I might have a dead-simple explanation for the different phase noise when locking at odd vs. even multiples of the free running frequency, by thinking about the locking phenomena in terms of the energy exchanged between the two oscillators, rather than looking at the voltage waveforms.
According to my theory, the lock "preference" in regards to odd/even multiples should not be affected by the non-linearity of the circuits. With or without non-linearities, the preference should still be there. However, you mention non-linearity as playing an important role, a role that might not be obvious without careful investigation, which makes me think I might be wrong trying to justify the locking preference in terms of energy.
Even divisions produce better PN results and wider lock range due to the complex behavior of the non-linear effects within the oscillator mechanics and best left to serious researchers to investigate (read deep dive into non-linear oscillator locking dynamics).
Was that remark mostly about the locking range, or about the phase noise at even vs odd multiples?
Anyway, the injection locking experiment looks so tempting to try right away.
Maybe not right now (being in the middle of something else), added to the wish list.
The weather is very hot here, and running the instruments for hours will make the room even hotter. I wonder if the experiment can be run in audio, with a big enough coil to make a 1kHz LC, then using the soundcard as AWG+DSO+SA. ("WaveForms", the software made by Digilent for their Analog Discovery, can also use the soundcard as a lab instrument
, with or without an Analog Discovery board).