Taming Chaos
GEM2505M
Frederick H. Willeboordse
[email protected]
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Bifurcations and Windows
Lectures 8 & 9
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Important Notice!
This is a double
lecture.
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Today’s Lecture
Bifurcations
Windows
Destinations
Crisis
Stability
The Story
We’ve seen what a bifurcation
diagram is.
How can we understand some
of its features?
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Bifurcations
Last time we had a first look
at the bifurcation diagram.
In high resolution …
the bewildering structure
can clearly be seen.
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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
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Bifurcations
As we can see in the bifurcation diagram,
something special happens at a = 0.75 so
let’s draw some cobwebs on or near this
value.
What happens here?
First Iterates
Spirals in
a = 0.6
Spirals slowly
a = 0.75
Spirals out
a = 0.9
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Bifurcations
Zooming into the spiral, we can notice
something ….
a = 0.9
a = 0.6
The slope of the intersection of the function plot and
the diagonal is changing.
Let us look at two lines crossing:
Q < 90
o
Q = 90
o
Q > 90
o
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Bifurcations
What kind of cobweb do
you think we obtain for the
graph to the right?
?
1.
2.
3.
4.
Q = 90
o
Spirals in
Spirals out
Period two
Chaotic
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Bifurcations
The Angle
Q < 90
o
Q = 90
o
Q > 90
o
The Cobweb
The Behavior
Spirals in
Stays Put
Spirals out
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Bifurcations
Next, let us investigate what happens to the
second composition near the first
bifurcation point.
Second Composition
Zigzags in
a = 0.6
1 Fixed Point
What happens here?
Only visible when
zooming in.
Zigzags in slowly
a = 0.75
1 Fixed Point
Zigzags in
a = 0.9
2 Fixed Points
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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.
First Composition
What happens here?
Second Composition
same fixed point, slope > 1
new period two fixed
point, slope < 1
a = 0.9
a = 0.9
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Bifurcations
Second Composition
If it’s fun once,
it’s fun twice!
a = 1.2
What happens here?
Forth Composition
Second Composition
same fixed points, slope > 1
a = 1.3
new period four fixed
points, slope < 1
a = 1.3
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Bifurcations
If it’s fun twice,
it’s fun thrice!
What happens here?
Eighth Composition
a = 1.38
a = 1.38
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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.
dk
dk+1
dk
= d = 4.6692
dk+1
for k to infinity
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Bifurcation Diagram
Feigenbaum constant
The constant d is called the
Feigenbaum constant.
Feigenbaum point
M. Feigenbaum
The point in the bifurcation diagram where the
period doubling reaches infinity is called the
Feigenbaum point.
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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
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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.
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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.
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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.
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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?
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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
a = 0.6
Third Composition
a = 1.3
Third Composition
a = 1.6
Clearly, there is only one fixed point. However, local
extrema are getting closer to the diagonal.
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Windows
Indeed at a = 1.75 the third composition touches the
diagonal. This creates three attracting fixed points.
Third Composition
a = 1.7
Third Composition
a = 1.75
Third Composition
a = 1.8
At a = 1.75, all the fixed points of 2n compositions are
repelling since we’re past the Feigenbaum point.
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Windows
Third Composition
After the window is
created, we back to
the previous story.
What happens here?
a = 1.752
Third Composition
a = 1.775
Sixth Composition
new period six fixed
points, slope < 1
a = 1.775
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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
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Windows
This can again be understood by examining the cobweb.
Third Composition
a = 1.790
Third Composition
a = 1.793
When starting from x0 = 0.0, the left path stays close while
the right path jumps all over the place.
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Windows
Zooming into the central part of the graph we see why.
Third Composition
Third Composition
a = 1.79
a = 1.793
At this point the orbit can escape. Hence when a reaches a
value where escape is possible, the window closes.
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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.
2.
3.
4.
The first iterate needs to be on the diagonal
The third iterate of the local minimum needs to be on
the diagonal
The sixth iterate needs to be on the diagonal
The sixth iterate of the local minimum needs to be on
the diagonal
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Crisis
When the sixth iterate is exactly on the diagonal.
Third composition
a = 1.79032749
For larger a, the paths
will escape this region,
for smaller a (until a =
1.75) the paths will
remain inside.
Why sixth? Since it’s two steps in the graph of the third
composition. I.e. we have 2 x 3 steps.
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Derivative & Slope
The slope of a function is given by it’s derivative. The
derivative of
is:
The derivative of this
function at this point is
given by the slope of this
line.
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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.
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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.
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Stability
Thus we see that a period one cycle is stable when the
absolute value of the slope is smaller than 1.
We can determine this graphically by
investigating the angle at which the
function plot intersects the diagonal.
What is the slope?
Or, and this is far more accurate of
course, we can evaluate the derivative at
the fixed point to find the slope.
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Key Points of the Day
Simple Map.
Amazing Properties!
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Think about it!
Can a crisis be a
good thing?
Crisis,
Sale,
Computer,
Simulation!
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References
http://www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson4/Demo1.html
http://www.expm.t.u-tokyo.ac.jp/~kanamaru/Chaos/e/
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