Periodic Behaviors and Graceful Transitions Between Them for Bimodal Systems: Example of a Jumper on a Trampoline

This letter sets up a simple model for the controlled jumps of a jumper on a trampoline. Such a model is a paradigm for the impulsive control of more general multi-mode multi-dimensional (M3D) systems. First, necessary and sufficient conditions for a motion to be periodic are set up and different types of periodic orbits, characterized by timing diagrams, are identified. Next, we characterize and identify graceful transitions between such orbits, using our raccordation framework. This generalizes notions of persistency of behavior to the class of impulsively controlled multi-mode multi-dimensional systems.