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A Verified Enclosure for the Lorenz Attractor (Rough Diamond)

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Book cover Interactive Theorem Proving (ITP 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9236))

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Abstract

A rigorous numerical algorithm, formally verified with Isabelle/HOL, is used to compute an accurate enclosure for the Lorenz attractor.

Accurately enclosing the attractor is highly relevant: a similar non verified computation is part of Tucker’s proof that the Lorenz attractor is chaotic in a rigorous mathematical sense. This proof settled a conjecture that Fields medalist Stephen Smale has put on his list of eighteen important mathematical problems for the twenty-first century.

F.Immler—Supported by the DFG RTG 1480 (PUMA).

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References

  1. de Figueiredo, L., Stolfi, J.: Affine arithmetic: concepts and applications. Numer. Algorithms 37(1–4), 147–158 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Girard, A., Le Guernic, C.: Zonotope/hyperplane intersection for hybrid systems reachability analysis. In: Egerstedt, M., Mishra, B. (eds.) HSCC 2008. LNCS, vol. 4981, pp. 215–228. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  3. Immler, F.: A verified algorithm for geometric zonotope/hyperplane intersection. In: Proceedings of the 2015 Conference on Certified Programs and Proofs, CPP 2015, pp. 129–136. ACM, New York (2015)

    Google Scholar 

  4. Immler, F.: Verified reachability analysis of continuous systems. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 37–51. Springer, Heidelberg (2015). http://dx.doi.org/10.1007/978-3-662-46681-0_3

    Google Scholar 

  5. Immler, F., Hölzl, J.: Ordinary differential equations. Archive of Formal Proofs, August 2015, Formal proof development. http://afp.sf.net/devel-entries/Ordinary_Differential_Equations.shtml

  6. Nipkow, T., Paulson, L.C., Wenzel, M. (eds.): Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  7. Tucker, W.: My thesis: the Lorenz attractor exists. http://www2.math.uu.se/~warwick/main/pre_thesis.html

  8. Tucker, W.: A rigorous ODE solver and Smale’s 14th problem. Found. Comput. Math. 2(1), 53–117 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

I would like to thank Florian Haftmann for providing the theories for parallelization with Isabelle/HOL and Makarius Wenzel for the underlying infrastructure for parallel combinators in Isabelle/ML.

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Authors and Affiliations

  1. Institut für Informatik, Technische Universität München, München, Germany

    Fabian Immler

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  1. Fabian Immler
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Correspondence to Fabian Immler .

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Editors and Affiliations

  1. Department of Informatics, King's College London, London, United Kingdom

    Christian Urban

  2. PLA University of Science and Technology, Nanjing, China

    Xingyuan Zhang

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Immler, F. (2015). A Verified Enclosure for the Lorenz Attractor (Rough Diamond). In: Urban, C., Zhang, X. (eds) Interactive Theorem Proving. ITP 2015. Lecture Notes in Computer Science(), vol 9236. Springer, Cham. https://doi.org/10.1007/978-3-319-22102-1_14

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  • DOI: https://doi.org/10.1007/978-3-319-22102-1_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22101-4

  • Online ISBN: 978-3-319-22102-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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