For a set of orbits with epicycles, we have a pattern which rotates with  ½. If the phases are all the same, we have the pattern on the left. If the phases increase systematically with radius we find a spiral pattern, where the oval orbits bunch up.
Density wave theory envisions a situation in which the very density enhancements caused by the synchronized epicycles perturb the orbits in just the way needed to maintain these same epicycles. Hence the entire system is stable and longlived.
Figure from BTI Fig 6.11
