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.
So again, HERE are the differential equations we're working with.
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.
So to use the language I'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.
Suppose we are somewhere HERE at a particular X,Y point.
(let me make that a little bigger).
Where am I going to go next?
Where will I go next in terms of X and Y?
I ask the equation. The equation determines the next values through this relationship.
And, I can think of dX/dt as telling me how much, in the direction of X change: does it go up or down?
ie does X increase or decrease, which means going THIS way or THAT way?
And similarly, Y can increase or decrease, which would mean going THIS way or THAT way.
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.
It tells me where to move. Where 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 could plug that in here and say "What are my directions? Where am I going?"
"What's my rate of change?" That might give me a different arrow over here.
I might have something different altogether.
So every point in space has a set of directions associated with it, given by these equations.
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 speed.
This might give a trajectory looking something like that.
Again, the main point is that X and Y determine subsequent values of X and Y: this is deterministic.
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 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's a crossing point. I'll draw that there in red.
What this equation says 'If you tell me X and Y ('f' is a deterministic function, 'g' is a deterministic function),
there is one and ONLY one dX/dt and dY/dt. That means there
is only one arrow indicating the direction of change from this [red] 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.
The conclusion is: in the phase plane, trajectories (eg things like THIS) cannot cross.
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,
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.