In this unit, I'll talk about fixed points and stability,two important concepts that we'll use throughout the course.But first I should mention that iterated functions are an example of a dynamical systemand the course is about dynamical systems so this is important,so I should say what dynamical systems are.I'll come back to this again, but for nowa dynamical system is a system that evolves forward in time according to a well defined and unchanging rule.And that's what we have with iterated functions:so we have a value, a number, that moves forwardchanges from iterate to iterate according to a well-defined rule.That's a function, a nice deterministic function: the same input gives the same output every timeand the rule doesn't change as we iterate.We just do the same thing over and over againusing the output of one step for the input for the next.So, iterated functions are a dynamical system.In the study of dynamical systems, we're often not particularly interested in the numbers of a particular orbit or itinerary,instead we're interested in understanding collections of orbits, many orbits all at once.More generally, we'd like to understand what sort of behaviour do we seein different types of dynamical systems.So this is the approach we'll take in this courseand we'll start that in this unit when we look at fixed points and stability.Let's start with an example.We'll consider again the squaring function: f(x)=x^2We know how to calculate an orbit for this functionwe just start with the number of the seed, in this case it's 1.1and then we square to get the next iterate 1.21we square that to get 1.46, and so on.We could choose a different seed, and we would get a different orbitlet's say instead of 1.1, I chose 1.2. I do the same thing, square again and again to get that orbitSo here's 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 chose a seed of 0.9I put 0.9 here, what will happen?Let's apply the function and see.For this initial condition the number gets smaller.It's getting closer and closer to 0.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'll square again and again to get the orbit.So again, when we square a number that's between 0 and 1 it gets smaller.Here we can see the numbers getting closer and closer to 0.So calculating the orbit, a particular orbit for a particular seedis 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'll use some graphical techniques that will help us see that betterand understand the function all at once.First, let me plot the time series plots. Let me show you the time series plots for these four orbits. 1.1, 1.2 they get bigger, for 0.9, 0.8 they get smaller.So here is the time series plot for that.You can see I have 4 different initial conditions, a square, a triangle, a circle and a diamondThe square is 1.2, you can see that's growing quite fast and goes off the graphThe triangles are 1.1, those grow. The circles are 0.9 and those are getting close to 0And 0.8 also gets close to 0.So this lets us see that numbers larger than 1 will get larger and largerand numbers between 0 and 1 we suspect, will get closer and closer to 0.