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 k<i>x up and m</i>g 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.