Alligood K., Sauer T., Yorke J.A. Chaos: An Introduction to Dynamical Systems
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O NE-DIMENSIONAL MAPS
a nonlinear effect, and the model given by g(x) is an example of a logistic growth
model.
Using a calculator, investigate the difference in outcomes imposed by the
models f(x)andg(x). Start with a small value, say x ⫽ 0.01, and compute f
k
(x)
and g
k
(x) for successive values of k. The results for the models are shown in
Table 1.1. One can see that for g(x), there is computational evidence that the
population approaches an eventual limiting size, which we would call a steady-
state population for the model g(x). Later in this section, using some elementary
calculus, we’ll see how to verify this conjecture (Theorem 1.5).
There are obvious differences between the behavior of the population size
under the two models, f(x)andg(x). Under the dynamical system f(x), the starting
population size x ⫽ 0.01 results in arbitrarily large populations as time progresses.
Under the system g(x), the same starting size x ⫽ 0.01 progresses in a strikingly
similar way at first, approximately doubling each hour. Eventually, however, a
limiting size is reached. In this case, the population saturates at x ⫽ 0.50 (one-
half million), and then never changes again.
So one great improvement of the logistic model g(x) is that populations
can have a finite limit. But there is a second improvement contained in g(x). If
n f
n
(
x
)
g
n
(
x
)
0 0.0100000000 0.0100000000
1 0.0200000000 0.0198000000
2 0.0400000000 0.0388159200
3 0.0800000000 0.0746184887
4 0.1600000000 0.1381011397
5 0.3200000000 0.2380584298
6 0.6400000000 0.3627732276
7 1.2800000000 0.4623376259
8 2.5600000000 0.4971630912
9 5.1200000000 0.4999839039
10 10.2400000000 0.4999999995
11 20.4800000000 0.5000000000
12 40.9600000000 0.5000000000
Table 1.1 Comparison of exponential growth model
f
(
x
) ⴝ 2
x
to logistic
growth model
g
(
x
) ⴝ 2
x
(1 ⴚ
x
).
The exponential model explodes, while the logistic model approaches a steady state.
4
1.2 COBWEB P LOT:GRAPHICAL R EPRESENTATION OF AN O RBIT
we use starting populations other than x ⫽ 0.01, the same limiting population
x ⫽ 0.50 will be achieved.
➮ COMPUTER EXPERIMENT 1.1
Confirm the fact that populations evolving under the rule g(x) ⫽ 2x(1 ⫺ x)
prefer to reach the population 0.5. Use a calculator or computer program, and try
starting populations x
0
between 0.0and1.0. Calculate x
1
⫽ g(x
0
),x
2
⫽ g(x
1
),
etc. and allow the population to reach a limiting size. You will find that the size
x ⫽ 0.50 eventually “attracts” any of these starting populations.
Our numerical experiment suggests that this population model has a natural
built-in carrying capacity. This property corresponds to one of the many ways
that scientists believe populations should behave—that they reach a steady-state
which is somehow compatible with the available environmental resources. The
limiting population x ⫽ 0.50 for the logistic model is an example of a fixed point
of a discrete-time dynamical system.
Definition 1.1 A function whose domain (input) space and range (out-
put) space are the same will be called a map.Letx be a point and let f beamap.
The orbit of x under f is the set of points 兵x, f(x),f
2
(x),...其. The starting point
x for the orbit is called the initial value of the orbit. A point p is a fixed point of
the map f if f(p) ⫽ p.
For example, the function g(x) ⫽ 2x(1 ⫺ x) from the real line to itself is a
map. The orbit of x ⫽ 0.01 under g is 兵0.01, 0.0198, 0.0388,...其, and the fixed
points of g are x ⫽ 0andx ⫽ 1 2.
1.2 COBWEB PLOT:GRAPHICAL
REPRESENTATION OF AN ORBIT
For a map of the real line, a rough plot of an orbit—called a cobweb plot—can be
made using the following graphical technique. Sketch the graph of the function
f together with the diagonal line y ⫽ x. In Figure 1.1, the example f(x) ⫽ 2x and
the diagonal are sketched. The first thing that is clear from such a picture is the
location of fixed points of f. At any intersection of y ⫽ f(x) with the line y ⫽ x,
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O NE-DIMENSIONAL MAPS
x.04
.04
.02
.02
.06
.08
f(x) = 2x
y = x
Figure 1.1 An orbit of
f
(
x
) ⴝ 2
x
.
