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?