# Strange behavior of $x_{n+1}=x_n +\lambda \sin x_n$

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.

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.

• 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 $x_{n+1}=x_n +\lambda\sin y_n$, $y_{n+1}=y_n + \lambda \sin x_n$. – Vincent Granville Jan 30 at 17:47
• The difference with Arnold's tongues seems to be that in my case, $x_n$ should not be interpreted as taken modulo $2\pi$. – Vincent Granville Jan 30 at 18:41
• 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). – Vincent Granville Jan 30 at 20:07