Linear and nonlinear systems

I am reading a very interesting book: "Sync – the emerging science of spontaneous order", by Steven Strogatz, published by Penguin Books in 2004.

On page 181 of the book, I found the best description of linearity (and nonlinearity) I have ever found. I thought I might share it with you.

The mathematician Stanislaw Ulam once said that calling a problem non-linear was like going to the zoo and talking about all the interesting non-elephant animals you see there. His point was that most animals are not elephants, and most equations are not linear. Linear equations describe simple, idealized situations where causes are proportional to effects, and forces are proportional to responses. If you bend a steel girder by two millimeters instead of one, it will push back twice as hard. The word linear refers to this proportionality: If you graph the deflection of the girder versus the force applied, the relationship falls on a straight line. (Here, linear does not mean sequential, as in “linear thinking,” plodding along, one thing after another. That’s a different use of the same word.)

Linear equations are tractable because they are modular: They can be broken into pieces. Each piece can be analyzed separately and solved, and finally all the separate answers can be recombined -literally added back together- to give the right answer to the original problem. In a linear system, the whole is exactly equal to the sum of the parts.

But linearity is often an approximation to a more complicated reality. Most systems behave linearly only when they are close to equilibrium, and only when we don’t push them too hard. A civil engineer can predict how a skyscraper will sway in the wind, as long as the wind is not too strong. Electrical circuits are completely predictable -until they get fried by a power surge. When a system goes nonlinear, driven out of its normal operating range, all bets are off. The old equations no longer apply.

Still, you shouldn’t get the idea that nonlinearity is dangerous or even undesirable. In fact, life depends on nonlinearity. In any situation where the whole is not equal to the sum of the parts, where things are cooperating or competing, not just adding up their separate contributions, you can be sure that nonlinearity is present. Biology uses it everywhere. Our nervous system is built from nonlinear components. The laws of ecology (to the extent we know them) are nonlinear. Combination therapy for AIDS patients -drug cocktails- are effective precisely because the immune response and the viral population dynamics are both nonlinear; the three drugs in combination are much more potent than the sum of the three of them taken separately. And human psychology is absolutely nonlinear. If you listen to your two favorite songs at the same time, you won’t get double the pleasure.

This synergistic character of nonlinear systems is precisely what makes them so difficult to analyze. They can’t be taken apart. The whole system has to be examined all at once, as a coherent entity.