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Differential Equations

, Volume 54, Issue 4, pp 525–538 | Cite as

Applicability of the Interval Taylor Model to the Computational Proof of Existence of Periodic Trajectories in Systems of Ordinary Differential Equations

  • N. M. Evstigneev
  • O. I. Ryabkov
Numerical Methods

Abstract

We consider the construction of the interval Taylor model used to prove the existence of periodic trajectories in systems of ordinary differential equations. Our model differs from the ones available in the literature in the method for describing the algorithms for the computation of arithmetic operations over Taylor models. In the framework of the current model, this permits reducing the computational expenditures for obtaining interval estimates on computers. We prove an assertion that permits establishing the existence of a periodic solution of a system of ordinary differential equations by verifying the convergence of the Picard iterations in the sense of embedding of the proposed Taylor models. An example illustrating how the resulting assertion can be used to prove the existence of a closed trajectory in the van der Pol system is given.

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References

  1. 1.
    Babenko, K.I., Osnovy chislennogo analiza (Foundations of Numerical Analysis), Moscow; Izhevsk: Inst. Komp. Issled., 2002.zbMATHGoogle Scholar
  2. 2.
    Moore, R.E., Methods and Applications of Interval Analysis, Philadelphia: SIAM, 1979.CrossRefzbMATHGoogle Scholar
  3. 3.
    Moore, R.E., Kearfott, R.B., and Cloud, M.J., Introduction to Interval Analysis, Philadelphia: SIAM, 2009.CrossRefzbMATHGoogle Scholar
  4. 4.
    Dobronets, B.S., Interval’naya matematika (Interval Mathematics), Krasnoyarsk: Krasnoyarsk Gos. Univ., 2004.Google Scholar
  5. 5.
    Sharyi, S.P., Konechnomernyi interval’nyi analiz (Finite-Dimensional Interval Analysis), Novosibirsk, 2010 (electronic book).Google Scholar
  6. 6.
    Babenko, K.I., On a problem of Gauss, Dokl. Akad. Nauk SSSR, 1978, vol. 238, no. 5, pp. 1021–1024.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Babenko, K.I. and Vasil’ev, M.M., On demonstrative computations in the problem of stability of plane Poiseuille flow, Sov. Math. Dokl., 1983, vol. 28, pp. 763–768.zbMATHGoogle Scholar
  8. 8.
    Shokin, Yu.I., Interval’nyi analiz (Interval Analysis), Novosibirsk: Nauka, 1981.Google Scholar
  9. 9.
    Kalmykov, S.A., Shokin, Yu.I., and Yuldashev, Z.Kh., Metody interval’nogo analiza (Methods of Interval Analysis), Novosibirsk: Nauka, 1986.zbMATHGoogle Scholar
  10. 10.
  11. 11.
    Rogalev, A.N., Upper and Lower Bounds of Sets of Solutions of Systems of Ordinary Differential Equations with Interval Parameters, Cand. Sci. (Phys.–Math.) Dissertation, Krasnoyarsk, 1996.Google Scholar
  12. 12.
    Nedialkov, N.S., Jackson, K.R., and Corliss, G.F., Validated solutions of initial value problems for ordinary differential equations, Appl. Math. Comput., 1999. vol. 105. no. 1. pp. 21–68.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Golitsyn, D.L., Ryabkov, O.I., and Burov, D.A., Algorithm for the numerical proof of the existence of periodic trajectories in two-dimensional non-autonomous systems of ordinary differential equations, Differ. Equations, 2013, vol. 49, no. 2, pp. 217–223.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Alfimov, G.L. and Zezyulin, D.A., Use of validated computations to compute vortex structures in the Bose–Einstein condensate, Nelineinaya Dinamika, 2009, vol. 5, no. 2, pp. 215–235.CrossRefGoogle Scholar
  15. 15.
    Nagatou, K., A computer-assisted proof on the stability of the Kolmogorov flows of incompressible viscous fluid, J. Comput. Appl. Math., 2004, vol. 169, pp. 33–44.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Watanabe, Y., A computer-assisted proof for the Kolmogorov flows of incompressible viscous fluid, J. Comput. Appl. Math., 2009, vol. 223, pp. 953–966.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cyranka, J. and Zgliczynski, P., Existence of globally attracting solutions for one-dimensional viscous Burgers equation with nonautonomous forcing–A computer assisted proof, SIAM J. Appl. Dyn. Syst., 2015, vol. 14, no. 2, pp. 787–821.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kearfott, R., Interval computations: Introduction, uses, and resources, Euromath Bull., 1996, vol. 2, no. 1, pp. 95–112.MathSciNetGoogle Scholar
  19. 19.
    Berz, M. and Makino, K., Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models, Reliab. Comput., 1998, vol. 4, pp. 361–369.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Berz, M. and Makino, K., Performance of Taylor model methods for validated integration of ODEs, in Lect. Notes Comput. Sci., 2006, vol. 3732, pp. 65–74.CrossRefGoogle Scholar
  21. 21.
    Lin, Y. and Stadtherr, M.A., Validated solutions of initial value problems for parametric ODEs, Appl. Numer. Math., 2007, vol. 57, pp. 1145–1162.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Berz, M. and Makino, K., Suppression of the wrapping effect by Taylor model-based verified integrators: Long-term stabilization by shrink wrapping, Int. J. Differ. Equ. Appl., 2005, vol. 10, no. 4, pp. 385–403.MathSciNetzbMATHGoogle Scholar
  23. 23.
    Corliss, G.F. and Rihm, R., Validating an a priori enclosure using high-order Taylor series, in Scientific Computing, Computer Arithmetic, and Validated Numerics, 1996, pp. 228–238.Google Scholar
  24. 24.
    Berz, M., Cosy Infinity Version 8 Reference Manual, NSCL Technical Report MSUCL-1088, Michigan, 1998.Google Scholar
  25. 25.
  26. 26.
    Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of Function Theory and Functional Analysis), Moscow: Nauka, 1972.zbMATHGoogle Scholar
  27. 27.
    Schauder, J., Der Fixpunktsatz in Funktionalraumen, Stud. Math., 1930, vol. 2, pp. 171–180.CrossRefzbMATHGoogle Scholar
  28. 28.
    Tychonoff, A., Ein Fixpunktsatz, Math. Ann., 1935, vol. 111, pp. 767–776.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Gilbarg, D. and Trudinger, N.S., Elliptic Partial Differential Equations of Second Order, Berlin: Springer-Verlag, 1983. Translated under the title Ellipticheskie differentsial’nye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Moscow: Nauka, 1989.CrossRefzbMATHGoogle Scholar
  30. 30.
    Lefschetz, S., Differential Equations: Geometric Theory, New York: Interscience, 1957. Translated under the title Geometricheskaya teoriya differentsial’nykh uravnenii, Moscow: Mir, 1961.zbMATHGoogle Scholar
  31. 31.
    Pilarczyk, P., Topological-numerical approach to the existence of periodic trajectories in ODE’s, in Proc. of the Fourth Int. Conf. on Dynam. Syst. and Differ. Equat. May 24–27. 2002, pp. 701–708.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Federal Research Center “Computer Science and Control” of the Russian Academy of SciencesMoscowRussia
  2. 2.Institute for Systems Analysis of the Russian Academy of SciencesMoscowRussia

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