Assignment week
The route to chaos via period doublings
Write a small report about the assignments listed below (could be hand-written, could
be typewritten). Add the computer scripts as an appendix. The answer to assignment 1 is always a picture.
1. The logistic map
Write a computer program (in Matlab) to reproduce the following figures from Chap-
ter 2 of the Lecture Notes:
(a) Figure 2.2: Draw the iterates f (x), f 2(x), . . . for λ = 0.683 and λ = 0.809
for an arbitrary initial value of x ∈ [0, 1]. Verify that the reached asymptotic
states are very different for the two figures. To get you started, the matlab code
(LOGISTIC map.m) is provided. In fact, you must now put in code what you
have already done on paper during last week’s lecture.
(b) Figure 2.4 and figure 2.6: Show repeated iterations of the function f (x) =
4λx(1 −x) where the value of λ is increased step by step. At each λ you can
start with an arbitrary value of x, N (many) times to let transients die, and then
iterate N times while drawing them. Already read the next question; it is about
the “many” times.
(c) Say “something” about the following: Near a bifurcation value λn it takes forever
(∞long) for iterates to reach the asymptotic state. To be more precise, the time
T (that is, number of iterates) diverges as T ∝|λ −λn|−1/2. Can you argue this
dependence ? (Of course, it must be related to the quadratic nonlinearity).
(d) Figure 2.7 and 2.8: Plot f 2n (λn, x) for the given n and λ in the Lecture notes to reproduce the sequence of the limit functions gr(x). The key point is that these figures should be the same (around x = 1/2) up to rescaling. Rescaling means enlarging horizontal and vertical axies of your figure. From this (that is trying…), estimate the universal number α. In fact, this is exactly what the renormalization argument does for you.
(e) Figure 2.12: Show the evolution of the spectrum via period doublings for n = 3,
n = 4 and n = 12. For figure (d) set the non-linearity parameter λ just beyond
λ∞ = 0.8924 . . ..
About spectra. You feed a time series of iterates I[i], i = 1, . . . , N to MAT-
LAB’s FFT function. The function provides two arrays, the real part Re[i]
of the Fourier transform, and the imaginary part Im[i]. Actually, your initial
times series is overwritten by the real part. Due to Nyquist, only Re[i], Im[i], i =
1, . . . , N/2 makes sense. For this assignment you plot the modulus of the Fourier
transfom, (Re[i]2 + Im[i]2)1/2. Of course, most of this you already knew.
2. An experiment
The picture below is the first claim to see the universal route to period doublings in
an experiment. This is a heat convection experiment, with the (reduced) Rayleigh
number R/Rc as nonlinearity parameter. (Convection first happened at Rc).
see: A. Libchaber, C. Laroche and S. Fauve, J. Physique, 43, L-211 (1982).
(a) From the (reduced) Raleigh numbers in pictures (B, C), make a prediction for
the Rayleigh number R/Rc of picture D. Discuss why this does not work equally
well for predicting the Rayleigh number R/Rc in picture C from those in A, B.
(b) The universal constant α is related to the amplitude of the subharmonics. The
scaling of the subharmonics in the spectrum is a mixture of α and α2, as in the
following expression
μ = 1
4
(
2α−2 + 2α−4)1/2
.
Think about the quadratic map, and explain why this is a mixture α and α2
(and ignore the numerical prefactors).
With α = 2.5029 . . ., the value μ = 0.15211 . . .. In power spectra, a decibel is
20 log10 A, where A is the amplitude ratio. Is this consistent with the observed
spectra ?
3. Numerics of superstable cycles
Let λn denote the value of λ at which the logistic map, f (x, λ) = λx(1 −x) has a
superstable cycle of period 2n.
(a) Write an implicit but exact formula for λn in terms of the point x = 1
2 and the function f (x, λ) = λx(1 −x). (You will find lots of inspiration in the lecture
notes, if not this very equation).
(b) Using a computer and the result of part (a), find λ2, λ3, …, λ7 to three significant
digits. Hint: formulate the problem as one where you have to find the zero of a
function. Alternatively, you could make pictures, zoom in, and find those zeros
by hand.
(c) Evaluate (λ3 −λ2)/(λ4 −λ3). This will approximate the other universal con-
stant of the period-doubling route to chaos. The problem is that success of this
assignment depends a little on success of the previous one. If necessary, you
could revert to (λ2 −λ1)/(λ3 −λ2). Be practical, this you would typically face
in an experiment.
