Classical mapping of kicked Harper model The Hamiltonian of kicked Harper model is given by 
$$
H=K\cos(p)+\left[K\cos(x)\sum_{n=-\infty}^{+\infty}\delta(t-n)\right]
$$
where $\delta$ function term represents the effect of very narrow pulses of an external field which kicks the system only at $t=\ldots,(n-1), n, (n+1)\ldots$ and between two neighbouring kick Hamiltonian reduces to $H=K\cos(p).$ 
How to derive the classical map from $(x_n,p_n)$ to $(x_{n+1},p_{n+1})$ after $t = n$ and show that it is area preserving. Also need to identify the fixed points and identify the stabilities.
I am confused that the Hamiltonian is in dimensionless form also my concepts related to mapping are quite weak. Can anybody explain, so that I could learn?
 A: I'm not going to derive the solution but will observe that is type of parametric resonance problem is discussed in many (more advanced) textbook on classical mechanics.  The solution is given
here for instance, if you need to check your calculation.
Given the solution:
$$
p_{n+1}=p_n+K\sin(x_n)\, ,\qquad x_{n+1}=x_n−K\sin(p_{n+1})\, ,
$$
it is easy to show that the area is preserved using exterior calculus. Area preservation means the area 2-form
$d p_{n+1}\wedge dx_{n+1}=d p_n\wedge dx_n$.  Thus
\begin{align}
dp_{n+1}&=dp_n+K\cos(x_n)dx_n\, ,\\
 dx_{n+1}&= dx_n-K\cos(p_{n+1})dp_{n+1}\, ,\\
&=dx_n-K\cos(p_{n+1})\left(dp_n+K\cos(x_n)dx_n\right)\, .
\end{align}
With this, and remembering $dp_k\wedge dp_k=dx_k\wedge dx_k=0$ and
$dx_k\wedge dp_k=-dp_k\wedge dx_k$ one rapidly finds
\begin{align}
d p_{n+1}\wedge dx_{n+1}&= dp_n\wedge dx_n(1-K^2\cos(p_{n+1})\cos(x_n))\\
&\qquad -K^2\cos(p_{n+1})\cos(x_n) dx_n\wedge dp_n\, ,\\
&=dp_n\wedge dx_n\, ,
\end{align}
which shows the area-preserving property of the map.
