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In the last video I presented two examples
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of a particular solution on a phase plane
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going from X and Y of (t) to an X/Y plot.
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In this video I'll discuss properties of the phase plane a little
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bit more generally.
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So again, HERE are the differential equations we're working with.
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The rate of change of X as a function of X and Y, and the rate of change of Y as a function of X and Y.
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So to use the language I've used before I would say that
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X and Y determine the rate of change of X and Y, and thus X and Y themselves.
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So, as with all the dynamical systems we've looked at,
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this is a deterministic dynamical system.
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X and Y determines the future. The present
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value of X and Y determine the future values of X and Y through this relationship.
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So let's think about what that would look like geometrically.
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Suppose we are somewhere HERE at a particular X,Y point.
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(let me make that a little bigger).
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Where am I going to go next?
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Where will I go next in terms of X and Y?
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I ask the equation. The equation determines the next values through this relationship.
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And, I can think of dX/dt as telling me how much, in the direction of X change: does it go up or down?
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ie does X increase or decrease, which means going THIS way or THAT way?
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And similarly, Y can increase or decrease, which would mean going THIS way or THAT way.
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And so I can think, if I'm at THIS point, I can plug into here, I get these two, and I can picture this as giving me an arrow, a direction.
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It tells me where to move. Where X and Y go next.
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So I can imagine maybe that's a little arrow. That's my direction.
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This little device tells me "go in this direction" at a certain speed.
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Then I could be at another point, and I could plug that in here and say "What are my directions? Where am I going?"
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"What's my rate of change?" That might give me a different arrow over here.
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I might have something different altogether.
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So every point in space has a set of directions associated with it, given by these equations.
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Knowing the value of a current point determines all the successive points that this goes to.
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One could picture this, if you want, as a whole bunch of arrows (technically a vector field),
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here on this plane, and that a particular trajectory or solution
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would follow the vector field like a particle drifting through a fluid, moving with speed.
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This might give a trajectory looking something like that.
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Again, the main point is that X and Y determine subsequent values of X and Y: this is deterministic.
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This fact of determinism has an important geometric consequence.
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In particular, it tells us that two curves in phase space (let's draw some axes again, to be clear...
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..there's an 'X,' there's a 'Y'), it says that two solutions in phase space can never cross.
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And here's why:
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Assume there's a crossing point. I'll draw that there in red.
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What this equation says 'If you tell me X and Y ('f' is a deterministic function, 'g' is a deterministic function),
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there is one and ONLY one dX/dt and dY/dt. That means there
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is only one arrow indicating the direction of change from this [red] point.
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But if we have two curves crossing like this, that violates this condition.
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If this was possible, then there would be two possible direction curves from this red dot.
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This would be ambiguous, and non-deterministic.
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But that's not the case. The equations are deterministic.
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So, this scenario HERE cannot happen.
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The conclusion is: in the phase plane, trajectories (eg things like THIS) cannot cross.
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We can have all kinds of other behaviors: it can loop around, it can cycle, it can go to a fixed point, it can off to infinity,
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but the only rule is that trajectories CAN'T cross.
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In the next video, I'll present a few more examples and we'll see one of the important consequences of this fact.
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