# 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 *x*_{n+1} v. x_{n} 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:

x_{n+1} |
= |
(x_{n} + c + (a/2) sin(2x_{n})) 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