So, here's another bifurcation diagram.
It's not for the logistic equation.
It's for a different differential equation.
And, I won't go into the origins of this
because the point of this exercise is just to get practice reading bifurcation diagrams.
So, there were two questions posed.
First is, what's the situation at h = 5?
So, in the bifurcation diagram we have some parameter that's changing.
Here, I called it h again.
Here, this would be p for population.
And, a bifurcation diagram is just a collection of a whole lot of phase lines.
So, when h is five, I can look for the phase line
by just ignoring everything except what is going on at 5.
I could imagine taking...
a pair of scissors
and slicing, cutting this piece out.
Like this.
Picking it up.
Moving it down here.
And I would get the following phase line.
So at h = 5, there is a single fixed point.
That's the solid line.
And it is stable because the arrows are pushing towards it.
So, at h = 5 there is one fixed point and it is stable.
What's going on somewhere over here at 10?
Again, I could imagine getting rid of the rest of it
just looking at a single slice.
The bifurcation diagram is many, many, many phase lines
all just stacked together in this direction.
So, I just go up to 10, and I grab the phase line
at that particular value,
and there it would look something like this:
There are two fixed points.
Sorry, three fixed points, excuse me.
So the middle fixed point here is unstable.
Arrows push away from it.
If you are at this fixed point and move to either side,
you don't return to this fixed point,
you head to another fixed point.
These two fixed points
those are stable.
Arrows are pointing towards them.
If you move a little bit away,
you get pushed back.
So this is the phase line associated with h = 10.
There are three fixed points.
Two stable ones,
and an unstable one in the middle.
So, these are bifurcation diagrams.
They are very commonly used to summarize the behavior
that a particular model, a differential equation, or an iterated function can do.
Very common device in dynamical systems
studying all sorts of dynamical systems, not just differential equations.
We'll see bifurcation diagrams again in the next unit.
which is about bifurcation diagrams for the logistic equation as an iterated function.
In the next video in this unit,
we'll leave bifurcation diagrams for a moment
and focus on the bifurcations themselves.
What's going on when we have these sudden transitions from one type of behavior to another?