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Let's do another example illustrating this qualitative approach to differential equations.
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So here is the example - the derivative of X, the rate of change of X, is some function of X,
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and this time I'm not even going to write down a formula for X,
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I'll just show the graph of f(X)- it's this blue curve.
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So different values of X, have a different values of the derivative.
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So remember a dynamical system is just
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a mathematical system that changes in time according to a well specified rule.
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and here, this plot here, is a representation of that rule,
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So this says if you're less than 2, the rule is: you should increase - your derivative is positive.
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If the derivative is positive - this blue curve is above the axis, then capital X is increasing,
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between 2 and 6, X decreases - the blue curve is negative, the derivative is negative,
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and greater than 6, X increases again - it increases because the derivative is positive
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- blue curve is above the x-axis.
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So right away we can draw a phase line for this differential equation.
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There are two equilibria, or fixed points...
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at 2 and 6, - so for the differential equation, we have an equilibrium fixed point, when the derivative is 0,
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because when a derivative is 0, the function is not changing, that happens at 2 and 6.
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In between 2 and 6, the derivative is negative - the blue curve is below the axis,
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so the quantity X is decreasing towards 2,
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below 2 - this is positive, so we're increasing;
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above 6, it's positive again, so it's increasing.
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So we have two different fixed points,
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- this is stable, or it's an attractor, 6 is unstable, or a repeller.
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We can also sketch solution curves to this differential equation.
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Start by drawing some axes...
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OK, so here are some axes ...
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- here's time - the units here are pretty arbitrary - just put something down here,
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and let's see - 2 is an attracting fixed point. If we start anywhere less than 2
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it'll increase rapidly, and then smoothly approach 2,
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so if I had an initial condition down here, you would probably do something like that...
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that must be approaching 2.
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If I'm somewhere between 2 and 6, I decrease until I hit 2,
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let's say I start just a little bit below 6, - 6 is an equilibrium point, a stable point,
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but if I'm a little bit below 6, my derivative is negative - that means I'll be decreasing,
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at first slowly, and then more rapidly, and then slowly again,
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and all of these curves - it's getting a little messy - but are approaching this fixed point at 2.
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May be I'll draw the equilibrium point on - as a dotted line; and then we have an equilibrium point at 6,
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and the equilibrium point here is unstable - make a mess of my figure here
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- points here would get pushed away - and these, let's draw these a little better, - problem bridging the two -
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So, these solution curves in purple, contain the same information in a sense, that this phase line does,
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so we can see we have a stable fixed point at 2, and it's attracting,
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and so a number of different orbits or solutions are all getting pulled in here,
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anything between - anything less than 6 actually, gets pulled here,
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and if we start a little bit above 6, we get pushed away.
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So again, start with a differential equation - differential equation is a rule
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for how X changes - that specifies the derivative for every X
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and from that we can quickly figure out which way things are moving
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- they're just moving to the right when this is positive, to the left when it's negative,
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then we can get our fixed points, and we can see our stability right away,
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and then we can take this plot for this plot and sketch some solutions.
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Let me do one more thing to illustrate the relationship between these purple curves
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- the solutions - and this, the phase line.
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I'm going to take the phase line, and turn it on its side, so it's pointing up instead of sideways.
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I do that.. here's the phase line... fixed point at 6... fixed point at 2...
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points between 6 and 2 go to 2... below 2 go up to 2... and here they go away like this.
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So if you put the phase line on its side, it tells you the direction that these purple solution curves are going,
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so here the purple solution curves are going down towards 2, down towards 2;
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here, they're going up away from 6; here they're going up towards 2.
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Before moving on to the next sub unit, I suggest that you practice these ideas on the quizzes that follow this lecture.
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These will give you a chance to become more familiar with these techniques and make sure you see how they're being used.