The Mandel brought set, ah, population of rabbits, thermal conviction in a fluid and the firing of neurons in your brain.

曼德柏集合

It's this one simple equation.

兔子種群、熱流體力學

This video is sponsored by fast hosts who are offering UK viewers the chance to win a trip to south by Southwest.

和腦神經活動在幾件事情中有何關聯？

If they can answer my question at the end of this video, so stay tuned for that.

是這個簡單的公式

Let's say you want to model a population of rabbits.

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If you have X rabbits this year, how many rabbits will you have next year?

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Well, the simplest model I can imagine where we just multiply by some number.

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The growth rate are, which could be say, too, and this would mean the population would double every year.

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And the problem with that is it means the number of rabbits would grow exponentially forever, so I can add the term one minus X to represent the constraints of the environment.

如果你想為兔子種群做一個模型

And here I am imagining the Population X is a percentage of the theoretical maximum, so it goes from 0 to 1, and as it approaches that maximum, then this term goes to zero, and that constrains the population.

如果你今年有X隻兔子

So this is the logistic map.

你明年會有幾隻兔子？

X n plus one is the population next year and X and is the population this year.

最簡單的答案就是乘以一個數字r

And if you graph the population next year versus the population this year, you see it is just an inverted parabola.

r代表成長率，可能設定2

It's the simplest equation you can make that has a negative feedback loop.

這代表種群數目每年加倍成長

The bigger the population gets over here, the smaller it'll be the following year.

不過這代表成長數目

So let's try an example.

會以指數型態永遠成長下去

Let's say we're dealing with a particularly active group of rabbits, so our equals 2.6.

所以我可以提供(1-X)

And then let's pick a starting population of 40% of the maximums of 400.4.

代表環境限制

Then times one minus 10.4, and we get 0.624 Okay, so the population increased in the first year.

這裡X可以代表

But what we're really interested in is the long term behavior of this population.

與理論最大值得所占比率，比率以0到1表示

So we can put this population back into the equation, and to speed things up, you can actually type 2.6 times.

如果X達到最大比率

Answer times one minus answer.

1-X會等於0

Get 0.61 The population dropped a little hit it again 0.619 point 613.617 point 615.616 point 615 And if I keep hitting, enter here, You see that the population doesn't really change.

這樣就可以限制種群數目，所以這就是

It has stabilized, which matches what we see in the wild.

單峰映象 (Logistic map)

Populations often remain the same as long as births and deaths are balanced.

Xn+1是明年的數目

Now I want to make a graph of this generation.

Xn是今年的數目，而如果你畫個圖

You can see here that it's reached an equilibrium value of 0.615 Now, what would happen if I change the initial population?

對比今年與去年

I'm just gonna move this slider here in what you see is the first few years change.

你會看到一個倒拋物線

But the equilibrium population remains the same, so we can basically ignore the initial population.

這個公式會讓你得到

So what I'm really interested in is how does this equilibrium population?

負反饋的圖形

Very depending on our the growth rate.

超過上限之後，數目就會逐漸下降

As you can see, if I lower the growth rate, the equilibrium population decreases.

看我們來看個例子

That makes sense.

讓我們假設針對一個單一的兔子群

And in fact, if our goes below one within the population drops and eventually goes extinct.

設定R等於2.6

So what I want to do is make another graph.

讓我們將起始種群比例

Where on the X axis I have are the growth rate and on the y axis I'm plotting the equilibrium population.

設定為理論最大的40%

The population you get after many, many, many generations.

乘以(1-0.4)

Okay, for low values of our we see, the population's always go extinct, so the equilibrium value is zero.

就會得到

But once our hits one, the population stabilizes onto a constant value, and the higher ours, the higher the equilibrium population.

0.624

So far, so good.

所以第一年的數目上升了

But now comes the weird part.

不過我們感興趣的是數目的長期表現

Once our passes three, the graph splits in two.

所以我們可以把這個數字代回公式裡

Why?

這裡可以簡單加速

What's happening?

設定2.6*Ans(1-Ans)

Well, no matter how many times you generate the equation, it never settles onto a single constant value.

得到0.61，數目掉了一些

Instead, it oscillates back and forth between two values.

再按一次

One year the population is higher, the next to your lower, and then the cycle repeats.

如果我一直將答案代回公式

The cyclic nature of populations is observed in nature to when you're there might be more rabbits and then fewer the next year and Maura again the year after.

你會看到數目不怎麼改變

As our continues to increase, the fork spreads apart, and then each one splits again.

它是穩定的，就像我們在野外觀察到的

Now, instead of oscillating back and forth between two values, populations go through a four year cycle before repeating.

在出生與死亡平衡的情況下

Since the length of the cycle or period has doubled, these are known as period, doubling by for cations and is our increases further their arm or period doubling by for cations.

種群數量會維持一定

They come faster and faster, leading two cycles of eight, 16 30 to 64 then at our equals, 3.57 chaos.

我想給你們看張圖

Population never settles down at all.

你可以看到它達到一個平衡點

It bounces around as if a random.

如果我們

In fact, this equation provided one of the first methods of generating random numbers on computers.

