Standard map

[1] It is constructed by a Poincaré's surface of section of the kicked rotator, and is defined by: where

The properties of chaos of the standard map were established by Boris Chirikov in 1969.

This map describes the Poincaré's surface of section of the motion of a simple mechanical system known as the kicked rotator.

The standard map is a surface of section applied by a stroboscopic projection on the variables of the kicked rotator.

However, this map is interesting from a fundamental point of view in physics and mathematics because it is a very simple model of a conservative system that displays Hamiltonian chaos.

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

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.

The phase-space of the standard map with the variation of the parameter from 0 to 5.19 ( in y axes, in x axes). Notice the appearance of a "dotted" zone, a signature of chaotic behavior .
Orbits of the standard map for K = 0.6.
Orbits of the standard map for K = 0.971635.
Orbits of the standard map for K = 1.2.
Orbits of the standard map for K = 2.0. The large green region is the main chaotic region of the map.
A single orbit of the standard map for K =2.0. Magnified close-up centered at , p = 0.666, of total width/height 0.02. Note the extremely uniform distribution of the orbit.