Isn't it true to say that a theoretical exponential becomes a practical sigmoid precisely because some property of the system has become saturated and gone nonlinear?
(Just trying to get this clear in my own head by writing it down.)
When a real world system goes non linear (like the Tacoma Narrows case) you don’t get a sigmoid but something catastrophic.
(If you graph the amplitude of the vibrations they increase and increase — and then — if it was a sigmoid they’d level out and stay at the max amplitude… but in reality they go to zero as there is no bridge left to vibrate.)
When a company is “growing exponentially” it may saturate the market and then the growth slows in a nice sigmoid function. That’s common. But if, for example, the investors insist that the company must maintain the growth at all costs… it breaks laws, gets destroyed and there’s no company left to grow. No exponential curve, no sigmoid, no signal at all.
Both the sigmoid and the total collapse are typical real world results of what a simple model would expect to be an unbounded exponential curve.
