| xn+1 | = | (xn + c + (a/2 ) sin(2xn)) mod 1 |
For values of a for which the invariant curve is smooth, we can construct a one dimensional return map xn+1 v. xn that is monotonic. The approach to the breakdown of the torus is signalled by this map becoming nonmonotonic. We can see this most easily in the limit b=0, when in fact the 2d map reduces to a one dimensional map:
| xn+1 | = | (xn + c + (a/2) sin(2xn)) mod 1 |
The return map for X is:
and is monotonic. We can investigate the iterations of this map using the 1D map applet (here the parameter b gives the external frequency which was c in the 2d map):
The quasiperiodic nature of the dynamics is shown by the power spectrum (increase the speed slider and iterate):
For a=1.2 (again adjusting c to keep the winding number close to 0.618) the 2D map is:
and a smooth fold develops around x=0.5. (The secondary folds that were visible for nonzero b are suppressed by the strong contraction for b=0.) The corresponding one-dimensional return map
is no longer monotonic, with a maximum at around x=0.9 and a minimum near x=0.1. The dynamicsis chaotic with the power spectrum:
showing broad band components.
Again frequency locking can occur giving a periodic orbit e.g. for a=0.8 changing the "external frequency" to 0.65 gives locking at a frequency of 2/3:
with power spectrum