The dotted line is a cobweb plot, a path that illustrates the production of a trajectory.
the input value x and the output f(x) are identical, so such an x is a fixed point.
Figure 1.1 shows that the only fixed point of f(x) ⫽ 2x is x ⫽ 0.
Sketching the orbit of a given initial condition is done as follows. Starting
with the input value x ⫽ .01, the output f(.01) is found by plotting the value
of the function above .01. In Figure 1.1, the output value is .02. Next, to find
f(.02), it is necessary to consider .02 as the new input value. In order to turn an
output value into an input value, draw a horizontal line from the input–output
pair (.01,.02) to the diagonal line y ⫽ x. In Figure 1.1, there is a vertical dotted
line segment starting at x ⫽ .01, representing the function evaluation, and then
a horizontal dotted segment which effectively turns the output into an input so
that the process can be repeated.
Then start over with the new value x ⫽ .02, and draw a new pair of vertical
and horizontal dotted segments. We find f(f(.01)) ⫽ f(. 02) ⫽ .04 on the graph
of f, and move horizontally to move output to the input position. Continuing in
this way, a graphical history of the orbit 兵.01,.02,.04,...其 is constructed by the
path of dotted line segments.
E XAMPLE 1.2
A more interesting example is the map g(x) ⫽ 2x(1 ⫺ x). First we find fixed
points by solving the equation x ⫽ 2x(1 ⫺ x). There are two solutions, x ⫽ 0
6
1.2 COBWEB P LOT:GRAPHICAL R EPRESENTATION OF AN O RBIT
C OBWEB P LOT
A cobweb plot illustrates convergence to an attracting fixed point of
g(x) ⫽ 2x(1 ⫺ x). Let x
0
⫽ 0.1 be the initial condition. Then the
first iterate is x
1
⫽ g(x
0
) ⫽ 0.18. Note that the point (x
0
,x
1
) lies on
the function graph, and (x
1
,x
1
) lies on the diagonal line. Connect
these points with a horizontal dotted line to make a path. Then find
x
2
⫽ g(x
1
) ⫽ 0.2952, and continue the path with a vertical dotted
line to (x
1
,x
2
) and with a horizontal dotted line to (x
2
,x
2
). An entire
orbit can be mapped out this way.
In this case it is clear from the geometry that the orbit we are follow-
ing will converge to the intersection of the curve and the diagonal,
x ⫽ 1 2. What happens if instead we start with x
0
⫽ 0.8? These are
examples of simple cobweb plots. They can be much more compli-
cated, as we shall see later.
0.1
0.5
0.5
g(x) = 2x(1-x)
x
y
Figure 1.2 A cobweb plot for an orbit of
g
(
x
) ⴝ 2
x
(1 ⴚ
x
).
The orbit with initial value .1 converges to the sink at .5.
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O NE-DIMENSIONAL MAPS
and x ⫽ 1 2, which are the two fixed points of g. Contrast this with a linear
map which, except for the case of the identity f(x) ⫽ x, has only one fixed point
x ⫽ 0. What is the behavior of orbits of g? The graphical representation of the
orbit with initial value x ⫽ 0.1 is drawn in Figure 1.2. It is clear from the figure
that the orbit, instead of diverging to infinity as in Figure 1.1, is converging to the
fixed point x ⫽ 1 2. Thus the orbit with initial condition x ⫽ 0.1 gets stuck, and
cannot move beyond the fixed point x ⫽ 0.5. A simple rule of thumb for following
the graphical representation of an orbit: If the graph is above the diagonal line
y ⫽ x, the orbit will move to the right; if the graph is below the line, the orbit
moves to the left.
E XAMPLE 1.3
Let f be the map of ⺢ given by f(x) ⫽ (3x ⫺ x
3
) 2. Figure 1.3 shows
a graphical representations of two orbits, with initial values x ⫽ 1.6and1.8,
respectively. The former orbit appears to converge to the fixed point x ⫽ 1asthe
map is iterated; the latter converges to the fixed point x ⫽⫺1.
-1
1
1
-1
1.8
x
y
1.6
f(x) =
3x - x
3
2
Figure 1.3 A cobweb plot for two orbits of
f
(
x
) ⴝ (3
x
ⴚ
x
3
) 2.
The orbit with initial value 1.6 converges to the sink at 1; the orbit with initial
value 1.8 converges to the sink at ⫺1.
