# 2D Circle Map - Demo 1

## The Trivial Limit

The 2D Circle map is

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

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

Since the behavior is periodic in *x* with period 1 we may truncate the values of *x* to the unit interval. Here we study the dissipative case *b<1*
First look at the trivial case *a=0*:

After a brief transient *y* relaxes to zero, and *x* satisfies the simple equation

If *c* is irrational (the value used is close to the Golden Mean), the successive points never lie on top of a previous point, and eventually will fill the whole interval. Since *x=1* is equivalent to *x=0* the orbit has the topology of a circle. If the map corresponds to a Poincare section of a 3-dimensional flow, the attractor will be a torus, corresponding to quasiperiodic motion.
For *c* rational (e.g. 0.6) successive points repeat, and the motion is periodic.

We can again understand the plot as arising from the flow on a torus or donut, but now the flow raps around the "tube" of the donut three times for every five revolutions around the "hole" giving a path that exactly repeats every 5 revolutions.

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Last modified Friday, February 11, 2000

Michael Cross