Consider a sequence $(x_n)$ satisfying $x_{n+1}=x_n +\lambda \sin x_n$.

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.

References

See Denis Serre's answer below. My discussion of the case $\lambda=8$, as well as the exact formula for $\alpha,\beta$, might be new. Other references include

• Chaotic Synchronization and Antisynchronization in Coupled Sine Maps. Maistrenko V. at al. International Journal of Bifurcation and Chaos, Vol. 15, No. 07, pp. 2161-2177 (2005). See here.
• Basins and Critical Curves Generated by A Family of Two-Dimensional Sine Maps. Nasr-Eddine Hamri, Yamina Soula. Electronic J. of Theoretical Physics 7, No. 24 (2010) 139–150. See here.