GEM2505M

1
Taming Chaos

• GEM2505M

Frederick H. Willeboordse frederik_at_chaos.nus.edu.s
g
2
Bifurcations and Windows

• Lectures 8 9

3
Important Notice!
This is a double lecture.
4
Todays Lecture

• Bifurcations
• Windows
• Destinations
• Crisis
• Stability

The Story
Weve seen what a bifurcation diagram is. How
can we understand some of its features?
5
Bifurcations
Last time we had a first look at the bifurcation
diagram. In high resolution the bewildering
structure can clearly be seen.
6
Bifurcations
Depending on the value of a, the logistic map can
have a periodicity of 1, 2 or more.
Why is that so?
In order to find out, let us inspect the cobweb
more closely.
Period 1
Period 2
7
Bifurcations
As we can see in the bifurcation diagram,
something special happens at a 0.75 so lets
draw some cobwebs on or near this value.
What happens here?
First Iterates
Spirals slowly
Spirals in
Spirals out
a 0.9
a 0.6
a 0.75
8
Bifurcations
Zooming into the spiral, we can notice something
.
a 0.6
a 0.9
The slope of the intersection of the function
plot and the diagonal is changing.
Let us look at two lines crossing
Q gt 90o
Q 90o
Q lt 90o
9
Bifurcations
What kind of cobweb do you think we obtain for
the graph to the right?
Q 90o
?

1. Spirals in
2. Spirals out
3. Period two
4. Chaotic

10
Bifurcations
The Angle
The Cobweb
The Behavior
Q lt 90o
Spirals in
Q 90o
Stays Put
Spirals out
Q gt 90o
11
Bifurcations
Next, let us investigate what happens to the
second composition near the first bifurcation
point.
What happens here?
Second Composition
Only visible when zooming in.
Zigzags in slowly
Zigzags in
Zigzags in
a 0.9
a 0.75
a 0.6
1 Fixed Point
1 Fixed Point
2 Fixed Points
12
Bifurcations
We can see that the slope of the first iterate
changes from being smaller than one to larger
than one, and that at the same time two new fixed
points of the second composition with a slope
smaller than one come into existence.
What happens here?
First Composition
Second Composition
same fixed point, slope gt 1
new period two fixed point, slope lt 1
a 0.9
a 0.9
13
Bifurcations
Second Composition
If its fun once, its fun twice!
What happens here?
a 1.2
Forth Composition
Second Composition
same fixed points, slope gt 1
new period four fixed points, slope lt 1
a 1.3
a 1.3
14
Bifurcations
If its fun twice, its fun thrice!
What happens here?
Eighth Composition
a 1.38
a 1.38
15
Bifurcation Diagram
Indeed for increasing nonlinearity, period
doublings continue up to infinity. However, the
distance between successive bifurcation points
decreases rapidly (as can be seen from the
bifurcation diagram).
In fact, the length ratio between successive
branches approaches a constant.
dk1
dk
dk
d 4.6692
dk1
for k to infinity
16
Bifurcation Diagram
Feigenbaum constant
The constant d is called the Feigenbaum constant.
M. Feigenbaum
Feigenbaum point
The point in the bifurcation diagram where the
period doubling reaches infinity is called the
Feigenbaum point.
17
Destinations
Let us return to the first iterate and small a.
We start with a certain value of x0 and see what
orbit it leads to. The question now is, do all
values of x0 lead to the same orbits?
a 0.5
Does the value of x0 matter?
?

