The Standard map

The area-preserving Standard (or Taylor-Chirikov) map is
    pn+1 = pn + K sin xn ,
    xn+1 = xn + pn+1   (mod 2π ).     (*)
One can derive these equations as the Poincare map for the kicked rotator - 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) ∑ n δ (t - nT) = p2/2 + K cos(x) ∑m ei 2π mt.
Rotator dynamics is reversible
    xn = xn+1 - pn+1,
    pn = pn+1 - K sin xn   (mod 2π ).
The map has reflection symmetries (x, p) → (-x, -p) and (π + x, p) → (π - x, -p), i.e. reflections with respect to the point (0, 0) and the center of the picture (π, 0). Due to mod 2π operator the "periodic" standard map is also invariant under translations p → p + 2πn. 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 + 2πn are similar.
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. Press Enter to set new K value. Coordinates x, p of the image center and its scales dx, dp are shown in the first text field.

Nearly integrable dynamics

For K = 0 the dynamics of rotator is integrable. Its moment pn = ω = const and its angle xn = nω   mod 2π. Thus it is the constant rotation. Every orbit lays on an invariant circle. When ω 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 δp = 4 K½.
E.g. the half-width is 0.632 for K = 0.1 . Click mouse in the left picture to get x, p. 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