In the last sub-unit, we looked at the Lotka-Volterra equations as our first example of a two-dimensional differential equation In this sub-unit, I'd like to look at two-dimensional differential equations more generally and focus on properties of the phase plane. And in this video, I'll start with a couple of examples. So a reminder of what we are working on here we're looking at differential equations of this form: so we have two variables, I'll call them x and y now instead of r and f or rabbits and foxes, any old variables, and this is a dynamical system it specifies how x and y change but because it's a differential equation it does so indirectly by telling us the the rates of change, the velocities of x and y, not directly giving x and y values. The rate of change of x is a function of x and y, the rate of change of y is a function of x and y, these could be different functions and note that x depends on y and y depends on x so we would say that these two different differential equations are "coupled" because they depend on each other. OK, so one can solve these equations and produce solutions using Euler's method or something like it And then one gets two solution curves. So let me show an example of that Here are two possible solution curves and if you did the quiz you stared at these before So this is X of t---X(t)---and this is Y of t.... Y(t).... and they both wiggle and then they are approaching zero so it looks like there is an attracting fixed point at zero so lets's think what the phase plane might look like. So I'll try to do a rough sketch of this then I'll show you the plot that I had a computer do. OK, so, draw some axes first ....... So this is Y ....and....this is X And I just want to get a general picture of the shape of this. So I start, X is -7 and Y is -3 So X is -7, Y is -3, that might--that's going to put me somewhere over here.. ...that's my starting point. So -Y, -X. And I know that I'm going to end up here and these wiggles indicate some kind of spiral and the big question is now which direction does the spiral go... So let's see--so the first thing that happens is Y increases and X is decreasing--X gets a little more negative while Y increases. So that's going to end up looking something---something like THAT. That sort of motion. So X is decreasing, moving to the left, alright, because X is going DOWN here, the value of X goes from -7 to -8, so X goes from -7 to -8, but Y is increasing, from -3...-2, to all the way up here--it looks like plus + 3--that's going to end up around THERE.... ...and then, it will spiral in like this. So this is a fixed point at 0, and it's stable because points get pulled in and we this sort of spiral thing...we can't really spiral in quite the same way in one dimension, but we can in two dimensions. So I think the best way to do this, to go from these shapes to this shape is to think about starting point, think about the ending point and then what might happen in the middle. Another thing you could do, you could sort of plot point by point so t=5, we have an X of--I dunno--around 4, and a Y of around -1.5 ---so maybe that's over here so anyway this isn't designed to be exactly to scale, but just give the general shape. Let me show you what a computer plot of this would look like: (Sneak that over here...try to get this all on the screen...there we go) And I should put arrows on this...my program doesn't do that automatically... So we have something spirally in to the origin this is a stable fixed point at X=0, Y=0 OK, so that was one example--let me do one more...... So here's another example-suppose we have these two solution curves-X is a function of t and Y is a function of t In this case again we have oscillatory behavior but the amplitude doesn't decrease So the amplitude and frequency are staying constant. So let's try to visualize this behavior, oscillations in X and oscillations in Y so if these were populations they would be perfectly cyclic What would that look like in the phase plane? So here are my axes, that is X against Y, no time on the phase plane And we start...it looks like I chose the same starting point as before...Y is -3, X is -7 So X is -7, Y is -3,.... I am going to start here... and then, let's see, X increases as Y decreases. Alright because I start Here-X is going UP, that means I expect this blue thing to move to the right and Y is going DOWN. So that motion is going to look something like that. When X is 0, Y looks to be about -5 Then, Y starts increasing, that means I'm going UP in this direction, X is still increasing We'll end up with motion that looks like this. So X is going from--boy--roughly -10 to 10, maybe that's 9.5 these are sort of min and max values for X, Y is going roughly between 4.5 and -4.5 So this would be an ellipse--it's not a circle because this is not the same as that. So the main point is that this type of motion where we have two quantities that are oscillating.... ...sinusoidally on a phase plane, will be some sort of ellipse or oval. So this sort of motion means that X is moving back and forth Y is moving back and forth and that those motions are in phase. Let me show you a computer version of this plot there it is...I'll put arrows on... so we can see X is going from a little bit less than 9, a little bit more than -10 between here, and the amplitude of the Y is about 4.5. And so this would just cycle around like this. So the main point of this is....let's see...so we'll be describing motion in phase space like this and in two-dimensional phase space, the thing to bear in mind is that this two-dimensional phase space plot is just two solutions graphed together without time.