改變一開始的數目

It was a way to get something unpredictable from a deterministic machine.

我會拖曳這一點

There is no pattern here, no repeating.

你會看到一開始的幾年改變了

Of course, if you did know the exact initial conditions, you could calculate the values exactly.

不過平衡點還是維持一定

So they're considered only pseudo random numbers Now.

所以我們基本上可以忽略起始狀態

You might expect the equation to be chaotic from here on out.

所以我真正感興趣的是

But as our increases border returns, there are these windows of stable, periodic behavior.

這個平衡狀態

Amid the chaos.

是怎麼被成長率R所影響的

For example, a R equals 3.83 There is a stable cycle with a period of three years, and his art continues to increase.

你可以看到如果我降低成長率

It splits into 6 12 24 and so on before returning to chaos.

整體平衡點會下降

In fact, this one equation contains periods of every length 37 51,000 and 52.

這很正常，事實上

Whatever you like, if you just have the right value of our looking At this bifurcation diagram, you may notice that it looks like a fractal.

如果R下降到比1還低

The large scale features look to be repeated on smaller and smaller scales.

整體數目會逐漸下降

And sure enough, if you zoom in, you see that it is in fact, a fractal.

直到種群滅絕

Arguably the most famous fractal is the Mandel brought set.

所以我想畫另一張圖

The plot twist here is that the bifurcation diagram is actually part of the mantle brought set.

以成長率R為X軸

How does that work?

而在Y軸

Well, quick recap on the mantel Brought said it is based on this iterated equation.

則代表種群平衡點

So the way it works is you pick a number, see any number in the complex plane and then start with Z equals zero and then iterated this equation over and over again.

種群數量在數個世代後的數目

If it blows up to infinity.

所以在低成長率R

Well, then the number C is not part of the set.

的情況下，我們可以看到總是走向滅絕

But if this number remains finite after unlimited iterations, well, then it is part of the Amanda brought set.

平衡點是0，但當R達到1

So let's try.

平衡點就會變成一個穩定的正數

For example, C equals one.

而R越高，平衡的數目也就越高

So we've got zero squared, plus one equals one.

目前為止都很正常

Then one squared plus one equals two, two squared, plus one equals +55 squared plus one equals 26.

不過事情開始變得奇怪

So pretty quickly.

當R越過3以後，圖像分裂

You can see that with C equals one.

成兩個

This equation is gonna blow up.

為什麼？發生什麼事？

So the number one is not part of the man who brought Set.

不管你疊代幾次公式

What if we try C equals negative one more than we've got Zero squared, minus one equals negative one negative one squared minus one equals zero.

它絕不會再穩定於一個數值

And so we're back to zero squared.

而是在兩個數值間擺盪

Minus one equals negative one.

有一年的數值比較高

So we see that this function is gonna keep oscillating back and forth between negative one and zero, and so it will remain finite, and so c equals negative One is part of the Mandel brought set.

下一年比較低，往復循環

Now, normally, when you see pictures of the mantle brought set, it just shows you the boundary between the numbers that caused this iterated equation to remain finite and those that cause it to blow up.

這種循環在自然環境中也會觀察得到

But it doesn't really show you how these numbers stay.

有一年的兔子可以比較多

Fine.

而下一年的就會比較少

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 generation actually takes.

當R繼續增加，兩點交叉越來越大

So if we look from the side, what you'll actually see is the bifurcation diagram.

然後兩點再次分裂

It is part of this Mandel brought set.

現在除了兩點

So what's really going on here?

數值擺盪，種群數目

Well, what this is showing us is that all of the numbers in the main cardio oId they end up stabilizing onto a single constant value.

種群數目變成四年循環

But the numbers in this main bulb will they end up oscillating back and forth between two values.

隨著循環加長與週期加倍

And in this bulb, the end of oscillating between four values, they've got a period of four and then ate and then 16 32 and so on.

分支的結果也加倍

And then you hit the chaotic part.

當R繼續增加，分支的結果

The chaotic part of the bifurcation diagram happens out here on what's called the needle of the man who brought Set where the metal brought set gets really thin.

也繼續加倍，分叉的成形越來越快

And you can see this medallion here that looks like a smaller version of the entire Manta brought set.

從8、16、32、64

Well, that corresponds to the window of stability in the bifurcation plot with a period of three.

直到R等於3.57

Now, the bifurcation diagram on Lee exists on the rial line because we only put real numbers into our equation.

渾沌

But all of these bulbs off of the main cardio oId well, they also have periodic cycles of for example, three or 45 And so you see these repeated ghostly images.

種群的數目永遠不會固定

If we look in the Z axis effectively there oscillating between these values as well.

隨意跳動出隨機的軌跡

Personally, I find this extraordinarily beautiful.

實際上這個公式提供了

But if you're more practically minded, you may be asking.

電腦的隨機數字產生器

But does this equation actually model populations of animals?

它是一種提供從有序機器中得到無序結果的方法

And the answer is yes, particularly in the controlled environment scientists have set up in labs.

它沒有固定的模式

What I find even more amazing is how this one simple equation applies to a huge range of totally unrelated areas of science.

