Equations are ordered, elegant mathematical constructs used to describe specific patterns. Can you imagine some formulas depict the very opposite: chaos and randomness? What's more intriguing, they also underpin several complicated theories behind natural phenomena.
A video presented on Veritasium used the following equation to describe an example of this type of miraculous mathematical paradox.
Xn+1 = rXn(1-Xn)
In this logistic model that describes how the change of an animal population, "r" denotes the growth rate, "Xn" the percentage of the maximum population at a certain year, and "Xn+1" the population of the year after. Given the population has access to a limited amount of space and natural resources, the equation uses "(1-Xn)" as a constrainer for how big or small the population can get.
Like many real-life species in nature, no matter what the size of the starting number is, the model predicts that the population fluates in the beginning but eventually reaches a stable equilibrium. If one plots the growth rate (r) against equilibrium (e), the curve would split into two paths once "r" reaches three. As "r" gets bigger, the original fork would split again and again. (This periodic doubling pattern is also know a bifurcation.) Once it reaches 3.75 and above, the value of "e" becomes random.
Why is this type of equations so important? Scientists have found the periodic doubling pattern in many places, such as water dripping from a loosely closed faucet, random patterns of spontaneous neural firing, arrhythmic hearts' response to electrical stimuli, and so much more.
Simple as these equations may look, the resulted mathematical dynamics can be immensely complicated. By taking a closer look at them and accepting chaos as a part of nature, we can have a better understanding of the nature of our world.
Source: Veritasium via Youtube