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Press Play or choose a preset.
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How to use
Pick a preset (the ball starts on the period-1 branch and is followed to the chosen Gamma) or press Drop ball for an arbitrary start. Space pauses, R re-seeds, D drops.
- Slide Gamma slowly from 1.1 upward and watch the hop pattern double at Gamma about 1.5 (e = 0.5), then double again and turn chaotic.
- Lower the restitution e and the whole cascade moves up in Gamma.
- Graph 2 is the impact map (phase against speed): a point for period 1, a few for periodic orbits, a thin strange structure for chaos.
- The bifurcation diagram (graph 3) is recomputed for the current e. The vertical line marks your Gamma.
Drag to orbit, wheel or pinch to zoom, or press Guided tour. Quality switches High/Low (also ?q=low).
The science
The table moves as s(t) = A sin(wt), and the shaking strength is the dimensionless Gamma = A w^2 / g, the peak table acceleration in units of g. Between impacts the ball is in free flight, z(t) = z0 + v t - g t^2/2. At an impact with table velocity u and incoming ball velocity v-, the restitution law gives
v+ = (1 + e) u - e v-
so every impact removes a fraction of the relative kinetic energy and the table puts it back. The pair (impact time, post-impact velocity) defines an exact two-dimensional map, found here by root-finding the flight equation. A ball can only leave the table if Gamma > 1. A periodic orbit that lasts exactly one table cycle needs
cos(phi*) = pi (1 - e) / ((1 + e) Gamma)
which requires Gamma >= pi(1-e)/(1+e) (1.047 for e = 0.5). In the variables (impact time, relative kinetic energy per mass after impact) the map contracts area by exactly e^2, which is why a strange attractor with a fractal cross-section lives in it. Raising Gamma at fixed e produces the classic cascade 1, 2, 4, ... then chaos with periodic windows (for example period 3 near Gamma = 2.4 and period 5 near 3.2 for e = 0.5).
Honest simplifications. The table is perfectly rigid and sinusoidal, the ball is a point mass, restitution is constant and independent of impact speed, there is no air drag or spin, and a chattering ball is treated as riding the table until the table accelerates down faster than g. Real balls show speed-dependent restitution. Which attractor you land on depends on the start: the presets follow the period-1 branch by continuation in Gamma, Drop ball starts from rest.
In-page checks: (1) the period-1 impact phase against the closed form, (2) flight time against the table period, (3) the finite-difference Jacobian determinant of the impact map against e^2, (4) the real-time sub-stepped simulator against the exact map.