Let's summarize and review the key results and ideas
from unit 2, on differential equations.
A differential equation is a type of dynamical system,
and we've looked at differential equation of this form,
again, the derivative of some variable,
is just a function of that variable.
Remember, a dynamical system is a system that changes in time,
according to a well specified rule,
and a differential equation is such a dynamical system.
The differential equation specifies the derivative of X,
as a function of X - that's the rule,
and the derivative, which if you haven't had calculus,
might be a new term,
- is just the instantaneous rate of change of X.
It's how fast X is changing, at a particular moment, or instant in time.
There are three main classes, or types of solution methods for differential equations:
first, is qualitative, or geometric techniques.
- there, we might sketch the right hand side of the differential equation,
and from that we can figure out fixed points, and their stability;
we can draw a phase line, as I've done here,
and we can sketch the general shape of solution,
the directions in which the solutions are going.
We can't however get an exact form for X of t,
we don't necessarily know how fast the solutions go,
but this method is good for giving an overall feel
for the long-term behaviour of solutions,
- where do orbits go? how many fixed points are there?
and what are there stabilities?
Another approach to solving a differential equation, is: computational.
So, I presented Euler's method,
and one could use Euler's method, or a fancier version - one of these Runge-Kutta methods,
to figure out to X of t, and one does so, step-by-step.
We start with an X value - we use that X value to figure out the derivative,
the derivative tells us the X value a little while later,
and we can go back here, plug X in, the new X,
and figure out the new derivative,
- a derivative we can use to figure out the X value a little later,
and so on, and so we're constantly swapping back and forth
between these two sides of the equation.
I think Euler's method gets at the heart of what a differential equation is:
a rule specifying how a quantity changes.
Computational techniques like this are reliable
and they work for all well posed equations.
It does require the use of some computer software, or spreadsheet.
These results are often called numerical results
because the end result is a table of numbers, not a formula,
but it's very easy to plot that table of numbers,
and get a feel for what the solutions are doing.
The last type of solution method is
one that I actually have only hinted at so far,
and that's a type of method called analytic.
So, here, the task is to find a formula
for the solution X of t, using calculus.
So calculus - there's a well-developed machinery
for finding derivatives, doing derivatives backwards, and so on.
So, depending on your point of view this can be
a lot of fun, or not so much fun at all.
I myself had mixed feelings about it.
My first differential equations class in college
felt like all the worst parts of calculus 2 coming back to haunt me.
Later on, in grad school, learning some other techniques
for solving differential equations: power series, and Laplace transforms,
- I enjoyed a lot, but anyway, it's a very different approach from the first two.
It uses all the machinery of calculus.
The bad news is, is that most nonlinear equations,
and that's what we'll be studying in this course by an large,
cannot be solved analytically.
In some cases, one can even prove that there isn't an analytical solution,
so, geometric, or numerical and computational solutions are necessary.
Moreover, even for equations that can be solved analytically,
doing so does not always lead to intuition, or understanding.
You might do a bunch of calculus tricks, and get a weird-looking formula,
and that's certainly a valuable thing,
but it might not give you a feel, or intuition,
for what the equations are doing.
In my view, many certainly not all, but many differential equations text books
place too much of an emphasis on analytic techniques.
This course will focus on qualitative and computational solutions,
- I think they're much better suited for dynamical systems and chaos,
particularly for an introductory course like this.
OK, let me review some of the key terminology from this unit,
and it's actually almost the same as the terminology from the previous one.
So, differential equations, just like iterated functions,
have fixed points, and a fixed point is a point that doesn't change.
In differential equations one often calls a fixed point an equilibrium point,
but it means the same thing.
A fixed point X is fixed, if it's derivative is 0.
If your derivative is 0, you aren't changing,
and if you aren't changing, you're a fixed point.
Fixed points have stability as well - they can be stable, or unstable.
A fixed point is stable if near by points
move closer to the fixed point when iterated,
or, perhaps I should say, if you have an initial condition
near the fixed point, and you solve the differential equation
it moves closer to that - to the fixed point.
A stable fixed point is also called an attracting fixed point,
or an attractor, or in differential equations, it is sometimes called a sink,
because you can imagine a lot of solutions all head into this one point,
so it looks like water going down a drain.
A fixed point is unstable if near by points move further away from it,
and an unstable fixed point is called a repelling fixed point, or a repellor,
- it's also called a source: you can imagine a lot of solution lines
or solution curves emanating from this repelling fixed point.
So, source and sink - I don't think I'll use those terms much,
- but they're pretty standard, so you might encounter them elsewhere.
Just as we did for iterated functions,
we can draw a phase line for differential equations,
and it lets us see, all at once, the long-term behaviour for all initial conditions.
In a phase line we lose time information, so for example here,
I know that solutions move towards 9, but I don't know how fast.
The phase line here is for a differential equation that has an attractor at 9.
- things are getting closer to 9, and a repellor at 1,
- things are getting pushed away from 1.
So, differential equations are a type of dynamical system,
and in dynamical systems, one of the goals of study,
is to classify, and characterize, the sorts of behaviours that we see.
So, for differential equations, what have we seen?
- well we've seen: Fixed points (stable and unstable).
- Orbits can approach a fixed point.
- Orbits can tend towards infinity,
or tend towards negative infinity - they can move off the ends of the phase line,
and that's pretty much it,
and additionally, an orbit cannot increase and then decrease,
- it's rate of change is only a function of X - it's value,
so that means that cycles or oscillations are not possible.
In the next several units we'll see that
differential equations are capable of doing some more interesting things
but we'll have to go to higher dimensions
in order to see this more interesting and exciting behaviour.