Bouncing Ball on a Vibrating Table: Period Doubling and Chaos (2D)

Setting up the lab bench...

Presets

Live readouts

Graphs

Table and ball height against time

Impact map: phase against post-impact speed

Bifurcation diagram: post-impact speed against Gamma

Flight-time return map (n against n+1)

What is happening now

Press Play or choose a preset.

Numerical checks

running...

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.

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.