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 pn = po
and xn = xo + 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
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 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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the standard map
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updated 7 September 2003