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?

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    $\begingroup$ Is the spring attached to the ground? $\endgroup$
    – user9886
    Commented Aug 10, 2012 at 8:22
  • $\begingroup$ Yes, the bottom end of the spring is fixed in place. $\endgroup$ Commented Aug 11, 2012 at 23:03

1 Answer 1


That's an impact system, and "ideal trampoline" is a good name. (Probably a more ideal one would be massless - i.e., there'd only be one mass $m$, the one not attached to the spring - but that has a boring dynamics.)

I'd expect the system to be chaotic, since some limiting versions of it are. It's well known that vibroimpact systems can be chaotic and, although there's no non-constant forcing here, in the limit of the attached mass $M$ being infinitely larger then the free one, $M$ can oscillate unperturbed and act as a periodically moving wall: that would be a version of the Fermi map, which is chaotic.

This suggests that, with respect to the mass ratio $r\equiv M/m$, the system should follow some route to chaos as $r$ varies in $[0,\infty)$, but it's not obvious to me at the moment whether this route would be period doubling or something else.


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