Oh, I see.
Don’t you think it is just the classical figure-8 behavior of eccentricity on almost frozen orbits?
Look at this post. The
osculating parameters in the special case of SSO and frozen eccentricity are really different from mean parameters. The eccentricity short periodic terms are larger than the mean eccentricity. This implies that the perigee revolves around orbit once per period. In this post, I explain that for such orbits “it is not the satellite that crosses its perigee once per orbit, it is the perigee that crosses the satellite once per orbit”. So getting two perigees per orbit is mind-blowing but true. It in fact happens as soon as you use J₂ (the global figure-8 shape), or J₂ + J₃ (the global figure-8 shape plus its shift along ey). What is counter-intuitive is that short periods are so large in this case, but it is because by design these orbits have collapsed all other terms down to zero, hence the short periodic terms show up.
Another consequence of this is that true anomaly does not increase steadily anymore. In fact, is oscillates and goes backward at some points! This is the result of two different effects: the large short period terms of eccentricity and the singularity of true anomaly for near singular orbits.