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
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
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  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
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 :)
"Symplectic Maps, Variational Principles, and Transport" Rev.Mod.Phys. 64, 795-848, (1992)
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Next: Homoclinic structures in
the standard map
updated 21 Sep 2003