A ball rolls in a symmetric double-well track and is shaken periodically at drive…

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Computational Physics, Ch Spectra of sampled signals

Problem

A ball rolls in a symmetric double-well track and is shaken periodically at drive frequency

f_d = 0.60\,\mathrm{Hz}

. Its position is recorded at

f_s = 50\,\mathrm{Hz}

for

N = 8192

samples, and three runs at increasing shake amplitude give three spectra, each computed with the same window and the same power normalisation used for the pendulum spectra. - Run P (periodic entrainment): a single line at

f_d

, with every other bin at or below

2\times10^{-7}

of the peak power. - Run D (period-doubled): lines at

f_d

and at an extra frequency below it, together with weaker lines at

1.5f_d

and

2.5f_d

; nothing else above the floor. - Run Ch (chaotic): a continuum spread over

0.05

5\,\mathrm{Hz}

, roughly flat from bin to bin, with the line at

f_d

standing

20\,\mathrm{dB}

above the local continuum level. (a) Give the frequency of the extra line in run D as a fraction of

f_d

, and name the phenomenon of the driven pendulum and logistic map that it is the spectral signature of. (b) For run Ch, a power ratio of

20\,\mathrm{dB}

means each continuum bin holds

1/100

of the power in the line at

f_d

. Using

f_s

N

and the stated continuum band, estimate the ratio of the total power summed over the whole continuum to the power in the single line at

f_d

, and say whether the result supports or contradicts the claim that a broad spectrum means a noisy measurement.

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