I just came up with a numerical result I can not verify yet analytically, numerically my software is also poor, but:
Consider circle map (Arnold map) :
\( \theta'= \theta+\Omega-{\frac{K}{2*\pi}}*sin(2\pi\theta) \)
Let \( \Omega=0.567143.. \)
And \( K= {\frac{\pi}{2*\Omega}.. \)
then map becomes:
\( \theta'= \theta+\Omega-{\frac{1}{4*\Omega}}*sin(2\pi\theta) \)
I did 1800 iterations with 50 digit accuracy ( This was my first try) and the resulting conjecture is:
\( lim (n->infinity) {\frac{\theta n}{n} = 1 \)
monotonically from below, no oscillations. So the resulting angle is 1 rad again. I was expecting it,as \( \Omega \) seems to be kind of a self frequency of a flow on unit circle? But than I do not know much about circle maps. Perhaps that is true for any \( \Omega \)? Not only Omega constant?
Ivars
Consider circle map (Arnold map) :
\( \theta'= \theta+\Omega-{\frac{K}{2*\pi}}*sin(2\pi\theta) \)
Let \( \Omega=0.567143.. \)
And \( K= {\frac{\pi}{2*\Omega}.. \)
then map becomes:
\( \theta'= \theta+\Omega-{\frac{1}{4*\Omega}}*sin(2\pi\theta) \)
I did 1800 iterations with 50 digit accuracy ( This was my first try) and the resulting conjecture is:
\( lim (n->infinity) {\frac{\theta n}{n} = 1 \)
monotonically from below, no oscillations. So the resulting angle is 1 rad again. I was expecting it,as \( \Omega \) seems to be kind of a self frequency of a flow on unit circle? But than I do not know much about circle maps. Perhaps that is true for any \( \Omega \)? Not only Omega constant?
Ivars