The Standard map

The area-preserving Standard (or Taylor-Chirikov) map is
    pn+1 = pn + K sin xn ,
    xn+1 = xn + pn+1   (mod 2p )
.     (*)
One can derive these equations as the Poincare map for the kicked rotator (or rotor) - a rigid rotated body subjected to an impulsive torque K sin(x) at moments nT . The Hamiltonian is (we put T = 1)
    H(p,x) = p2/2 + K cos(x) Sn d (t - nT) = p2/2 + K cos(x) Sm ei 2p mt .

Controls: Click mouse to plot new (red) orbit. Click mouse with Alt/Ctrl to zoom In/Out. Hold the Shift key to zoom in the vertical direction only. It is number of points in an orbit. Number of generated random orbits is shown in the last field. Press Enter to set new values. Click "Rand" and "Clear" while you get more accurate (random!) picture. Coordinates x, p of the image center and its scales dx, dp are shown in the first text field.
Rotator dynamics is reversible
    xn = xn+1 - pn+1 ,
    pn = pn+1 - K sin xn   (mod 2p ) .

The map has reflection symmetries (x, p) -> (-x, -p) and (p + x, p) -> (p - x, -p) , i.e. reflections with respect to the point (0, 0) and the center of the picture (p, 0) .

Due to mod 2p operator the "periodic" standard map is also invariant under translations p -> p + 2p n . Therefore the map has the vertical translation symmetry and can be thought as acting on a torus. You can test to the left, that all orbits with p + 2p n are similar.

Nearly integrable dynamics

For K = 0 , the dynamics of rotator is integrable. Its moment pn = w = const and its angle xn = nw   mod 2p . Thus it is the constant rotation. Every orbit lays on an invariant circle. When w is rational every orbit is periodic, otherwise they are quasiperiodic and densely cover the circle
For small K we get resonant island at p = 0 (in the center of the left picture). Taking K and p small implies [1] that the differences in (*) can be replaced approximately by derivatives
    dx/dt = p,     dp/dt = K sin x .
It is the nonlinear pendulum equations with the Hamiltonian
    H = p2/2 + K sin x .
The separatrix H = K bounds the resonant island. Its width is
    max Dp = 4 K1/2.
E.g. the half-width is 0.632 for K = 0.1 . Click mouse in the left picture to get x, p in the status bar. As K increases the width of the island grows more slowly then predicted by the pendulum approximation. (Really, the story is much more intricate as you can see on the right picture :)

[1] J.D.Meiss "Symplectic Maps, Variational Principles, and Transport" Rev.Mod.Phys. 64, 795-848, (1992)

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updated 21 Sep 2003