GEM2505M

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Taming Chaos

• GEM2505M

Frederick H. Willeboordse frederik_at_chaos.nus.edu.s
g
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Quantifying the Dynamics

• Lecture 10

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Important Notice!
Special QA session. See announcement on web!
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Todays Lecture
The Story
Weve seen how we can understand some of the main
features of the bifurcation diagram. How can we
quantify some of these features?

• Lyapunov Exponents
• Homoclinic Points
• Intermittency
• Fractals The Logistic Map

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Stability
Thus far we considered the stability of a single
fixed point. How about the stability of a
period-k orbit?
The most straightforward answers is that it is
determined by the slope of the kth composition as
we have seen before.
But, in mathematical rather than graphical terms,
what do we do?
What is, e.g., the stability of a period 2 orbit?
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Stability
The second composition of the logistic map is
given by
The slope is given by the derivative which is
(using the chain rule)
But what do we see? The term in brackets is just
x1! And therefore, the slope of the second
composition is given by
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Stability
If its funny once, its funny twice!
This same procedure works of course also for
higher iterates and we can conclude that the
stability of a period-n orbit is given by
Again the absolute value must be smaller than one
in order for the orbit to be attracting.
Since we know from the bifurcation diagram that
x0 xn-1 change for different nonlinearities, we
can wonder whether there is a most stable orbit.
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Second composition
Stability
Super-stable orbits
Indeed there is
Slope 0
The smallest absolute value is of course 0. Hence
an orbit which has a fixed point with a slope of
0 (a horizontal line) is the most stable orbit
and therefore called super-stable.
As
is a product and a unequal to zero for all
periodic orbits with a period larger than 1, we
can immediately infer that a super-stable orbit
contains the point x 0.
A super-stable orbit contains the point x 0
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Stability
Super-stable orbits
And as before, we can find these graphically
right away by identifying where the periodic
orbits in the bifurcation diagram intersect the
x-axis.
Points on super-stable orbits
How many super-stable orbits are there?
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Lyapunov Exponents
Of course, if we talk about stability, we would
like to have some kind of a number, a quantifier,
that can tell us in a relative sense how stable
an orbit is.
As such one could think that the product of
derivatives would provide such a quantifier. This
is not really the case, however, since the number
of terms in the product depends on the
periodicity and in the case of a chaotic orbit
would be infinite.
One option would be to divide by the number of
terms.
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Lyapunov Exponents
However When considering sensitive dependence
on initial conditions one can see that errors
grow exponentially fast.
Both axes linear
Y-axis log, x-axis linear
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Lyapunov Exponents
Put differently .
Now we have a straight line.
Both axes linear
Both axes linear
Take the log of the data
Take the log of the data, anddivide by the
x-value.
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Lyapunov Exponents
Put differently we see that if something grows
exponentially in time, then the log of that
something divided by the time remains constant.
Therefore we could argue that a reasonable
quantifier for the stability l of an orbit is
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Lyapunov Exponents
Since
We obtain
And if the orbit is not periodic, we should take
the limit
or more generally
l is called the Lyapunov exponent of the orbit.
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Lyapunov Exponents
Not in exam
The preceding few slides are plausible enough but
do not really stress the fundamental connection
between the Lyapunov exponent and the derivative
or the growth of an error.In order to do so, let
us choose a somewhat different approach.
The difference between two initially nearby
orbits can be expressed as
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Lyapunov Exponents
Not in exam
Dividing both sides by e we obtain
Note This is the first derivative of the
function f n
for
In other words
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Lyapunov Exponents
Not in exam
According to the chain rule we have
And consequently
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Lyapunov Exponents
Not in exam
Dropping the approximate and taking the log
Which, after reversing the order, taking the
limit, dividing by n and changing the ln of a
product to a sum of ln, again becomes
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Lyapunov Exponents
Recall
Periodic orbits
If we have a period-k orbit, the Lyapunov
exponent becomes
a 0.75
E.g. for period 2 we have
with x1 and x2 the two periodic points.
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Lyapunov Exponents
We have seen that for increasing a, the orbits
bifurcate. What would the Lyapunov exponent be
exactly at a bifurcation point? (e.g. a 0.75)
?
What would the Lyapunov exp. be?

