Chirikov standard map: KAM tori and the breakdown of integrability

What you are seeing: the simplest area-preserving map that exhibits the transition from regular motion to chaos. At each step we kick a rotator and let it free-rotate; the result is a discrete Hamiltonian dynamics on the torus $(\theta, p) \in [0, 2\pi)^2$. The rule is $p_{n+1} = p_n + K\sin(\theta_n)$, $\theta_{n+1} = \theta_n + p_{n+1}$ (both mod $2\pi$).

At $K = 0$ the dynamics is integrable: every horizontal line $p = \text{const}$ is invariant, and trajectories fill it densely if $p/2\pi$ is irrational or close to it in a Diophantine sense. As $K$ grows, the KAM theorem guarantees that "sufficiently irrational" tori survive while rational tori shatter into chains of islands and surrounding chaos. The most persistent torus has the golden-mean winding number $(\sqrt 5 - 1)/2$ and breaks last, at the critical value $K_\text{crit} \approx 0.9716$. Above this value, orbits can in principle diffuse arbitrarily far in $p$.

Click anywhere on the plot to seed a new orbit at that $(\theta, p)$. Each color is a separate initial condition. Drag the $K$ slider and watch the regular curves break up.