The Bifurcation Theory — The Logistic Curve

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 the brain?

It’s this one single equation.

It’s the simplest equation you can make that has a negative feedback loop. 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 and then times (1–0.4) 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 “Ans” times (1 — “Ans”), get 0.61. So the population dropped a little. Hit it again, 0.619, 0.613, 0.617, 0.615, 0.616, 0.615, and 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.

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. So 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 = 1. So, we’ve got ⁰² + 1 = 1.

Then ¹² + 1 = 2

²² + 1 = 5, ⁵² + 1 = 26.

So pretty quickly you can see that, with C = 1, this equation is going to blow up So the number 1 is not part of the Mandelbrot set. What if we try C = -1? Well, then we’ve got ⁰² — 1 = -1, (-1)² — 1 = 0, and so we’re back to ⁰² — 1 = -1. So we see that this function is going to keep oscillating back and forth between -1 and 0, and so it will remain finite, and so C = -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 H these numbers stay finite.

So 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 well 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 4, and then 8 and then 16, 32 and so on, and 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, and 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 3. 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, and so 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?” And the answer is: yes! Particularly in controlled environments, 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 Libchaber. 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. And 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 and 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 2. And as he continued to increase the temperature, he saw the period-doubling again. Now he had 4 different temperatures before the cycle repeated. And then 8.

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 and 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 with a periodic beat, and then it went into a 2-cycle (2 beats close together), and then a 4-cycle (4 different beats before it repeated), 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 the 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: “da-dip”, “da-dip”, “da-dip”, And 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.

Now I wanna 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 the ratio closed in on this number four point six six nine which is now called the Feigenbaum constant.

The bifurcations come faster and faster, but in a ratio that approaches this fixed value And 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 so you could use x_(n + 1) = sin(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’s sparked a revolution and 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.

The Figures and graphs above are results of a python program, which can be downloaded from the following link. https://github.com/ssinha2103/Chaos

You can also visit this youtube video for reference:-

https://www.youtube.com/watch?v=ovJcsL7vyrk&t=91s

That's all.

Thank you for being patient and going through all these.

If you wish to contact, you can find me here — https://www.linkedin.com/in/ssinha2103/