Conference abstracts

Session B1 - Computational Dynamics

July 14, 15:30 ~ 16:00

Numerical study of wide periodic windows for the quadratic map

Zbigniew Galias

AGH University of Science and Technology, Poland   -   galias@agh.edu.pl

Periodic windows for the map $f(x)=ax(1-x)$ are studied numerically.

An efficient method to find very accurate rigorous bounds for the endpoints of the periodic window with a given symbol sequence is presented. The method is based on applying the interval Newton method to find positions of bifurcation points of corresponding periodic orbits. The method is capable to handle periodic windows with periods up to several millions. We discuss how to select an initial point for the Newton method to improve the convergence speed. The method is applied to find all $1\,966\,957\,258$ periodic windows with periods $2\leq p\leq 36$ and show that the total width of these windows is above $0.611834003131$. Positions of periodic windows' endpoints are found with the accuracy of more than 60 decimal digits.

A heuristic algorithm to locate wide periodic windows based on the results obtained for periodic windows with low periods is described. Periodic windows are classified as primary windows and period-tupling windows. Candidates for symbol sequences corresponding to wide primary windows are constructed iteratively from shorter symbol sequences by insertion of a single symbol and by substitution of a single symbol by two-symbol sequences. Symbol sequences corresponding to wide period-tupling windows are constructed iteratively from symbol sequences of primary windows and period-tupling windows found previously. The algorithm is used to find the majority of wide periodic windows with periods $p\geq 37$.

From the results concerning periodic windows found it follows that the measure of the set of regular parameters in the interval $[3,4]$ is above $0.613960137$. Using these results, we estimate that the true value of the measure of the set of regular parameters is close to $0.6139603$.

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