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I am trying to plot the stroboscopic map of the classical kicked rotor, which is characterized by the equations: $$p_{n+1} = p_n - \frac{dV}{dx}|_{x=x_n}$$ $$x_{n+1} = x_n +p_{n+1}$$ where $x_n$ is on a ring of radius $1/(2\pi)$ and $V = -\frac{K}{(2\pi)}\cos(2\pi x)$

What I did was I took a random $(x_0, p_0)$ in the range $(0,1)$ for both $(x, p)$ initially, then calculated $(x_1, p_1)$ where I checked if $x>1 \text{ or }x<0$ and changed the $x$ value such that it lies in the bound $[0, 1]$, I continued this for a fixed number of times say 1000. Note: I didn't do anything for $p$ since there are no constraints on $p$.

I am getting satisfactory results for $K = 0.5$, near-integrable region, and $K = 3.5$ mixed region but for $K=8$, which is the chaotic region. I am not getting the required result.

Can anyone tell me where my approach is wrong? Any link to reference material would be beneficial.

$$K = 0.5$$

K=0.5

$$K = 3.5$$

K=3.5

$$K = 8$$

K=8

As seen in the $K = 8$ case, it should be chaotic, and the dots should fill the entire space, but the dots are loosely packed.

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For the less perturbed system, the KAM curves limit the diffusion of the orbits which therefore remain in the region plotted, whereas for the strongly chaotic case they're free to diffuse to higher values of $|p|$.

But the system has a symmetry, as described by its Wikipedia entry (my emphasis):

$p_{n}$ is not periodic as in the standard map. However, one can directly see that two rotators with same initial angular position $\theta_{0}$ but shifted dimensionless momentum $p_{0}$ and $p_{0}+2\pi l$ (with $l$ an arbitrary integer) will have the same exact stroboscopic dynamics, but with dimensionless momentum shifted at any time by $2\pi l$ (this is why stroboscopic phase portraits of the kicked rotator are usually displayed in a single momentum cell $p\in [-\pi ,\pi]$).

as well as in Scholarpedia:

Due to the periodicity of $\sin x$ the dynamics can be considered on a cylinder (by taking $x \mod 2\pi$) or on a torus (by taking both $x,p\mod 2\pi$).

So, unless you're investigating, say, momentum diffusion, you're also free to keep $p$ bounded to $[0,1]$ for your plots, and the highly chaotic orbits will be dense as expected.

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