This document explores iteration of the quadratic logistic map equation f(x) = ax(1-x) through various methods. It finds the fixed points and critical point of the equation. Cobweb plots are used to visualize the behavior of iterations for different values of a and x0, finding cases where the fixed points are attractors or repellers. A Feigenbaum diagram shows the emergence of periodic cycles and eventually chaos as a increases. While the behavior is predictable for a < 3, beyond this value it bifurcates into multiple branches representing complex periodic behavior before descending into chaos.

1. Iteration of Quadratic Equations - Draft
Thomas Jeffs
March 17, 2016
1 Introduction
Iteration lies at the heart of many concepts and methods in mathematics. For example, it can be used to solve
equations in image processing, fractal generation, Fibonnaci sequences, and many algorithms of all sorts.
Previously we explored iterating through a linear function of the form fpxq “ ax ` b, where a represented
the slope and b represented the y-intercept. In that case, we found that a had little effect on the convergence
or divergence of the series of iterations. However, x0 (the chosen initial value) was the deciding factor in
determining the long term behavior of the iteration sequence. In this article, we will explore iteration again,
but this time we will examine a quadratic function. Recall that quadratic functions are functions that follow
the form
fpxq “ ax2
` bx ` c (1)
We will be exploring a specific quadratic formula known as the logistics map, which is used to model
populations with a carrying capacity with a rational population rate:
fpxq “ axp1 ´ xq (2)
This form will help us demonstrate how why quadratic functions are more complicated and how even a
seemingly simple quadratic function can become chaotic very quickly.
Quadratic iteration is more complicated because it has the possibility of having multiple fixed points.
Recall, that ξ can be called a fixed point of fpxq if and only if fpξq “ ξ. The maximum number of fixed
points in a function fpxq is determined by the highest exponent of x. For instance, fpxq “ ax2
`bx`c has a
maximum of two fixed points. While it’s technically correct to say that it will have exactly two fixed points,
it’s possible in some cases, that the two fixed points are the same, thus leaving only one fixed point.
2 Results
2.1 Initial Findings
We begin the exploration by finding some of the more common, yet critical, values for the logistics map. To
find the points where the function fpxq “ axp1 ´ xq crosses the x-axis, we set the function equal to zero and
solve for x.
fpxq “ axp1 ´ xq
0 “ axp1 ´ xq
This leaves us with two factors, ax and p1 ´ xq, next we set each of these equal to zero and solve.
ax “ 0 1 ´ x “ 0
x “ 0 x “ 1
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2. Therefore for the quadratic equation fpxq “ axp1 ´ xq, intercepts the x-axis at x “ t0, 1u. Because we want
to observe the behavior of the logistics map as iterates, we will focus on the domain x “ r0, 1s
Next, we will find the critical points of the logistics map. Recall that critical points are points on a curve
with a gradient of zero, that is Bf
By “ 0, however, because the logistics map is defined only in terms of x, this
results in f1
pxq “ 0. In the case of the logistics map, there is only one curve, so we will only see one critical
point.
fpxq “ ax ´ ax2
f1
pxq “ a ´ 2ax
0 “ a ´ 2ax
0 “ ap1 ´ 2xq
0 “ 1 ´ 2x
x “
1
2
Therefore, x “ 1
2 is the only critical point for the logistics map. Next, we’ll find the fixed points ξ of the
logistics map.
To find the fixed points of the logistics map, we set the functions fpξq “ ξ and solve for ξ.
fpξq “ aξp1 ´ ξq
ξ “ aξp1 ´ ξq
ξ “ aξ ´ aξ2
0 “ pa ´ 1qξ ´ aξ2
0 “ rpa ´ 1q ´ aξsξ
Setting each term equal to zero allows us to see the fixed points of the general equation.
ξ “ 0 pa ´ 1q ´ aξ “ 0
aξ “ pa ´ 1q
ξ “
pa ´ 1q
a
For this quadratic function fpxq, the fixed points will always be:
ξ “ t0,
a ´ 1
a
u (3)
Lastly, we will attempt to define a general equation for the iteration of the logistics map. Using the
abbreviated form for iteration, xn “ apxn´1 ´ x2
n´1q,
x0 “ x0
x1 “ apx0 ´ x2
0q x1 “ apx0 ´ x2
0q
x2 “ apx1 ´ x2
1q x2 “ arapx0 ´ x2
0q ´ a2
px0 ´ x2
0q2
s
x3 “ apx2 ´ x2
2q x3 “ ararapx0 ´ x2
0q ´ a2
px0 ´ x2
0q2
s ´ a2
rapx0 ´ x2
0q ´ a2
px0 ´ x2
0q2
s2
s
expand x3 x3 “ ara2
r´ax4
0 ` 2ax3
0 ` p´a ´ 1qx2
0 ` x0s ´ r´ax4
0 ` 2ax3
0 ` p´a ´ 1qx2
0 ` x0s2
s
After just 3 iterations, it’s clear that the general solution for xn would be far too complex to evaluate
easily. In order to observe the behavior of the iterations over time, we’ll use a different method. We will use
a cobweb plot. A cobweb plot is a special tool which displays the values of an iteration sequence as they
appear on the graph of the function itself. The graph is developed by graphing the function fpxq, along
with the line y “ x. Lastly, for each iteration value we plot a line joining segments px0, 0q to px0, x1q, then
px1, x1q to px1, x2q and so on until observations can be made.
