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\title{\bf A verified computational technique to locate chaotic regions
of a
Henon system}
\authorinheadline{Bal\'{a}zs B\'{a}nhelyi and Tibor Csendes}
\titleinheadline{Verified chaos identification}
\author{Bal\'{a}zs B\'{a}nhelyi \inst{1}, Tibor Csendes \inst{2}}
\institute{
University of Szeged, Institute of Informatics \\
\mbox{e-mail: banhelyi@inf.u-szeged.hu}\\ [4mm]
\and
University of Szeged, Institute of Informatics\\
\mbox{e-mail: csendes@inf.u-szeged.hu}\\%[4mm]
}
\maketitle
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\begin{abstract}
We present a computer assisted proof of the existence of a horseshoe of
the
5-th iterate classical H\'{e}non map (${\cal H}(x,y) = (1+y-ax^2, bx)$).
An
earlier, published theorem \cite{Galias} gives three geometrical
conditions to
be fulfilled by all points of the solution region, given by 2
parallelograms.
We analyze these conditions separately and in case when all of them hold
true,
the proof is ready. The method applies interval arithmetic and recursive
subdivision. This verified technique proved to be fast, and we can use it
in a
framework program.
To find a region that fulfills the respective conditions, the program
combines
a global optimization procedure and our interval arithmetic \cite{CXSC}
based
checking technique described earlier. The algorithm obtains the H\'{e}non
map and
the parallelogram parameters and checks whether these fulfill the
conditions.
If not then it provides a penalty for this structure for the optimization
procedure. The penalty is zero if the structure doesn't brake any
conditions.
In this way we have obtained an optimization problem with six parameters.
If
the program finds the optimum and it is zero, then the search is
successful,
and we were able to locate a region where the investigated H\'enon map
instance has a chaotic behaviour.
The obtained coordinates of the lower parallelogram vertices for the
H\'enon
transformation parameters $a =1.939838$, $b=0.39146881$ are
$$x_a = 0.33298647, \; x_b = 0.49115518, \; x_c = 0.50960044, \; {\rm and}
\; x_d
= 0.59020179$$
\noindent with $y_0 = 0.01$, $y_1 = 0.28$, and $\tan \alpha = 2.0$. \\
\noindent In addition to the above result, we have extended the result of
\cite{Galias} in the sense that instead of their H\'enon parameter values
of
$a = 1.4$ and $b = 0.3$, we have determined a set of those parameter
values
that cause the chaotic behavior for the ${\cal H}^7$ transformation with
the
same parallelograms. The obtained intervals were $a \in [1.377599,
1.401300]$
and $b \in [0.277700, 0.310301]$. The technique by which this result was
obtained is an earlier interval optimization procedure able to solve
tolerance
optimization problems \cite{Annals}.
\vspace{3 mm}
\noindent The authors are grateful to Barnabas Garay (BME, Budapest,
Hungary) and Mih\'{a}ly
G\"{o}rbe (GAMF, Kecskem\'{e}t, Hungary) for their contribution and
support.
\vspace{5 mm}
\noindent{\bf Categories and Subject Descriptors:} Education and other
fields
of Applied Informatics
\vspace{5 mm}
\noindent{\bf Key Words and Phrases:} Chaos, Henon-map, verified method,
interval arithmetic
\end{abstract}
\begin{thebibliography}{x}
\itemsep=-3pt
\small
\bibitem{Annals} Csendes, T., Z.B. Zabinsky, and B.P. Kristinsdottir:
Constructing large feasible suboptimal intervals for constrained nonlinear
optimization. Annals of Operations Research, 58:279-293, 1995.
\bibitem{CXSC} C-XSC Languages home page: \\
{\tt
http://www.math.uni-wuppertal.de/org/WRST/index$\underline{~}$en.html}
\bibitem{Galias}
Galias, Z. and P. Zgliczynski. Computer assisted proof of chaos in the
Lorenz
equations. Physica D, 115:165-188, 1998.
\end{thebibliography}
\noindent{\Large\bf Postal addresses}
\bigskip
\noindent{\small\em
\parbox{67 truemm}{\noindent {\bf Bal\'{a}zs B\'{a}nhelyi } \\
University of Szeged,\\
Institute of Informatics,\\
H-6701 Szeged, P.O. Box 652,\\
Hungary}
\parbox{67 truemm}{\noindent {\bf Tibor Csendes} \\
University of Szeged,\\
Institute of Informatics,\\
H-6701 Szeged, P.O. Box 652,\\
Hungary}
}
\end{document}