Abstract
We propose a numerical procedure for obtaining a mathematically reliable long-term solutions of chaotic dynamical systems. The procedure is based on the multiple-precision Taylor series method and the globally grid refinement method. The globally grid refinement method is used for verification of the computed solution and is a more rigourous verification than one in the works of other authors. We use as a test problem the classical Rossler system with the standard parameters. The GMP library is used for our multiple-precision floating point arithmetic.
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References
Liao, S.: On the reliability of computed chaotic solutions of non-linear differential equations. Tellus A: Dyn. Meteorol. Oceanogr. 61(4), 550–564 (2008)
Kehlet, B., Logg, A.: Quantifying the computability of the Lorenz system. arXiv preprint arXiv:1306.2782 (2013)
Kalitkin, N.N., Koryakin, P.V.: Numerical Methods, Vol. 2: Methods of Mathematical Physics (2013)
Wang, P., Li, J.: On the relation between reliable computation time, float-point precision and the Lyapunov exponent in chaotic systems. arXiv preprint arXiv:1410.4919 (2014)
Rossler, O.E.: An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)
Jorba, A., Zou, M.: A software package for the numerical integration of ODEs by means of high-order Taylor methods. Exp. Math. 14(1), 99–117 (2005)
Barrio, R., et al.: Breaking the limits: the Taylor series method. Appl. Math. Comput. 217(20), 7940–7954 (2011)
Moore, R.E.: Methods and Applications of Interval Analysis. Society for Industrial and Applied Mathematics (1979)
Multipurpose Information and Computing Complex of the Laboratory of IT of JINR. http://hlit.jinr.ru. Accessed 31 Jan 2020
Wang, P., Liu, Y., Li, J.: Clean numerical simulation for some chaotic systems using the parallel multiple-precision Taylor scheme. Chin. Sci. Bull. 59(33), 4465–4472 (2014). https://doi.org/10.1007/s11434-014-0412-5
Acknowledgements
We thank the Laboratory of Information Technologies of JINR, Dubna, Russia for the opportunity to use the computational resources of the HybriLIT Heterogeneous Platform. The work of the first author is supported by the Grant No DO1-271/16.12.2019 of the Ministry of Education and Science of Bulgaria.
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Hristova, R., Hristov, I. (2023). Computing Reliable Solutions of Chaotic Dynamical Systems Using Multiple-precision Arithmetic. In: Georgiev, I., Kostadinov, H., Lilkova, E. (eds) Advanced Computing in Industrial Mathematics. BGSIAM 2019. Studies in Computational Intelligence, vol 1111. Springer, Cham. https://doi.org/10.1007/978-3-031-42010-8_10
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DOI: https://doi.org/10.1007/978-3-031-42010-8_10
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