Every bounce leaves a trace.
The structure behind the motion.
Compare return maps, local stability, recurrence and spectral structure. Every diagnostic states how it samples the system.
The collision ledger
| Impact | Ball | Time s | Before m/s | After m/s | Impulse N·s | Phase rad | Contact mode |
|---|
From motion to mathematics.
From a falling ball to a nonsmooth dynamical system. Derivations use upward-positive coordinates and the cosine table convention of your source code.
A landscape of impacts.
Sweep the exact impact map and watch a periodic branch split into a richer set of collision states.
Sweep configuration
180 impacts discarded, then 48 recorded at each Γ. A capture resets the state to lift-off. This finite sample reveals branches, not a definitive chaos classification.
The high-bounce approximation predicts a different period-doubling threshold. The lines above use the exact local period-1 condition; admissibility and competing attractors still depend on the initial state.
A small system, carefully modeled.
The workbench connects your exploration code to a physical, inspectable experiment. These are the sources and modeling decisions behind it.