Name: 這個方程將改變你對世界的看法（邏輯圖）。 (This equation will change how you see the world (the logistic map))
Uploaded: 2021-01-14T10:26:17.000Z
Duration: 18 min 39 s

What's the connection between a dripping faucet?

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

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.

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 where we just multiply by some number.

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

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.

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.

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.

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

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

Then times one minus 10.4, 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.

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.

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.

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

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?

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

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

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

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.

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.

So what I want to do is make another graph.

Where on the X axis I have are 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 our we see, the population's always go extinct, so the equilibrium value is zero.

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

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

Well, no matter how many times you generate 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 to your lower, and then the cycle repeats.

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.

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.

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.

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

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.

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

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.

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

For example, a R equals 3.83 There is a stable cycle with a period of three years, and his art 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 37 51,000 and 52.

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.

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

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

Arguably the most famous fractal is the Mandel brought set.

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

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

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.

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.

平衡點就會變成一個穩定的正數

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

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

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

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.

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.

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.

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.

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.

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

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.

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這個方程將改變你對世界的看法（邏輯圖）。 (This equation will change how you see the world (the logistic map))

eventually

incredibly

constant

period