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I have a discrete dynamical System


with a paramteter $\lambda>0$, and where all $x_{n}$s are in [0,1]. $f$ is actually a larger algorithm, and a bit difficult to write down in two lines, but maybe someone can already point me into a direction based on what I describe here...

The behavior I obtain is the following: $\lambda_1>\lambda_2>0$ are two values. For any $\lambda>\lambda_1$, the series of $x_n$ converges to an asyp. stable fixed point. For $\lambda_1>\lambda>\lambda_2$, the series of $x_n$ goes toward a periodic state for large $n$, with apparently the same period for all these $\lambda$, but a different "amplitude", which increases with decreasing $\lambda$, cf. the figure. For $\lambda<\lambda_2$ there is no apparent whatsoever anymore.

Extract of the course of $x_n$ for large $n$, for three different parameters. Fig:[Extract of the course of $x_n$ for large $n$, for three different parameters.]

The only thing this reminds me of are the dynamics the logistic map, going from convergence over periodicity to chaos. But the periodicity here is quite different, the period is always the same. Plus, the fact that the $\{x_n\}_n$ seem to lie on a continuous periodic curve for large $n$ is remarkable.

Have you ever seen a similar behavior anywhere before? I'd be happy for any reference, maybe it can provide me with a different point of view on the dynamics of this system.

Thank you!

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The way you have defined your system, it seems to me it is a higher dimensional system, since $x_{n+1}$ depends not only on $x_n$ but $x_{n-1},x_{n-2}$ etc. – nonlinearism Jun 13 '13 at 14:50
yes ... so? ... – robster Jun 17 '13 at 9:57

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