What's the connection between a dripping faucet, the Mandelbrot Set, a population of rabbits, thermal convection in a fluid, and the firing of neurons in your brain?

That's the lead question asked in the following under-19 minute video by Veritasium exploring the logistic map, in which very complex and chaotic outcomes follow from very seemingly simple math relationships.

If you'd like to play around with the logistic map for modeling the population of rabbits over time, using the equation:

X_{n+1} = rX_{n}(1 - X_{n})

We've built a simple tool to make it easier to cycle through the outputs for whatever parameter values you choose to enter. If you're reading this article on a site that republishes our RSS news feed, please click here to access a working version of this tool our site.

For added fun, we'll note that bifurcated behavior has also been observed in stock prices. A 2014 paper by David Nawrocki and Tonis Vaga describes that scenario:

We propose a bifurcation model of market returns to describe transitions between an 'over-reaction' mean regressive state and 'under-reaction' trend persistent states. Since July 1929, the Dow Jones Industrial Average has exhibited non-stationary state transition behavior, including: (1) mean regressive behavior during crisis situations during the Great Depression of the 1930s and again in the crisis of 2008 when the availability of credit was interrupted; (2) strongly bifurcated, or trend persistent behavior from the 1940s through 1975; and (3) more efficient behavior since 1975. The bifurcation dynamic evident in the pre-1975 era is somewhat enhanced by conditional volume and moderate volatility. The bifurcation model is used to develop a quantitative measure of the degree of market efficiency, which indicates that the market has become more efficient, i.e. less trend persistent, since 1975 with the advent of negotiated commissions and computerized trading techniques. Similar findings are presented for the S&P 500 index and the CRSP Value Weighted Index, which represent large capitalization markets.

There's also a 2016 paper by Marzena Kozlowska *et al.* that points to the "flickering" behavior in stock prices as an early warning signal when the market nears a "bifurcated catastrophic transition", or "tipping point".

We came across both these papers while investigating potential explanations for why long winning streaks and especially why long losing streaks have become less frequent over time. One thing led to another and suddenly we were immersed in the math of rabbit population growth and Mandelbrot sets, which is pretty interesting in and of itself.

Welcome to the world of complexity!...