沒有重複

The first major experimental confirmation came from a fluid dynamics ist named Lib Chamber.

當然如果你知道起始狀態

He created a small rectangular box with mercury inside, and he used a small temperature ingredient to induce conviction just to counter rotating cylinders of fluid inside his box.

你就可以計算之後的結果

That's all the box was large enough for, And of course, he couldn't look in and see what the fluid was doing.

所以這只被當作偽隨機數字

So he measured the temperature using a probe in the top, and what he saw was a regular spike, a periodic spikes in the temperature.

你可能會覺得這個結果就從此渾沌了

That's like when the logistic equation converges on a single value.

但當R繼續增加，有序的狀態又回來了

But as he increased the temperature, radiant ah wobble developed on those rolling cylinders at half the original frequency, the spikes in temperature were no longer the same height.

這裡又可以觀察到固定周期的變化

Instead, they went back and forth between two different heights.

在渾沌之間產生，例如當R

He had achieved period, too, and as he continued to increase the temperature, he saw Period doubling again.

等於3.83，就會產生一個固定循環週期

Now he had four different temperatures before the cycle repeated and then ate.

以三年為單位

This was a pretty spectacular confirmation of the theory in a beautifully crafted experiment.

當R繼續增加，又變成六年一個週期

But this was only the beginning.

12、24，直到結果又變回混沌

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.

事實上，這個公式包含

It's amazing in these papers to see the bike for cation diagram emerged, albeit a bit fuzzy, because it comes from real world data.

所有長度，3750、1052

In another study, scientists gave rabbits a drug that sent their hearts into fib relation.

不管你想要哪種

I guess they felt there were too many rabbits out there.

只要你有適當的R值

I mean, if you don't know what fibrillation is, is where your heart beats in an incredibly irregular way and doesn't really pump any blood.

看著這張分叉的圖

So if you don't fix it, you die.

你可能會發現它看起來像個碎形

But what they found was on the path defibrillation.

所有的大圖都是由重複小圖所組成

They found the period doubling route to chaos.

當然

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

如果你放大檢視，你會發現它真的

Behavior.

是一個碎形

No, it was really cool about this study was they monitor the heart in real time and used chaos theory to determine when to apply electrical shocks to the heart.

最著名的碎形是曼德柏集合

To return it to period is city, and they were able to do that successfully.

之前展示的分叉繪圖

So they used chaos to control a heart and figure out a smarter way to deliver electric shocks to set it beating.

其實是曼德柏集合的一部份

Normally again, that's pretty amazing.

怎麼會這樣

And then there is the issue of the dripping faucet.

簡單說明曼德柏集合

Most of us, of course, think of dripping faucets is very regular periodic objects.

它是以這個公式所產生的

But ah, lot of research has gone into finding that once the flow rate increases a little bit, you get period doubling.

產生的方式是你選一個隨意數字C

So now the drips come to a time T tip, and eventually from a dripping faucet, you can get chaotic behaviour just by adjusting the flow rate, and you think like what really is a faucet.

放在複數平面上，以Z等於0為起始條件

Well, there's constant pressure, water and a constant size aperture.

然後反覆疊代

And yet what you're getting is chaotic dripping, so this is a really easy, chaotic system you can experiment with at home.

如果它發散至無限

Go open a tap just a little bit and see if you can get a periodic dripping in your house.

那這個數字C就不是集合的一部份

The bifurcation diagram pops up in so many different places that it starts to feel spooky.

但是

I wanna tell you something that'll make it seem even spookier.

如果這個數字在無限次疊代中維持有限

There was this physicist, Mitchell Feigenbaum, who was looking at when the bike for cations occur.

那它就是集合的一部份

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.

讓我們測試C=1

The bike occassions come faster and faster, but in a ratio that approaches this fixed value.

[看圖公式]

And no one knows where this constant comes from.

[看圖公式]

It doesn't seem to relate to any other known physical constant, so it is itself a fundamental constant of nature.

[看圖公式]

What's even crazier is that it doesn't have to be the particular form of the equation I showed you earlier.

[看圖公式]

Any equation that has a single hump.

很快的你可以看到當C=1

If you iterated the way that we have, so you could use X m plus one equals sine X, for example, if you iterated that one again and again and again, you'll also see by for cations.

這個公式將會爆炸

Not only that, but the ratio of when those by for cations occur will have the same scaling 4.669 Any single hump function iterated.

所以1就不是曼德柏集合的一部份

We'll give you that fundamental constant.

如果我們測試C=-1呢

So why is this?

[看圖公式]

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.

[看圖公式]

In 1976 the biologist Robert May I wrote a paper in nature about this very equation.

[看圖公式]

It sparked a revolution in people looking into this stuff.

所以我們可以看到

I mean, that paper's been sighted thousands of times.

這個公式的結果反覆在-1與0之間變換

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.

所以這會維持有限，所以C就是集合的一部份

And I still think that today we don't really teach this way.

通常你看到曼德柏集合的圖

I mean, we teach simple equations and simple outcomes because those are the easy things to do and those of the things that make sense.