I am wondering how you are inducing your period ? Are you on birth control pills by any chance .. perhaps the kind where you don't get a period so you are now "forcing" a withdrawal bleed ? If this is the case then you are not really inducing a period as the withdrawal bleeding is not a real period, the kind that comes with ovulation. I therefore do not think this will have any effect on your cyst.
If the cyst is mostly anechoic and it is indeed quite small (the size of a normal ovarian follicle), then most likely it will be gone by the time you have your next ultrasound. My experience is quite different, unfortunately, but most likely benign as well. I have an almost 9 cm simple cyst on my left ovary which I have been watching for about 18 months now. It keeps growing at a very slow rate so I am having it removed with surgery. I don't really have a choice at this point. Fortunately, it's behaviour over the past year and a half is very telling as to whether or not it is dangerous and its "looks" on the ultrasound has everyone convinced that it is benign (ie. thin walled, simple cyst on one ovary). Let us hope they are correct for my sake
Anyway, sometimes these damned things don't go away on their own and they have to be removed. I don't know if it had anything to do with it but I had been on birth control pills for about twenty years non stop (I don't have children) before we found the problem. Needless to say that I have gone off them and never plan to ever go on any other form of hormonal "therapy" (ha ha that's a good one .. "therapy") ever again.
Come back and let us know how you are doing. I am sure that your cyst will have resolved by your next ultrasound.
All the best.