mathematics//dynamical systems//superexponential growth
Superexponential growth is growth whose rate itself increases over time, faster than any fixed exponential, and it is the regime between ordinary exponential growth and a finite-time singularity. An exponential grows by the same factor in every interval; a superexponential grows by a larger factor in each interval than in the one before.
Superexponential growth is growth whose rate itself increases over time, faster than any fixed exponential, and it is the regime between ordinary exponential growth and a finite-time singularity. An exponential grows by the same factor in every interval; a superexponential grows by a larger factor in each interval than in the one before.
A simple example shows that it can stay finite. If the monthly growth rate rises steadily (10%, then 20%, then 30%, so g(t)=0.1 tg(t) = 0.1,tg(t)=0.1t per month), the level is
x(t)=x0 e0.05 t2x(t) = x_0\, e^{0.05\,t^2}x(t)=x0e0.05t2
which is brutal (about e20e^{20}e20, half a billion times larger, after 20 months) and still finite at every date.
Seeing growth rates rise is evidence of superexponential growth, never of a singularity. The two only separate when the rate diverges, which a few years of data cannot show.
In a loop that improves itself in rounds, gains per round that grow faster than proportionally give superexponential and even doubly exponential growth without any singularity.