# 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 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.

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

## Further reading

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