Consider the function . Sketch this function and plot the first few points of its
orbit , i.e. plot the points , , etc...,
for the starting values , and . Describe what
is happening to the orbit of
for each of these starting values.
Solution:
:
The orbit of under is . The point
is a fixed point of .
:
The orbit of under
is .
The iterates of are
getting smaller in magnitude on each iteration and approaching zero.
:
The orbit of under is . The
iterates of are getting
larger in magnitude on each iteration and approaching infinity.
Let .
Find all of the fixed points of .
Solution: To find the fixed points we need to solve . This has one solution,
, so has just one fixed point at .
Consider the family of functions defined by where is a constant and . Determine all of the fixed
points of .
Hint: You may end up with different fixed points depending on the
value of .
Solution: To find the fixed points of we solve , treating as a constant. For most values of , this has one solution, , however we can see that when
, then all are solutions. Thus,
the fixed points of are
, and , . Notice that when , , for which all are clearly fixed
points.
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Consider the function . Sketch
and on the same set of axes
and show graphically that has
two fixed points. Label these fixed points on your sketch as and such that .
Use a graphical method (i.e. cobweb diagram) to help determine
the behaviour of various orbits starting near both and . Use your diagram to make an
educated guess as to the nature (attractive, repelling, or neither) of
each fixed point.
Now consider the family of functions where is a constant. For what values of do fixed points of exist? Some sketches of the graphs
of for various values of
may help, but they are not
necessary.
Solution:
From the graph, we can see that intersects the line twice, and thus has two fixed
points.
Cobweb diagram:
From our cobweb diagram we see that the iterates of are attracted towards the fixed
point , so we can guess
that this is an attractive fixed point. On the other hand, the iterates
of move away from the fixed
point , so this is likely
to be a repelling fixed point.
To find the fixed points of we need to solve . Using the
quadratic formula, this has solutions . This has (real) solutions only when the argument of the
square root is greater than (or equal to) zero, i.e. Thus, has fixed points when . Notice that when , has one fixed point (only one
solution to the quadratic formula) and when , has two fixed points (two
solutions to the quadratic formula).
Let . Find all
fixed points and periodic points of period two of .
Solution: First, let’s find any fixed points by solving
. This has just
one solution, , so has one fixed point at .
Now, let’s find any periodic points of period two. We need to solve
. So we need to solve . Rearranging, this
gives ,
which has three solutions and . Since is a fixed point, the remaining
two points must form a two cycle. Thus, and are the points of period two of .
CHALLENGE Let . Find all fixed points and periodic points of period
two of .
Solution: First, let’s find the fixed points. Using the
quadratic formula, this gives two solutions, , which are the two fixed
points of .
Next, we want to solve for any periodic points of period two. First
we find , and then solve . Factoring the cubic part of this expression
could be difficult, but luckily we know that the fixed points of are also solutions to . This means
that must
be a factor. Thus we have The two new solutions (which are not fixed
points of the original function) are and . Thus we must
have the two cycle .
CHALLENGE Consider the function . Show that has an infinite number of fixed
points.
Solution: To find the fixed points of we must solve Considering the
unit circle (or a graph of ), we see that this is true when
is an odd multiple of degrees, i.e.
We can write this as is
a fixed point of when for any
integer . Since the set of all
integers is an infinite set, this results in an infinite number of fixed
points for .