Grade 11/12 Math Circles
Dynamical Systems and Fractals Part 1
Discrete-Time Dynamical
Systems
Put simply, a dynamical system is any system which changes
over time. The study of dynamical systems is an important part of
applied mathematics and allows us to understand and predict the
long-term behaviour of many physical processes. When time is measured in
discrete units (as opposed to continuous units) then we have what is
called a discrete-time dynamical system. As a simple example,
consider computing the compound interest on an investment (compound
interest means that the interest earned is based on the current value of
the investment). If the interest is compounded (added) annually, then it
makes sense to define as the
value of the investment in year .
If we start with some initial amount of money, , after one year elapses the value of
the investment will be where is the interest rate. Similarly, after
a second year elapses, the investment will be worth Are you starting to
see a pattern? After years, the
investment is worth If we define a function, , then we can write a model
for the value of the investment after years as . The value of the
investment in year depends on the
value from the previous year, year , and this dependence is defined by
the function .
This was a fairly simple example. There are many processes, both
natural and man-made, which can be modelled as discrete-time dynamical
systems. A few common examples include: population growth of plants and
animals measured annually, radioactive decay, and bacteria growth. In
order to study the dynamics of a system governed by the relation , we must understand the
behaviour of the function when
it is applied iteratively (multiple times) to some initial state.
Iterations of
Functions
Let’s consider a function .
We can think of functions as "machines" which map inputs to outputs.
If we want to model the long term behaviour of a discrete-time
dynamical system defined by the relation then we will need to
consider applying multiple
times, or iterating .
Given some initial state (initial input) , the iterates of under the function are defined as follows.
Definition (Iterates of )
Let be a function such that
is in the domain of . Then we can define the
following:
is the first iterate of under .
is the second iterate of under .
is the nth iterate of under .
The collection of these iterates is
called the orbit of under .
The orbit of a function can look very different depending on the
starting value that we choose. Let’s look at some examples!
Example 1
Consider the function
with initial value .
The first iterate is .
The second iterate is .
The third iterate is .
Continuing this process, the orbit of under is .
We see that the iterates of
continue to get smaller and smaller, eventually approaching zero.
Example 2
Once again, let , and
this time consider the initial value .
The first iterate is .
The second iterate is .
The third iterate is .
Continuing this process, the orbit of under is .
The value is what is called
a fixed point of .
Example 3
Finally, let’s consider the initial value , again with .
The first iterate is .
The second iterate is .
The third iterate is .
Continuing this process, the orbit of under is
We see that the iterates of
continue to get larger and larger, eventually approaching infinity.
Fixed Points
In the previous section, we saw that a specific input value, namely
the value , caused the orbit
of the function to get
"stuck" repeating a single value over and over. This type of a special
value is called a fixed point of the function .
Definition (Fixed Point)
Let be a function and be in the domain of . Then is a fixed point
of if .
A function may have multiple fixed points, one fixed point, or no
fixed points at all. We can find the fixed points of the function by solving the equation for all possible
solutions which are in the
domain of .
Example 4
Let’s go back to the function from our previous examples. We know that this function has
at least one fixed point since . Let’s find out if there are any others!
Set and
solve. This has two
solutions: and , both of which must be
fixed points of .
Let’s check!
Now it’s your turn!
Exercise 1
Find all of the fixed points of the function .
Exercise 1 Solution
Set and
solve. This has two
solution: and , both of which must be
fixed points of .
Let’s check!
Finding the fixed points of
is equivalent to finding the intersections of the function with the line . Let’s look once again at the
function which we now
know has two fixed points at and . These
two fixed points correspond to the two intersections with the line seen in the figure below.
When studying the dynamics of a system governed by the function , we are often interested in how the
iterates of behave near its
fixed points. Are they attracted towards a fixed point? Do they move
away from the fixed point, never coming close to it? Or maybe they
bounce around a fixed point, but never touch it? Using a figure like the
one shown above, we can use a graphical approach to help determine the
behaviour of some iterates of .