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3. a x0 Limit ξ “ 0 ξ “ a´1
a
0 .5 x “ 0 Attractor Repeller
0.5 .5 x “ 0 Attractor Repeller
1.0 .5 x « 0 Attractor Repeller
1.5 .5 x « 0.35 Attractor Repeller
2.0 .5 x “ .5 Repeller Repeller
2.5 .5 x « .66 Repeller Attractor
3.0 .5 x « .66 Repeller Attractor
3.5 .5 Divergent Repeller Attractor
4.0 .5 x “ 1 Repeller Repeller
Table 1: Observations of the Cobweb plots in the ranges a “ r0, 4s and holding x0 “ 0.5.
2.2 Cobwebs
Now that we have some important values for the logistics map, we’ll start to evaluate how the function
behaves as we iterate the function using chosen initial values for x0 and a. We will use cobweb graphs to
help illustrate how the iteration behaves over time. In order to understand the results, it’s important to
understand the following definitions:
Definition Attractor - A set of numerical values toward which a system tends to move towards, for a range
of system conditions. See Figure 1 for an example of an attractor.
Definition Repeller - A set of numerical values which a system tends to move away from over time, for a
range of system conditions.
In our case, we will be examining the attraction or repulsion from the fixed points, ξ “ 0, a´1
a , using
different initial values for a and x0.
As we explored the cobweb graphs of several different values, we found that the value for x0, inside the
domain of x0 “ r0, 1s, didn’t have an effect on whether or not the iteration would converge, or diverge.
Instead, the value of x0, only affected how quickly the iteration sequence would converge or diverge to its
attraction/repellent points. When x0 was close to the attracting fixed point, it would use fewer iterations to
converge. However, when x0 is outside of the specified domain, the iteration will diverge.
The results lead to an exploration of when a fixed point becomes an attractor or repeller. Upon exami-
nation, the attraction property is based on the slope at the fixed point. To find the slope, we evaluate the
derivative of fpxq at the fixed points ξ “ 0, a´1
a .
fpxq “ ax ´ ax2
f1
pxq “ a ´ 2ax
f1
p0q “ a ´ 2ap0q f1
p
a ´ 1
a
q “ a ´ 2ap
a ´ 1
a
q
f1
p
a ´ 1
a
q “ a ´ 2pa ´ 1q
f1
p0q “ a f1
p
a ´ 1
a
q “ 2 ´ a
As we can see, the slope at the fixed point x “ 0 is reflected as the value of a, while the slope at the fixed
point x “ a´1
a follows the linear equation m “ 2 ´ a. Focusing on the slope at the non-zero fixed point,
we can see that when 1 ă a ă 3, the fixed point will be an attractor. This contradicts the observations
that we made in Table 1, where the non-zero fixed point didn’t show attraction until a “ 2.5. This is one
of the limitations of the cobweb graph, the math shows conclusively when the non-zero fixed point will be
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4. Figure 1: Cobweb graph were a “ 2.9 and x0 “ 0.45.
an attractor, but the cobweb graph can’t show the attraction property unless the value of x0 is sufficiently
far enough away from the fixed point to demonstrate attraction or repulsion. We can further prove that the
a “ r1, 3s will attract to the non-zero fixed point by proving that any attracting fixed point has a specified
interval:
Claim: If |f1
pξq| ă M ă 1, Then ξ is an attractor. Where M is a constant.
Proof:
lim
xÑξ
|
fpxq ´ fpξq
x ´ ξ
| ă M ă 1
lim
xÑξ
|
fpxq ´ ξ
x ´ ξ
| ă M ă 1
This implies, by the definition of limit, that there is an interval[I] of values containing ξ, such that the
equation is true. Thus,
|
fpxq ´ ξ
x ´ ξ
| ă M
|fpxq ´ ξ| ă M|x ´ ξ|, for all x in I
˝ QED
As defined, M ă 1, meaning that the two sides will get closer and closer as time moves on. Thus showing
that the fixed point ξ will be an attractor.
