2D Circle Map - Demo 5

One Dimensional Return Map

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
First look at this limit in the two dimensional representation for a=0.8:

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 dynamics

is 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

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Last modified Tuesday, December 14, 1999
Michael Cross