# Hamiltonian Chaos - Demo 1

## 2D Circle Map: Dissipative ......

The 2D Circle Map is defined by the equations

x_{n+1} |
= |
x_{n} + c - y_{n+1}/2 |

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

For *b=0.5* the map is dissipative. We are interested in the breakdown of the single attractor corresponding to an invariant torus as the nonlinearity is increased. In the following 3 demonstrations click on the running plot to start from a new initial condition: since a transient of length 100 is not plotted, you should see the dynamics returns to the same attractor.

For a=0.8.....

.....the curve is smooth.

As *a* increases the curve crinkles, e.g. for *a=1.0 .....*:

For *a=1.2 .....*

.....the curve has become a fractal, as you can see by enlarging portions of the plot or calculating dimensions, and the motion is chaotic.

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Last modified Sunday, March 5, 2000

Michael Cross