8
1.3 STA B ILITY O F F IXED P OINTS
Fixed points are found by solving the equation f(x) ⫽ x.Themaphas
three fixed points, namely ⫺1, 0, and 1. However, orbits beginning near, but not
precisely on, each of the fixed points act differently. You may be able to convince
yourself, using the graphical representation technique, that initial values near ⫺1
stay near ⫺1 upon iteration by the map, and that initial values near 1 stay near 1.
On the other hand, initial values near 0 depart from the area near 0. For example,
to four significant digits, f(.1) ⫽ 0.1495,f
2
(.1) ⫽ 0.2226,f
5
(.1) ⫽ 0.6587, and
so on. The problem with points near 0 is that f magnifies them by a factor
larger than one. For example, the point x ⫽ .1 is moved by f to approximately
.1495, a magnification factor of 1.495. This magnification factor turns out to be
approximately the derivative f
(0) ⫽ 1.5.
1.3 STABILITY OF FIXED POINTS
With the geometric intuition gained from Figures 1.1, 1.2, and 1.3, we can describe
the idea of stability of fixed points. Assuming that the discrete-time system exists
to model real phenomena, not all fixed points are alike. A stable fixed point has
the property that points near it are moved even closer to the fixed point under
the dynamical system. For an unstable fixed point, nearby points move away as
time progresses. A good analogy is that a ball at the bottom of a valley is stable,
while a ball balanced at the tip of a mountain is unstable.
The question of stability is significant because a real-world system is con-
stantly subject to small perturbations. Therefore a steady state observed in a
realistic system must correspond to a stable fixed point. If the fixed point is unsta-
ble, small errors or perturbations in the state would cause the orbit to move away
from the fixed point, which would then not be observed.
Example 1.3 gave some insight into the question of stability. The derivative
of the map at a fixed point p is a measure of how the distance between p and a
nearby point is magnified or shrunk by f. That is, the points 0 and .1 begin exactly
.1 units apart. After applying the rule f to both points, the distance separating
the points is changed by a factor of approximately f
(0). We want to call the fixed
point 0 “unstable” when points very near 0 tend to move away from 0.
The concept of “near” is made precise by referring to all real numbers within
a distance
⑀
of p as the epsilon neighborhood N
⑀
(p). Denote the real line by ⺢.
Then N
⑀
(p) is the interval of numbers 兵x 僆 ⺢ : |x ⫺ p| ⬍
⑀
其. We usually think
of
⑀
as a small, positive number.
Definition 1.4 Let f be a map on ⺢ and let p be a real number such that
f(p) ⫽ p. If all points sufficiently close to p are attracted to p,thenp is called a
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O NE-DIMENSIONAL MAPS
sink or an attracting fixed point. More precisely, if there is an
⑀
⬎ 0suchthat
for all x in the epsilon neighborhood N
⑀
(p), lim
k→
⬁
f
k
(x) ⫽ p,thenp is a sink. If
all points sufficiently close to p are repelled from p,thenp is called a source or a
repelling fixed point. More precisely, if there is an epsilon neighborhood N
⑀
(p)
such that each x in N
⑀
(p) except for p itself eventually maps outside of N
⑀
(p),
then p is a source.
In this text, unless otherwise stated, we will deal with functions for which
derivatives of all orders exist and are continuous functions. We will call this type
of function a smooth function.
Theorem 1.5 Let f be a (smooth) map on ⺢, and assume that p is a fixed
point of f.
1. If |f
(p)| ⬍ 1, then p is a sink.
2. If |f
(p)| ⬎ 1, then p is a source.
Proof: P
ART 1. Let a be any number between |f
(p)| and 1; for example, a
could be chosen to be (1 ⫹ |f
(p)|) 2. Since
lim
x→p
|f(x) ⫺ f(p)|
|x ⫺ p|
⫽ |f
(p)|,
there is a neighborhood N
(p) for some
⬎ 0sothat
|f(x) ⫺ f(p)|
|x ⫺ p|
⬍ a
for x in N
(p), x ⫽ p.
In other words, f(x) is closer to p than x is, by at least a factor of a (which is
less than 1). This implies two things: First, if x 僆 N
⑀
(p), then f(x) 僆 N
(p); that
meansthatifx is within
of p,thensoisf(x), and by repeating the argument, so
are f
2
(x),f
3
(x), and so forth. Second, it follows that
|f
k
(x) ⫺ p| ⱕ a
k
|x ⫺ p| (1.1)
for all k ⱖ 1. Thus p is a sink.