2.3 K-cycles
Table 1 showed an interesting anomaly around a “ 3.5, the cobweb graph appeared to be divergent, because
the ”legs” of the graph didn’t seem to be moving towards a single value. This behavior is defined as k-cycles,
where k represents the number of values that the iteration sequence seems to be ”settling” on. An example
of a 3-cycle can be seen in Figure 3, it’s clear that the graph isn’t moving towards one single value. Instead,
the sequence has ”settled” on 3 values that continue to recur in the iteration sequence. In the case of Figure
3, the value of a “ 3.835 produces a sequence with resulting values of
fp3.835q “ r¨ ¨ ¨ , 0.4945144, 0.9586346, 0.1520743, 0.4945144, 0.9586346, 0.1520743, ¨ ¨ ¨ s
As we’ll see later, this 3-cycle is a bit of a unicorn.
4

5. Figure 2: Cobweb graph were a “ 0.75 and x0 “ 0.5.
Figure 3: Cobweb graph were a “ 3.835 and x0 “ 0.5, this is an example of a 3-cycle.
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6. Figure 4: Feigenbaum diagram for a “ r0, 4s.
Figure 5: Feigenbaum diagram for a “ r2.8, 4s.
2.4 Feigenbaum
To explore the number of k-cycles each possible a value can produce, and to see a clearer picture of what
happens to the quadratic iterations after a “ 3 we turn to what’s known as a Feigenbaum diagram. This
diagram displays the values of x versus the value of a.
Figure 4 shows the Feigenbaum diagram for our logistics map. The diagram helps us to see that in certain
intervals, the behavior is very predictable. As you can see, the behavior meets our expectations right up
until a “ 3, then it bifurcates into two branches. A short time later, at around a “ 3.4 it bifurcates again.
This continues until there are so many branches that it’s impossible to determine what’s going on, this is
called Chaos. It’s clear that there are 3 distinct areas of the diagram based on the range of a.
In the range a “ r0, 1s, we see that the value of x is constant, in fact, it’s constant x “ 0. If we look back
at the original logistics map iteration sequence, we can see that for values of a ă 1, the sequence coefficient
a gets smaller and smaller. As we get further into the iterations, the resultant number gets smaller and
smaller. Also, we can observe the slope at the fixed point f1
p0q “ a, meaning that when ´1 ą a ą 1, then
the fixed point x “ 0 is an attractor.
The next interval a “ r1, 3s behaves exactly as expected. As we defined earlier, the non-zero fixed point
is an attractor when a falls inside this interval. Thus, each point of convergence is mapped following the line
that represents x “ a´1
a , hence the curved shape.
The last interval is the most interesting, a “ r3, 4s is the beginning of the k-cycle period, shown in Figure
6

7. Figure 6: Feigenbaum diagram for a “ r3.825, 3.85s.
5. Each distinct point represents a value for x for each given a value. Traveling up the graph, it’s possible
to see how many cycles a chosen value of a will produce. For example, if we select a “ 3.1, we should see
that it is a 2-cycle and that it will settle on two values over time.
fp3.1q “ r¨ ¨ ¨ , 0.7645665, 0.5580141, 0.7645665, 0.5580141, 0.7645665, 0.5580141, ¨ ¨ ¨ s
As expected, the values for x bounce between 0.7645665 and 0.5580141.
Looking further along the axis, we see a point where the bifurcation seems to fold back in on itself, this
section has been highlighted in Figure 6. This is where our magic unicorn of the 3-cycle can be found. As we
know, bifurcation is basically a function line splitting into two directions. So, you might think that given a
number of bifurcations, you shouldn’t see any odd sets. But this figure demonstrates a set of 3 and 5-cycles.
3 Summary
As we’ve seen, even seemingly simple quadratic equations can produce very unexpected results. In this case
we evaluated the logistic map and saw that it was easy to predict the behavior of the functions iterations
within a very small range. Anything after a “ 4 diverged almost immediately, except in the cases where
x0 “ a´1
a , ni which case, the sequence converged immediately. I presume that the limit of a “ 4 is related
to the position of the function. For instance, if the function were shifted higher in the y-axis, I believe this
would increase the highest possible value of a. It seems that a rational relationship exists between the line
y “ x and fpxq. And that this ratio represents the maximum value of a. This is an excellent introduction to
chaos and how easily it can rear it’s ugly head. We were able to observe that fixed points are very important
in quadratic iteration. If we were to expand beyond powers of 2 quadratics, we would see that the number
of possible fixed points is delegated by the highest order exponent. For example, x3
will have a maximum of
3 fixed points. All fixed points act similar to magnetic poles, you can’t have two attractors or two repellers
nearest to each other. I assume that the value of a and it’s position relative to an attractor and repeller will
determine how quickly it will break down into chaos. But alas, we will have to explore that another time.
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