This equation will change how you see the world (the logistic map)

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? It's this one simple equation. This video is sponsored by Fast Hosts, who are offering UK viewers the chance to win a trip to South by Southwest if they can answer my question at the end of this video, so stay tuned for that.

Let's say you want to model a population of rabbits. If you have X rabbits this year, how many rabbits will you have next year? Well, the simplest model I can imagine is where we just multiply by some number, the growth rate R, which could be, say, 2. This would mean the population would double every year.

The problem with that is it means the number of rabbits would grow exponentially forever. So, I can add the term 1 minus X to represent the constraints of the environment. Here, I'm imagining the population X is a percentage of the theoretical maximum, so it goes from 0 to 1. As it approaches that maximum, then this term goes to 0, and that constrains the population.

So, this is the logistic map: Xn plus 1 is the population next year, and Xn is the population this year. If you graph the population next year versus the population this year, you see it is just an inverted parabola. It's the simplest equation you can make that has a negative feedback loop. The bigger the population gets over here, the smaller it'll be the following year.

So, let's try an example. Let's say we're dealing with a particularly active group of rabbits. So, R equals 2.6, and then let's pick a starting population of 40% of the maximum, so 0.4. Then, we times 1 minus 0.4, and we get 0.624. Okay, so the population increased in the first year, but what we're really interested in is the long-term behavior of this population.

So, we can put this population back into the equation, and to speed things up you can actually type "2.6 times answer times 1 - answer" and get 0.61. So, the population dropped a little. Hit it again: 0.619, 0.613, 0.617, 0.615, 0.616, 0.615.

If I keep hitting Enter here, you see that the population doesn't really change. It has stabilized, which matches what we see in the wild: populations often remain the same as long as births and deaths are balanced.

Now, I want to make a graph of this iteration. You can see here that it's reached an equilibrium value of 0.615. Now, what would happen if I change the initial population? I'm just going to move this slider here, and what you see is the first few years change, but the equilibrium population remains the same.

So, we can basically ignore the initial population. What I'm really interested in is how does this equilibrium population vary depending on R, the growth rate? As you can see, if I lower the growth rate, the equilibrium population decreases. That makes sense. In fact, if R goes below 1, well then the population drops and eventually goes extinct.

What I want to do is make another graph where on the x-axis I have R, the growth rate, and on the y-axis I'm plotting the equilibrium population—the population you get after many, many, many generations.

Okay, for low values of R, we see the populations always go extinct, so the equilibrium value is zero. But once R hits 1, the population stabilizes onto a constant value, and the higher R is, the higher the equilibrium population. So far so good, but now comes the weird part.

Once R passes 3, the graph splits in two. Why? What's happening? Well, no matter how many times you iterate the equation, it never settles onto a single constant value. Instead, it oscillates back and forth between two values. One year the population is higher, the next year lower, and then the cycle repeats.

The cyclic nature of populations is observed in nature, too. One year there might be more rabbits and then fewer the next year, and more again the year after. As R continues to increase, the fork spreads apart, and then each one splits again. Now, instead of oscillating back and forth between two values, populations go through a four-year cycle before repeating.

Since the length of the cycle or period has doubled, these are known as period doubling bifurcations. As R increases further, there are more period doubling bifurcations that come faster and faster, leading to cycles of 8, 16, 32, 64, and then at R equals 3.57, chaos. The population never settles down at all; it bounces around as if at random.

In fact, this equation provided one of the first methods of generating random numbers on computers. It was a way to get something unpredictable from a deterministic machine. There is no pattern here, no repeating. Of course, if you did know the exact initial conditions, you could calculate the values exactly, so they are considered only pseudo-random numbers.

Now, you might expect the equation to be chaotic from here on out, but as R increases, order returns. There are these windows of stable periodic behavior amid the chaos. For example, at R equals 3.83, there is a stable cycle with a period of 3 years, and as R continues to increase, it splits into 6, 12, 24, and so on before returning to chaos.

In fact, this one equation contains periods of every length: 3, 750, 1052, whatever you like, if you just have the right value R. Looking at this bifurcation diagram, you may notice that it looks like a fractal. The large-scale features look to be repeated on smaller and smaller scales.

Sure enough, if you zoom in, you see that it is in fact a fractal. Arguably, the most famous fractal is the Mandelbrot set. The plot twist here is that the bifurcation diagram is actually part of the Mandelbrot set.

How does that work? Well, a quick recap on the Mandelbrot set: it is based on this iterated equation. The way it works is you pick a number C, any number in the complex plane, and then start with Z equals 0 and then iterate this equation over and over again.

If it blows up to infinity, well then the number C is not part of the set. But if this number remains finite after unlimited iterations, well then it is part of the Mandelbrot set. So, let's try, for example, C equals 1.

So, we've got 0 squared plus 1 equals 1, then 1 squared plus 1 equals 2, 2 squared plus 1 equals 5, 5 squared plus 1 equals 26. So, pretty quickly you can see that with C equals 1, this equation is going to blow up. So, the number 1 is not part of the Mandelbrot set.

What if we try C equals negative 1? Well then, we've got 0 squared minus 1 equals negative 1, negative 1 squared minus 1 equals 0, and so we're back to 0 squared minus 1 equals negative 1. So, we see that this function is going to keep oscillating back and forth between negative 1 and 0, and so it'll remain finite.

C equals negative 1 is part of the Mandelbrot set. Now normally when you see pictures of the Mandelbrot set, it just shows you the boundary between the numbers that cause this iterated equation to remain finite and those that cause it to blow up. But it doesn't really show you how these numbers stay finite.

What we've done here is actually iterated that equation thousands of times and then plotted on the z-axis the value that that iteration actually takes. So, if we look from the side, what you'll actually see is the bifurcation diagram—it is part of this Mandelbrot set.

