- 2 ngày trước
Danh mục
🤖
Công nghệPhụ đề
00:00What's the connection between a dripping faucet, the Mandelbrot set, a population of rabbits,
00:07thermal convection in a fluid, and the firing of neurons in your brain? It's this one simple
00:14equation. This video is sponsored by Fast Hosts, who are offering UK viewers the chance to win a
00:21trip to South by Southwest if they can answer my question at the end of this video, so stay tuned
00:26for that. Let's say you want to model a population of rabbits. If you have X rabbits this year, how
00:36many rabbits will you have next year? Well the simplest model I can imagine is where we just
00:40multiply by some number, the growth rate r, which could be say 2, and this would mean the population
00:46would double every year. And the problem with that is it means the number of rabbits would grow
00:51exponentially forever. So I can add the term 1 minus X to represent the constraints of the
00:57environment. And here I'm imagining the population X is a percentage of the theoretical maximum, so it
01:04goes from 0 to 1. And as it approaches that maximum, then this term goes to 0, and that constrains the
01:11population. So this is the logistic map. Xn plus 1 is the population next year, and Xn is the population this
01:21year. And if you graph the population next year versus the population this year, you see it is just an
01:28inverted parabola. It's the simplest equation you can make that has a negative feedback loop. The bigger
01:34the population gets over here, the smaller it'll be the following year. So let's try an example.
01:39Let's say we're dealing with a particularly active group of rabbits, so r equals 2.6, and then let's pick a
01:49starting population of 40% of the maximum, so 0.4, and then times 1 minus 0.4, and we get 0.624. Okay,
02:02so the population increased in the first year. But what we're really interested in is the long-term
02:07behavior of this population. So we can put this population back into the equation. And to speed
02:12things up, you can actually type 2.6 times answer times 1 minus answer, get 0.61. So the population
02:22dropped a little. Hit it again, 0.619, 0.613, 0.617, 0.615, 0.616, 0.615. And if I keep hitting enter here,
02:35you see that the population doesn't really change. It has stabilized, which matches what we see in the
02:41wild. Populations often remain the same as long as births and deaths are balanced. Now, I want to
02:46make a graph of this iteration. You can see here that it's reached an equilibrium value of 0.615.
02:53Now, what would happen if I changed the initial population? I'm just going to move this slider
02:58here, and what you see is the first few years change, but the equilibrium population remains the
03:07same. So we can basically ignore the initial population. So what I'm really interested in
03:14is how does this equilibrium population vary depending on r, the growth rate? Well, as you can
03:21see, if I lower the growth rate, the equilibrium population decreases. That makes sense. And in fact,
03:28if r goes below 1, well, then the population drops and eventually goes extinct. So what I want to do
03:37is make another graph where on the x-axis, I have r, the growth rate, and on the y-axis, I'm plotting
03:46the equilibrium population, the population you get after many, many, many generations. Okay, for low values
03:53of r, we see the populations always go extinct. So the equilibrium value is 0. But once r hits 1,
03:59the population stabilizes onto a constant value. And the higher r is, the higher the equilibrium
04:05population. So far, so good. But now comes the weird part. Once r passes 3, the graph splits in 2.
04:16Why? What's happening? Well, no matter how many times you iterate the equation, it never settles
04:23onto a single constant value. Instead, it oscillates back and forth between two values. One year the
04:30population is higher, the next year lower, and then the cycle repeats. The cyclic nature of populations
04:36is observed in nature too. One year there might be more rabbits, and then fewer the next year,
04:41and more again the year after. As r continues to increase, the fork spreads apart, and then each one
04:48splits again. Now, instead of oscillating back and forth between two values, populations go through a
04:54four-year cycle before repeating. Since the length of the cycle or period has doubled, these are known as
05:01period doubling bifurcations. And as r increases further, there are more period doubling bifurcations.
05:07They come faster and faster, leading to cycles of 8, 16, 32, 64, and then at r equals 3.57. Chaos.
05:17The population never settles down at all. It bounces around as if at random. In fact,
05:23this equation provided one of the first methods of generating random numbers on computers.
05:28It was a way to get something unpredictable from a deterministic machine. There is no pattern here,
05:34no repeating. Of course, if you did know the exact initial conditions, you could calculate the values
05:39exactly, so they are considered only pseudo-random numbers. Now, you might expect the equation to be
05:45chaotic from here on out, but as r increases, order returns. There are these windows of stable periodic
05:53behavior amid the chaos. For example, at r equals 3.83, there is a stable cycle with a period of three
06:00years. And as r continues to increase, it splits into 6, 12, 24, and so on before returning to chaos.
