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In the last video I presented two examples 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 bit more generally.
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So here, again, are the differential equations we're working with
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the rate of change of X is a function of X and Y, and the rate of change of Y is a function of X and Y.
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So, to use the language that we used before, I would say that X and Y determine the rate of change
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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 value of X and Y---determine
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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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So suppose we are somewhere here, at a particular X, Y point
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..a particular X, Y value.
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So I could say, alright, where am I going next? Where do I go next in terms of X and Y?
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Well, I ask the equation--the equation determines the next values through this relationship
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and I could think of dX/dt as telling me how much and the direction of X change-does it go up or down, which in this case
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i. e., does X increase or decrease, which means going this way,
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or that way. Similarly Y can increase or decrease meaning going this way or that way
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So I can think, if I am at this point, I plug into here
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I get these these two--and I picture this as giving me an arrow,
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a direction--it tells me where to move, where do 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
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and say "what are my directions, where am I going, what is my rate of change?"
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That might give me a different arrow over here
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or I might have something different altogether.
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So every point in space has a set of directions associated with it
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given by these equations.
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So 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--
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technically a vector field---here on this plane and that a particular
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trajectory or solution would follow the vector field like a particle drifting through a fluid moving with speed.
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So this might give a trajectory....something like that.
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So, again, the main point is that X and Y determine subsequent values of
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X and Y --this is deterministic.
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So 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 on again to be clear--
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--if there's an X, there's a Y--
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it says that two solutions in phase space can never cross.
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And here's why: assume there IS a crossing point
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I'll draw that there in red. So, what this equation says is:
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"if you tell me X and Y (f is a deterministic function, g is a deterministic function)
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"if you tell me X and Y, there is one and only one dX/dt and dY/dt
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that means there is one and ONLY one arrow
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indicating the direction of change from this 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
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from this red dot....this would be ambiguous and
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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...and the conclusion is
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that in the phase plane trajectories, i.e. things like this, cannot cross.
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So we can have all sorts of other behaviors--it can loop around--
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--it can cycle--it can go to a fixed point--it can go 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 present a few more examples and we'll see
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one of the important consequences of this fact.