Can a passive hopper hop forever?

It is now understood that stable gait can be exhibited by passive walking m odels. The energy losses associated with collisions in such passive models play a central role in achieving stable gait, We study a simple passive mod el for hopping. The model consists of a two-mass, one-spring system. The on ly mode of energy dissipation is through the plastic collisions with the ri gid ground, We show that there exist solutions that involve lossless inelas tic collisions and thus lead to incessant hopping. The global dynamics of t he hopping model is studied by constructing a one-dimensional map. We show that the fixed points of the one-dimensional map exhibit one-way stability. The consequences of an infinite number of fixed points and their one-way s tability on the dynamics is also studied and it is shown that there exists a nested basin of attraction for each fixed point of the system, making the fate of an individual orbit unpredictable.