Dara, I always need to work against a physical explanation.
I have learned from experience that it is common to be accused of numerology or "trendology" if one doesn't have a physical basis.

"Maybe the dynamical system has ’beats’ like waves canceling some parts of other waves."

Yet that is essentially what constitutes Bloch waves -- a set of waveforms that are allowed based on the constraints of the system. Within the math abstraction of sloshing, standing waves develop which interfere with each other.

Adding delay differencing adds another layer of complexity. I am not even certain that the 2-year delay manifests as a periodicity.

In digital signal processing a delay difference is a $1 - 1/Z$ term, which is a building block for a filter. And unless I am mistaken, this should eliminate 2-year periodicities because it acts as a notch filter. Think about it, if one samples a sin wave of 2 year period and takes the difference at any two points separated by two years, one gets a zero result !