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 your brain
it's this one simple equation this video
is sponsored by fast hosts who are
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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 is
where we just multiplied by some number
the growth rate R which could be say 2
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 1 minus X to
represent the constraints of the
environment and here I'm 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 0 and that constrains
the population so this is the logistic
map xn plus 1 is the population next
year and xn 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 R equals two point six and
then let's pick a starting population of
40% of the maximum so point four and
then times 1 minus 0.4 and we get 0.62
four 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
two point six times answer times one -
answer get point six one so the
population dropped a little hit it again
point six one nine point six one three
point six one seven point six one five
point six one six point six one five 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 iteration you can
see here that it's reached an
equilibrium value of point six one five
now what would happen if I change the
initial population I'm just going to
move this slider here and 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 vary depending on are the
growth rate so as you can see if I lower
the growth rate the equilibrium
population decreases that makes sense
and in fact if R goes below one well
then the population drops and eventually
goes extinct so what I want to do is
make another graph where on the x axis I
have R 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
R we see the populations always go
extinct so the equilibrium value is zero
but once our hits 1 the population
stabilizes on to a constant value and
the higher R is the higher the
equilibrium population
so far so good but now comes the weird
part
once our passes three the graph splits
in two why what's happening well no
matter how many times you iterate the
equation it never settles on to a single
constant value instead it oscillates
back and forth between two values one
year the population is higher the next
year lower and then the cycle repeats
the cyclic nature of populations is
observed in nature too one year there
might be more rabbits and then fewer the
next year and more 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 bifurcation z' and as R
increases further there are more period
doubling bifurcation z' they come faster
and faster leading to cycles of 8 16 32
64 and then at R equals three point five
seven chaos the population never settles
down at all it bounces around as if at
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 are considered
only pseudo-random numbers now you might
expect the equation to be chaotic from
here on out but as R increases order
returns there are these windows of
stable periodic behavior amid the chaos
for example at R equals 3 point 8 3
there is a stable cycle with a period of
3 years and as R 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 3750 1052 whatever you like if
you just have the right value
are 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
Mandelbrot set the plot twist here is
that the bifurcation diagram is actually
part of the Mandelbrot set how does that
work well 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 equals 1 so we've got 0
squared plus 1 equals 1 then 1 squared
plus 1 equals 2 2 squared plus 1 equals
5 5 squared plus 1 equals 26 so pretty
quickly you can see that with C equals 1
this equation is going to blow up so the
number 1 is not part of the Mandelbrot
set what if we try C equals negative 1
well then we've got 0 squared minus 1
equals negative 1 negative 1 squared
minus 1 equals 0 and so we're back to 0
squared minus 1 equals negative 1 so we
see that this function is going to keep
oscillating back and forth between
negative 1 and 0 and so it'll remain
finite and so C equals negative 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 how 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 they end up stabilizing on to 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 they end up
oscillating between four values they've
got a period of four and then eight 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 three 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 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
dynamicists named Lib Taber 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 two 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 eight 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 out with a
periodic beat and then it went into a
two cycle two beats close together and
then a four cycle four different beats
before it repeated again and eventually
a periodic behavior now it 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 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 to tip to tip 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 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 and I want to 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 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 xn plus 1 equals sine 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 for point six
six nine 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
papers 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 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 are 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 Gleeks 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 hey this video is
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