1:21

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

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

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

Â Just to keep things easy,

Â let's stay in the linear realms and

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

Â How about that?

Â And the dy over dt = -x- y + 2.

Â They can be nonlinear as well.

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

Â understanding than doing the algebra and the actual physical computations.

Â So let's draw our phase plane down here.

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

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

Â So let's start in the same way.

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

Â Dx over dt = 0 gives

Â us 0 = 3x- y- 2.

Â So this is actually not just the equation for one or

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

Â In this case, a line.

Â So if we solve this equation,

Â we have y = 3x- 2.

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

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

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

Â every one we go over, we go up three.

Â So, let's draw that line.

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

Â This is called not a fixed point but a nullcline.

Â In this case, the x-nullcline.

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

Â What about dy over dt?

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

Â We can do the same thing.

Â 0 = -x- y + 2, and

Â that gives us the line for

Â the y-nullcline, y =- x + 2.

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

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

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

Â This is the y-nullcline.

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

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

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

Â are vertical because they have no x component.

Â And all of the arrows along the y-nullcline would

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

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

Â When our system state is along the x-nullcline,

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

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

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

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

Â 5:59

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

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

Â the y derivative with respect to time are zero.

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

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

Â So now we know the set of system states

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

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

Â And we know the system states,

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

Â 6:50

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

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

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

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

Â So that works.

Â If you plug in x=1 to the equation for

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

Â So this is indeed the intersection point.

Â This is the single fixed point of our system.

Â 8:07

We know that the sine of the x derivative,

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

Â And that is because in order for

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

Â You also know that the sign of the y derivative,

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

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

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

Â bounded by the nullclines are constant.

Â So what do I mean by this?

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

Â Let's an easy one.

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

Â So what are dx over dt and

Â dy over dt at the point 4,0?

Â Well, dx over dt is 3 times

Â 4- 0- 2 = 10.

Â And dy over dt = -4- 0 + 2 = -2.

Â 10:48

No matter where you choose in region B,

Â the arrows will point down and to the left.

Â So that was region A, that was region B.

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

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

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

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

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

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

Â 11:44

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

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

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

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

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

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

Â 12:12

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

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

Â If your dynamical system is nonlinear,

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

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

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

Â