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.


The phase space of the standard map, \(\mathbb{T}^2\), is represented by the black canvas below. 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.


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

Further reading

  1. Boris Chirikov and Dima Shepelyansky. Chirikov standard map. From Scholarpedia.
  2. Standard Map. From Wikipedia.
  3. Eric Weisstein. Standard Map. From MathWorld—A Wolfram Web Resource.
Konstantinos Efstathiou
Konstantinos Efstathiou
Mathematics Dynamical Systems