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