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.


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.

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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.

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

  • $\begingroup$ 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 Jan 30 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 Jan 30 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 Jan 30 at 20:07

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