In this explorable you can investigate the dynamics of a famous two-dimensional, time discrete map, known as the standard or Chirikov–Taylor map, one of the most famous dynamical systems that exhibits deterministic chaos. It is almost identical to the “Kicked Rotator”, an idealized physical system in which a rotating rod is periodically kicked by an external force.

Click a few times on the empty display panel below and keep on reading….

This is how it works:

The system is defined by the iterative equations

[\begin{align} p _{n+1} &= p _n+K\sin(x_n) \mod 2\pi\newline x _{n+1} &= x _n +p _{n+1} \mod 2\pi \end{align}]

for the two variables $x$ and $p$. Given an intial condition $(x_0,p_0)$, the map generates a sequence $(x_n,p_n)$ with $n=0,1,2…$ that defines an orbit. An intuitive way of visualizing an orbit is as a sequence of dots in the two-dimensional box, where the x-axis corresponds to x-coordinate (duh) and the y-axis to the p-coordinate. This is the dynamical system’s phase space. Because of the modulo $2\pi$ the points are all limited to the range $[0,2\pi]$ along both axes.

The constant $K$ is the crucial parameter of the system. When $K=0$, for example, the $p_n$ coordinate remains fixed $p_n=p_0$ and $x_n$ advances a constant step to the right (modulo $2\pi$) every step. When $K>0$ the situation is more complicated. You can vary the parameter $K$ using the slider. Initially this parameter is $K=0.5$.

When you click on a point in phase space you can select the initial condition $(x_0,p_0)$ and an orbit of 1000 steps is generated. You can continue to select additional initial conditions to generate a family of orbits.

You should be able to see orbits that fill periodic patterns. Before you start changing $K$ make sure to pick a dozen or so intitial conditions. The last chosen orbit is always the red one.

At $K=0.5$ the orbits should fill smooth wave like patterns or quasi-elliptic closed orbits. When you increase $K$ you should be able to observe areas in phase space in which this periodicity is lost and orbits are chaotically scattered. The chaotic sea increases when you increase $K$ further. Observe what happens as you slowly vary $K$. You can always add additional orbits by clicking at selected points in phase space.

The standard map is almost identical to the so-called kicked rotator, which has a physical interpretation, e.g. an idealized, frictionless pendulum that is subjected to short but intense gravitational pulses that, depending on the angular state of the pendulum, transfer angular momentum to the system.