Skip to main content
Log in

A Verified Optimization Technique to Locate Chaotic Regions of Hénon Systems

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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

  2. T. Csendes (1988) ArticleTitleNonlinear parameter estimation by global optimization – efficiency and reliability Acta Cybernetica 8 361–370

    Google Scholar 

  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. T. Csendes Z.B. Zabinsky B.P. Kristinsdottir (1995) ArticleTitleConstructing large feasible sub-optimal intervals for constrained nonlinear optimization Annals of Operations Research 58 279–293 Occurrence Handle10.1007/BF02096403

    Article  Google Scholar 

  5. C-XSC Languages home page (2005), http://www.math.uni-wuppertal.de/org/WRST/index_en.html

  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. Z. Galias P. Zgliczynski (2001) ArticleTitleAbundance of homoclinic and heteroclinic orbits and rigorous bounds for the topological entropy for the Hénon map Nonlinearity 14 909–932 Occurrence Handle10.1088/0951-7715/14/5/301

    Article  Google Scholar 

  8. R. Klatte U. Kulisch A. Wiethoff C. Lawo M. Rauch (1993) C-XSC - A C++ Class Library for Extended Scientific Computing Springer-Verlag Heidelberg

    Google Scholar 

  9. B.P. Kristinsdottir Z.B. Zabinsky T. Csendes M.E. Tuttle (1993) ArticleTitleMethodologies for tolerance intervals Interval Computations 3 133–147

    Google Scholar 

  10. B.P. Kristinsdottir Z.B. Zabinsky M.E. Tuttle T. Csendes (1996) ArticleTitleIncorporating manufacturing tolerances in optimal design of composite structures Engineering Optimization 26 1–23

    Google Scholar 

  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

  12. Neumaier A. (2004) Complete search in continuous global optimization and constraint satisfaction. Acta Numerica 271–369

  13. A. Neumaier T. Rage (1993) ArticleTitleRigorous chaos verification in discrete dynamical systems Physica D 67 327–346 Occurrence Handle10.1016/0167-2789(93)90169-2

    Article  Google Scholar 

  14. T. Rage A. Neumaier C. Schlier (1994) ArticleTitleRigorous verification of chaos in a molecular model Physical Review E. 50 2682–2688 Occurrence Handle10.1103/PhysRevE.50.2682

    Article  Google Scholar 

  15. H. Ratschek J. Rokne (1988) New Computer Methods for Global Optimization Ellis Horwood Chichester

    Google Scholar 

  16. P. Zgliczynski (1997) ArticleTitleComputer assisted proof of the horseshoe dynamics in the Hénon map Random & Computational Dynamics 5 1–17

    Google Scholar 

  17. P. Zgliczynski (2003) ArticleTitleOn smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach Journal of Differential Equations 195 271–283 Occurrence Handle10.1016/j.jde.2003.07.009

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tibor Csendes.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Csendes, T., Garay, B.M. & Bánhelyi, B. A Verified Optimization Technique to Locate Chaotic Regions of Hénon Systems. J Glob Optim 35, 145–160 (2006). https://doi.org/10.1007/s10898-005-1509-9

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-005-1509-9

Keywords

Navigation