| xn+1 | = | xn + yn+1/2 |
| yn+1 | = | yn + a sin(2xn) |
| xn+1 | = | xn + yn+1/2 |
| yn+1 | = | yn + a sin(2xn) |
For a=0 the system is integrable, with the orbits simply
| xn+1 | = | xn + y/2 |
The orbits are straight lines spanning the x range corresponding to the tori of the Hamiltonian dynamics. The winding number is simply the value of y. Note that for an initial condition with a rational y only discrete points on the torus will be visited, but this is a set of measure zero, and most initial conditions will be irrational (except for the finite accuracy of the computer representation of the numbers) and the dynamics will fill out the full curve.