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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
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a little bit more generally.
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Here 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
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Y is a function of X and Y.
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So, to use the language we've used before,
I would say that X and Y determine the
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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, this is a deterministic
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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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So I could say where am I going next?
Where will 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
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relationship and I could think of dX/dt as
telling me how much and the direction of
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X change - does it go up or down, which
in this case, i.e. does X increase or
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decrease, which means going this way,
or that way.
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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 I get these two and I
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can picture this as giving me an arrow,
a direction. It tells me where to move,
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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
plug that in here and say, "what are my
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directions, where am I going,
what is my rate of change?"
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That might give me a different arrow
over here or I might have something
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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
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that this goes to.
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One could picture this if you want as a
whole bunch of arrows, technically a
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vector field, here on this plane and that
a particular trajectory or solution would
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follow the vector field like a particle
drifting through a fluid
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moving with a 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
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on again to be clear - there's an X,
there's a Y - it says that two solutions
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in phase space can never cross.
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And here's why: assume there is a crossing
point. I'll draw that there in red.
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What this equation says is: "if you tell
me X and Y (f is a deterministic function,
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g is a deterministic function)"if you tell
me X and Y, there is one and only one
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dX/dt and dY/dt." That means there is one
and only one arrow indicating the
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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 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 that in the phase
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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, it can
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cycle, it can go to a fixed point,
it can go off to infinity, but the only
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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
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important consequences of this fact.