✎ E XERCISE T1.1
Show that inequality (1.1) holds for k ⫽ 2. Then carry out the mathematical
induction argument to show that it holds for all k ⫽ 1, 2, 3,...
10
1.3 STA B ILITY O F F IXED P OINTS
✎ E XERCISE T1.2
Use the ideas of the proof of Part 1 of Theorem 1.5 to prove Part 2.
Note that the proof of part 1 of Theorem 1.5 says something about the rate
of convergence of f
k
(x)top. The fact that |f
k
(x) ⫺ p| ⱕ a
k
|x ⫺ p| for a ⬍ 1is
described by saying that f
k
(x) converges exponentially to p as k →
⬁
.
Our definition of a fixed-point sink requires only that the sink attract some
epsilon neighborhood (p ⫺
⑀
,p⫹
⑀
) of nearby points. As far as the definition
is concerned, the radius
⑀
of the neighborhood, although nonzero, could be
extremely small. On the other hand, sinks often attract orbits from a large set
of nearby initial conditions. We will refer to the set of initial conditions whose
orbits converge to the sink as the basin of the sink.
With Theorem 1.5 and our new terminology, we can return to Example
1.2, an example of a logistic model. Setting x ⫽ g(x) ⫽ 2x(1 ⫺ x) shows that the
fixed points are 0 and 1 2. Taking derivatives, we get g
(0) ⫽ 2andg
(1 2) ⫽ 0.
Theorem 1.5 shows that x ⫽ 1 2 is a sink, which confirms our suspicions from
Table 1.1. On the other hand, x ⫽ 0 is a source. Points near 0 are repelled from
0 upon application of g. In fact, points near 0 are repelled at an exponential
magnification factor of approximately 2 (check this number with a calculator).
These points are attracted to the sink x ⫽ 1 2.
What is the basin of the sink x ⫽ 1 2 in Example 1.2? The point 0 does
not belong, since it is a fixed point. Also, 1 does not belong, since g(1) ⫽ 0
and further iterations cannot budge it. However, all initial conditions from the
interval (0, 1) will produce orbits that converge to the sink. You should sketch
a graph of g(x) as in Figure 1.1 and use the idea of the cobweb plot to convince
yourself of this fact.
There is a second way to show that the basin of x ⫽ 1 2is(0, 1), which
is quicker and trickier but far less general. That is to use algebra (not geometry)
to compare |g (x) ⫺ 1 2| to |x ⫺ 1 2|. If the former is smaller than the latter, it
means the orbit is getting closer to 1 2. The algebra says:
|g(x) ⫺ 1 2| ⫽ |2x(1 ⫺ x) ⫺ 1 2|
⫽ 2|x ⫺ 1 2||x ⫺ 1 2| (1.2)
Now we can see that if x 僆 (0, 1), the multiplier 2|x ⫺ 1 2| is smaller than one.
Any point x in (0, 1) will have its distance from x ⫽ 1 2 decreased on each itera-
tion by g. Notice that the algebra also tells us what happens for initial conditions
outside of (0, 1): they will never converge to the sink x ⫽ 1 2. Therefore the
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O NE-DIMENSIONAL MAPS
basin of the sink is exactly the open interval (0, 1). Informally, we could also
say that the basin of infinity is (⫺
⬁
, 0) 傼 (1,
⬁
), since the orbit of each initial
condition in this set tends toward ⫺
⬁
.
Theorem 1.5 also clarifies Example 1.3, which is the map f(x) ⫽ (3x ⫺
x
3
) 2. The fixed points are ⫺1, 0, and 1, and the derivatives are f
(⫺1) ⫽ f
(1) ⫽
0, and f
(0) ⫽ 1.5. By the theorem, the fixed points ⫺1 and 1 are attracting fixed
points, and 0 is a repelling fixed point.
Let’s try to determine the basins of the two sinks. Example 1.3 is already
significantly more complicated than Example 1.2, and we will have to be satisfied
with an incomplete answer. We will consider the sink x ⫽ 1; the other sink has
very similar properties by the symmetry of the situation.