First, let’s choose a starting point near the fixed point . To find the first iterate
we simply draw a straight line up from to the point . To find the next iterate,
we need to determine where lies
on the -axis, so we draw a line over to the line which gives us the point . Drawing a line straight down
will intersect with the point
on the -axis, and from here we can find the point . Continuing the process, we
find , , and so on, as shown in the figure
below.
We can repeat this process using several starting values near the
fixed point which
results in the next figure. This is called a cobweb
diagram.
The cobweb diagram shown above indicates that iterates near the fixed
point tend to
approach . We might say that the
iterates are attracted to the fixed point .
What about the other fixed point ? Using the same process, we
can draw a cobweb diagram for points near .
We see from our diagram that when we start near the fixed point the iterates move away from
the fixed point. On the left hand side the iterates get smaller and
smaller, once again approaching the fixed point , whereas on the right hand
side, the iterates get larger and larger, moving away from both fixed
points. We say that the iterates are repelled away from .
Attractive and Repelling
Fixed Points
In the previous section we saw an example of both an attractive and a
repelling fixed point. Let’s formalize what this means, starting with
the definition of an attractive fixed point.
Definition (Attractive Fixed Point)
A fixed point is an
attractive fixed point of if there exists an interval
surrounding , say with , such that for all
, the iterates of approach in the limit as approaches infinity (). Put in simple
terms, this means that as gets
large, the iterates get
arbitrarily close to .
This could look something like this:
or like this:
Example 5
Once again we look at . We saw graphically that the fixed point at is an attractive fixed point. To
show that this is true, let’s define the interval which clearly contains
the point . Notice that all
satisfy .
This means that if we choose any , .
Repeating this argument, we see that , i.e. the iterates are decreasing in magnitude towards
the fixed point .
Technically, a bit more work needs to be done to show that the
sequence does in fact converge to the fixed point , but that’s beyond the scope of this
lesson.
Now, what about repelling fixed points?
Definition (Repelling Fixed Point)
A fixed point is a
repelling fixed point of if there exists some interval around
, say with , such that for all
, , we have that . This
means that the function, , maps
the point further away from the
fixed point .
Notice how the definition of a repelling fixed point only depends
on the behaviour of the first iteration of , NOT the long term behaviour.
Example 6
Let’s look at one
last time. Our cobweb diagram indicated that the second fixed point at
is a repelling fixed point. To
show that this is true, let’s define the interval which contains . For any , we have Therefore is a repelling fixed point.
Periodic Points
Another interesting behaviour we might come across when studying the
iteration dynamics of a function are orbits which exhibit periodic
behaviour, effectively jumping back and forth between a finite
number of values. Let’s look at an example!
Example 7
Consider the function and the initial value . We can see that and . Therefore the orbit of under is . We
say that the pair of points is a
two-cycle of .
Definition (Periodic Points)
The point is a
periodic point of period of if the following is true
If is a periodic point of
period , then the set is called an -cycle of .
We can use the first bullet point in the definition above in order to
solve for the periodic points of period of a function. Let’s take another look
at the function .
Example 8
Let’s say that we want to find all of the two cycles of the function
. This means we
need to find the fixed points of . But what is ?
To find the fixed points we solve , i.e. . This has
solutions and
. The point is a fixed point of the
original function (we could
also call it a periodic point of period one) and the points form a two-cycle, as
we saw previously.
Now it’s your turn!
Exercise 2
Given that ,
find the periodic points of period two of .
Hint: You may want to find the fixed points of first.
Exercise 2 Solution
First, find the fixed points of by solving . This has solutions and . so these must be our fixed
points.
Now we want to solve for the fixed points of . Setting gives . This is true for all
values of . Does this mean
that all values of are
periodic points of period two of ? Almost!
Since the points
and are fixed points
of , they cannot also be
periodic points of period two. We also need to be careful and consider
the domain of , which excludes
the point (since is undefined when
). This means that cannot be a periodic point of . What we are left with is that all
except for , and are periodic points of period two of
. We could write the set of
periodic points of period two of as .