PHY411. PROBLEM SET 5
1. A chaotic map on the circle
Definition We recall Delaney’s definition of a chaotic map. Let D be a metric
space. The function f : D → D is chaotic if
(a) The periodic orbits of F are dense in D.
(b) The function f is topologically transitive. In other words f mixes the set
really well.
(c) The function f exhibits extreme sensitivity to initial conditions.
Recall that a function f : D → D is topologically transitive if for all open sets
U, V in D there is an x in U and a natural number n such that f n (x) is in V .
Consider the map on the circle θ ∈ [0, 1] with
g(θ) = 2θ
mod 1
Show that this map is chaotic. (In particular show a,b, you need not show c).
2. Numerical Investigation on the width of Arnold Tongues
Consider the sine-circle map
θn+1 = [θn + Ω −
K
sin(2πθn )]
2π
mod 1
The map is symmetric about Ω = 1/2 so the width of the tongue with W = 1/3
is the same as that at W = 2/3. The width of an Arnold tongue with rational
winding number W = p/q is of order K −q .
For K = 1, is the width of the Arnold tongue with rational winding number
W = 1/5 equal to that with W = 2/5? Numerically compute the widths of
the two tongues to see if they are different. Check your numerical accuracy by
confirming that the tongue at W = 3/5 is approximately the same width as that at
2/5. These two should be the same width by symmetry about 1/2. The difference
1
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PHY411. PROBLEM SET 5
between these two measurements you can use as an estimate of your numerical
error. You should only conclude that the tongues at W = 1/5 and W = 2/5 are a
different size if they are different by a larger amount than the computed difference
between the tongues for W = 2/5 and 3/5.
Here is a code that computes the winding numbers in the Devil’s staircase.
(But this routine is not high enough precision to use here)
http://astro.pas.rochester.edu/∼aquillen/phy411/pylab/devilstair.py
A higher precision set of routines is here: (you need this routine!)
http://astro.pas.rochester.edu/∼aquillen/phy411/pylab/devil stair h.py
A routine that calls the higher precision routines to compute widths of tongues
is here: (you need this one!)
http://astro.pas.rochester.edu/∼aquillen/phy411/pylab/tongue.py
Figure 1. Winding or rotation number as a function of Ω for K = 1,
also known as the devil’s staircase. I computed this with K = 1 − 10−3
and using 300 iterations of the map.
3. On Newton’s method - Stable sets
PHY411. PROBLEM SET 5
3
Given a function f (x), the mapping
Nf (x) = x −
f (x)
f 0 (x)
can give an efficient way to iteratively find a root of f from a starting initial x
that lies near the root. This is known as Newton’s method. However, Nf does
not always converge to the nearest root and Nf can also have chaotic orbits. The
function Nf is particularly badly behaved if f 0 (x) has roots.
Consider the function
f (x) =
1
−1
x
that has a single root.
(a) What is the map Nf (x)?
(b) What are the fixed points for Nf ?
(c) To which point (or ∞ or −∞) does Newton’s method converge for x ∈ (0, 1)?
(d) To which point (or ∞ or −∞) does Newton’s method converge for x ∈ (1, 2)?
(e) To which point (or ∞ or −∞) does Newton’s method converge for x < 0?
(f) To which point (or ∞ or −∞) does Newton’s method converge for x > 2?
(g) How does Nf behave for x = 0, 1, 2?
(h) Over what region does Newton’s map converge to the root of f ?
Hint: It helps to construct cobweb plots for Nf .
4. On Topological conjugacy
Consider the map from the complex plane to the complex plane
g(z) = az + b
where a, b are complex numbers.
Show that g(z) is topologically conjugate to f (z) with
f (z) = cz
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PHY411. PROBLEM SET 5
where c is a complex number. In other words find an invertible map h(z) such
that
f (h(z)) = h(g(z))
or in other words
f ◦h=h◦g
or
h−1 ◦ f ◦ h = g
.
Hint: try an affine map.
When two maps are topologically conjugate, then periodic orbits in one map
correspond to periodic orbits in the other map.
5. On Area Preserving maps Consider the area preserving map
xn+1 = xn + aInγ
In+1 = In + sin xn+1
(1)
(2)
with x ∈ [0, 2π]. a) Show that the map is area preserving.
1
b) Show that with Wn = a gamma In that the map can be written in the form
Wn+1
xn+1 = xn + Wnγ
= Wn + K sin xn+1
(3)
(4)
and find K.
c) For what value of γ is this map the standard map?
Also I have some code that computes this map and it shows some nice stuff, in
particular the Kepler map has γ = −1.5 and is quite pretty.
6. Box dimension of the Baker maps’ attractor
The Baker map
(
(cxn , 2yn )
(xn+1 , yn+1 ) =
(1 + c(xn − 1), 1 + 2(yn − 1))
for
for
yn ≤ 1/2
yn > 1/2
For c < 1/2 the map is not area preserving and there is an attractor.
(5)
PHY411. PROBLEM SET 5
5
Minkowski dimension or box-counting dimension is a way of measuring the
fractal dimension of a set. Suppose that N () is the number of boxes of side
length required to cover the set.
The box dimension is:
log N ()
→0 log(1/)
Compute the box dimension for the attractor of the Baker’s map with c = 1/3.
dimbox = lim
If the limit for the box dimension does not exist, one may still take the limit
superior and limit inferior, which respectively define the upper box dimension
and lower box dimension. The upper box dimension is sometimes called the
entropy dimension, Kolmogorov dimension, Kolmogorov capacity, limit capacity
or upper Minkowski dimension, while the lower box dimension is also called the
lower Minkowski dimension. The upper and lower box dimensions are strongly
related to the more popular Hausdorff dimension. Only in very special settings is
it important to distinguish between the three.