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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.

Generalizations

I added a lot of material in this article. It mostly deals with the basins of attractions in the 2-dimensional case. The picture below (taken from that article) features some of these basins.

enter image description here

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.
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This is exactly the dynamics studied by V. I. Arnold, which exhibits what is known as Arnold's tongues. See this link.

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  • $\begingroup$ Thank you. Short answer, but straight to the point. I will read the literature on this topic. I may add some beautiful pictures obtained when considering the 2-D system $x_{n+1}=x_n +\lambda\sin y_n$, $y_{n+1}=y_n + \lambda \sin x_n$. $\endgroup$
    – Vincent Granville
    Commented Jan 30, 2021 at 17:47
  • $\begingroup$ The difference with Arnold's tongues seems to be that in my case, $x_n$ should not be interpreted as taken modulo $2\pi$. $\endgroup$
    – Vincent Granville
    Commented Jan 30, 2021 at 18:41
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    $\begingroup$ Note that $\lambda=-8$ yields same behavior as $\lambda=8$. Chaotic behavior of $x_n$ begins when $|\lambda|$ is large enough, around $\lambda=4.60334$ possibly depending on $x_0$ (not that far from the first Feigenbaum constant, in the cases I tested). $\endgroup$
    – Vincent Granville
    Commented Jan 30, 2021 at 20:07

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