1:21

We'll have one equation for dx over dt, the time derivative of x,

and we'll have another equation for dy over dt, the time derivative of y.

And each of these can be a function of both x and y.

Just to keep things easy,

let's stay in the linear realms and

let's say that the dx over dt is 3x- y- 2.

How about that?

And the dy over dt = -x- y + 2.

They can be nonlinear as well.

But I'd like to have this video be focused more on the dynamical systems

understanding than doing the algebra and the actual physical computations.

So let's draw our phase plane down here.

So in one dimension, the important parts of our picture had to with

the fixed points, where the derivative of x with respect to time was equal to zero.

So let's start in the same way.

Let's find all the places where dx over dt = 0.

Dx over dt = 0 gives

us 0 = 3x- y- 2.

So this is actually not just the equation for one or

two or three points, this is the equation for an entire curve.

In this case, a line.

So if we solve this equation,

we have y = 3x- 2.

So everywhere along this line, dx over dt = 0.

So, let's draw that line on our graph.

Well, the y-intercept is at -2 and for

every one we go over, we go up three.

So, let's draw that line.

This is the line where the dx over dt is equal to 0.

This is called not a fixed point but a nullcline.

In this case, the x-nullcline.

The x-nullcline is the line along which dx over dt = 0.

What about dy over dt?

Let's find all the places where dy over dt equal zero.

We can do the same thing.

0 = -x- y + 2, and

that gives us the line for

the y-nullcline, y =- x + 2.

So this gives us the equation for the y-nullcline.

So that's got a slope of -1 and it should go through

2 on the y-intercept so we get something like that.

This is the y-nullcline.

So everywhere along the y-nullcline, the dy over dt = 0.

So if we were filling in all the arrows on our phase plane,

we would find that all of the arrows along the x-nullcline

are vertical because they have no x component.

And all of the arrows along the y-nullcline would

be horizontal because they have no y component because dy over dt=0.

So the nullclines tell us a little bit about our system.

When our system state is along the x-nullcline,

it will not feel any desire to move in the x direction because dx over dt = 0.

When our system is on the y-nullcline, it will not feel any

desire to move in the y direction because dy over dt = 0.

Are there any system states in which our system will not want to move at all?

5:59

And the intersection of the two nullclines is called a fixed point.

For the very reason that both the x derivative with respect to time and

the y derivative with respect to time are zero.

So if our system is sitting at a fixed point, it will not feel any push at all.

So the fixed points are the intersection points of the nullclines.

So now we know the set of system states

that won't feel any push in the y direction.

We know the set of system states that won't feel any push in the x direction.

And we know the system states,

which is just one in this case, where the system won't feel any push at all.

6:50

Well, you'd just find the intersection of the two

lines which, in this case, is at 1, 1.

So if you plug 1, 1 into the first equation for

the x-nullcline, you get 3 times 1- 2 = 1.

So that works.

If you plug in x=1 to the equation for

the y-nullcline, you get -1 + 2 = 1.

So this is indeed the intersection point.

This is the single fixed point of our system.

8:07

We know that the sine of the x derivative,

dx over dt, cannot change as long as you are in that region.

And that is because in order for

the sign to change, you would have to cross the x-nullcline.

You also know that the sign of the y derivative,

with respect to time, cannot change, because you are not allowed to

cross the y-nullcline if you are forced to stay in region A.

So this tells us that the signs of the derivatives in a region

bounded by the nullclines are constant.

So what do I mean by this?

Let's pick the point, some point in region A.

Let's an easy one.

How about right here where x=4 and y=0?

So what are dx over dt and

dy over dt at the point 4,0?

Well, dx over dt is 3 times

4- 0- 2 = 10.

And dy over dt = -4- 0 + 2 = -2.

10:48

No matter where you choose in region B,

the arrows will point down and to the left.

So that was region A, that was region B.

We'll mark region A as down in to the right,

B is down to the left and you can do the same thing with C and D.

And if you just pick a point in C and D,

you will find that all the arrows in C point up and to the left and

all of the arrows in D points up and to the right.

So all of the arrows in C will point up and to the left.

11:44

This generally tells us how our system will behave if its state is in

any of the regions we've outlined or along the nullclines or at the fixed point.

So if it's an A, our system will move down and to the right.

If our system is in region B, it will move down and to the left.

If our system is in region C, it will move up and to the left.

And if our system us in region D, it will move up and to the right.

12:12

So how we did this was we found the nullclines, we found the fixed points and

then we filled in the directions of the arrows in A, B, C, and D.

If your dynamical system is nonlinear,

so you have some more complicated function for dx over dt and

dy of t in terms of x and y, your nullclines won't be straight lines,

they might be fancy curves and you might have multiple fixed points.