So, I'm trying to fit my statistical model. I have real data, and I have an ODE that predicts the results of the real data. Both have strong periodic components. The ODE is fixed via dimensional analysis to have period 2, or maybe $2+\epsilon$ for some small epsilon, I haven't thought about whether the perturbation effects perturb the frequency, but in any case, it's very close to 2. There's a statistical parameter that's a timescale for each data series where I rescale the real data's time points to try to match this fixed period, this tells me how long a certain meaningful process takes in my real data. In MCMC inevitably the period will not be dead spot on. Whenever you have two periodic signals subtracted from each other, you wind up with a signal that's close to the average period modulated by a signal that is close to 1/difference in periods (or frequency is the difference in frequencies).

I've been trying to model the errors as gaussian processes to form a likelihood, but I've been ignoring this periodic component. If the periods are exactly aligned, the signal goes away, but even a tiny fraction of a percent different and the signal shows up. I should really allow the errors to have this periodic component but constrain its size to be small. I'm working on that now.

One thing that's important is that near the "peaks" the model should fit much better, whereas near the "troughs" I expect it to fit less well, but it's not very important. So I need to anchor the gaussian processes to have small total variance near times 0,2,4... while maintaining the periodic signal and some high frequency noise. Gaussian processes are pretty flexible fortunately.