Standard map

Picture a rotor that turns freely, except that once per period it receives a sharp kick whose strength depends on its angle. Write down its angle $θ$ and angular momentum $p$ just after each kick and you get the standard map, also called the Chirikov-Taylor map after Boris Chirikov and John Taylor. It is the simplest model of how the invariant curves of a regular Hamiltonian system break up as a perturbation grows, and its single parameter $K$ takes it from almost regular motion to almost complete chaos.

What you see

The square is the phase space: $θ$ runs from 0 to $2 π$ across, $p$ from 0 to $2 π$ down, and both wrap around at the edges. About 4,000 orbits start on an even grid, one dot each, and the map is applied to all of them once per frame. Only the current positions are drawn. The dots are colored in eight bands by their starting value of $p$ . On an invariant curve a band of dots keeps its order and smears out along the curve; in a chaotic region the colors mix. The λ₁ (Lyapunov) readout is an estimate of the largest Lyapunov exponent per iteration for one orbit that starts near the corner $(0, 0)$ , where the map has an unstable fixed point.

Integrable (K=0.2): almost every orbit stays on a smooth invariant curve. Rows of dots slide sideways at different speeds, since $p$ sets how far $θ$ advances per kick. Around the stable fixed point $(π, 0)$ lies an island of closed curves, which appears split between the top and bottom edges because $p$ wraps around.

Chaos onset (K=0.97): just below the critical value $K_{c} \approx 0.9716$ found by John Greene in 1979. At $K_{c}$ the last invariant curve that runs all the way across the square breaks; its winding number is the golden mean. Above $K_{c}$ a chaotic orbit can travel through every value of $p$ .

Mixed (K=1.5): a large island around $(π, 0)$ with smaller islands nearby, surrounded by a connected chaotic sea.

Chaotic sea (K=3.0): most of the square is chaotic and the colors mix quickly. The island around $(π, 0)$ has shrunk but survives.

Global chaos (K=6.0): the fixed point at $(π, 0)$ lost its stability at $K = 4$ , and no large island is left. The colors mix faster still.

The rules

p_{n + 1} = p_{n} + K \sin θ_{n}, θ_{n + 1} = θ_{n} + p_{n + 1},

both taken modulo $2 π$ . The Jacobian determinant is 1, so the map preserves area: there is no attractor, and the dots never crowd onto a smaller set. It has two fixed points. The one at $(0, 0)$ is unstable for every $K > 0$ . The one at $(π, 0)$ is stable for $0 < K < 4$ , since the trace of its Jacobian, $2 - K$ , then lies between $- 2$ and 2.

Parameters

K (nonlinearity): the kick strength, 0 to 10 (default 0.97). At $K = 0$ every orbit moves on a horizontal line.