In this unit, I will talk about fixed points
and stability.
Two important concepts we will use
throughout the course.
But first, I should mention
that iterated functions are an example
of a dynamical system
and the course is about dynamical systems.
So, this is important.
So I should say what dynamical systems are.
And I'll come back to this again.
But, for now:
Dynamical system is a system
that evolves in forward forward in time
according to a well defined and unchanging rule.
And that is what we have for iterated functions.
So, we have a value, a number, that moves forward
changes from iterate to iterate.
According to a well defined rule.
That is the function. A nice deterministic function.
The same input gives the same output each time.
And the rule doesn't change as we iterate.
We are doing the same thing over and over again.
Using the output of one step
for the input of the next
So, iterated functions are a dynamical system
In the study of dynamical systems,
We are often not particularly interested in the numbers
of a particular orbit or itinerary.
Instead we are interested collections of orbits,
many orbits all at once.
And more generally we would like to understand
What sort of behaviors do we see in different types
of dynamical systems?
So this is the approach we will take in this course
and so we will start that in this unit
When we look at fixed points
and stability
Let's start with an example.
We will consider the squaring function.
F(x) = x^2.
We know how to calculate an orbit for this function.
We just start with the number, the seed.
In this case I choose 1.1.
And then we square it to get the next iterate.
1.21.
We square that to get 1.46.
And so on.
We could choose a different seed.
And we would get a different orbit.
Let's say instead of 1.1, I choose 1.2.
I would do the same thing
Square again and again.
To get that orbit.
So here is the orbit for the seed 1.2.
Note that both orbits get larger
when you square a number larger than 1.
The number gets larger.
So these numbers will continue to grow
Let's try another one.
Suppose I choose a seed of 0.9.
I'll put 0.9 here.
What will happen?
Well let's apply the function and see.
For this initial condition,
the number gets smaller.
It's getting closer and closer to zero.
If you square a number between 0 and 1,
It gets smaller not larger.
Let's try one more initial condition.
One more seed.
I'll try 0.8.
And I will square again and again to get the orbit.
So again when we square a number between 0 and 1,
it gets smaller.
Here we can see the number gets close and closer to zero.
So, calculating the orbit
a particular orbit for a particular seed is not too difficult.
It just requires a little bit of calculator work.
But it doesn't let us get a sense of the big picture.
What does this function do?
So we will use some graphical techniques
that will help us see this better
and understand the function all at once.
First, let me plot the time series plots
for these four orbits.
1.1 and 1.2 they get bigger.
0.9 and 0.8 they get smaller.
So here is the time series plot for that.
You can see I have four different initial conditions.
A square, a diamond --sorry
-- a sqaure, a triangle, a circle, and a diamond.
The square that is 1.2.
We can see that growing quite fast.
It goes off the graph.
The triangles are 1.1
Those grow.
The circles are 0.9 and those are getting close to zero.
And then 0.8 also gets close to zero.
So this lets us see
the numbers larger than 1 will get larger and larger
And numbers between 0 and 1
we suspect will get closer and closer to zero.