You would expect the sequence $x_n$ to depend on $x_0$ and to exhibit a chaotic, Brownian-type behavior, and indeed it does pretty much all the time.

However, if $\lambda=8$ (also true if $\lambda$ is very close to $8$), we have $x_n \sim \pm 2\pi n$. The sign depends on the initial value $x_0$. Assuming $x_0=2$ and $\lambda=8$, we have $x_{2n}-x_{2n-1}\sim \alpha=7.939712...$ and $x_{2n-1}-x_{2n-2}\sim \beta=-1.65653...$ with $\alpha + \beta = 2\pi$. Also, $\alpha$ is solution of $$2\pi=\alpha +\alpha\cos\alpha -\sqrt{\lambda^2-\alpha^2}\sin\alpha.$$

I am wondering if this non-chaotic behavior also happens with other values of the parameter $\lambda$, and when the sign alternates (depending on $x_0$) in the asymptotic formula $x_n \sim \pm 2\pi n$. The sign is very sensitive to $x_0$. Are there other unexpected (non-chaotic) behavior for this sequence, depending on $\lambda$ and $x_0$? For instance, if $x_0$ is large (say $x_0=67$) and $1<\lambda<3$, then $x_n$ converges very rapidly so the sequence looks flat. If $x_0=67, \lambda=7.99$, we have the expected chaotic behavior. If $x_0=67, \lambda=8$ we have the behavior described earlier. And with $\lambda>8.02$ we are back to chaotic behavior. Now if $x_0=67, \lambda=4$, then $x_n$ stays in a flat, narrow band, constantly oscillating.