So, what's really going on here? Well, what this is showing us is that all of the numbers in the main cardioid end up stabilizing onto a single constant value. But the numbers in this main bulb will, they end up oscillating back and forth between two values, and in this bulb, they end up oscillating between four values.

They've got a period of four, and then eight, and then sixteen, thirty-two, and so on. Then you hit the chaotic part. The chaotic part of the bifurcation diagram happens out here on what's called the needle of the Mandelbrot set, where the Mandelbrot set gets really thin.

You can see this medallion here that looks like a smaller version of the entire Mandelbrot set. Well, that corresponds to the window of stability in the bifurcation plot with a period of three. Now, the bifurcation diagram only exists on the real line because we only put real numbers into our equation, but all of these bulbs off of the main cardioid, well, they also have periodic cycles of, for example, 3 or 4 or 5.

You see these repeated ghostly images. If we look in the z-axis, effectively, they're oscillating between these values as well. Personally, I find this extraordinarily beautiful, but if you're more practically minded, you may be asking, "But does this equation actually model populations of animals?" The answer is yes, particularly in the controlled environment scientists have set up in labs.

What I find even more amazing is how this one simple equation applies to a huge range of totally unrelated areas of science. The first major experimental confirmation came from a fluid dynamicist named Lib Taber.

He created a small rectangular box with mercury inside, and he used a small temperature gradient to induce convection: just two counter-rotating cylinders of fluid inside his box—that's all the box was large enough for. Of course, he couldn't look in and see what the fluid was doing, so he measured the temperature using a probe in the top.

What he saw was a regular spike—a periodic spike in the temperature—that's like when the logistic equation converges on a single value. But as he increased the temperature gradient, a wobble developed on those rolling cylinders at half the original frequency.

The spikes in temperature were no longer the same height; instead, they went back and forth between two different heights. He had achieved period two, and as he continued to increase the temperature, he saw period doubling again. Now, he had four different temperatures before the cycle repeated, and then eight.

This was a pretty spectacular confirmation of the theory in a beautifully crafted experiment, but this was only the beginning. Scientists have studied the response of our eyes and salamander eyes to flickering lights. What they find is a period doubling that, once the light reaches a certain rate of flickering, our eyes only respond to every other flicker.

It's amazing in these papers to see the bifurcation diagram emerge, albeit a bit fuzzy because it comes from real-world data. In another study, scientists gave rabbits a drug that sent their hearts into fibrillation. I guess they felt there were too many rabbits out there.

I mean, if you don't know what fibrillation is, it's where your heart beats in an incredibly irregular way and doesn't really pump any blood. So, if you don't fix it, you die. But what they found was on the path to fibrillation, they found the period doubling route to chaos.

The rabbits started out with a periodic beat, and then it went into a two-cycle, two beats close together, and then a four-cycle, four different beats before it repeated again, and eventually aperiodic behavior.

Now, what was really cool about this study was they monitored the heart in real time and used chaos theory to determine when to apply electrical shocks to the heart to return it to periodicity, and they were able to do that successfully. So, they used chaos to control a heart and figure out a smarter way to deliver electric shocks to set it beating normally again. That's pretty amazing.

And then there is the issue of the dripping faucet. Most of us, of course, think of dripping faucets as very regular periodic objects, but a lot of research has gone into finding that once the flow rate increases a little bit, you get period doubling.

So now, the drips come two at a time: tip, tip, tip. Eventually, from a dripping faucet, you can get chaotic behavior just by adjusting the flow rate. And you think, like, what really is a faucet? Well, there's constant pressure water and a constant size aperture, and yet what you're getting is chaotic dripping.

So, this is a really easy chaotic system you can experiment with at home. Go open a tap just a little bit and see if you can get a periodic dripping in your house. The bifurcation diagram pops up in so many different places that it starts to feel spooky.

And I want to tell you something that'll make it seem even spookier. There was this physicist, Mitchell Feigenbaum, who was looking at when the bifurcations occur. He divided the width of each bifurcation section by the next one, and he found that ratio closed in on this number: 4.669, which is now called the Feigenbaum constant.

The bifurcations come faster and faster, but in a ratio that approaches this fixed value. No one knows where this constant comes from; it doesn't seem to relate to any other known physical constant, so it is itself a fundamental constant of nature.

What's even crazier is that it doesn't have to be the particular form of the equation I showed you earlier. Any equation that has a single hump, if you iterate it the way that we have—even if you could use Xn plus 1 equals sine X, for example—if you iterate that one again and again and again, you will also see bifurcations.

Not only that, but the ratio of when those bifurcations occur will have the same scaling: 4.669. Any single-hump function iterated will give you that fundamental constant. So why is this?

Well, it's referred to as universality because there seems to be something fundamental and very universal about this process, this type of equation, and that constant value. In 1976, the biologist Robert May wrote a paper in Nature about this very equation.

It sparked a revolution in people looking into this stuff. I mean, that paper's been cited thousands of times, and in the paper, he makes this plea that we should teach students about this simple equation because it gives you a new intuition for ways in which simple things, simple equations, can create very complex behaviors.

I still think that today we don't really teach this way. I mean, we teach simple equations and simple outcomes because those are the easy things to do, and those are the things that make sense. We're not gonna throw chaos at students, but maybe we should. Maybe we should throw at least a little bit, which is why I've been so excited about chaos, and I am so excited about this equation.

Because you know how did I get to be 37 years old without hearing of the Feigenbaum constant? Ever since I read James Gleick's book Chaos, I have wanted to make videos on this topic, and now I'm finally getting around to it. Hopefully, I'm doing this topic justice because I find it incredibly fascinating, and I hope you do too.

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