Is there a general theory of locking simple (two-transistor or two- tube or two-whatever) RC multivibrators to a sine wave or square wave reference? Especially when frequency multiplication or division is done in the process?

What I'm thinking, is that someone must've worked out some theory that says if you have a multivibrator of a certain natural frequency, and that if you inject a sine wave of a given amplitude at a base or a cathode or whatever with a slightly different frequency, that you will be able to lock if the natural frequency is 10% high or 5% or low or whatever, and if the amplitude injected is bigger, then the lock range is bigger, etc.

I'm using some very vague terms above. I've explored the topic using Spice and clearly there's some more general math rather than "run a metric buttload of Spice simulations" because there are obvious patterns in the regions of stability. The patterns are kind of pretty when you are locking to a harmonic or subharmonic.

But rather than running a metric buttload of Spice simulations, I believe there must be a more general theory about amplitude of injected reference and range of lock.

Maybe this is some class of PLL with very broad/nonexistent loop filter.

My old-fashioned references that talk about phase locking multivibrators, or phantastatrons, like the MIT Radiation Lab books, give some general guidance but not really the formulas I believe must exist.

Any clues for me?

Tim.