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