If an integrable quasiperiodic system is slightly perturbed so that it becomes nonintegrable, only a finite number of n-map cycles remain as a result of mode locking. One will be elliptical and one will be hyperbolic.

Surrounding the elliptic fixed point is a region of stable map orbits which circle it, as illustrated above in the standard map with K=1.5. As the map is iteratively applied, the island is mapped to a similar structure surrounding the next point of the elliptic cycle. The map thus has a chain of islands, with the fixed point alternating between elliptic (at the center of the islands) and hyperbolic (between islands). Because the unperturbed system goes through an infinity of rational values, the perturbed system must have an infinite number of island chains.

See also

Map Orbit, Mode Locking, Quasiperiodic Function

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Cite this as:

Weisstein, Eric W. "Island." From MathWorld--A Wolfram Web Resource.

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