Cyclotron sequences

Question

If one considers how the particle's energy grows in the (idealized) cyclotron, one gets the following sequence of numbers $$E_1=1+2V, \;\;E_n=E_{n-1}+2V\cos{[2\pi(E_1+\ldots E_{n-1})]},\;n\ge 2. \tag{1}$$ For $V\ll 1$, one can find an approximate expression $$E_n\approx 1+\sqrt{\frac{V}{\pi}}\,\frac{\mathrm{sn}(2\sqrt{\pi V}\,n,1/\sqrt{2})}{\mathrm{dn}(2\sqrt{\pi V}\,n,1/\sqrt{2})}, \tag{2}$$ where $\mathrm{sn}$ and $\mathrm{dn}$ are Jacobi elliptic functions. Have numeric sequences of this type been investigated in the literature? In particular, is it possible to improve the approximation (2) for not very small values of V?

I'm also interested in the sequence $$E_1=1+V, \;\;E_n=E_{n-1}+(-1)^{n-1}V\cos{[\pi(E_1+\ldots E_{n-1})]},\;n\ge 2, \tag{3}$$ which is closely related to (1) for small $V$.

Answer 1

This would appear to be identical to https://en.m.wikipedia.org/wiki/Standard_map, with the identifications: $x_n = 2 \pi \Sigma E_n + \pi$, $p_n = 2\pi E_n$, $ K = V/\pi$.

Physically, your idealized cyclotron may be same as the kicked rotator, so generalizations you are interested in might already be explored under that name.