06:09In fact, this one equation contains periods of every length. 37, 51, 1052, whatever you like if you
06:17just have the right value of r. Looking at this bifurcation diagram, you may notice that it looks like
06:25a fractal. The large-scale features look to be repeated on smaller and smaller scales. And sure
06:32enough, if you zoom in, you see that it is in fact a fractal. Arguably the most famous fractal is the
06:40Mandelbrot set. The plot twist here is that the bifurcation diagram is actually part of the Mandelbrot set.
06:48How does that work? Well, quick recap on the Mandelbrot set. It is based on this iterated
06:55equation. So the way it works is you pick a number c, any number in the complex plane, and then start
07:02with z equals zero, and then iterate this equation over and over again. If it blows up to infinity,
07:09well then the number c is not part of the set. But if this number remains finite after unlimited
07:15iterations, well then it is part of the Mandelbrot set. So let's try for example c equals one.
07:22So we've got zero squared plus one equals one, then one squared plus one equals two,
07:28two squared plus one equals five, five squared plus one equals 26. So pretty quickly you can see that
07:35with c equals one, this equation is going to blow up. So the number one is not part of the Mandelbrot set.
07:42What if we try c equals negative one? Well then we've got zero squared minus one equals negative one,
07:49negative one squared minus one equals zero, and so we're back to zero squared minus one equals negative
07:54one. So we see that this function is going to keep oscillating back and forth between negative one
07:59and zero, and so it'll remain finite. And so c equals negative one is part of the Mandelbrot set.
08:07Now normally when you see pictures of the Mandelbrot set, it just shows you the boundary between
08:13the numbers that cause this iterated equation to remain finite and those that cause it to blow up.
08:19But it doesn't really show you how these numbers stay finite. So what we've done here is actually
08:26iterated that equation thousands of times and then plotted on the z-axis the value that that iteration
08:34actually takes. So if we look from the side, what you'll actually see is the bifurcation diagram.
08:44It is part of this Mandelbrot set. So what's really going on here? Well, what this is showing us is that
08:54all of the numbers in the main cardioid, they end up stabilizing onto a single constant value.
09:01But the numbers in this main bulb, well, they end up oscillating back and forth between two values.
09:09And in this bulb, they end up oscillating between four values. They've got a period of four,
09:15and then eight, and then 16, 32, and so on. And then you hit the chaotic part. The chaotic part
09:21of the bifurcation diagram happens out here on what's called the needle of the Mandelbrot set,
09:27where the Mandelbrot set gets really thin. And you can see this medallion here that looks like a
09:33smaller version of the entire Mandelbrot set. Well, that corresponds to the window of stability in the
09:39bifurcation plot with a period of three. Now, the bifurcation diagram only exists on the real line
09:46because we only put real numbers into our equation. But all of these bulbs off of the main cardioid,
09:54well, they also have periodic cycles of, for example, three or four or five. And so you see these
10:03repeated ghostly images. If we look in the z-axis, effectively, they're oscillating between these
10:11values as well. Personally, I find this extraordinarily beautiful. But if you're more practically minded,
10:23you may be asking, but does this equation actually model populations of animals? And the answer is
10:30yes, particularly in the controlled environment scientists have set up in labs. But what I find
10:36even more amazing is how this one simple equation applies to a huge range of totally unrelated areas
10:44of science. The first major experimental confirmation came from a fluid dynamicist named Libchaber.
10:52He created a small rectangular box with mercury inside, and he used a small temperature gradient
11:00to induce convection. Just two counter-rotating cylinders of fluid inside his box. That's all the
11:07box was large enough for. And of course, he couldn't look in and see what the fluid was doing,
11:11so he measured the temperature using a probe in the top. And what he saw was a regular spike,
11:18a periodic spike in the temperature. That's like when the logistic equation converges on a single value.
11:25But as he increased the temperature gradient, a wobble developed on those rolling cylinders
11:30at half the original frequency. The spikes in temperature were no longer the same height.
11:36Instead, they went back and forth between two different heights. He had achieved period two.
