Mathematical Singularity - Finite-time Singularity

A finite-time singularity occurs when one input variable is time, and an output variable increases towards infinite at a finite time. These are important in kinematics and PDEs – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically the simplest finite-time singularities are power laws for various exponents, of which the simplest is hyperbolic growth, where the exponent is (negative) 1: More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses (using t for time, reversing direction to so time increases to infinity, and shifting the singularity forward from 0 to a fixed time ).

An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of kinetic energy is lost on each bounce, the frequency of bounces becomes infinite as the ball comes to rest in a finite time. Other examples of finite-time singularities include the Painlevé paradox in various forms (for example, the tendency of a chalk to skip when dragged across a blackboard), and how the precession rate of a coin spun on a flat surface accelerates towards infinite, before abruptly stopping (asd studied using the Euler's Disk toy).

Hypothetical examples include Heinz von Foerster's facetious "Doomsday's Equation" (simplistic models yield infinite human population in finite time).