2D Circle Map - Demo 1

The Trivial Limit

The 2D Circle map is

xn+1 = xn + c - yn+1/2
yn+1 = b yn - a sin(2xn)
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

xn+1 = xn + c
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