11:43And as he continued to increase the temperature, he saw period doubling again. Now he had four
11:50different temperatures before the cycle repeated. And then eight. This was a pretty spectacular confirmation
11:57of the theory in a beautifully crafted experiment. But this was only the beginning. Scientists have
12:04studied the response of our eyes and salamander eyes to flickering lights. And what they find is a period
12:12doubling. That once the light reaches a certain rate of flickering, our eyes only respond to every other
12:19flicker. It's amazing in these papers to see the bifurcation diagram emerge, albeit a bit fuzzy,
12:26because it comes from real world data. In another study, scientists gave rabbits a drug that sent
12:33their hearts into fibrillation. I guess they felt there were too many rabbits out there. I mean,
12:37if you don't know what fibrillation is, it's where your heart beats in an incredibly irregular way and
12:42doesn't really pump any blood. So if you don't fix it, you die. But what they found was on the path to
12:47fibrillation, they found the period doubling route to chaos. The rabbit started out with a periodic beat,
12:55and then it went into a two cycle, two beats close together, and then a four cycle, four different
13:01beats before it repeated again, and eventually aperiodic behavior. Now, what was really cool
13:07about this study was they monitored the heart in real time and used chaos theory to determine when
13:13to apply electrical shocks to the heart to return it to periodicity. And they were able to do that
13:19successfully. So they used chaos to control a heart and figure out a smarter way to deliver electric
13:25shocks to set it beating normally again. That's pretty amazing. And then there is the issue of the
13:32dripping faucet. Most of us, of course, think of dripping faucets as very regular periodic objects,
13:39but a lot of research has gone into finding that once the flow rate increases a little bit,
13:45you get period doubling. So now the drips come two at a time. And eventually, from a dripping
13:52faucet, you can get chaotic behavior just by adjusting the flow rate. And you think, like,
13:57what really is a faucet? Well, there's constant pressure water and a constant size aperture,
14:04and yet what you're getting is chaotic dripping. So this is a really easy chaotic system you can
14:11experiment with at home. Go open a tap just a little bit and see if you can get a periodic dripping in
14:18your house. The bifurcation diagram pops up in so many different places that it starts to feel spooky.
14:25Now, I want to tell you something that'll make it seem even spookier. There was this physicist,
14:31Mitchell Feigenbaum, who was looking at when the bifurcations occur. He divided the width of each
14:38bifurcation section by the next one, and he found that ratio closed in on this number, 4.669,
14:47which is now called the Feigenbaum constant. The bifurcations come faster and faster, but in a ratio
14:54that approaches this fixed value. And no one knows where this constant comes from. It doesn't seem to
15:01relate to any other known physical constant. So it is itself a fundamental constant of nature.
15:09What's even crazier is that it doesn't have to be the particular form of the equation I showed you
15:16earlier. Any equation that has a single hump, if you iterate it the way that we have, so you could use
15:23xn plus 1 equals sin x, for example. If you iterate that one again and again and again, you will also see
15:30bifurcations. Not only that, but the ratio of when those bifurcations occur will have the same scaling,
15:394.669. Any single hump function iterated will give you that fundamental constant.
15:48So why is this? Well, it's referred to as universality, because there seems to be
15:56something fundamental and very universal about this process, this type of equation,
16:03and that constant value. In 1976, the biologist Robert May wrote a paper in Nature about this very
16:11equation. It sparked a revolution in people looking into this stuff. I mean, that paper's been cited
16:16thousands of times. And in the paper, he makes this plea that we should teach students about this
16:23simple equation because it gives you a new intuition for ways in which simple things, simple equations,
16:31can create very complex behaviors. And I still think that today we don't really teach this way. I mean,
16:41we teach simple equations and simple outcomes because those are the easy things to do. And
16:46those are the things that make sense. We're not going to throw chaos at students, but maybe we should,
16:53maybe we should throw at least a little bit, which is why I've been so excited about chaos. And I'm so
16:58excited about this equation because, you know, how did I get to be 37 years old without hearing of the
17:04Feigenbaum constant? Ever since I read James Gleick's book, Chaos, I have wanted to make videos on this
17:12topic. And now I'm finally getting around to it. And hopefully I'm doing this topic justice because
17:19I find it incredibly fascinating and I hope you do too.
17:26Hey, this video is supported by viewers like you on Patreon and by Fast Hosts. Fast Hosts is a UK-based
17:33web hosting company whose goal is to support UK businesses and entrepreneurs at all levels,
17:38providing effective and affordable hosting packages to suit any need. For example, they provide easy
17:44registration for a huge selection of domains with powerful management features included. Plus,
17:50they offer hosting with unlimited bandwidth and smart SSD storage. They ensure reliability and security
17:57using clustered architecture and data centers in the UK. Now, if you are also in the UK,
18:03you can win two tickets to South by Southwest, including flights and accommodation. If you can
18:07answer my techie test question, which is which research organization created the first website?
18:14If you can answer that question, then enter the competition by clicking the link in the description
18:19and you could be going to South by Southwest courtesy of Fast Hosts. Their data centers are based
18:24alongside their offices in the UK. So whether you go for a lightweight web hosting package or a fully
18:30fledged dedicated box, you can talk to their expert support teams 24-7. So I want to thank Fast Hosts
18:35for supporting Veritasium and I want to thank you for watching.
Được khuyến cáo
1:30:04
|
Sắp Tới
21:12
23:29
35:21
33:59
31:54
6:06
18:25
31:22
31:32
20:06
Hãy là người đầu tiên nhận xét