Standard map

Main properties

For ${\displaystyle K=0}$ the map is linear and only periodic and quasiperiodic orbits are possible. When plotted in phase space (the θ–p plane), periodic orbits appear as closed curves, and quasiperiodic orbits as necklaces of closed curves whose centers lie in another larger closed curve. Which type of orbit is observed depends on the map's initial conditions.

Nonlinearity of the map increases with K, and with it the possibility to observe chaotic dynamics for appropriate initial conditions. This is illustrated in the figure, which displays a collection of different orbits allowed to the standard map for various values of ${\displaystyle K>0}$. All the orbits shown are periodic or quasiperiodic, with the exception of the green one that is chaotic and develops in a large region of phase space as an apparently random set of points. Particularly remarkable is the extreme uniformity of the distribution in the chaotic region, although this can be deceptive: even within the chaotic regions, there are an infinite number of diminishingly small islands that are never visited during iteration, as shown in the close-up.

Circle map

The standard map is related to the circle map, which has a single, similar iterated equation:

${\displaystyle {\theta }_{n+1}={\theta }_n+\mathrm{\Omega }-K\sin ({\theta }_n)}$

as compared to

${\displaystyle {\theta }_{n+1}={\theta }_n+p_n+K\sin ({\theta }_n)}$

${\displaystyle p_{n+1}={\theta }_{n+1}-{\theta }_n}$

for the standard map, the equations reordered to emphasize similarity. In essence, the circle map forces the momentum to a constant.