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.