*draft*
# Resonances

There are also resonances, corresponding to the periodic orbits with
"frequency" *(m, n)*, where *m* and *n* are integers.
This means that the orbit rotates *m* times around the cylinder in
*n* iterations. I.e. for "extended" map *p*_{n} = p_{o}
and *x*_{n} = x_{o} + 2p m .
You see *1/2, 1/3, 1/4, 2/3* resonances below. Note that *1/3* and
*2/3* resonances are symmetric with respect to the
*(p, p)* point
(in the center of the picture).

*Controls:* Click mouse with *Shift* to get one step of an
orbit.
Each resonance consists of a chain of *n* islands and each island
has a structure similar to the pendulum.
Perturbation theory implies that the width of the *m/n* resonance
grows as *K*^{ n/2} for *K* small. At the center of the
island, and at the cusp of the separatrix, are periodic orbits with
frequency *m/n* . Typically there appear to be only two such periodic
orbits.
Orbits trapped in an island move successively from one island to another,
following the periodic orbit (they skip *m-1* islands each step). Thus
there is an entire region of phase space that has frequency *m/n* .

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*updated* 7 September 2003