First, cobweb plots (see Figure 1.3) convince us that the interval I
1
⫽
(0,
3) of initial conditions belongs to the basin of x ⫽ 1. (Note that f(
3) ⫽
f(⫺
3) ⫽ 0.) So far it is similar to the previous example. Have we found the
entire basin? Not quite. Initial conditions from the interval I
2
⫽ [⫺2, ⫺
3)
map to (0, 1], which we already know are basin points. (Note that f(⫺2) ⫽ 1.)
Since points that map to basin points are basin points as well, we know that the
set [⫺2, ⫺
3) 傼 (0,
3) is included in the basin of x ⫽ 1. Now you may be
willing to admit that the basin can be quite a complicated creature, because the
graph shows that there is a small interval I
3
of points to the right of x ⫽ 2that
mapintotheintervalI
2
⫽ [⫺2, ⫺
3), and are therefore in the basin, then a
small interval I
4
to the left of x ⫽⫺2 that maps into I
3
, and so forth ad infinitum.
These intervals are all separate (they don’t overlap), and the gaps between them
consist of similar intervals belonging to the basin of the other sink x ⫽⫺1. The
intervals I
n
get smaller with increasing n, and all of them lie between ⫺
5and
5. Since f(
5) ⫽⫺
5andf(⫺
5) ⫽
5, neither of these numbers is in
the basin of either sink.
✎ E XERCISE T1.3
Solve the inequality |f (x) ⫺ 0| ⬎ |x ⫺ 0|,wheref (x) ⫽ (3x ⫺ x
3
) 2. This
identifies points whose distance from 0 increases on each iteration. Use
the result to find a large set of initial conditions that do not converge to
any sink of f .
There is one case that is not covered by Theorem 1.5. The stability of a
fixed point p cannot be determined solely by the derivative when |f
(p)| ⫽ 1(see
Exercise 1.2).
12
1.4 PERIODIC P OINTS
So far we have seen the important role of fixed points in determining the
behavior of orbits of maps. If the fixed point is a sink, it provides the final state for
the orbits of many nearby initial conditions. For the linear map f(x) ⫽ ax with
|a| ⬍ 1, the sink x ⫽ 0 attracts all initial conditions. In Examples 1.2 and 1.3,
the sinks attract large sets of initial conditions.
✎ E XERCISE T1.4
Let p be a fixed point of a map f . Given some
⑀
⬎ 0, find a geometric
condition under which all points x in N
⑀
(p) are in the basin of p.Use
cobweb plot analysis to explain your reasoning.
1.4 PERIODIC POINTS
Changing a, the constant of proportionality in the logistic map g
a
(x) ⫽ ax(1 ⫺ x),
can result in a picture quite different from Example 1.2. When a ⫽ 3.3, the fixed
points are x ⫽ 0andx ⫽ 23 33 ⫽ .
69 ⫽ .696969 ...,both of which are repellers.
Now that there are no fixed points around that can attract orbits, where do they
go? Use a calculator to convince yourself that for almost every choice of initial
condition, the orbit settles into a pattern of alternating values p
1
⫽ .4794 and
p
2
⫽ .8236 (to four decimal place accuracy). Some typical orbits are shown in
Table 1.2. The orbit with initial condition 0.2 is graphed in Figure 1.4. This
figure shows typical behavior of an orbit converging to a period-2 sink 兵p
1
,p
2
其.It
is attracted to p
1
every two iterates, and to p
2
on alternate iterates.
There are actually two important parts of this fact. First, there is the apparent
coincidence that g(p
1
) ⫽ p
2
and g(p
2
) ⫽ p
1
. Another way to look at this is that
g
2
(p
1
) ⫽ p
1
;thusp
1
is a fixed point of h ⫽ g
2
. (The same could be said for p
2
.)
Second, this periodic oscillation between p
1
and p
2
is stable, and attracts orbits.
This fact means that periodic behavior will show up in a physical system modeled
by g. The pair 兵p
1
,p
2
其 is an example of a periodic orbit.
Definition 1.6 Let f beamapon⺢. We call p a periodic point of period
k if f
k
(p) ⫽ p,andifk is the smallest such positive integer. The orbit with initial
point p (which consists of k points) is called a periodic orbit of period k. We will
often use the abbreviated terms period-k point and period-k orbit.
Notice that we have defined the period of an orbit to be the minimum
number of iterates required for the orbit to repeat a point. If p is a periodic point
of period 2 for the map f,thenp is a fixed point of the map h ⫽ f
2
. However, the
converse is not true. A fixed point of h ⫽ f
2
may also be a fixed point of a lower
13