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A verified optimization technique to locate chaotic regions of Hénon systems. (English) Zbl 1103.90076
Summary: We present a new verified optimization method to find regions for Hénon systems where the conditions of chaotic behaviour hold. The present paper provides a methodology to verify chaos for certain mappings and regions. We discuss first how to check the set theoretical conditions of a respective theorem in a reliable way by computer programs. Then we introduce optimization problems that provide a model to locate chaotic regions. We prove the correctness of the underlying checking algorithms and the optimization model. We have verified an earlier published chaotic region, and we also give new chaotic places located by the new technique.
90C26Nonconvex programming, global optimization
37D45Strange attractors, chaotic dynamics
C-XSC 2.0
[1]Bánhelyi, B., Csendes, T. and Garay B.M., Optimization and the Miranda approach in detecting horseshoe-type chaos by computer. Manuscript, submitted for publication. Available at www.inf.u-szeged.hu/csendes/henon2.pdf
[3]Csendes, T., Bánhelyi B. and Hatvani L., Towards a computer-assisted proof for chaos in a forced damped pendulum equation. Manuscript, submitted for publication. Available at www.inf.u-szeged.hu/csendes/jcaminga.pdf
[4] · Zbl 0843.90107 · doi:10.1007/BF02096403
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[6]Dellnitz, M. and Junge, O. (2002), Set oriented numerical methods for dynamical systems. Handbook of dynamical systems, Vol. 2, North-Holland, Amsterdam, pp. 221–264.
[7] · Zbl 1067.37052 · doi:10.1088/0951-7715/14/5/301
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[11]Markót, M.C. and Csendes, T. A New Verified Optimization Technique for the ”Packing Circles in a Unit Squre” Problems. Accepted for publication in the SIAM J. on Optimization. Available at www.inf.u-szeged.hu/csendes/publ.html
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[13] · Zbl 0783.58047 · doi:10.1016/0167-2789(93)90169-2
[14] · doi:10.1103/PhysRevE.50.2682
[17] · Zbl 1043.35034 · doi:10.1016/j.jde.2003.07.009