# Hamiltonian Chaos - Demo 3

## Standard Map: the Integrable Limit

The standard map is the area preserving (*b=1*) 2D circle map with *c=0*, and is given by
x_{n+1} |
= |
x_{n} + y_{n+1}/2 |

y_{n+1} |
= |
y_{n} + a sin(2x_{n}) |

For *a=0* the system is integrable, with the orbits simply

x_{n+1} |
= |
x_{n} + y/2 |

with *y* a constant determined by the initial point. Run the demonstration and click at various points on the running plot to start from new initial conditions.

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.

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Last modified Thursday, March 2, 2000

Michael Cross