Bánhelyi, Balázs

Bánhelyi Balázs

University of Szeged

A verified computational technique to locate chaotic regions of aHénon systemBalázs BánhelyiWe present a computer assisted proof of the existence of a horseshoe of the 5th iterate clas-sical Hénon map       	  . An earlier, published theorem [3, 4] gives threegeometrical conditions to be fulfilled by all points of the solution region, given by 2 parallel-ograms. We analyze these conditions separately and in case when all of them hold true, theproof is ready. The method applies interval arithmetic and recursive subdivision. This verifiedtechnique proved to be fast, and we can use it in a framework program. To find a region thatfulfills the respective conditions, the program combines a global optimization procedure andour interval arithmetic [2] based checking technique described earlier. The algorithm obtainsthe Hénon map and the parallelogram parameters and checks whether these fulfill the c ondi-tions. If not then it provides a penalty for this structure for the optimization procedure. Thepenalty is zero if the structure doesn’t brake any conditions. In this way we have obtained anoptimization 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 Hénonmap instance has a chaotic behaviour. The obtained coordinates of the lower parallelogramvertices for the Hénon transformation parameters       are         and 	  with        and tan  In addition to the above result, we have extended the result of [4] in the sense that in steadof their Hénon parameter values of    and   , we have determined a set of thoseparameter values that cause the chaotic behavior for the transformation with the same par-allelograms. The obtained intervals were     and    .The technique with which this result was obtained is an earlier interval optimization procedureable to solve tolerance optimization problems [1]. The author is grateful to Barna Garay (BME,Budapest) and Mihály Görbe (GAMF, Kecskemét, Hungary) for their contribution and support.References[1] Csendes, T., Z.B. Zabinsky, and B.P. Kristinsdottir: Constructing large feasible suboptimalintervals for constrained nonlinear optimization. Annals of Operations Research, 58:279293,1995.[2] CXSC Languages home page: http://www.math.uni-wuppertal.de/org/WRST/index_en.html[3] Galias, Z. and P. Zgliczynski. Computer assisted proof of chaos in the Lorenz equations.Physica D, 115:165188, 1998.[4] Zgliczynski, P.: Computer assisted proof of the horseshoe dynamics in the Hénon map.Random and Computational Dynamics 5(1997) 117.27

A verified computational technique to locate chaotic regions of a Hénon system : [abstract]

http://acta.bibl.u-szeged.hu/59734/1/cscs_2004_027.pdf

A verified computational technique to locate chaotic regions of a Hénon system

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