Section 8.3 is entitled “Qualitative Analysis.”
Spoiler alert: Solving a system of equations exactly is a
rare occasion indeed. Therefore, we’ll need tools to analyze our systems, as to
not leave them alone in a corner, allowing them to rot and decay.
As it turns out, we’ll probably be looking at a lot of
qualitative analysis in the future. I mean, the systems we can solve are merely
a handful of examples of an ocean full of systems of unsolvable equations. The
sooner we start, the sooner we finish, right?
Right. Probably. Maybe. I don’t actually know, but let’s
just get on with it.
Okay, so what we’re going to be talking about for this
section mirrors what we’ve been talking about in previous sections. Perhaps you
remember section 2.7, where we talked about exactness and uniqueness. Or
perhaps you remember the section we discussed equilibrium points and solutions
(I believe it was section 2.9, but I don’t have that section number memorized;
remembering 2.9 was pure dumb luck).
Let’s just jump right into the theorems, shall we?
“Suppose the function f(t,
x) is defined and continuous in the
region R and that the first partial derivatives of f are also continuous in R. Then given any point (t0, x0)
∈
R, the initial value problem
has a unique solution defined in an interval contained t0.
Furthermore, the solution will be defined at least until the solution curve t →
(t, x(t)) leaves the region R”
(348).
This nicely puts exactness and uniqueness into one theorem
for us, as to save paper (and our time). Also, you should notice that one of
the only differences between this theorem and the theorems for exactness and
uniqueness back in 2.7 is that the functions are vector valued.
An important geometric fact that arises from this theorem is
that two solution curves in a phase space for an autonomous system (i.e. x’ = f(x)) cannot meet at a
point until the curves coincide. However, this does not apply to non-autonomous
systems, so just remember what you’re talking about and when.
Here’s a theorem regarding exactness and uniqueness for
linear systems:
“Suppose hat A = A(t) is an n × n matrix and f(t) is a column vector and that the
components of both are continuous functions of t in an interval (α, β). Then,
for any t0 ∈ (α, β), and for any y0 ∈ Rn, the inhomogeneous system
with the initial condition
has a unique solution defined for all t ∈
(α, β)” (349).
If the coefficients are constants, then the domain of the
system is the entire real line. This means solutions to linear systems with
constant coefficients exist on all R.
Now, let’s move onto equilibrium solutions and points.
Let’s look back at our Lotka-Volterra predator-prey model
(if you haven’t read 8.2, then I encourage you to stop reading right about now
and go read my summary. You could also just Google it and find some webpage. I
don’t really mind what you do).
Recall that to find equilibrium solutions, we set our
right-hand side equal to zero and solve. For the first equation, you would get
F = 0 or A = a/b. In the phase plane, the solution set would be the union of
these two lines. This would be called the F-nullcline. Thus, the solution set
to the second equation would be the union of S = 0 and F = c/d. This solution
set would be called the S-nullcline.
An equilibrium point is where both of these equations are
equal to zero. These are the points that are in the intersection of the two
nullclines. Therefore, there are two equilibrium points: (0, 0)T and
(c/d, a/b)T. The equilibrium solutions are
Arbitrarily for an autonomous system x’ = f(x), there will be a vector x0 which f(x0)
= 0, which will be called an equilibrium point. The function x(t) = x0 will be called an equilibrium solution.
Here are some websites also summarizing stuff like this: http://www.sosmath.com/diffeq/system/qualitative/qualitative.html
I encourage you to look at nullclines. They’re actually
pretty cool. For now, I’ll leave you with 8.3, and 8.4 will be up sometime in the
future.
Also, I may or may not write up a pre-test section about
important things that will be on my test (supposedly stuff up until and
including 7.5). This might be a thing, but it might not. In any event, just
know that I probably won’t be summarizing any sections next week. I have to
study and prepare, rather than get ahead.
I’ll see you when I see you.
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