# The Standard map

The area-preserving Standard (or Taylor-Chirikov) map is

*p*_{n+1} = p_{n} + K sin x_{n },

x_{n+1} = x_{n} + p_{n+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) = p*^{2}/2 + K cos(x)
∑_{ n} δ (t - nT) = p^{2}/2 + K cos(x)
∑_{m} e^{i 2π mt}.

Rotator dynamics is reversible

*x*_{n} = x_{n+1} - p_{n+1},

p_{n} = p_{n+1} - K sin x_{n}
(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 *p*_{n} = ω = const and its
angle *x*_{n} = 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 = p*^{2}/2 + K sin x.

The separatrix *H = K* bounds the resonant island. Its width is

*max δp = 4 K*^{½}.

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)

Contents
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Next: Homoclinic structures in
the standard map

*updated* 21 Sep 2003