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 system in 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 forward in time according
to a well def find an unchanging Rule
and that's what we have with iterated
functions so we have a value a number
that moves forward changes from iterate
to iterate according to a well-defined
rule that's the function a nice
deterministic function the same input
gives the same output every time and the
rule doesn't change as we iterate we're
just 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're
often not particularly interested in the
numbers of a particular orbit or
itinerary instead we're interested in
understanding uh collections of orbits
many orbits all at once and more
generally would like to understand what
sort of behavior U do we see in
different types of dynamical
systems so this is the approach we'll
take in this course and 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
ofx =
x^2 we know how to calculate an orbit
for this function we just start with a
number the seed in this case I chose 1.1
and then we square square 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 chose
1.2 I'd do the same thing Square again
and again to get that
orbit so here's the orbit for the seed
1.2 note that both orbits get larger
when you square a number larger than one
the number gets larger so these numbers
will continue to grow let's try another
one suppose I chose a seed of
0.9 I put 0.9
here what will happen well let's apply
the function in
C for this initial condition the number
gets smaller it's getting closer and
closer to zero if you square a number
between zero and one 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 zero and one it gets
smaller here we can see the number is
getting closer and closer to
zero so calculating the orbit a
particular orbit for particular seed is
not too difficult it just requires a
little bit of calculator work but it
doesn't let us uh get a sense of the big
picture what does this function
do so um we'll use some graphical
techniques that'll help us see this
better and 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 0.9 0 8 they get
smaller so here's the time series plot
for that you can see I have four
different initial conditions a square a
diamond uh sorry square a triangle a
circle and a diamond the square that's
1.2 we can see that growing quite fast
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 that numbers larger
than one will um get larger and larger
and numbers between zero and one we
suspect will get closer and closer to
zero