Suppose we have a mass attached to the top of an ideal (linear and massless) spring oriented vertically in a uniform gravitational field, and on top of that mass there is another mass resting on it. The two masses are not attached at all, so they will lose contact with each other as the normal force is about to become negative. Also suppose that once the two masses separate and collide again, they undergo perfectly elastic collisions.
First of all, is there a name for systems like this? It seems like an "ideal trampoline" to me but searching for that doesn't yield much. Has anyone ever discussed it in a book?
Second of all, is this system chaotic? For sufficiently small oscillations, of course, the masses remain in contact the whole time and you get simple harmonic oscillation, but above some threshold the free mass will keep bouncing off the spring-attached mass and it's quite nontrivial to figure out what eventually happens. Do you get interesting things like period doubling?