In this optional sub unit
I'll present the bifurcation diagram
for a different differential equations
and this will lead us to
the phenomenon of Hysteresis or path dependence
we will see that in a second.
We will start with this differential equation
dx/dt. I'll use x this time
instead of P.
because this doesn't really represent a population
is rx plus x cubed minus x to the fifth
So r is now our parameter.
Before it was h.
This time we will use r.
So, we will build up the bifurcation diagram
piece by piece
by letting r be different values
plotting the right hand side of this
and seeing what the function looks like
and making a phase line
So here is what we have if r equals one
Down, up, and down
So there are three fix points.
Because the line crosses the x axis three times
here, here, and here.
So, three fixed points.
One, two, three.
make a note this is for r equals one
When this function is negative.
This is a derivative.
The derivative is negative.
X is decreasing
When positive we are increasing.
Negative decreasing, positive increasing
So, this function has three fixed points.
There is an unstable fixed point at zero.
and there are two stable fixed points
out here a little bit more than
one away from the origin.
So that's the situation when R equals 1.
If I decrease R and make it a little
bit negative.
This curve gets a little wiggle in it.
and it starts to look like this
So the curve gets steeper
but it aquires a little wiggle in here.
So let's calculate
let's figure out the phase line for this
here we have five fixed points.
1,2,3,4,5
equilibria class of five.
and they kind of scrunch together
That's going to be a little challenging
for me to draw.
Ok, so there are the fixed points.
1,2,3,4,5.
the function is positive
so we are moving to the right
negative in here, then positive
negative,positive, negative.
R equals zero point two.
So I see three stable fixed points.
Here, here, and here in the middle
So you have probably noticed
a stable fixed point occurs
when the line crosses the axis from top to bottom.
So that happens here, here, and here.
So we have these two unstable fixed points
here and here.
When the line goes from below to above.
So, five fixed points, three are stable
and two are unstable.
This is the story for minus 0.2
the last r we will look at
is r equals minus 0.4
R is a little bit more negative here.
and what happens is
these bumps straighten out.
So this bump and this bump
get pulled up and down.
And we end up with this.
So here, the phase line is kind of simple
almost boring again
So we have one fixed point.
So we had five but four of them disappeared.
And we are just left with this one.
at the origin.
And it keeps --it's stability
so we had a little hard to see.
We had four and here we had one.
but this one the one at the origin remains.
Ok, so we have three phase lines.
So we can connect them
Sort of glue them all together
and see what the bifurcation diagram might look like.
so as before I"m going to slice off.
these phase lines.
and let's take a look.
Here is R equals 1
Here is R equals minus 0.2
And I should've written here
this was r equals minus 0.4
Here is the what we have.
So from these phase lines
it might not be immediately clear
what the entire bifurcation diagram looks like
we might want to do a few more phase lines
For immediate R values.
try an R of 0. a R of -.1
A r of +.1, and so on
But rather than take the time to do that.
Let me sketch what this looks like
and then I'll show you a neater drawing
of the bifurcation
diagram
Since the main goal is to get this bifurcation diagram
and then look at it and learn about Hysteresis
so let me just draw a few things on here.
So I'm going to use blue
for an unstable fixed point
and so it turns out I have a line
of unstable fixed points here.
Wait sorry those are stable.
Oh, dear how can I recover from this
this was going to be blue
Maybe it's red and blue, purple, or it looks mostly red
So these are stable.
It's just the wrong color
It's stable the arrows are going in
and then we also have some stable fixed points
Here and here. Here and Here.
and these are going to look like this.
And this one is going to come down like this.
and then we will have unstable fixed point here.
and this line connects up here.
So that's our bifurcation diagram
It's not the best picture in the world.
To me, it kind of looks like
a fish like a salmon that's throwing up.
Which you know.
is not what I intended.
but this is the bifracation diagram.
So we have stable points in red
and unstable points in blue.
And hopefully you can see how the blue and red lines
line up with these fixed points.
And this vommiting fish looking this.
