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 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