The Mandel brought set, ah, population of rabbits, thermal conviction in a fluid and the firing of neurons in your brain.
It's this one simple equation.
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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.
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.
So this is the logistic map.
X n plus one is the population next year and X and is the population this year.
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.
So let's try an example.
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.
So we can put this population back into the equation, and to speed things up, you can actually type 2.6 times.
Answer times one minus answer.
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.
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.
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?
Very depending on our the growth rate.
As you can see, if I lower the growth rate, the equilibrium population decreases.
That makes sense.
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.
So far, so good.
But now comes the weird part.
Once our passes three, the graph splits in two.
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.
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.
Population never settles down at all.
It bounces around as if a 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're considered only pseudo random numbers Now.
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.
Amid the chaos.
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.
How does that work?
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.
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 Amanda brought set.
So let's try.
For example, C equals one.
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 pretty quickly.
You can see that with C equals one.
This equation is gonna blow up.
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.
And so we're back to zero squared.
Minus one equals negative one.
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.
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.
So if we look from the side, what you'll actually see is the bifurcation diagram.
It is part of this Mandel brought set.
So what's really going on here?
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.
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 man who brought Set where the metal brought set gets really thin.
And you can see this medallion here that looks like a smaller version of the entire Manta brought set.
Well, that corresponds to the window of stability in the bifurcation plot with a period of three.
Now, the bifurcation diagram on Lee exists on the rial line because we only put real numbers into our equation.
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.
If we look in the Z axis effectively there 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 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 dynamics ist named Lib Chamber.
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.
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 spikes in the temperature.
That's like when the logistic equation converges on a single value.
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.
Instead, they went back and forth between two different heights.
He had achieved period, too, 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 ate.
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 bike for cation diagram emerged, albeit a bit fuzzy, because it comes from real world data.
In another study, scientists gave rabbits a drug that sent their hearts into fib relation.
I guess they felt there were too many rabbits out there.
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.
So if you don't fix it, you die.
But what they found was on the path defibrillation.
They found the period doubling route to chaos.
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.
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.
To return it to period is city, 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 is very regular periodic objects.
But ah, 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 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.
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.
I wanna tell you something that'll make it seem even spookier.
There was this physicist, Mitchell Feigenbaum, who was looking at when the bike for cations 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 bike occassions 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 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.
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.
We'll 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 I 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 sighted 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.
And 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 of 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 Greeks Book Chaos, I have wanted to make videos on this topic, and now I'm finally getting around to it.
And hopefully I'm doing this topic justice because I find it incredibly fascinating and I hope you do too.
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