• Depends on a (not all bif. points have the same
  l)
• 1 or minus 1
• 0
• 1/2k with k the periodicity just before the bif.

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Lyapunov Exponents
Super-stable orbits
Super-stable orbits go through 0. Consequently,
the Lyapunov exponent is given by
Points on super-stable orbits
E.g.
Why? Since the ln of 0 is minus infinity (and all
the other terms are finite).
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Lyapunov Exponents
Versus a
1
2
3
4
Similarly to the bifurcation diagram, we can plot
l versus a.

• Second bifurcation
• Period 4 super-stable orbit
• Third bifurcation
• Period 3 super-stable orbit

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Lyapunov Exponents
We just saw that the Lyapunov exponent of a
super-stable orbit is minus infinity. Yet in the
graph of the Lyapunov exponent versus the
smallest exponent is around minus 2.5.
?
Why would that be?

• Our calculation is wrong
• The graph is always wrong
• The resolution of the graph is limited
• There is no infinity in the real world

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Homoclinic Points
Homoclinic points were discovered by Henry
Poincaré in his studies of the solar system. In a
similar form they also exist in the logistic map.
Third Iterate
a 1.75
Plot exactly touches diagonal
From the left, zigzags in to fixed point (cannot
pass it)
From the right, zigzags away from fixed point.
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Homoclinic Points
The point where the plot touches the diagonal is
a so-called saddle point which is both attracting
and repelling, depending on the side from which
it is approached.
Here, homoclinic points are all those points on
the repelling side (i.e. right hand side) of the
saddle that when iterated will eventually end up
on the saddle via the attracting side.
Note Here we do not have stable and unstable
manifolds since these require two or more
dimensions.
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Intermittency
When the plot is very close to touching but does
not actually touch the diagonal yet, a small
channel is left.
a 1.7498
a 1.7496
a 1.7498
Every third time step is plotted.
While passing through this channel, the x-values
of the orbit do not change much leading to
laminar looking sections in the time series.
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Intermittency
Route to chaos
Histogram
a 1.7496
Starting form the opening point of the period
three window (a 1.75), when decreasing the
non-linearity a, the length of the laminar
regions decreases from infinitely long to very
short.
Hence this is an alternative route to chaos as
compared to the period-doubling route to chaos
discussed previously.
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Fractal
Histogram for a 2.0
Fractals in the logistic map
The orbit of the logistic map at a 2.0 is not
fractal as can readily be seen from the histogram
to the right.
However, there are fractal structures in the
bifurcation diagram. For example the set of
super-stable points.
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Fractal
accumulation pointa 1.401155
Fractal dimension
Another fractal may be at the accumulation point
where the orbit is neither periodic nor chaotic.
Some estimates are that D(a a) 0.538.
Conceptually, how can one understand this? If one
approaches the accumulation point from the
chaotic side (starting at say a 1.6), one can
see that first there are two bands, then four,
eight, etc. this is similar to the construction
of the Cantor set.
1.60
a
1.37
Remove
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Fractal
or enlarged
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Fractal
Relationship to theMandelbrot set
x 0.25
x -2.0
The Logistic map can be written as
Which is exactly the real part of the iterative
map used for the Mandelbrot set.
Period three window
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Key Points of the Day

• Stability
• Lyapunov Exponents
• Intermittency

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Think about it!

• Is nature based on stability or instability?

Stable, House, Cards, Unstable!
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References
http//www.cmp.caltech.edu/mcc/Chaos_Course/Lesso
n4/Demo1.html
http//www.expm.t.u-tokyo.ac.jp/kanamaru/Chaos/e/