Testing data for regular oscillations This is a little hard to explain, but I would like to test data for periodic oscillations, but not necessarily oscillations of the same amplitude. for example (crudely!): 

So basically I want to test whether something is happening periodically, but not necessarily the same thing. I'm not too sure how to approach this; I don't think autocorrelation is appropriate given the potential differences in oscillations. Perhaps look at frequencies in the Fourier domain?
[In case anyone's wondering - I'm not Banksy]
 A: Two most obvious options are spectral analysis and autocorrelations. In fact, they're very related concepts. The autocorrelation process such as $y_t=c+\phi y_{t-1}+\varepsilon_t$ would imply certain spectrum. See, for instance, Yule-Walker estimation of spectral densities.
Historically, autocorrelation was used for finite samples initially. The reason is that spectral analysis, such as Fourier, assumes that infinite series. Whenever you have short series at low frequencies Fourier analysis will have artifacts from both sampling frequency and the sample size. In these cases autocorrelation approach, such as Yule-Walker, will behave better.
Here, I'm lumping wavelet analysis with Fourier into the spectral analysis. Wavelets can work better when shape of your waves is very different from typical sine waves, for which Fourier is best suited, but you have to pick the appropriate wavelet pattern.
So, I would suggest starting with the simplest tools of exploratory analysis such as periodograms and correlograms. They'll give you an idea of the possible frequencies in the data, at least the lower frequencies. 
There's also a technique that I used to employ in astrophysics data, but forgot what is its name. Imagine you have your plot on a transparency. You cut your plot into pieces, then stack them on top of the other, so that you see all parts overimposed on each other. You loop through different number of cuts, and if there's a repeating pattern it'll emerge in one of the cuts. It's very similar to correlogram but it's more visual.
Here's the example. I have the periodic wave, and added the noise and also the change in the amplitude. The graph in the middle shows the stacked cuts at the periodicity of the sine wave, while the graph on the bottom is some other periodicity. In the middle you see that there's some kind of pattern. I used to work with light intensity coming from the stars. When the stars rotate, the intensity may change. So, we tried to detect the rotation speed by detecting the periodicity of light.

The MATLAB code:
rng(0)
t=1:0.1:40;
x=sin(t').*(1+sin(t'/pi))+randn(size(t'))/3;

subplot(3,1,1)
plot(x,'.')

subplot(3,1,2)
p=round(2*pi/0.1);
k=floor(length(x)/p);
m=reshape(x(1:p*k),p,k)
% plot(sum(m,2),'.')
plot(m,'.k')

subplot(3,1,3)
p=75;
k=floor(length(x)/p);
m=reshape(x(1:p*k),p,k)
% plot(sum(m,2),'.')
plot(m,'.k')

If your amplitudes are systematically increasing/decreasing, you'll probably run into variable variance. There are ways of dealing with this such as log-transform and Box-Cox transform.
