carole Schiffman and Hal Varian for their comments and suggestions.