I'd like to conclude this first unit by summarizing what we've done so far.
The main topic of this unit is Iterated Functions.
I begin by introducing or reminding you about functions. A function is a rule that takes a number as an input and then outputs another number.
I've encouraged you to think about a function as an action.
A function is something that does something to a number, and outputs a new number.
This is a standard picture for that: x is the standard input, something happens inside this box
- this action is called f, and it produces a new number called f of x, x after f has done its "thing".
The functions that we will study are deterministic. That means that the output is determined only by the input,
and if you put the same input in the function several times, you'll get the same output everytime.
Then, we studied iterating functions. So we iterate a function by turning it into a feedback loop:
we took a static situation, a function, and we made it dynamic by looping it around.
So we start with some number, we apply f to it, we get a new number, then we use that number as input, apply f again and so on.
An iterated function is an example of a dynamical system, a system that evolves in time according to a well-defined unchanging rule.
We iterate a function by applying it again and again to a number - it's a repetitious process.
The number we start with is known as the seed or the initial condition, and is usually denoted x-naught or x-zero.
When you iterate a function, you get a sequence of numbers, and this sequence is called an itinerary or an orbit.
It's also sometimes called a time series or a trajectory.
Individual iterates are denoted by x-t, where t is the time step.
For example, x-5 would be the fifth iterate.
A useful way to visualize an itinerary is with a time series plot.
So here's a time series, I plotted this itinerary and we can see that the function increases
and then approaches what appears to be a fixed point around 4.3.
A fixed point of a function is a number that is not changed when the function acts on it.
For example, 1 is a fixed point of the function f of x equals x squared, because f of 1 is 1, 1 squared is 1.
More generally, x is a fixed point if f of x equals x;
if you input x to the function and what you get out happens to be the same thing you put in.
There're different types of fixed points: stable and unstable.
The fixed point is stable, if nearby points move closer to the fixed point when they're iterated.
A stable fixed point is also called an attracting fixed point, or an attractor.
A fixed point is unstable, if nearby points move further away from the fixed point when they're iterated.
An unstable fixed point is also called a repelling fixed point or a repellor.
The stability of fixed points can be summarized graphically in these pictures.
So here's a stable or attracting fixed point - the fixed point is this point here on the line
- and this shows that nearby points are pulled towards it. And it's an attractor.
And here's an unstable fixed point. Here's the fix point, nearby orbits are pushed away from it;
it's unstable or repelling. This picture can help us think about it.
A stable fixed point is like a marble on the bottom of a bowl: if we push this to either side, it returns to its original position.
An unstable fixed point is like a marble perched on top of a ball.
If we push it a little bit, it moves to the right or to the left and does not return to the fixed point.
Usually when studying dynamical systems, we 're interested in stable behavior, as this is what we're most likely to observe.
Unstable behavior doesn't last long.
The phase line lets us see all at once the behavior of all initial conditions.
In a phase line, time information is not included. We can tell what direction the orbits go, but not how fast they go.
An example, here's the phase line for the squaring function. This phase line shows that seeds larger than 1 tend towards infinity,
1 is an unstable fixed point, seeds between 0 and 1 approach 0, and 0 is a stable fixed point.
In dynamical systems, one of the main goals is to classify or characterize the types of behavior seen in particular classes of dynamical systems.
So far, we've examined one class or type of dynamical system, iterated functions, and we've seen the following sorts of behaviors:
fixed points, both stable and unstable - we've seen that orbits can approach a fixed point, get pulled towards an attractor,
orbits can keep on growing, they can tend towards infinity, get larger and larger
and they can also tend towards negative infinity, they keep getting more and more negative, moving more and more left on the number line.
But this isn't the end of the story. We'll encounter more types of behavior soon.
So at this point, I hope that iterated functions seemed - if not yet simple - then certainly simple-minded and definitely repetitious.
All iterated functions involve is choosing a number and then applying a function to that number
over and over and over and...over again, and see what happens.
It's a pretty simple process really, but we will see in the weeks ahead that this simple repetitious process
is capable of producing some surprising and complex results.
So this brings Unit 1 to an end. A reminder that I encourage you to practice these ideas and techniques on the quizes and in the homework.
If you have any questions about the quizes or homework, or any other course material,
please post your questions in the forum on the course website.
See you next week in Unit 2.