So let me draw another nicer version of this diagram
and we will analyze that.
And learn about Hysteresis.
So here is a slightly neater version
of the bifucation diagram.
From the previous screen.
And I'll be focusing on the positive x-values
I've only drawn arrows on here.
So we have a line of stable fixed points.
Attractors.
and we have here in blue a line of unstable
fixed points repellers.
Unstable here, and stable here.
So, let's imagine let's sort of talk through
a scenario with this.
That the parameter starts off somewhere off here.
And we have a postive x value.
We are going to get pulled to this attractor
and now imagine the parameter
is going to be decreasing
who knows in this case.
I don't know if there is a clear physical or analogue
or something but whatever R is. It decreases
So as R decreases then the equilibrium
value decreases.
then we decrease R some more
and the equilibrium value decrease some more
then we move down along here.
And this looks alot like
what happened when we were increasing
the fishing rate in the logistic differential equation
So we move down here,
R continues to decrease
R continues to decrease
R continues to decrease
until we get here.
And then this fixed point
this attractor up here disappears
It's gone.
It decrease a little bit more.
The quantity of x whatever it is
is going to get pulled down here to zero.
And so then,
perhaps we like this positive thing
is good
zero is bad
maybe this is growth rate of the economy
or some fishing, some number of fish
or something
and we zip down here.
Then we might say "Uh-oh, we crashed"
"We better increase R."
and so we will increase R.
but this red point down here is stable.
it's attracting
And so we don't automatically jump up to here.
because this is stable.
We move a little bit
We get pushed back.
So then we would increase R,
We will increase R,
we will increase R still.
More, until we get a little bit over here.
Then. this fixed point loses it's stability.
We go from Red to blue.
and then we will jump back up to here.
So again, we are seeing jumps
But this time there is a new feature.
Which is as follows:
Suppose we wanted to know if R was around here.
Whatever that is -0.2
What stable behavior would we observe in this model
and the answer is,
it would depend on not just
on the R value, but where one came from.
and this is the idea of the Hysteresis.
Let me draw a picture sort of to illustrate.
or outline the story I just told.
So thinking of this portion of the bifurcation diagram
I guess I'll just make a really rough sketch of this.
So, I could move down this way
then I come to this collapse point
and I go down here.
Then, I would increase until here.
and then I would jump back up
and could go in either direction here.
So, so this is to connect it r = 0
So this system so has path dependence.
So what would you observe at this r value
Well it depends not just on the R value.
but on the path to get there.
If you reach this R value,
the one where my finger is
from above, from the right
Then you would be up here.
Here on this diagram.
If you approached this R value from below
having going beyond this and sort of falling off that cliff
then you will be down here at zero
this is called hysteresis or path dependence.
So the term of this behavior
is hysteresis or path dependence.
So that the equilibrium property
the oberseved behavior of this differential equation
this model
depends not only on R.
It looks like it only depends on R.
If you tell me what R is.
I can solve the differential equation
I can tell you what X would end up being.
But in the situation where you have multiple attractors
and they are arranged like this
knowing R is not enough
you need to know where R came from.
It depends not just on R.
But on the path R took.
This is surprising and interesting
I think because path dependence
is a type of memory
The value of the population
whatever this is
in a sense remembers where its been.
It's not obvious at all that this equation has memory
built into it
This says the growth rate,
the change of X and this number R.
So it's a type of memory or history
that get introduced into a differential equation
as a result of this bifurcation
this particular structure in a bifurcation diagram
like this.
I don't know that this
is common or ubiquitous in differential equations
But it's not uncommon either
But you don't need a tremendous complicated equation
to get this behavior
So this is another type-I guess--
of bifucations
Two bifucations.
There is a bifurcation here
and a bifurcation there.
and taken together
those two bifurcations lead to this path dependence.
So, again to underscore it one more time
We have a simple differential equation
something that is continuous, smooth, differential,
doesn't have any memory built in
and we can have a system behave in jumps
and that develops a memory or path dependence
So that's the idea behind.
Hysteresis or path dependence