The 2D Circle Map is defined by the equations
|xn+1||=||xn + c - yn+1/2|
|yn+1||=||b yn - a sin(2xn)|
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.
.....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.