# Hamiltonian Chaos - Demo 2

## 2D Circle Map: .... and Area Preserving.

Now look at the same parameters except *b=1*. For each demonstration click on various points of the running plot to start from new initial conditions.

You should find a rich variety of behavior for each set of parameters with

- Tori (appearing as curves traversing the plot) that are remnants of the tori of the integrable system
*a=0*. These curves have irrational winding number, i.e. quasiperiodic motion.
- "Island chains" of elliptical orbits (tori) surrounding discrete points giving finite winding number corresponding to frequency locked orbits.
- Chaotic orbits that are most easily found in the hyperbolic regions between two elliptical islands.

The structure is very rich: if you enlarge various regions of the plot you will find the same range of behavior repeated on smaller and smaller scales. (Remember you can reset the scales by clicking outside a stopped plot.) As *a* increases the chaotic orbits become more evident and more of the tori break down.
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Last modified Sunday, March 5, 2000

Michael Cross