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\begin{document}
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\title {On parallel computation of the bifurcation points for the logistic map
% \footnote{Work partially supported by grant 0-01-00200
% from the Russian Foundation for Basic
% Research.}
}
\author{Yuri A. Blinkov}
\affiliation{
Department of Mathematics and Mechanics, \\
Saratov State University, \\
410012 Saratov, Russia}
\email{BlinkovUA@info.sgu.ru}
\maketitle
The logistic map is a discrete-time analogue of the continuous
logistic Verhulst's equation and has the form of a
one-parameter nonlinear recurrence relation
\begin{equation}
\label{ch4.4:LogisticMap}
x_{n+1} =\mu x_n(1-x_n),\quad \mu>0\,.
\end{equation}
Detecting the bifurcation values of parameter $\mu$ is a hard
computational problem~\cite{Kotsireas:2004:ECB}.
It is described by the following system of polynomial equations
\begin{equation}
\label{ch4.4:LogisticMap1}
\left\lbrace
\begin{array}{l}
x_2 =\mu x_1(1-x_1\,,)\\
x_3=\mu x_2(1-x_2)\,,\\
\ldots \\
x_n=\mu x_{n-1}(1-x_{n-1})\,,\\
x_1=\mu x_n(1-x_n)\,,\\
\mu^n \prod\limits_{k=1}^{n}(1-2x_k)=1\,.
\end{array}
\right.
\end{equation}
From the system~(\ref{ch4.4:LogisticMap1}) one can compute a polynomial in $\mu$ by doing elimination of the variables $\{x_1,\ldots,x_n\}$. This can be achieved by computing a Gr\"{o}bner basis for a degree compatible term order
and then by constructing a univariate polynomial in $\mu$ that belongs to the ideal generated by the system~(\ref{ch4.4:LogisticMap1}). The last construction is done by the linear algebra methods applied to the ideal as a vector space generated by the power products in $\{x_1,\ldots,x_n,\mu\}$.
In doing so, computation of roots of the univariate polynomial obtained is a much more simple problem to detect the values of $\mu$ directly from the multivariate system~(\ref{ch4.4:LogisticMap1}).
We discuss the algorithmic aspects of this approach the specialized computer
algebra system \textsl{GINV}~\cite{Blinkov:2008:SCA} to compute the bifurcation
polynomial in $\mu$ for $n=9$. Among the real roots of this polynomial we found
the bifurcation point $\mu=3.687196\ldots$.
\vspace*{-1cm}
\begin{thebibliography}{1}
\bibitem{Blinkov:2008:SCA}
Yu.~A. Blinkov and V.~P. Gerdt.
\newblock Specialized computer algebra system {GINV}.
\newblock {\em Programming and Computer Software}, 34(2):112--123, 2008.
\verb|http://invo.jinr.ru/ginv/index.html|
\bibitem{Kotsireas:2004:ECB}
I.~S. Kotsireas and K.~Karamanos.
\newblock Exact computation of the bifurcation point $b_4$ of the logistic map
and the bailey-broadhurst conjectures.
\newblock {\em Journal Bifurcation and Chaos}, 14:2417--2423, 2004.
\end{thebibliography}
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\end{document}