The goal of this segment is to introduce
you to Ordinary Differential Equations
or ODEs; what they look like and how they
relate to dynamical systems. Much of what
I'll talk about today, is covered in
depth in my notes about ODEs which are
available on the webpage. Now, Ordinary
Differential Equation may sound scary, but
if you know what a derivative is you'll
have no trouble with this. An ODE express
some relationships between derivatives
of an unknown function like this. This is
a very simple ODE. Here, the unknown
function is x of t and the derivative is
taken with respect to time that's the dt
down there. You can take derivatives with
respect to other variables as you probably
saw in your calculus classes. You also
probably saw the shorthand notations that
people use for derivatives; x dot and
x prime. And often, btw people leave out
the independent variable from all of this.
I will do that a lot since we are
interested in dynamical systems-those that
vary with time and that means that my
derivatives are almost always with respect
to time. Now, to solve this ODE you play
Sherlock Holmes. We don't know x of t but
we know that its 1st derivative = 1. And
then we have to figure out what x of t
could make that statement true. Here, the
answer is pretty easy. x of t is all the
functions of time that have a slope pf 1.
All of these are functions that have a
slope of 1 so all of those lines satisfy
that differential equation. In functional
form these are all written like this.
So, to find the exact solution of an ODE
you also need to know something to pin
down which curve it is. What is x of t at
t=0 for example. Here's a slightly harder
example. Let's say that x double prime
that is the second derivative of function
x of t = minus the function x and lets say
we also know that at time equals 0, x is 1
This is an ODE. This is an initial
condition (IC). Now, Sherlock Holmes again
What function is the negative of its own
second derivative. Well, for example sine
or cosine. Which one does the initial
condition tell you it is. Cosine. Because
cosine of 0 = 1 whereas sine of 0 = 0. So
we know the the solution to this
differential equation is cosine of t. Now
there's lots more codified ways to solve
Ordinary Differential Equations, which you
learn in an ODE class. Our emphasis will
not be on analytic solutions like the ones
we just wrote down. Btw Analytic Solutions
are Closed Form Solutions. You can write
them down with a finite amount of pencil
symbols or chalk or whatever. The reason
that we're not gonna emphasize ODEs that
have an Analytic Solutions is because any
ODE that can be solved analytically is by
definition not chaotic, and I'll come back
to that. But first, I want to talk a lil
bit about why ODEs. They are very good at
capturing physics, chemistry, biology,
economics and so on and so forth. To show
you how this works I'm gonna go through a
quick example that you may remember from
your physics classes. It is a mass on a
spring. Now, the mass is m and you all
know that means there is a force down in
the amount of m*g pulling on the spring.
This particular spring is a very simple
spring. It pulls back on you or pushes on
you in proportion to how far you deform it
from its equilibrium position where it
wants to hang. Lets say the end of the
spring would be here if there weren't any
mass hanging on it. But its actually kind
of here because the mass is pulling on it.
This deformation is called x and if the
deformation is x then the spring is going
to pull that away in proportion to the
deformation x. Now, if kx up and mg down
are not in balance then the mass will
accelerate and the physics that describes
that is F=m*a. So, we can start writing
our equation now. We can say if m*g down
minus k*x up is not equal to 0 then the
mass is gonna move. And the last piece we
need in order to figure out the
Differential Equation for this is to look
at 'a' and realize that acceleration = the
first derivative of velocity which = 2nd
derivative of position and that means that
we're done. We can write this with
everything on the left hand side of the =
sign if we want like this. I can also
divide through by m to get the x double
prime term by itself like this. Now, this
equation actually looks an awful lot like
one of the examples we just did a couple
of minutes ago. It k = m = 1 and g = 0
then, this equation looks like this and we
already know what the solutions to this
equation are. They are sines and cosines.
Which one it is depends on where the mass
is at t = 0. Lets say its x=1, then the
solution looks like this-a cosine. Now
this is the position of the mass. What do
you think the velocity looks like. By
definition, the velocity is the derivative
of the position and the derivative of a
cosine is a sine. And this makes sense if
you think about being on a swing when you
are all the way out of the end, your
velocity=0 and then as you come back in
on the swing as you go through the very
middle of this swing where its hanging
down at the bottom, that's when you are
going the fastest. One last point here,
the way we have written things, the mass
will oscillate forever. In the real world
that does not happen. Instead the
oscillation will die out because of
friction and it dies out faster if the
friction is higher. So really there should
be another term in the differential
equation to model this. It should look
something like minus beta times velocity.
So the overall equation should look
something like this.