Analytic enclosure of the fundamental matrix solution

Roberto Castelli; Jean-Philippe Lessard; Jason D. Mireles James

Applications of Mathematics (2015)

  • Volume: 60, Issue: 6, page 617-636
  • ISSN: 0862-7940

Abstract

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This work describes a method to rigorously compute the real Floquet normal form decomposition of the fundamental matrix solution of a system of linear ODEs having periodic coefficients. The Floquet normal form is validated in the space of analytic functions. The technique combines analytical estimates and rigorous numerical computations and no rigorous integration is needed. An application to the theory of dynamical system is presented, together with a comparison with the results obtained by computing the enclosure in the C s category.

How to cite

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Castelli, Roberto, Lessard, Jean-Philippe, and Mireles James, Jason D.. "Analytic enclosure of the fundamental matrix solution." Applications of Mathematics 60.6 (2015): 617-636. <http://eudml.org/doc/271817>.

@article{Castelli2015,
abstract = {This work describes a method to rigorously compute the real Floquet normal form decomposition of the fundamental matrix solution of a system of linear ODEs having periodic coefficients. The Floquet normal form is validated in the space of analytic functions. The technique combines analytical estimates and rigorous numerical computations and no rigorous integration is needed. An application to the theory of dynamical system is presented, together with a comparison with the results obtained by computing the enclosure in the $C^s$ category.},
author = {Castelli, Roberto, Lessard, Jean-Philippe, Mireles James, Jason D.},
journal = {Applications of Mathematics},
keywords = {rigorous numerics; fundamental matrix solution; Floquet theory; analytical category; rigorous numerics; fundamental matrix solution; Floquet theory; analytical category},
language = {eng},
number = {6},
pages = {617-636},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Analytic enclosure of the fundamental matrix solution},
url = {http://eudml.org/doc/271817},
volume = {60},
year = {2015},
}

TY - JOUR
AU - Castelli, Roberto
AU - Lessard, Jean-Philippe
AU - Mireles James, Jason D.
TI - Analytic enclosure of the fundamental matrix solution
JO - Applications of Mathematics
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 6
SP - 617
EP - 636
AB - This work describes a method to rigorously compute the real Floquet normal form decomposition of the fundamental matrix solution of a system of linear ODEs having periodic coefficients. The Floquet normal form is validated in the space of analytic functions. The technique combines analytical estimates and rigorous numerical computations and no rigorous integration is needed. An application to the theory of dynamical system is presented, together with a comparison with the results obtained by computing the enclosure in the $C^s$ category.
LA - eng
KW - rigorous numerics; fundamental matrix solution; Floquet theory; analytical category; rigorous numerics; fundamental matrix solution; Floquet theory; analytical category
UR - http://eudml.org/doc/271817
ER -

References

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  1. Cabré, X., Fontich, E., Llave, R. de la, 10.1512/iumj.2003.52.2245, Indiana Univ. Math. J. 52 (2003), 283-328. (2003) Zbl1034.37016MR1976079DOI10.1512/iumj.2003.52.2245
  2. Cabré, X., Fontich, E., Llave, R. de la, The parameterization method for invariant manifolds II: Regularity with respect to parameters, Indiana Univ. Math. J. 52 (2003), 329-360. (2003) Zbl1034.37017MR1976080
  3. Cabré, X., Fontich, E., Llave, R. de la, 10.1016/j.jde.2004.12.003, J. Differ. Equations 218 (2005), 444-515. (2005) Zbl1101.37019MR2177465DOI10.1016/j.jde.2004.12.003
  4. Castelli, R., Lessard, J.-P., A method to rigorously enclose eigenpairs of complex interval matrices, Internat. Conf. Appl. Math. In Honor of the 70th Birthday of K. Segeth Academy of Sciences of the Czech Republic, Institute of Mathematics, Prague (2013), 21-31. (2013) MR3204427
  5. Castelli, R., Lessard, J.-P., 10.1137/120873960, SIAM J. Appl. Dyn. Syst. (electronic only) 12 (2013), 204-245. (2013) Zbl1293.37033MR3032858DOI10.1137/120873960
  6. Castelli, R., Lessard, J.-P., James, J. D. Mireles, 10.1137/140960207, SIAM J. Appl. Dyn. Syst. (electronic only) 14 (2015), 132-167. (2015) MR3304254DOI10.1137/140960207
  7. Day, S., Lessard, J.-P., Mischaikow, K., 10.1137/050645968, SIAM J. Numer. Anal. 45 (2007), 1398-1424. (2007) Zbl1151.65074MR2338393DOI10.1137/050645968
  8. Gameiro, M., Lessard, J.-P., 10.1016/j.jde.2010.07.002, J. Differ. Equations 249 (2010), 2237-2268. (2010) Zbl1256.35196MR2718657DOI10.1016/j.jde.2010.07.002
  9. Hungria, A., Lessard, J.-P., James, J. D. Mireles, Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach, (to appear) in Math. Comput. (2015). 
  10. Lessard, J.-P., James, J. D. M., Reinhardt, C., 10.1007/s10884-014-9367-0, J. Dyn. Differ. Equations 26 (2014), 267-313. (2014) MR3207723DOI10.1007/s10884-014-9367-0
  11. James, J. D. Mireles, Mischaikow, K., 10.1137/12088224X, SIAM J. Appl. Dyn. Syst. (electronic only) 12 (2013), 957-1006. (2013) MR3068557DOI10.1137/12088224X
  12. Rump, S. M., INTLAB---INTerval LABoratory, T. Csendes Developments in Reliable Computing SCAN-98 conference, Budapest. Kluwer Academic Publishers Dordrecht (1999), 77-104, http://www.ti3.tu-harburg.de/rump/. (1999) Zbl0949.65046
  13. Yakubovich, V. A., Starzhinskij, V. M., Linear Differential Equations with Periodic Coefficients, Vol. 1, 2, Wiley, New York Halsted, Jerusalem (1975). (1975) Zbl0308.34001MR0364740

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