1. Yes
2. No
3. Sometimes
4. Depends

18
Destinations
For small a the answer is no. But for large a,
things are a bit more subtle.
Roughly, there are 4 possibilities
a 0.5
a 1.0
a 1.44
a 2.0
All points have exactly the same orbit.
All points have the same orbit though it may be
shifted.
All points have different orbits though there is
a gap.
All points have completely different orbits.
19
Destinations
What is going on becomes a bit clearer if we look
at the second composition.
a 1.0
a 1.0
We see that depending on the value of x0 the
orbit goes to a different fixed point.
20
Destinations
Indeed, we can graphically determine where the
possible x0 go.
Points on
go to
Points on
go to
a 1.0
Hence we see that there are basically three
regions. Two go to the red point and one to the
green point.
21
Bifurcation Diagram
We just saw that there is a period doubling
cascade to infinity.
Window
From the bifurcation diagram, it is also clear
that there are windows (periodic regions beyond
the Feigenbaum point).
Why would that be?
22
Windows
Thus far we investigated iterates 1,2,4,8 etc.
But there is no reason not to consider 3,5,6,7
etc. as well!
Third Composition
Third Composition
Third Composition
a 0.6
a 1.6
a 1.3
Clearly, there is only one fixed point. However,
local extrema are getting closer to the diagonal.
23
Windows
Indeed at a 1.75 the third composition touches
the diagonal. This creates three attracting fixed
points.
Third Composition
Third Composition
Third Composition
a 1.7
a 1.8
a 1.75
At a 1.75, all the fixed points of 2n
compositions are repelling since were past the
Feigenbaum point.
24
Windows
Third Composition
After the window is created, we back to the
previous story.
What happens here?
a 1.752
Sixth Composition
Third Composition
same fixed points, slope gt 1
new period six fixed points, slope lt 1
a 1.775
a 1.775
25
Windows
If we look at the bifurcation diagram starting
form the third iterate closely, we see that at
some stage something special happens.
It abruptly ends here
26
Windows
This can again be understood by examining the
cobweb.
Third Composition
Third Composition
a 1.790
a 1.793
When starting from x0 0.0, the left path stays
close while the right path jumps all over the
place.
27
Windows
Zooming into the central part of the graph we see
why.
Third Composition
Third Composition
a 1.793
a 1.79
At this point the orbit can escape. Hence when a
reaches a value where escape is possible, the
window closes.
28
Crisis
Third Composition
The event that leads to the closing of the window
is called a crisis. The value of a for which is
occurs can be determined graphically.
a 1.793
How?
?

1. The first iterate needs to be on the diagonal
2. The third iterate of the local minimum needs to
   be on the diagonal
3. The sixth iterate needs to be on the diagonal
4. The sixth iterate of the local minimum needs to
   be on the diagonal

29
Crisis
When the sixth iterate is exactly on the
diagonal.
Third composition
For larger a, the paths will escape this region,
for smaller a (until a 1.75) the paths will
remain inside.
a 1.79032749
Why sixth? Since its two steps in the graph of
the third composition. I.e. we have 2 x 3 steps.
30
Derivative Slope
The slope of a function is given by its
derivative. The derivative of
is
The derivative of this function at this point is
given by the slope of this line.
31
Slope Fixed Point
The fixed point of the first iterate can be
obtained by solving the equation
In other words we need to solve
(use )
Use
However only one of these fixed points is in the
interval -1,1
Period 1 fixed point we need.
32
Stability
Now that we know what the value of the fixed
point x is, we can insert this into the
derivative to obtain the slope of the function
plot at this fixed point.
Of course we can draw this and indeed, at a
0.75, the absolute value of the slope becomes
larger than 1.
33
Stability
Thus we see that a period one cycle is stable
when the absolute value of the slope is smaller
than 1.
What is the slope?
We can determine this graphically by
investigating the angle at which the function
plot intersects the diagonal.
Or, and this is far more accurate of course, we
can evaluate the derivative at the fixed point to
find the slope.
34
Key Points of the Day

• Simple Map.
• Amazing Properties!

35
Think about it!

• Can a crisis be a good thing?

Crisis, Sale, Computer, Simulation!
36
References
http//www.cmp.caltech.edu/mcc/Chaos_Course/Lesso
n4/Demo1.html
http//www.expm.t.u-tokyo.ac.jp/kanamaru/Chaos/e/