I think part of the confusion comes from the fact that "period doubling" can mean two rather different things, which are probably related but this is not completely known. The first is a purely observational meaning, which happens when you have a clear periodic pulsation in a star, but the amplitude shows a higher-lower-higher-lower pattern that implies a "double" period to see the full cycle. Then there's the theoretical meaning, which is a step in the approach to chaos.
The way the approach to deterministic chaos often works is, you have a simple system that shows a simple period. But then you increase the degree of nonlinearity in the system, and you see bifurcations into something that requires double the number of iterations to be cyclic. If each iteration represents a cycle time, like the time to go around an orbit or the time for a pulsation, then that means double the period. Maybe you started with an interative mapping that converges to a fixed point, but then you increased the nonlinearity and it turned into a limit cycle where the iterations go back and forth between two points.
Now, it's not immediately obvious the connection between that and the amplitude modulation of a simple cycle, and I think many of the sources are purposefully ambiguous about that connection because nonlinear dynamics is still a mysterious field. But the basic idea seems to be, if you have a linear system, it has uncoupled pulsation modes that are higher harmonics of the lowest frequency "fundamental" mode. But if you have a nonlinear system, you can have coupling between those modes, which end up looking like amplitude modulation in a lower harmonic due to resonant coupling with a higher harmonic. If the coupling has an odd-integer relationship, like a frequency of 2 coupling to a frequency of 3, then if the first oscillation in the frequency 2 mode couples resonantly with the frequency 3 harmonic, then the second oscillation will not couple well, it will hit the harmonic when it is 3/2 of the way through a cycle, so is out of phase. Then the third cycle of the frequency 2 mode will hit the frequency 3 mode when it has gone through 3 full cycles and is back into resonance. So that gives the "period doubling" where the amplitude of the frequency 2 harmonic gets a boost every second time around, owing to its ability to receive energy from the higher mode as well.
The connection between that and period doubling bifurcations is not entirely obvious either, but it seems that the idea is that if you have nonlinearity, the period doubling you find on the path to chaos might be related to this kind of resonant interaction between the simple behavior you see, and higher frequency behaviors that could exist but only make their presence known as a kind of residue from how they interact with the behavior you do see. In other words, if the linearity is weak, some simple oscillation gets excited, but if you crank up the nonlinearity, that simple oscillation starts to overlap with more complicated behaviors that require more cycles to return to the original resonance relationship. Crank up the nonlinearity even more, and all you see are the complicated behaviors, you have complete chaos. The period doubling in variable stars represents only a small amount of nonlinearity, it's not chaotic and the relatively simple nature of the resonances allow for a large amplitude to build up and allow the stars to be classified as variable in the first place.
To do more research into this, you might look up the "Blazhko effect," or how period doubling also applies to other variable stars. It sounds like an active area of research where there are still more questions than answers, but Kepler data has the promise of producing the necessary information to solve it.