Let's do another example illustrating this qualitative approach to differential equations.
So here is the example - the derivative of X, the rate of change of X, is some function of X,
and this time I'm not even going to write down a formula for X,
I'll just show the graph of f(X)- it's this blue curve.
So different values of X, have a different values of the derivative.
So remember a dynamical system is just
a mathematical system that changes in time according to a well specified rule.
and here, this plot here, is a representation of that rule,
So this says if you're less than 2, the rule is: you should increase - your derivative is positive.
If the derivative is positive - this blue curve is above the axis, then capital X is increasing,
between 2 and 6, X decreases - the blue curve is negative, the derivative is negative,
and greater than 6, X increases again - it increases because the derivative is positive
- blue curve is above the x-axis.
So right away we can draw a phase line for this differential equation.
There are two equilibria, or fixed points...
at 2 and 6, - so for the differential equation, we have an equilibrium fixed point, when the derivative is 0,
because when a derivative is 0, the function is not changing, that happens at 2 and 6.
In between 2 and 6, the derivative is negative - the blue curve is below the axis,
so the quantity X is decreasing towards 2,
below 2 - this is positive, so we're increasing;
above 6, it's positive again, so it's increasing.
So we have two different fixed points,
- this is stable, or it's an attractor, 6 is unstable, or a repeller.
We can also sketch solution curves to this differential equation.
Start by drawing some axes...
OK, so here are some axes ...
- here's time - the units here are pretty arbitrary - just put something down here,
and let's see - 2 is an attracting fixed point. If we start anywhere less than 2
it'll increase rapidly, and then smoothly approach 2,
so if I had an initial condition down here, you would probably do something like that...
that must be approaching 2.
If I'm somewhere between 2 and 6, I decrease until I hit 2,
let's say I start just a little bit below 6, - 6 is an equilibrium point, a stable point,
but if I'm a little bit below 6, my derivative is negative - that means I'll be decreasing,
at first slowly, and then more rapidly, and then slowly again,
and all of these curves - it's getting a little messy - but are approaching this fixed point at 2.
May be I'll draw the equilibrium point on - as a dotted line; and then we have an equilibrium point at 6,
and the equilibrium point here is unstable - make a mess of my figure here
- points here would get pushed away - and these, let's draw these a little better, - problem bridging the two -
So, these solution curves in purple, contain the same information in a sense, that this phase line does,
so we can see we have a stable fixed point at 2, and it's attracting,
and so a number of different orbits or solutions are all getting pulled in here,
anything between - anything less than 6 actually, gets pulled here,
and if we start a little bit above 6, we get pushed away.
So again, start with a differential equation - differential equation is a rule
for how X changes - that specifies the derivative for every X
and from that we can quickly figure out which way things are moving
- they're just moving to the right when this is positive, to the left when it's negative,
then we can get our fixed points, and we can see our stability right away,
and then we can take this plot for this plot and sketch some solutions.
Let me do one more thing to illustrate the relationship between these purple curves
- the solutions - and this, the phase line.
I'm going to take the phase line, and turn it on its side, so it's pointing up instead of sideways.
I do that.. here's the phase line... fixed point at 6... fixed point at 2...
points between 6 and 2 go to 2... below 2 go up to 2... and here they go away like this.
So if you put the phase line on its side, it tells you the direction that these purple solution curves are going,
so here the purple solution curves are going down towards 2, down towards 2;
here, they're going up away from 6; here they're going up towards 2.
Before moving on to the next sub unit, I suggest that you practice these ideas on the quizzes that follow this lecture.
These will give you a chance to become more familiar with these techniques and make sure you see how they're being used.