Summary
We present a novel return map approach for studying the global dynamics of a vibro-impact (VI) pair, that is, a ball moving in a harmonically forced capsule. Computationally efficient short-time realizations divide the state space according to different dynamics. A small collection of reduced piece-wise polynomial approximations yields a composite map capturing transients and reproducing bifurcation sequences of the full system. Valuable for cobweb analysis, this framework inspires auxiliary maps based on the extreme bounds of the maps, yielding global dynamics of energetically favorable states. We apply this framework to understand stochastically enhanced bi-stability for parameter regions exhibiting grazing and noise-sensitive biases. Results are relevant for recent designs of VI-based energy harvesters and nonlinear energy transfer.
