Creative Commons LicenseStandard map by Konstantinos Efstathiou is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.

Standard map

The standard map is an area-preserving diffeomorphism from the two-dimensional torus \(\mathbb{T}^2\) to itself. Given coordinates \((x,y) \in \mathbb{T}^2\) the standard map can be expressed as \[ y' = y + \frac{K}{2\pi} \sin(2 \pi x), \quad x' = x + y'. \] Both \(x\) and \(y\) are defined modulo 1.

Instructions

The phase space of the standard map, \(\mathbb{T}^2\), is represented by the black canvas at the right. The horizontal coordinate is \(x\) and the vertical one is \(y\). The range of both is \([0,1)\).

Clicking inside the canvas draws an orbit with N iterations starting at that point. Orbits are drawn with a randomly chosen color. The button draws 100 orbits with random initial conditions and N iterations. Changing the value of K will also clear the canvas before any more orbits are drawn.

Further reading

  1. Boris Chirikov and Dima Shepelyansky. Chirikov standard map. From Scholarpedia. http://www.scholarpedia.org/article/Chirikov_standard_map.
  2. Standard Map. From Wikipedia. http://en.wikipedia.org/wiki/Standard_map.
  3. Eric Weisstein. Standard Map. From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/StandardMap.html.