It will surely do one thing: give me a headache trying to figure out how the geometry should be...I place a fuel injector and a spark and somthing like a flamholder in the cycling duct, in order to re-accelerate the cycled air. will this do anything?
Basically, you are building two ram jets, coupled in such a way that they each provide each other with ram pressure. What you are doing is trading fuel efficiency for low speed operation, which in some cases is a really nice trade to do.
You will most certainly need to provide some air flow to start the contraption, but that's also manageable.
What makes me somewhat cautious is that my next train of thoughts took me to a setup with two ram jets side by side, with crosswise tubes providing the feedback between them, but at that point I realize that the construction, although visually quite different from your original concept, was conceptually almost identical, except for the the engines feeding each other instead of themselves.
This clearly needs some more thinking. There may be some possibilities, but they require some work to find.