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