Hi everyone. Welcome to unit 2 of the course.
In this unit I'll introduce another type of dynamical system: differential equations
but before I do I want to say a bit more about iterated functions.
I'll quickly review some of the ideas from last unit,
and more importantly, I'll underscore some concepts in dynamical systems
that will be useful as we start thinking about differential equations.
So, let's get started.
So, let's return again to iterated functions,
and I want to use iterated functions to introduce a little bit of new terminology and language
that will be useful when we look at differential equations.
So, an iterated function is a function that's turned into a feedback loop.
So, for a concrete example: suppose that the function is 1/2 X + 3
and I choose a seed of 2
Then, what's the first iterate? Well, I need to apply the function to the number
So, 1/2 of 2 is 1 , 1 + 3 is 4
What's the next value? X2, the second iterate. The function tells me.
I apply the function to 4
A 1/2 of 4 is 2 , 2 + 3 is 5
and I can keep going, you're familiar with this process now,
and I can form the itinerary.
So the function determines the itinerary.
So let me write this in a slightly different way.
So, in this notation, this tells me that the next X, the next value in the orbit
is equal to the current value Xn after the function is applied to it.
So, I can think of this equation, which really says in symbols what this picture up here says in picture,
that the function gives you the next iterate, or orbit,
You can think of the function as a rule applied again and again that specifies the orbit or the itinerary.