Periodic motions that affect earth axis tilt

Question

It is said that the earth axis changes its tilt with a period of 41 000 years. So used the image

https://upload.wikimedia.org/wikipedia/commons/1/17/Obliquity_berger_0_to_1000000.png

removed the grid so it became like this

Then I used the following script

function rasterwave(imagefile)
    data=imread(imagefile);
    x=[1:size(data,1)]';
    X=repmat(x, 1, size(data,2));
    colsum=sum(data);
    y=sum(data(:, [1:size(data,2)]').*X)./colsum;
    Y=abs(fft(y));
    plot(Y(2:length(Y)/2), '.-')
end

To extract a spectrum (Frequency and intensity in arbitrary units):

It looks like there are two dominant frequencies in the spectrum, or maybe there is one peak followed by a band. Can these frequencies somehow be derived from the orbital periods of other planets?

Update:

Since the original image marks the timescale, it is actually possible to compute a sample rate, and plot the period on the x axis:

function rasterwave(imagefile, max_duration)
    data=imread(imagefile);
    x=[1:size(data,1)]';
    X=repmat(x, 1, size(data,2));
    colsum=sum(data);
    y=sum(data(:, [1:size(data,2)]').*X)./colsum;
    Y=abs(fft(y));
    fs=length(Y)/max_duration;
    f=linspace(0, fs, length(Y));
    plot(1.0./f(2:length(f)/2), Y(2:length(Y)/2), '.-')
end

Now we see the 41 000 year period together with some 53 000 year cycle.

Answer 1

Short answer: There are effects, but probably too small to appear in your graph (even though it's not clear what scale your graph is).

The orientation of the Earth's poles are generally divided into components of Precession and Nutation. Precession has a cycle of about 26,000 years, nutation has a much shorter cycle of about 18.6 years. Precession is the larger of the two (by far), and generally modeled on effects of the Sun and Moon. Nutation is much smaller, and only has an effect of about 9 arcseconds.

This image shows the combined effects of the two. The larger circle is due to precession, while the short period wiggles on the larger circle are the nutation effects.

But, nutation can be broken down into both Luni-Solar and Planetary components. The planetary components are very small, and often ignored. For example, the IAU has defined and IAU2000A a nutation model including the Sun, Moon, and planetary effects, and IAU2000B a model that only includes the most prominent Sun and Moon effects.

You can see the breakdown in the source code for IAU's Standards of Fundamental Astronomy software. The full implementation is in nut00a.c, and the reduced version is in nut00b.c

The difference between the new models is a few milli-arcseconds. As depicted in this IAU publication. So it's not likely any graph displayed at a modern screen resolution would show effects of both precession and planetary nutation at the same time.

Further details are available in (among other sources) "THE IAU 2000A AND IAU 2006 PRECESSION-NUTATION THEORIES AND THEIR IMPLEMENTATION"