Multimodal map and complex basin of attraction of a simple hopper

In this paper, we study the global dynamics of a simple passive mechanical model for hopping. The hopper is a two-mass, single-spring system constrained to move in the vertical direction (under gravity) above a rigid ground. The hopper model and its basic dynamics including the existence of incessant hopping motions have been reported elsewhere. Here, we extend the study to investigate the global dynamics of the hopper. The global map of the hopper is multimodal. We construct an approximate analytic map near the fixed points of the map and show that the fixed points exhibit one-way stability. We also show that the map is invariant under the inversion of the mass ratio of the hopper. Next, we construct the global basin of attraction of these fixed points and show that their structure is highly complex and retains form at finer scales. This structure of the basin of attraction contains regions where the fate of an arbitrary initial condition becomes unpredictable.