Computing geometric Lorenz attractors with arbitrary precision
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- by D. S. Graça, C. Rojas and N. Zhong PDF
- Trans. Amer. Math. Soc. 370 (2018), 2955-2970 Request permission
Abstract:
The Lorenz attractor was introduced in 1963 by E. N. Lorenz as one of the first examples of strange attractors. However, Lorenz’ research was mainly based on (non-rigorous) numerical simulations, and, until recently, the proof of the existence of the Lorenz attractor remained elusive. To address that problem some authors introduced geometric Lorenz models and proved that geometric Lorenz models have a strange attractor. In 2002 it was shown that the original Lorenz model behaves like a geometric Lorenz model and thus has a strange attractor.
In this paper we show that geometric Lorenz attractors are computable, as well as show their physical measures.
References
- V. S. Afraĭmovič, V. V. Bykov, and L. P. Sil′nikov, The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR 234 (1977), no. 2, 336–339 (Russian). MR 0462175
- V. Araujo, M. J. Pacifico, E. R. Pujals, and M. Viana, Singular-hyperbolic attractors are chaotic, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2431–2485. MR 2471925, DOI 10.1090/S0002-9947-08-04595-9
- Vítor Araújo and Maria José Pacifico, Three-dimensional flows, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 53, Springer, Heidelberg, 2010. With a foreword by Marcelo Viana. MR 2662317, DOI 10.1007/978-3-642-11414-4
- Vasco Brattka, Peter Hertling, and Klaus Weihrauch, A tutorial on computable analysis, New computational paradigms, Springer, New York, 2008, pp. 425–491. MR 2762094, DOI 10.1007/978-0-387-68546-5_{1}8
- M. Braverman and M. Yampolsky, Non-computable Julia sets, J. Amer. Math. Soc. 19 (2006), no. 3, 551–578. MR 2220099, DOI 10.1090/S0894-0347-05-00516-3
- Stefano Galatolo, Mathieu Hoyrup, and Cristóbal Rojas, Statistical properties of dynamical systems—simulation and abstract computation, Chaos Solitons Fractals 45 (2012), no. 1, 1–14. MR 2863582, DOI 10.1016/j.chaos.2011.09.011
- Stefano Galatolo, Mathieu Hoyrup, and Cristóbal Rojas, A constructive Borel-Cantelli lemma. Constructing orbits with required statistical properties, Theoret. Comput. Sci. 410 (2009), no. 21-23, 2207–2222. MR 2519306, DOI 10.1016/j.tcs.2009.02.010
- D. S. Graça, N. Zhong, and J. Buescu, Computability, noncomputability and undecidability of maximal intervals of IVPs, Trans. Amer. Math. Soc. 361 (2009), no. 6, 2913–2927. MR 2485412, DOI 10.1090/S0002-9947-09-04929-0
- Daniel S. Graça, Ning Zhong, and Jorge Buescu, Computability, noncomputability, and hyperbolic systems, Appl. Math. Comput. 219 (2012), no. 6, 3039–3054. MR 2992003, DOI 10.1016/j.amc.2012.09.031
- John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
- John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. MR 556582, DOI 10.1007/BF02684769
- Morris W. Hirsch, Stephen Smale, and Robert L. Devaney, Differential equations, dynamical systems, and an introduction to chaos, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 60, Elsevier/Academic Press, Amsterdam, 2004. MR 2144536
- E. N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci. 20 (1963), 130–141.
- Stefano Luzzatto, Ian Melbourne, and Frederic Paccaut, The Lorenz attractor is mixing, Comm. Math. Phys. 260 (2005), no. 2, 393–401. MR 2177324, DOI 10.1007/s00220-005-1411-9
- Jacob Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Astérisque 261 (2000), xiii–xiv, 335–347 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755446
- C. Rojas, Randomness and ergodic theory: An algorithmic point of view, PhD thesis, École Polytechnique and Università di Pisa, 2008.
- Steve Smale, Mathematical problems for the next century, Math. Intelligencer 20 (1998), no. 2, 7–15. MR 1631413, DOI 10.1007/BF03025291
- Warwick Tucker, A rigorous ODE solver and Smale’s 14th problem, Found. Comput. Math. 2 (2002), no. 1, 53–117. MR 1870856, DOI 10.1007/s002080010018
- Marcelo Viana, Stochastic dynamics of deterministic systems, volume 21, IMPA Rio de Janeiro, 1997.
- Klaus Weihrauch, Computable analysis, Texts in Theoretical Computer Science. An EATCS Series, Springer-Verlag, Berlin, 2000. An introduction. MR 1795407, DOI 10.1007/978-3-642-56999-9
- Lai-Sang Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys. 108 (2002), no. 5-6, 733–754. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. MR 1933431, DOI 10.1023/A:1019762724717
- Ning Zhong and Klaus Weihrauch, Computability theory of generalized functions, J. ACM 50 (2003), no. 4, 469–505. MR 2146883, DOI 10.1145/792538.792542
Additional Information
- D. S. Graça
- Affiliation: Faculdade de Ciências e Tecnologia, Universidade do Algarve, C. Gambelas, 8005-139 Faro, Portugal – and – Instituto de Telecomunicações, Portugal
- Email: dgraca@ualg.pt
- C. Rojas
- Affiliation: Departamento de Matemáticas, Universidad Andres Bello, República 498, 2do piso, Santiago, Chile
- Email: crojas@mat-unab.cl
- N. Zhong
- Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025
- Email: ning.zhong@uc.edu
- Received by editor(s): October 5, 2016
- Received by editor(s) in revised form: December 21, 2016, and February 15, 2017
- Published electronically: October 31, 2017
- Additional Notes: The first author was partially supported by Fundação para a Ciência e a Tecnologia and EU FEDER POCTI/POCI via SQIG - Instituto de Telecomunicações through the FCT project UID/EEA/50008/2013.
The second author was partially supported by projects Fondecyt 1150222, DI- 782-15/R Universidad Andres Bello and Basal PFB-03 CMM-Universidad de Chile. - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 2955-2970
- MSC (2010): Primary 03D78; Secondary 37D45, 37A35
- DOI: https://doi.org/10.1090/tran/7228
- MathSciNet review: 3748590